     James Joseph Sylvester
Collected Mathematical Papers, Vol. I
           Book pp. 1–608
                                                1.
  Analytical Development of Fresnel’s Optical Theory of
                       Crystals
   [Philosophical Magazine, XI. (1837), pp. 461–469, 537–541; XII. (1838),
                             pp. 73–83, 341–345]
                                                                                          p. 1
   The following is, I believe, the first successful attempt to obtain the full
development of Fresnel’s Theory of Crystals by direct geometrical methods.
Hitherto little has been done beyond finding and investigating the properties of
the wave surface, a subject certainly curious and interesting, but not of chief
importance for ordinary practical purposes. Mr Kelland, in a most valuable
contribution to the Cambridge Philosophical Transactions 1 , has incidentally
obtained the difference of the squares of the velocities of a plane front in terms of
the angles made by it with the optic axes. I have obtained each of the velocities
separately, and in a form precisely the same for biaxal as for uniaxal crystals.
   I have also assigned in my last proposition the place of the lines of vibration
in terms of the like quantities, and that in a shape remarkably convenient for
determining the plane of polarization when the ray is given. For at first sight
there appears to be some ambiguity in selecting which of the two lines of vibration
is to be chosen when the front is known. If p be the perpendicular from the
centre of the surface of elasticity let fall upon the front, ι1 , ι2 the angles made by
the front with the optic planes, ε1 , ε2 the angles between its due line of vibration
and the optic axes, I have shown that
                         s                                      s
                             b2 − p2 sin ι1                         b2 − p2 sin ι2
              cos ε1 =              ·       ,        cos ε2 =              ·       ,
                             a2 − c2 sin ι2                         a2 − c2 sin ι1
so that all doubt is completely removed. The equation preparatory to obtaining
the wave surface is found in Prop. 6 by common algebra, without any use of
the properties of maxima and minima, and various other curious relations are
discussed.
   Without the most careful attention to preserve pure symmetry, the expressions
could never have been reduced to their present simple forms.                     p. 2


      Analytical Reduction of Fresnel’s Optical Theory of
                           Crystals.

                                     Index of Contents.

   In Proposition 1, a plane front within a crystal being given, the two lines of
vibration are investigated.
  1
      See Lond. and Edinb. Phil. Mag. Vol. x. p. 336.


                                                2
     In Proposition 2 it is shown that the product of the cosines of the inclinations
of one of the axes of elasticity to the two lines of vibration, is to the same for
either other axis of elasticity in a constant ratio for the same crystal; and the
two lines of vibration are proved to be perpendicular to each other.
     In Proposition 3, a line of vibration being given, the front to which it belongs
is determined; and it is proved that there is only one such, and consequently any
line of vibration has but one other line conjugate to it.
     In Proposition 4, certain relations are instituted between the positions of, and
velocities due to, conjugate lines.
     In Proposition 5, the angles made by the front with the planes of elasticity
are found in terms of the velocities only.
     In Proposition 6, the above is reversed.
     In Proposition 7, the position of the planes in which the two velocities are
equal (viz. the optic planes) is determined.
     In Proposition 8, the position of a front in respect to the optic axes is expressed
in terms of the velocities.
     In Proposition 9, the problem is reversed, and it is shown that if v1 , v2 be the
two normal velocities with which any front can move perpendicular to itself, and
ι1 , ι2 the angles which it makes with the optic planes, then

                                      ι1 + ι2                    ι1 + ι2
                                                2                       2
                    v12 = a2 sin                      + c2 cos                  ,
                                         2                          2
                                                 2                        2
                                       ι1 − ι2                  ι1 − ι2
                                                           
                    v22 = a2       sin                +c2
                                                            cos                 .
                                          2                        2
   In the 10th the angle made by a line of vibration with the axes of elasticity is
expressed in terms of the two velocities of the front to which it belongs.
   In the 11th Proposition the velocity due to any line of vibration is expressed
in terms of the angles which it makes with the optic axes, viz.

                          v 2 − b2 = (a2 − c2 ) cos ε1 cos ε2 .

   In the 12th Proposition ε1 , ε2 are separately expressed in terms of ι1 , ι2 .
   In the Appendix I have given the polar or rather radio-angular equation to
the wave surface, from which the celebrated proposition of the ray flows as an
immediate consequence.                                                            p. 3



                                       Proposition 1.

   If
                                      lx + my + nz = 0                              (a)


                                                  3
be the equation to a given front, to determine the lines of vibration therein.
   It is clear that if x, y, z be any point in one of these lines, the force acting
on a particle placed there when resolved into the plane must tend to the centre.
Consequently the line of force at x, y, z must meet the perpendicular drawn upon
the front from the origin. Now the equation to this perpendicular is
                                            X   Y   Z
                                              =   =                                            (1)
                                            l   m   n
and the forces acting at x, y, z are a2 x, b2 y, c2 z parallel to x, y, z, so that the
equation to the line of force is
                                     X −x   Y −y  Z −z
                                           = 2   = 2 .                                         (2)
                                      a2 x   b y   c z
From (2) we obtain

                                 b2 yX − a2 xY = (b2 − a2 )xy,                                 (3)
                                    c zY − b yZ = (c − b )yz,
                                      2      2               2   2
                                                                                               (4)
                                    a2 xZ − c2 zX = (a2 − c2 )zx.                              (5)

Hence

     (b2 − a2 )xyn + (c2 − b2 )yzl + (a2 − c2 )zxm
                                 = b2 y(nX − lZ) + c2 z(lY − mX) + a2 x(mZ − nY );

but by equations (1)

                  lZ − nX = 0,             mX − lY = 0,              nY − mZ = 0

therefore
                                      n             l            m
                         (b2 − a2 )     + (c2 − b2 ) + (a2 − c2 ) = 0.                         (b)
                                      z             x            y
Also we have
                                          nz + lx + my = 0                                     (a)
therefore
                                          z             x
                                                                         
       (b − a )n + (c − b )l + nl (c − b ) + (b2 − a2 )
        2     2   2        2    2     2          2       2
                                                                              = (a2 − c2 )m2
                                          x             z
or
                        1 n 2
              2
              z                                                      oz
(c − b )
  2      2
                    +      (c − b2 )l2 + (b2 − a2 )n2 − (a2 − c2 )m2    + (b2 − a2 ) = 0.
              x         nl                                            x
And in like manner interchanging b, y, m with c, z, n
                        1 n 2
              2
              y                                                      oy
(b2 − c2 )          +      (b − c2 )l2 + (c2 − a2 )m2 − (a2 − b2 )n2    + (c2 − a2 ) = 0.
              x         ml                                            x

                                                     4
                                                                                                     p. 4
                 y1 z1         y2 z 2                                      y z
                                     
  Hence if         ,   ,         ,          be the two systems of values of , , then
                 x1 x1         x2 x2                                       x x
                         Y   y1 Z   z1                Y   y2 Z   z2
                                                                       
                           =   ,  =                     =   ,  =
                         X   x1 X   x1                X   x2 X   x2
are the two lines of vibration required.


                                        Proposition 2.

  By last proposition it appears that
                                        y1 y2  c2 − a2
                                              = 2                                              (c)
                                        x1 x2   b − c2
and
                                        z 1 z2  b2 − a2
                                               = 2                                             (d)
                                        x1 x2    c − b2
therefore
                                y1 y2 + z1 z2  c2 − b2
                                              = 2      = −1
                                    x1 x2      b − c2
therefore
                                   x1 x2 + y1 y2 + z1 z2 = 0.
And therefore the two lines of vibration are perpendicular to each other.
  N.B. Equations (c) and (d) must not be overlooked.


                                        Proposition 3.
                                                  y1 z 1
  A line of vibration is given (that is             ,    are given) and the position of the
                                                  x1 x1
front is to be determined.
   Let lx + my + nz = 0 be the front required, then lx1 + my1 + nz1 = 0, and
                                   l             m            n
                     (b2 − c2 )      + (c2 − a2 ) + (a2 − b2 ) = 0.
                                  x1             y1           z1
Eliminating n we get
                     x1              z1                          y1              z1
                                                                                   
      l (a2 − b2 )      − (b2 − c2 )            + m (a2 − b2 )      − (c2 − a2 )          =0
                     z1              x1                          z1              y1
therefore
                  l   x1 (a2 − b2 )y12 − (c2 − a2 )z12
                    =
                  m   y1 (b2 − c2 )z12 − (a2 − b2 )x21
                      x1 a2 (x21 + y12 + z12 ) − (a2 x21 + b2 y12 + c2 z12 )
                    =                                                        .
                      y1 b2 (x21 + y12 + z12 ) − (a2 x21 + b2 y12 + c2 z12 )


                                                  5
                                                                                             p. 5
   If now we make x21 + y12 + z12 = 1

                                a2 x21 + b2 y12 + c2 z12 = v12

and therefore
                                    l   x1 a2 − v12
                                      =   ·
                                    m   y1 b2 − v12
and in like manner
                                    l   x1 a2 − v12
                                      =   ·         ;
                                    n   z1 c2 − v12
therefore
                   (a2 − v12 )x1 x + (b2 − v12 )y1 y + (c2 − v12 )z1 z = 0
is the equation required.


                                    Proposition 4.

    l l
     , having each only one value, shows that only one front corresponds to
    m n
the given line of vibration. Let x2 , y2 , z2 , v2 correspond to x1 , y1 , z1 , v1 for the
conjugate line of vibration, then the equation to the front may be expressed
likewise by
                (a2 − v22 )x2 x + (b2 − v22 )y2 y + (c2 − v22 )z2 z = 0,
so that
                      (a2 − v12 )x1   (b2 − v12 )y1   (c2 − v12 )z1
                                    =               =               .
                      (a2 − v22 )x2   (b2 − v22 )y2   (c2 − v22 )z2


                                    Proposition 5.

   To find ω, ϕ, ψ, the angles made by the front with the planes of elasticity in
terms of v1 , v2 .
   By the last proposition

                             (a2 − v12 )2 x21
(cos ω)2 =
           (a2 − v12 )2 x21 + (b2 − v12 )2 y12 + (c2 − v12 )2 z12
                                          (a2 − v12 )(a2 − v22 )x1 x2
          = 2                                                                                   .
           (a − v1 )(a2 − v2 )x1 x2 + (b2 − v12 )(b2 − v22 )y1 y2 + (c2 − v12 )(c2 − v22 )z1 z2
                   2           2


Now, by Proposition 2,
                               x1 x2    y1 y2    z1 z2
                                      = 2     = 2
                              c2 − b2  a − c2  b − a2

                                              6
                                                                                             p. 6
    therefore (cos ω)2

                                    (a2 − v12 )(a2 − v22 )(c2 − b2 )
=
 (a2 − v12 )(a2 − v22 )(c2 − b2 ) + (b2 − v12 )(b2 − v22 )(a2 − c2 ) + (c2 − v12 )(c2 − v22 )(b2 − a2 )
       (a2 − v12 )(a2 − v22 )(c2 − b2 )
= 4 2
 a (c − b2 ) + b4 (a2 − c2 ) + c4 (a2 − b2 )
 (a2 − v12 )(a2 − v22 )
= 2                     .
  (a − b2 )(a2 − c2 )
Similarly,
                                         (b2 − v12 )(b2 − v22 )
                            (cos ϕ)2 =                          ,
                                         (b2 − a2 )(b2 − c2 )
                                           (c2 − v12 )(c2 − v22 )
                            (cos ψ)2 =                            .
                                           (c2 − a2 )(c2 − b2 )


                                    Proposition 6.

    To find v1 , v2 in terms of ω, ϕ, ψ.
    By the last proposition
              (cos ω)2             a2                             1
                       =                         − v22 · 2                    ,
                2
              a − v1 2   (a − b )(a − c )
                            2     2     2     2         (a − b )(a2 − c2 )
                                                               2

              (cos ϕ)2             b2                            1
                       =                         − v22 · 2                   ,
                2
               b − v12   (b2  − a 2 )(b2  − c2 )        (b − a 2 )(b2 − c2 )

              (cos ψ)2             c2                            1
                       = 2                       − v22 · 2                   ,
                2
              c − v1 2   (c − b )(c2 − a2 )
                                 2                      (c − a2 )(c2 − b2 )
therefore
                         (cos ω)2 (cos ϕ)2 (cos ψ)2
                                  + 2       + 2       = 0.
                         a2 − v12   b − v12   c − v12
Just in the same way
                         (cos ω)2 (cos ϕ)2 (cos ψ)2
                                  + 2       + 2       = 0,
                         a2 − v22   b − v22   c − v22

so that v12 , v22 are the two roots of the equation

                         (cos ω)2 (cos ϕ)2 (cos ψ)2
                                  + 2      + 2      = 0.
                         a2 − v 2   b − v2   c − v2
Cor. Hence the equation to the wave surface may be obtained by making

                         (cos ω)x + (cos ϕ)y + (cos ψ)z = v,

                                               7
                                                                                                           p. 7
   or if we please to apply Prop. 5, we may make
  s                                 s
        (a2 − v12 )(a2 − v22 )          (b2 − v12 )(b2 − v22 )
                               x+                              y
        (a2 − b2 )(a2 − c2 )            (b2 − a2 )(b2 − c2 )
                                                                   s
                                                                       (c2 − v12 )(c2 − v22 )
                                                              +                               z = v1 ,
                                                                       (c2 − a2 )(c2 − b2 )
or, if we please2 ,
  s                                 s
        (a2 u2 − 1)(a2 − v 2 )           (b2 u2 − 1)(b2 − v 2 )
                               x+                               y
         (a2 − b2 )(a2 − c2 )             (b2 − a2 )(b2 − c2 )
                                                                   s
                                                                       (c2 u2 − 1)(c2 − v 2 )
                                                             +                                z = 1.
                                                                        (c2 − a2 )(c2 − b2 )


                                          Proposition 7.

  To find when v1 = v2 .
  By Prop. 4,
                  x1 (v12 − a2 )   y1 (v12 − b2 )   z1 (v12 − c2 )
                                 =                =                .                                 (θ)
                  x2 (v22 − a2 )   y2 (v22 − b2 )   z2 (v22 − c2 )
Hence when v1 = v2 we have, generally speaking,
                                  x1     y1     z1
                                     =      = .
                                  x2     y2     z2
Now
                            x1 x2 + y1 y2 + z1 z2 = 0;
therefore x21 + y12 + z12 would = 0, which is absurd.
   The only case therefore when v1 can = v2 is when one of those terms of
                          0                                       x1    z1   0
equation (θ) becomes : thus suppose v1 = b, then we have             =     = , and
                          0                                       x2    z2   0
                          x1     y1
we can no longer infer       = .
                          x2     y2
   Let now (ω1 , ϕ1 , ψ1 )(ω2 , ϕ2 , ψ2 ) be the two systems of values which ω, ϕ, ψ
assume when v1 = v2 = b, then applying the equation of Prop. 5 we have
                                    s                              s
                                         a2 − b2                       a2 − b2
                         cos ω1 =                      cos ω2 =
                                         a2 − c2                       a2 − c2
                               cos ϕ1 = 0                  cos ϕ2 = 0
                                    s                              s
                                         b2 − c2                       b2 − c2
                         cos ψ1 =                      cos ψ2 =                ,
                                         a2 − c2                       a2 − c2
   2
       See below, p. 27. Ed.


                                                   8
so that b must correspond to the mean axis.                                                    p. 8



                                         Proposition 8.

   ι1 , ι2 being the angles made by the front with the optic planes, to find ι1 , ι2 in
terms of v1 , v2 .
   By analytical geometry

             cos ι1 = cos ω · cos ω1 + cos ϕ · cos ϕ1 + cos ψ · cos ψ1
                        s                              s
                            (v12 − a2 )(v22 − a2 )         a2 − b2
                    =                              ·
                            (a2 − b2 )(a2 − c2 )           a2 − c2
                            s                              s
                                (v12 − c2 )(v22 − c2 )         c2 − b2
                        +                              ·
                                (c2 − a2 )(c2 − b2 )           c2 − a2
                         q                                 q
                            {(v12 − a2 )(v22 − a2 )} +          {(v12 − c2 )(v22 − c2 )}
                    =
                                                      a2 − c2
and similarly

             cos ι2 = cos ω · cos ω2 + cos ϕ · cos ϕ2 + cos ψ · cos ψ2
                        q                                  q
                            {(v12 − a2 )(v22 − a2 )} −          {(v12 − c2 )(v22 − c2 )}
                    =                                                                      .
                                                   a2 − c2



                                         Proposition 9.

   To find v1 , v2 in terms of ι1 , ι2 .
   By the last proposition

                                   (v12 − a2 )(v22 − a2 ) − (v12 − c2 )(v22 − c2 )
              cos ι1 · cos ι2 =
                                                    (a2 − c2 )2
                                   (a4 − c4 ) − (a2 − c2 )(v12 + v22 )
                                 =
                                               (a2 − c2 )2
                                   (a + c ) − (v12 + v22 )
                                      2    2
                                 =
                                           a2 − c2
therefore
                        v12 + v22 = a2 + c2 − (a2 − c2 ) cos ι1 cos ι2 .




                                                  9
Again,

      (sin ι1 )2 · (sin ι2 )2 = 1 − (cos ι1 )2 − (cos ι2 )2 + (cos ι1 )2 (cos ι2 )2
                                     (v12 − a2 )(v22 − a2 ) + (v12 − c2 )(v22 − c2 )
                             =1−2·
                                                       (a2 − c2 )2
                                (a2 + c2 )2 − 2(a2 + c2 )(v12 + v22 ) + (v12 + v22 )2
                              +
                                                     (a2 − c2 )2
                              v − 2v v + v
                               4      2  2    4
                             = 1 2 1 22 2 2
                                 (a − c )
                                                                                        p. 9
  therefore
                             v12 − v22 = (a2 − c2 ) sin ι1 · sin ι2
but
                     v12 + v22 = (a2 + c2 ) − (a2 − c2 ) cos ι1 cos ι2
therefore
                           a2 + c2 a2 − c2
                     v12 =         −        cos(ι1 + ι2 )
                              2          2
                                  ι1 + ι2 2           ι1 + ι2 2
                                                          
                         = a2 sin           + c2 cos
                                     2                    2

                          a2 + c2 a2 − c2
                     v22 =        −        cos(ι1 − ι2 )
                             2          2
                                 ι1 − ι2 2           ι1 − ι2 2
                                                         
                        = a2 sin           + c2 cos            .
                                    2                    2
Thus for uniaxal crystals where ι1 + ι2 = 180◦

                                            v12 = a2

                                 v22 = a2 (cos ι)2 + c2 (sin ι)2 .
   Cor. Hence we may reduce the discovery of the two fronts into which a plane
front is refracted on entering a crystal to the following trigonometrical problem.

                                                P
                                                                O
                                                        F

                             I                          G        H

                                                    A          C
                                             Fig. 1.


                                                10
   Let a sphere be described about any point in the line in which the air front
intersects the plane of incidence. Let the great circle P I denote the latter plane,
IF the former, OA, OC also great circles, the planes of single velocity. Suppose
IGH to be one of the refracted fronts intersecting OA, OC in G and H, then
                    (a2 + c2 ) − (a2 − c2 ) cos(G + H)    (sin P IF )2
                                                       =               .
                               2(vel. in air)2           (sin P IGH)2
The double sign will give rise to two positions of the refracted front IGH.
   The propositions which follow are perhaps more curious than immediately
useful.                                                                     p. 10



                                     Proposition 10.

    To determine the portion of a line of vibration in terms of the two velocities
of its corresponding front.
                                                          y1 z1
    We have here to determine the quantities                  ,     (of Prop. 1) in terms of
                                                          x1 x1
v1 , v2 , or on putting x21 + y12 + z12 = 1, x1 , y1 , z1 are to be found in terms of v1 , v2 .
    By Prop. 3
                                           l            m           n
                        x1 : y1 : z1 :: 2         : 2           : 2
                                        a − v1 b − v1 c − v12
                                              2             2

and by Prop. 5
                       l2 : m2 : n2 :: (b2 − c2 )(a2 − v12 )(a2 − v22 )
                                       : (c2 − a2 )(b2 − v12 )(b2 − v22 )
                                       : (a2 − b2 )(c2 − v12 )(c2 − v22 );
therefore

  x21 : y12 : z12
                                        a2 − v22              2
                                                          2 b − v2
                                                                    2               2
                                                                                2 c − v2
                                                                                          2
                           :: (b2 − c2 ) 2       : (c2
                                                       − a )          : (a 2
                                                                             − b )          .
                                        a − v12              b2 − v12              c2 − v12
Let α, β, γ be the angles made by the given line of vibration with the elastic
axes, then
                                        x21
                     (cos α)2 =
                                  x21 + y12 + z12
                              = (b2 − c2 )(a2 − v22 )(b2 − v12 )(c2 − v12 )
divided by

  (b2 − c2 )(a2 − v22 )(b2 − v12 )(c2 − v12 ) + (c2 − a2 )(b2 − v22 )(c2 − v12 )(a2 − v12 )
                                                + (a2 − b2 )(c2 − v22 )(a2 − v12 )(b2 − v12 )


                                               11
and therefore
                             (b2 − c2 )(a2 − v22 )(b2 − v12 )(c2 − v12 )
                         =
                             (v12 − v22 )(a2 − b2 )(b2 − c2 )(c2 − a2 )
(where it is to be observed that the reduction of the denominator is simply the
effect of a vast heap of terms disappearing under the influence of contact with
the magic circuit (a2 − b2 ), (b2 − c2 ), (c2 − a2 ), a simpler instance of which was
seen in Proposition 5).
   In fact the coefficient of v 4 · v 2

                             = (b2 − c2 ) + (c2 − a2 ) + (a2 − b2 )

                                              =0
                                                                                        p. 11
   that of v12 · v22

                             = (c2 + b2 )(c2 − b2 )
                               + (a2 + c2 )(a2 − c2 )
                               + (b2 + a2 )(b2 − a2 )
                             = (c4 − b4 ) + (a4 − c4 ) + (b4 − a4 )
                             = 0.

The term in which neither v1 nor v2 enters

                       = a2 b2 c2 {(b2 − c2 ) + (c2 − a2 ) + (a2 − b2 )}

                                              = 0.
The coefficient of

                  −v12 = a2 · (b4 − c4 ) + b2 · (c4 − a4 ) + c2 · (a4 − b4 )

and that of

               v22 = b2 c2 · (c2 − b2 ) + c2 a2 · (a2 − c2 ) + a2 b2 · (b2 − a2 )

each of which
                             = (a2 − b2 ) · (b2 − c2 ) · (c2 − a2 ).
Hence
                                      v12 − b2 (a2 − v22 )(c2 − v12 )
                         (cos α)2 =            ·                      ,
                                      v12 − v22 (a2 − b2 )(a2 − c2 )
in like manner (cos β)2 = &c. and

                                      v12 − b2 (c2 − v22 )(a2 − v12 )
                         (cos γ)2 =            ·                      .
                                      v12 − v22 (c2 − b2 )(c2 − a2 )



                                               12
                                        Proposition 11.

   ε1 , ε2 being the angles between any line of vibration and the optic axes, required
the velocity due to that line in terms of ε1 , ε2 .
   By analytical geometry,

                          cos ε1 = cos α · cos ϕ1 + cos γ · cos ψ1

                          cos ε2 = cos α · cos ϕ1 − cos γ · cos ψ1
therefore

      cos ε1 · cos ε2 = (cos α)2 (cos ϕ1 )2 − (cos γ)2 (cos ψ1 )2
                                            (a2 − v22 )(c2 − v12 ) − (c2 − v22 )(a2 − v12 )
                                        (                                                     )
                       v 2 − b2
                      = 21      ·
                       v1 − v22                             (a2 − c2 )2
                       v12 − b2 (a2 − c2 )(v22 − v12 )
                      =            ·
                       v12 − v22     (a2 − c2 )2
                       b2 − v12
                      = 2        .
                       a − c2
Hence
                             v12 = b2 − (a2 − c2 ) cos ε1 cos ε2
and in like manner, for the conjugate line of vibration

                            v22 = b2 − (a2 − c2 ) cos ε′1 cos ε′2 .
                                                                                                       p. 12



                                        Proposition 12.

   To find ε1 , ε2 in terms of ι1 , ι2 .


   (cos ε1 )2 + (cos ε2 )2 = 2(cos α)2 · (cos ϕ1 )2 + 2(cos γ)2 · (cos ψ1 )2
                                                 (a2 − v22 )(c2 − v12 ) + (c2 − v22 )(a2 − v12 )
                                             (                                                     )
                              v 2 − b2
                           = 2 21
                              v1 − v22                           (a2 − c2 )2

but by Prop. 9

                                         ι1 + ι2                   ι1 + ι2
                                                  2                       2
                      v12 = a2       sin                +c 2
                                                               cos
                                            2                         2
                                                   2                        2
                                        ι1 − ι2                    ι1 − ι2
                                                              
                      v22 = a2 sin                      + c2 cos
                                           2                          2

                                                   13
therefore
                                   b2 − v12
(cos ε1 )2 + (cos ε2 )2 =                                        multiplied by
                            (a2 − c2 ) sin ι1 · sin ι2
                                           h                2                2                  2                2 i
                            2(a2 − c2 )2       cos ι1 +ι
                                                      2
                                                        2
                                                                 sin ι1 −ι
                                                                        2
                                                                          2
                                                                                   + cos ι1 −ι
                                                                                            2
                                                                                              2
                                                                                                       sin ι1 +ι
                                                                                                              2
                                                                                                                2


                                                                  (a2 − c2 )2
                                   b2 − v12
                      =                                {(sin ι1 )2 + (sin ι2 )2 }
                            (a2 − c2 ) sin ι1 · sin ι2

and we have seen that
                                                            b2 − v12
                                   cos ε1 cos ε2 =
                                                            a2 − c2
therefore                                      s
                                                   b2 − v12 sin ι1 + sin ι2
                    cos ε1 + cos ε2 =                      ·√                 ,
                                                   a2 − c2    sin ι1 · sin ι2
                                               s
                                                   b2 − v12 sin ι1 − sin ι2
                    cos ε1 − cos ε2 =                      ·√                 ,
                                                   a2 − c2    sin ι1 · sin ι2
therefore                               v(
                                        u b2 − v 2 sin ι1
                                        u                 )
                               cos ε1 = t 2     1
                                                   ·
                                                 2 a −c          sin ι2
                                        v(
                                        u b2 − v 2 sin ι2
                                        u                 )
                               cos ε2 = t       1
                                                   ·
                                           2     2 a −c          sin ι1
and in like manner                     v(
                                       u b2 − v 2 sin ι1
                                       u                 )
                                    ′
                               cos ε = t       2
                                                 ·
                                    1
                                                   a2 − c2 sin ι2
                                       v(
                                       u b2 − v 2 sin ι2
                                       u                 )
                                    ′
                               cos ε = t       2
                                                 ·
                                    2
                                                   a2 − c2 sin ι1
where v1 , v2 for the sake of neatness are left unexpressed in terms of ι1 , ι2 . p. 13
  This is the simplest form by which the position of the lines of vibration can
be denoted.
  Cor. From the last proposition it appears that
                                        cos ε1   sin ι1
                                               =        .
                                        cos ε2   sin ι2
Hence we may construct geometrically for the two planes of polarization.




                                                    14
                              F
                                               E

                          I                             G


                                                   K

                                    P
                                        Fig. 2.

   Let I, K be the projections of the two optic axes on a sphere, E the projection
of the normal to the front, P the projection of one line of vibration; then
                                  cos P K   sin KE
                                          =         .
                                  cos P I    sin IE
Draw F EG the circle of which P is the pole, meeting P K, P I produced in G
and F .
  Then
                            cos P K = sin KG,
and
                                   cos P I = sin IF,
therefore
                                  sin KG     sin KE
                                           =
                                   sin F I    sin IE
therefore
                                   sin KG   sin IF
                                          =
                                   sin KE   sin IE
therefore
                                  sin KEG = sin IEF
therefore KEG = IEF or 180◦ − IEF . But P EF = P EG, therefore EP bisects
either the angle IEK or the supplement to it.
   These two positions of EP give the two planes of polarization. The construc-
tion is the same as that given in Mr Airy’s tracts, and originally proposed, I
believe, by Mr MacCullagh.                                                      p. 14


                                    Addendum.

  If in the equation of Prop. 6, viz.

                       (cos ω)2 (cos ϕ)2 (cos ψ)2
                                + 2      + 2      =0
                       a2 − v 2   b − v2   c − v2

                                          15
                         1 1 1 1
we change a, b, c, v into , , , , and consider v to be the length of a line drawn
                         a b c v
perpendicular to the plane
                          cos ω · x + cos ϕ · y + cos ψ · z = 0,
the equation to the extremity thereof must be
                  a2 r2 (cos ω)2 b2 r2 (cos ϕ)2 c2 r2 (cos ψ)2
                                +              +               = 0,
                     a2 − r2        b2 − r2        c2 − r2
where ω, ϕ, ψ denote the angles between the radius vector r, and the axes of
x, y, z, so that the equation may be written
                          a2 x2        b2 y 2      c2 z 2
                                   +          +            = 0,
                        a2 − r2 b2 − r2 c2 − r2
which is that of the wave surface.
   But we have seen that
                                         2                        2
                                ι 1 ± ι2                    ι1 ± ι2
                                                       
               v = c cos
                 2    2
                                              + a sin
                                                 2
                                                                        ,
                                    2                          2
therefore the equation to the wave surface may be written
                                              2                     2
                          1     cos ι1 ±ι2
                                               sin ι1 ±ι2
                            =          2
                                           +          2
                                                          ,
                         r2         c2             a2
where ι1 , ι2 denote the angles between the radius vector v and the two lines
                                                            1 1 1
which would be the optic axes if a, b, c were changed into , , so that if e be
                                                            a b c
the inclination of either to the mean axis of elasticity
                                   v           s
                                   u 1
                                   u 2 − 12   c a2 − b2
                           cos e = t a   b
                                         1  =           ,
                                         a2
                                            − c12        b   a2 − c2
                                   v            s
                                   u 1
                                   u 2 − 12   a   b2 − c2
                           sin e = t b  1
                                         c
                                            =             .
                                        a2
                                           − c12         b   a2 − c2
These lines I shall call by way of distinction the prime radii3 .                  p. 15
     Cor. 1. If r1 , r2 be the two values of r corresponding to the same values of
ι1 , ι2 we have
                 1    1   1                                    ι1 + ι2
                                 (                 2                    2 )
                                          ι1 − ι2
                                                             
                    − 2 = 2           cos                − cos
                  2
                 r1  r2  c                   2                    2
                                1                       ι1 + ι 2
                                    (                                     2 )
                                        ι1 − ι2 2
                                                                
                            + 2     sin           − sin
                               a           2               2
                              1   1
                                    
                           = 2 − 2 sin ι1 · sin ι2 ,
                              c   a
   3
     Upon the authority of Professor Airy I have appropriated the term optic axes to the lines
normal to the fronts of single velocity.


                                              16
which proves the celebrated problem of two rays having a common direction in a
crystal.
   Cor. 2. The intersection of any concentric sphere with the wave surface is
found by making r constant. Hence ι1 ± ι2 becomes constant, and therefore
rι1 ± rι2 = constant. Hence the curve of intersection is the locus of points, the
sum or difference of whose distances from two poles when measured by the arcs
of great circles is constant; the poles being the points in which the prime radii
pierce the sphere.
   In three cases these spherico-ellipses or spherico-hyperbolas become great
circles:
   (1) When ι1 ± ι2 = the angle between the two poles, in which case the curve
of intersection is the great circle which comprises the two poles.
   (2) When ι1 − ι2 = 0, when the locus is a great circle perpendicular to the
former and bisecting the angle between the optic axes.
   (3) When ι1 + ι2 = 180◦ , when the locus is a great circle perpendicular to the
two above, and bisecting the supplemental angle between the two axes.
   Various other properties may be with the greatest simplicity deduced from the
radio-angular equation. The hurry of the press leaves me time only to subjoin
the following

                                  Proposition.

   To find the inclination of the radius vector to the tangent plane, in terms of
the angles which the radius vector makes with the prime radii.
   Let O be the centre of the wave surface, OA, OB the two prime radii, OP
any radius vector. Let OP = v, P OA = ι1 , P OB = ι2 , and let the inclination of
the planes P OA, P OB = µ; then
                                           2                  2
                        1     sin ι1 +ι2
                                                  cos ι1 +ι2

                          2
                            =       2
                                     2
                                                +      2
                                                         2
                                                                    ,
                        r         a                   c
(taking only the positive sign for the sake of brevity).                              p. 16
   Let OQ, OR be the two adjacent radii vectores, so assumed that
                    QOA = P OA,            QOB = P OB + δι2 ,
                    ROB = P OB,             ROA = P OA + δι1 ,
and let p, q, r, a, b be the projections of P, Q, R, A, B on a sphere of which O is
the centre, then it is clear that
                            qpa = 90◦ ,          rpb = 90◦ ;
draw qm perpendicular to pb, then pm = δι2 , and therefore
                                 pm        pm       δι2
                        pq =           =         =       .
                               sin pqm   sin apb   sin µ

                                           17
In like manner
                                                        δι1
                                           pr =              .
                                                       sin µ
Now the angle QP O
                                          p
                                                                 qr
                                      m




                                      a                 b

                                               Fig. 3.


                                      r · P OQ         r · pq
                           = tan−1             = tan−1 dr     ;
                                     OQ − OP           dι δι2         2

also
             d(1/r2 )            1    1         ι1 + ι2 2
                                 (                                        )
                         d
                                                       
                      =             −       cos
               dι2      dι2      c2 a2              2
                               1   1        ι1 + ι 2       ι1 + ι 2
                                                               
                           =− 2 − 2     sin            cos            ;
                               c   a           2              2
therefore
                            dr    1   1   1
                                                           
                                 = r2 2 − 2 sin(ι1 + ι2 ),
                           r dι2  4   c  a
therefore
                                     r2       1    1
                                                           
                     cot QP O =                  − 2 sin(ι1 + ι2 ) sin µ.
                                     4        c2  a
In like manner
                                     r2       1    1
                                                           
                     cot RP O =                  − 2 sin(ι1 + ι2 ) sin µ,
                                     4        c2  a
therefore
                                          QP O = RP O.
Also it is clear that rpq = apb = µ. And to find the inclination of OP to RP Q,
we have only to describe a sphere of which P is the centre, and intersecting
P Q, P R, P O in Q′ , R′ , O′ .
   Then R′ O′ Q′ = µ, and
                                                       1   1
                                          (                                     )
                                              r2
                                                                 
             ′   ′     ′    ′        −1
            O Q = O R = cot                              −   sin(ι1 + ι2 ) sin µ .
                                              4        c2 a2

                                                   18
                                                 O′




                               R′                N         Q′

                                          Fig. 4.


                                                                                          p. 17
   Draw O′ N perpendicular to R′ Q′ , then O′ N measures the inclination of the
radius vector to the tangent plane4 .
   And
                                            µ
                                 Q′ O′ N = ,
                                            2
therefore
                              µ
                          cos = tan O′ N · cot O′ Q′ ,
                              2
therefore
                                         cot O′ Q′
                             cot O′ N =            ,
                                           cos µ2
and therefore
                            1   1   1    µ
                                                 
                  cot O′ N = r2   −   sin · sin(ι1 + ι2 ).
                            2   a2 c2    2
Let AOB the angle between the optic axes = 2e, then by mere trigonometry
                               s
                                    sin e + ι1 −ι   sin e − ι1 −ι
                                                 2
                                                                2
                                                                        
                          µ
                       sin =                   2               2
                                                                   ,
                          2                 sin ι1 · sin ι2

therefore the tangent of the inclination between the radius vector and the normal
                                            s
           1   1    1                            sin e + ι1 −ι   sin e − ι1 −ι
                                                            2
                                                                             
                                                                              2
          = r2    − 2 sin(ι1 + ι2 )                         2               2
                                                                                .
           2   a2  c                                     sin ι1 · sin ι2

In like manner the tangent of the inclination between the same radius vector
and the normal at the other point of the wave-surface pierced by it
                                              s
         1       1   1                            sin e + ι1 +ι   sin e − ι1 +ι
                                                             2
                                                                              2
                                                                                  
        = (r1 )2   −   sin(ι1 − ι2 )                         2               2
                                                                                 .
         2       a2 c2                                    sin ι1 · sin ι2
    O is the projection of the ray and R′ O′ of the tangent plane. Therefore O′ N being
   4 ′

perpendicular to R′ Q′ represents their inclination.


                                            19
    We may, in the same way, find the inclination of the tangent plane to either of
the prime radii, and to the plane which contains them both, in terms of ι1 and
ι2 ; the former by a remarkably elegant construction; but the final expressions do
not present themselves under the same simple aspect.
    If we call ϕ the angle between the ray and the front, we may still further
reduce by substituting for r2 its values in terms of ι1 , ι2 and we shall obtain
                                           2(c2 − a2 )
                        cot ϕ =
                                  c2 tan ι1 +ι
                                            2 + a cot 2
                                              2    2   ι1 +ι2

           s
                        ι1 + ι2         ι1 + ι2
                                                                        
          ×     sin e +         sin e −                 · cosec ι1 · cosec ι2 .
                           2               2
                                                                                      p. 18
   And if π1 , π2 be the inclinations of the normal to the two prime radii, it may
be shown that
                                                              µ
                      cos π1 = cos ϕ sin ι1 ∓ sin ϕ cos ι1 sin ,
                                                              2
                                                              µ
                      cos π2 = cos ϕ sin ι2 ± sin ϕ cos ι2 sin .
                                                              2
                                  µ
   Cor. 1. For uniaxal crystals = 90◦ and ι1 + ι2 = 180◦ , so that the tangent
                                  2
of the inclination of normal to radius vector
                     1           1    1
                                       
                    = r2 ·          − 2 sin 2ι    for one point,
                     2           a2  c
and
                                 = 0 for the other.
   Cor. 2. For every point in the circular section which passes through the poles
    µ
sin = 0, and for the other two circular sections ι1 ± ι2 = 0 or 180◦ .
    2
   Therefore every point in the three circular sections is an apse.
                                 1     1
   Cor. 3. When a nearly = c, 2 − 2 is very small; and therefore the normal
                                a      c
and radius vector very nearly coincide.
   Cor. 4. Referring to fig. 4 we see that O′ N bisects the angle R′ O′ Q′ . Now
R′ O, Q′ O are respectively perpendicular to the planes passing through O′ and
the optic axes; and therefore the meridian plane as we may term it, that is, the
plane containing both the ray and the normal, always bisects the angle formed
by the two planes drawn through the ray and the two optic axes.
   Cor. 5. When
                                   ι1 or ι2 = 0,
                                  ι2 or ι1 = e.
                                   0
And therefore ϕ assumes the form , which indicates that the extremities of the
                                   0
four prime radii are singular points.

                                            20
   In concluding for the present it behoves me to state that one step has been
omitted in the foregoing paper5 , viz. the actual performance of the eliminations
which lead to the rectilinear equation to the wave-surface. But Mr Archibald
Smith’s elegant and brief Memoir in the Cambridge Philosophical Transactions 6
of last year leaves nothing to be desired further on that head.                    p. 19
   That I have not exhibited it in its proper place (Prop. 6) arises only from my
respect to the principle of literary propriety. With this important blank supplied
the Analytical Theory may be pronounced to be complete.
   For all errors and imperfections in what precedes my excuse must be press of
time and a total want of the materials to be derived from consulting works of
reference.


   Since writing the above I have had an opportunity of reading the paper of our
living Laplace inserted as part of the Third Supplement to his System of Rays
in the Transactions of the Royal Irish Academy, in which the principal foregoing
results are obtained by aid of a more refined and transcendental analysis.
   The nature of the four singular points is there discussed and the existence of
four circles of plane contact demonstrated.
   The former may be very easily shown thus: when ι1 is very small ι2 =
2e − ι1 cos ψ very nearly, ψ denoting the inclination of the plane in which e is
reckoned to the plane in which ι2 is reckoned.
   Hence
  1         1 1      1     1 1        1
 2                                      
          =      +       −         −      cos{2e − ι1 (cos ψ ± 1)}
  r         2 a2 c2        2 a2 c2
            1 1      1     1 1        1             1 1       1
                                                           
          =      +       −         −      cos 2e −          −     sin 2e(cos ψ ± 1)ι1
            2 a2 c2        2 a2 c2                  2 a2 c2
            1     1  q
          = 2− 2       {(a2 − b2 )(b2 − c2 )} (cos ψ ± 1)ι1 ,
            b   b ac
therefore
                                                     !1/2           !1/2        
                        1               b2                  b2                  
                r = b 1 + (cos ψ ± 1) 1 − 2                     −1          ι1       .
                        2               a                   c2                  

Take ψ constant and let the abscissae and ordinates be reckoned respectively
along and perpendicular to the prime ray.
   Then
                       y                   q
                  ι1 =     nearly, and r = y 2 + x2 = x,
                       x
  5
      See below, p. 27. Ed.
  6
      Vol. vi. Also Phil. Mag. April, 1838, p. 335. Ed.



                                                 21
or, if we change the origin to the other extremity of the prime ray,
                                      y
                               ι1 =     ,   r = b − x,
                                      b
so that the equation becomes
                                        s
                    1                              b2
                                                           2
                  x                                             b
                                                                      
                 − = (cos ψ ± 1)                 1− 2              −1      .
                  y 2                              a            c2
                                                                                    p. 20
   Hence at each singular point the surface is touched by a cone, the equation to
the generating line of which is given by the above, the extreme angle between it
and the prime ray being
                            s                    
                              
                                    b2  2
                                         b
                                               
                      cot−1     1−         − 1 .
                                            a2       c2

   When b = a, ψ always = π2 and the cone returns into a plane.
   Again, let us suppose that the position of any perpendicular from the centre
is given, and that of the corresponding radius vector required.
   Let OA, OB 7 denote what we have termed the optic axes, but which it will
be more agreeable to analogy to term the prime perpendiculars from centre,
and let OP be the given normal. Take OQ, OR contiguous perpendiculars from
centre in planes P OQ, ROP , perpendicular to P OA, P OB respectively, then the
inclination of the two former will be the same as that of the two latter, and may
be termed µ.
   Let ι1 , ι2 now denote the angles P OA, P OB respectively, then

                       QOA = ι1 ,           QOB = ι2 + δι2 ,

                       ROA = ι1 + δι1 ,              ROB = ι2 .

   The ray will be found by joining O with the intersection of three planes drawn
at P, Q, R, perpendicular to OP, OQ, OR, respectively.
   Now from Prop. 9 it appears that
                    v(
                                 +                     +
                    u                  2                    2 )
                             ι     ι               ι     ι
                                             
                               1     2               1     2
               OP = t a2 sin              + c2 cos
                    u
                                                                 ,
                                       2                          2

using only one sign for the sake of simplicity, which we may do by throwing the
ambiguity upon the way in which ι1 or ι2 is measured, also
                                                 d · OP
                            OQ = OP +                   δι2 ,
                                                   dι2


                                            22
                                                           T
                              L
                                                                       Q
                                             R                 P




                                                   O

                                                 Fig. 5.


                                                        d · OP
                                   OR = OP +                   δι1 .
                                                          dι1
 Let δι1 = δι2 , then it is clear that OQ = OR, and the intersection of the two
planes perpendicular to OQ, OR is therefore a line perpendicular to the plane
QOR, and to the line which bisects the angle QOR.
   In fact if we draw QT, RT perpendicular to OQ, OR respectively in the plane
QOR, the intersection in question passes through T and is perpendicular to OT ;
also
                                           1
                                               
                          OT = OQ · sec      ROQ = OQ
                                           2
to the first order of smallness.                                                            p. 21
   Now it is easy to see (just as on p. 16) that

                                                         δι1
                                        ROP =                 ,
                                                        sin µ
and also
                                                         δι2
                                        QOP =                 ,
                                                        sin µ
therefore ROP = QOP and therefore P OT is perpendicular to QOR.
   Hence the problem is reduced to finding L the intersection of two lines T L, P L
drawn in the same plane P OT .
   Now because OT L, OP L are each right angles, a circle may be made to pass
through L, T, P, O.
   Hence the angle
                                                               OP × P OT
                           P LO = P T O = tan−1
                                                               OT − OP

                                                                            µ cos 2 µ
                                                                         δι2      1
                         OP × P OR · cos 12 µ                      OP × sin
               = tan−1            d·OP
                                                        = tan−1         d·OP
                                                                                        ,
                                   dι2 δι2                               dι2 δι2
  7
      OA, OB are not expressed in the figure.


                                                   23
and
                             OL = OP · sec P OL.
Also the position of the plane P OL is known, and therefore the radius is
completely determined in magnitude and position.
   It may be worth while also to remark that the above constructions enable
us to form a series of equations between the magnitude of the radius and its
inclinations to the two prime perpendiculars.
   In fact, if we call π1 , π2 the two inclinations in question
                                                                  µ
                  cos π1 = cos P OL cos ι1 ± sin P OL sin ι1 · sin ,
                                                                  2
                                                                  µ
                  cos π2 = cos P OL cos ι2 ∓ sin P OL sin ι2 · sin ,
                                                                  2
and of course if we call the angle between the two prime normals 2E
                           s
                               sin E + ι1 +ι   sin E − ι1 +ι
                                            2
                                                           2
                                                             
                      µ
                   sin =                  2               2
                                                              .
                      2                 sin ι1 sin ι2
                                                                   0
   Cor. 1. When ι1 or ι2 = 0, tan P OL assumes the form              which may be
                                                                   0
interpreted analogously to the method used in the reverse problem, but may be
more elegantly illustrated by
   Cor. 2. Which is that the meridian plane P OT (that is, the plane in which
both normal and radius lie) bisects the angle formed by ROP, QOP , and therefore p. 22
   that formed by the planes drawn through the normal and the two prime
normals to which these two are perpendicular.
   Now we have found (Cor. 4, page 18), that it also bisects the angle formed
by the two planes passing through the radius and the two prime radii. Hence
when the ray is given, we may find by the easiest geometry the normal and the
tangent plane, and vice versâ.
   Thus suppose (N, N ′ )(R, R′ ) to be the projections of the prime perpendiculars
and prime radii on a sphere concentric with the wave surface.
   Let n be the projection of any given perpendicular on the same sphere; join
nN, nN ′ ; bisect N nN ′ by nM , which will be the meridian plane.
   Draw from R′ , R′ T V perpendicular to nM and make R′ T = T V . Produce
RV to meet M n in r, then RrM = R′ rM , and therefore r is the projection of
the radius. Just in the same way when r is given we may find n.
   Now suppose n to come to N , then the position of the meridian plane nM
becomes indeterminate, and r from a point becomes a locus, subject to the
condition that R′ rN = RrN . From r draw rD perpendicular to rN .
   Then it is clear that because rN bisects RrR′
                         sin RD    sin Rr     sin RN
                                 =         =            ,
                         sin R D
                              ′    sin R r
                                        ′    sin R′ N ′

                                        24
                                            r


                                            n
                                                    T
                           V

                       R                                          R′
                                    N     M             N′

                                         Fig. 6.




                                                r




                       R                                          D
                                     N                       R′

                                         Fig. 7.


and therefore D is a fixed point and N D a fixed length, and

                           cos rN D = tan rN · cot N D;

therefore the projection of the locus of r upon a plane drawn at N perpendicular
to the line joining N with the centre O is given by the equation

                               ρ = ON · cot N D · cos θ,

N being the origin and the projection of N D the prime radius; which is the
equation to a circle passing through N , and whose diameter = ON cot N D.
   Hence at the extremity of each prime perpendicular the tangent plane meets
the surface in a circle passing through that extremity and whose radius = 12 b cot a,
a being to be found from the equation
                            sin(2E + a)   sin(E + e)
                                        =            ,
                                sin a     sin(E − e)
that is
                           tan(E + a) = (tan E)2 cot e.
                                                                                        p. 23


                                           25
    Just in the same way it may be shown that the trace of the perpendiculars to
the tangent planes of the surface at the point where it is pierced by any prime
radius upon a plane perpendicular to that radius at its extremity, is also a circle
passing through it, and curved in an opposite direction from the circle of plane
contact nearest to it.
    Hence the enveloping cone at these points may be described as being perpen-
dicular to the circular cone, formed by drawing lines from the centre to the above
described circle; that is every generating line of the one will be perpendicular to
the generating line which it meets of the other.
    More generally it easily appears from fig. 6 that if a series of great circles
(representing meridian planes) be taken intersecting the great circle N RR′ N ′ in
a fixed point, a plane perpendicular to the radius passing through that point, will
intersect the cone of rays as well as the cone of perpendiculars corresponding to
those meridian planes, in two circles. So that there exist an indefinite number of
circular cones of rays corresponding to circular cones of perpendiculars touching
each other in a line lying in the plane containing the extreme axes, and having
their circular sections perpendicular to that line.
    The cusps are explained by the cone of rays degenerating into a right line,
and the circles of plane contact by the cone of perpendiculars so degenerating.
    Furthermore I observe in conclusion that when a ray is given it follows from
the general geometrical construction above that there will be two meridian planes
according as we take R with R′ , or with a point 180◦ from R′ , and consequently
these two planes will be perpendicular to each other.
    And similarly when a normal is given there will be two meridian planes
perpendicular to each other.
    Thus the planes passing through any radius and the two normals at the points
where it pierces the wave surface, are perpendicular to each other, as are also
the two planes passing through any normal and its two corresponding radii.
    Moreover a glance at fig. 2 will show that the two lines of vibration corre-
sponding to any front lie respectively in the two meridian planes passing through
the perpendicular to that front or, in other words, the intersection of a plane
drawn through either ray belonging to a front perpendicular thereunto is always
a line of vibration in that front.
    This has been noticed, I think, by Sir William Hamilton for the particular
case of the singular points.
    As two fronts belong to every ray, so two rays pertain to every front. And
from what has been said above it appears that the two lines of vibration in any
front are the projections of its two rays upon its own plane.                       p. 24


                                     Note 1.

   In the paper above, it is shown that the meridian plane, that is, the plane

                                        26
containing the ray and normal, always passes through a line of vibration in the
corresponding point. Now the line of force called into action by a displacement
in the line of vibration clearly lies in this very plane; for the resolved part of it
lies in the line of vibration itself.
   Harmony and analogy concur in suggesting that as two of these four lines are
perpendicular to each other, so are also the other two, or in other words, that
the ray is always perpendicular to the direction of unresolved force.
   The following investigation verifies this conjecture.
   Let x, y, z be the coordinates of a point taken at distance unity from the origin
and in any line of vibration; then the cosines of the angles made by the line of
force with the axes are as a2 x : b2 y : c2 z respectively.
   Let ω be the inclination between the line of vibration and the line of force,
then
                           a2 x · x + b2 y · y + c2 z · z            a2 x2 + b2 y 2 + c2 z 2
      cos ω = p                                                    =                          .
                     {(a4 x2 + b4 y 2 + c4 z 2 )(x2 + y 2 + z 2 )}    a4 x2 + b4 y 2 + c4 z 2
                                                                     p


Let                                 q
                                      a4 x2 + b4 y 2 + c4 z 2 = P,
then
                                          P 2 = v 4 (sec ω)2 .
  Now let α, β, γ be the angles of inclination between the coordinate planes
and the front in which the line of vibration lies, and λ some quantity to be
determined. I have shown in Prop. 3 that if

                                        λ cos α = (a2 − v 2 )x,

then will
                                        λ cos β = (b2 − v 2 )y,
and
                                        λ cos γ = (c2 − v 2 )z;
therefore

        λ2 = a4 x2 + b4 y 2 + c4 z 2 − 2v 2 (a2 x2 + b2 y 2 + c2 z 2 ) + v 4 = P 2 − v 4 .

Again,

                     (cos α)2     (cos β)2     (cos γ)2
                (                                                )
          λ 2
                                +            +                       = x2 + y 2 + z 2 = 1;
                    (a2 − v 2 )2 (b2 − v 2 )2 (c2 − v 2 )2

therefore
                          1       (cos α)2     (cos β)2     (cos γ)2
                               =             +            +             .
                      P 2 − v4   (a2 − v 2 )2 (b2 − v 2 )2 (c2 − v 2 )2

                                                  27
Now
                         1                  1                1
                                =                        =      (cot ω)2 .
                     P 2 − v4       v 4 (sec ω)2 − v 4       v4
                                                                                           p. 25
   And in Mr Smith’s investigation of the form of the wave surface (already
alluded to8 ) by great good fortune I find ready to my hand

                (cos α)2      (cos β)2      (cos γ)2         1
                           +             +             = 2 2         ,
               (a − v )
                 2    2  2   (b − v )
                               2    2  2   (c − v )
                                             2    2  2  v (r − v 2 )

r being the radius vector to the point whose tangent plane is parallel to the
point in question.
   Hence
                                v4            v2        v2
                  (cot ω)2 = 2 2         =         =         ,
                            v (r − v 2 )   r2 − v2   r2 − p2
p being the length of the perpendicular from the centre upon the tangent plane,
for p = v.
   Hence (cot ω)2 = the square of the cotangent of the angle between radius
vector and normal.
   Or, in other words, the line of force is as much inclined to the line of vibration
as the ray is to the normal.
   Now the normal is perpendicular to the line of vibration, and all four lines lie
in one plane.
   Therefore the ray is perpendicular to the line of force.        Q. E. D.
   I may be allowed to conclude this long paper with a summary of some of the
most remarkable consequences which I have extricated from Fresnel’s hypothesis.
   (1) The two meridian planes corresponding to any given radius are perpendic-
ular to each other9 .
   (2) So are the two corresponding to any given normal.
   (3) Every meridian plane bisects the angle formed by two planes drawn through
the radius and the two prime radii.
   (4) It also bisects the angle formed by two planes drawn through the normal
and the two prime normals.
   (5) Each meridian plane contains one line of vibration and the corresponding
line of force.
   (6) The ray is perpendicular to the line of force.
   All these conclusions, except the fourth, are, I believe, original.                p. 26
   The theory of external and internal conical refraction follows immediately as
a particular consequence from the third and fourth combined as already shown;
  8
    See above, p. 18.
  9
    I have defined the meridian plane to be that which contains radius vector and normal
belonging to the same point.


                                             28
the same propositions also enable us to draw a tangent plane to any point of the
wave surface by mere Euclidean geometry. May not some of these conclusions
serve to suggest to physical inquirers the question, Has the theory been started
from the most natural point of view?

            Note 2.               Investigation 10 of the Wave Surface.

   Since the appearance of the preceding parts, I have succeeded in completing
the self-sufficiency of my method by deducing the equation to the wave surface
from the expressions given in Prop. 5 for the angles between a front and the
principal planes in terms of its two velocities. If these angles be ω, ϕ, ψ, and the
two velocities v1 , v2 we found
                                                s
                                                    (a2 − v12 )(a2 − v22 )
                                   cos ω =                                 ,
                                                    (a2 − b2 )(a2 − c2 )
                                                s
                                                    (b2 − v12 )(b2 − v22 )
                                   cos ϕ =                                 ,
                                                    (b2 − a2 )(b2 − c2 )
                                                s
                                                    (c2 − v12 )(c2 − v22 )
                                   cos ψ =                                 .
                                                    (c2 − a2 )(c2 − b2 )
Let the tangent plane to the wave surface be written
                          cos ω     cos ϕ     cos ψ
                                ·x+       ·y+       · z = 1, 11                                      (α)
                           v1        v1        v1
then
                                                      ϕ                      ψ
                           d cos
                              v1
                                 ω
                                                d cos
                                                   v1                  d cos
                                                                          v1
                                      2 x +               2 y +               2 z = 0,         (β)
                                  1                     1                     1
                          d       v1            d       v1            d       v1

                              d cos ω     d cos ϕ     d cos ψ
                                       x+          y+         z = 0.                                 (γ)
                              d(v2 ) 2    d(v2 ) 2    d(v2 )2
Let12                 s
                1         a2 − v12                           q                                 1
                                   = ξ,                          (a2 − b2 )(a2 − c2 ) =          ,
                v1        a2 − v22                                                             A
                      s
                 1        b2 − v12                           q                                 1
                                   = η,                          (b2 − a2 )(b2 − c2 ) =          ,
                 v1       b2 − v22                                                             B
                      s
                 1        c2 − v12                           q                                 1
                                   = ζ,                          (c2 − a2 )(c2 − b2 ) =          ,
                 v1       c2 − v22                                                             C




                                                             29
                                                                                                             p. 27
     then equation (γ) becomes

                                       Aξx + Bηy + Cζz = 0,                                           (1)

and equation (β)
                                   Aa2    Bb2    Cc2
                                       x+     y+     z = 0,                                           (2)
                                    ξ      η      ζ
and equation (α) may be written under two forms, viz.

                   (a2 − v22 )Aξx + (b2 − v22 )Bηy + (c2 − v22 )Cζz = 1,                              (3)

or                            !                          !                      !
                       a2     A            b2     B                       c2     C
                           −1   x+             −1   y+                        −1   z = 1.             (4)
                       v12    ξ            v12    η                       v12    ζ
From (1)
                                        Aξx + Bηy = −Cζz.                                             (5)
From (2)
                                    Aa2    Bb2     Cc2
                                        x+     y=−     z.                                             (6)
                                     ξ      η       ζ
From (3) and (1)
                               A(a2 − c2 )ξx + B(b2 − c2 )ηy = 1.                                     (7)
From (2) and (4)
                                                x              y
                               A(a2 − c2 )        + B(b2 − c2 ) = c2 .                                (8)
                                                ξ              η
From (5) and (6)

                                                                               η      ξ
                                                                                       
                   C 2 c2 z 2 − B 2 b2 y 2 − A2 a2 x2 = ABxy a2                  + b2   .             (9)
                                                                               ξ      η

From (7) and (8)

                                                                          η   ξ
                                                                               
 c2 − B 2 (b2 − c2 )2 y 2 − A2 (a2 − c2 )2 x2 = ABxy                        +   (a2 − c2 )(b2 − c2 ). (10)
                                                                          ξ   η
  10
    This investigation supplies the step which Mr Tovey was desirous should appear in the
Magazine. [Phil. Mag. March, 1838, p. 261. Ed.]
 11
    In lieu of v1 we might write
                           r v2 in the denominator without affecting the result.
                                  n                             o
                                       a2
                                        2 − 1       (a2 − v22 )
                       cos ω           v1
  12
       Observe, that         = p                        , and so on for the rest.
                        v1       {(a2 − b2 )(a2 − c2 )}




                                                         30
From (9) and (10)

                                                      ξ
                 AB(a2 − b2 )(a2 − c2 )(b2 − c2 )xy
                                                      η
                    = a2 c2 − (a2 − c2 )(b2 − c2 )C 2 c2 z 2
                      − {a2 (b2 − c2 )2 − b2 (a2 − c2 )(b2 − c2 )}B 2 y 2
                      − {a2 (a2 − c2 )2 − a2 (a2 − c2 )(b2 − c2 )}A2 x2
                    = a2 c2 − c2 z 2 − c2 y 2 − a2 x2 .                                (11)

From (11), interchanging (a, x, ξ) with (b, y, η) we have
                                               η
          AB(b2 − a2 )(b2 − c2 )(a2 − c2 )xy     = b2 c2 − c2 z 2 − c2 x2 − b2 y 2 .   (12)
                                               ξ

Finally, from (11) and (12) we have

{a2 c2 − (a2 − c2 )x2 − c2 (x2 + y 2 + z 2 )}{b2 c2 − (b2 − c2 )y 2 − c2 (x2 + y 2 + z 2 )}

                               = (a2 − c2 )(b2 − c2 )x2 y 2 ,
that is
                (x2 + y 2 + z 2 )(a2 x2 + b2 y 2 + c2 z 2 ) − a2 (b2 + c2 )x2
                    − b2 (a2 + c2 )y 2 − c2 (b2 + a2 )z 2 + a2 b2 c2 = 0
the equation required.




                                             31
                                             2.
                  On the Motion and Rest of Fluids
              [Philosophical Magazine, XIII. (1838), pp. 449–453]
                                                                                       p. 28
M. Ostrogradsky’s memoir on this subject inserted in the Scientific Memoirs
seems to have excited much attention, and has been made the occasion of some
annotations13 by a distinguished writer in the Philosophical Magazine. Mr Ivory’s
recent papers in the same periodical must still more tend to invest with a new
interest all such speculations. It seems to me desirable therefore to present the
theory of fluids in all the simplicity of which it is susceptible.
   I consider a fluid as a collection of particles subject to some law of relative
position other than that of rigidity. These particles by their mutual actions
maintain the connections of the system. As to the law of force between them
we know nothing; but I assume it is a general principle of nature, that for each
instant of time the sum of the internal actions (reckoned by the product of each
particle into the square of the space due to the internal force acting on it) is
a minimum. This in fact is Gauss’s principle of least restraint. We may if we
please split this principle into two parts; that is to say, assume that the internal
system of forces is always such as if acting alone would keep the fluid at rest;
and then again assume that any equilibrating system of forces must be subject
to the law of virtual velocities. I say assume, because it is impossible à priori to
prove this.
   Lagrange’s so-called demonstration is unworthy of his name, and (albeit
sanctioned by the powerful oral authority of an ex-Cambridge Professor) contrary
alike to sense and honesty. It is better therefore at once to proceed upon Gauss’s
principle. It might easily be shown that this is in effect tantamount in all cases
to D’Alembert’s and Lagrange’s principles combined.
   Before entering upon the investigation I may call attention to one point of
great analytical interest, and relating to the difficult subject of the algebraical
sign, viz. that if the density of a point (x, y) in any circumscribed space be
expressed by the quantity du  dx + dy so that the mass is
                                   dv


                                  du                        dv
                       ZZ                      ZZ          
                            dx dy            +        dx dy    ,
                                  dx                        dy
that is not equivalent to         Z                                                    p. 29

                                       (u dy + v dx),

that is if we please
                                        dy  dx
                                Z                    
                                       u +v    ds,
                                        ds  ds
   Phil. Mag. May, 1838, p. 385. Ed.
  13




                                             32
(where s is for clearness’ sake and to avoid double limits taken an element of the
bounding curve) as at first sight it might appear to be, but is in fact equal to

                                       dy  dx
                                  Z              
                                      u −v    ds.
                                       ds  ds

I shall demonstrate this point in the next number14 of the Magazine. It at
first caused me some trouble in conducting the annexed inquiry. I shall also
take occasion at some other time to revert to a new species (as I believe) of
partial differential equations; that is to say, where there are fewer of them
than of the principal variables, which may be called therefore Indeterminate
Partial Differential Equations. A complete solution of one of these appears in
the subjoined

                                      Investigation.

   For the sake of simplicity I take an incompressible fluid. The method is nowise
different for a fluid of varying density.
   Let ∆x, ∆y, ∆z be any displacement undergone by a particle at the point
x, y, z parallel to the axes x, y, z respectively; it is easily shown that to satisfy
the condition of invariability of mass we must have

                                d∆x d∆y d∆z
                                    +    +    = 0.                               (α)
                                 dx   dy   dz
One relation between u, v, w the velocities parallel to x, y, z is obtained immedi-
ately by putting uδt, vδt, wδt, for ∆x, ∆y, ∆z, which gives
                                  du dv dw
                                    +   +    = 0,                                 (1)
                                  dx dy   dz
as usual.
   Again, if X, Y, Z be the impressed forces, and X1 , Y1 , Z1 the internal forces
acting on any particle parallel to the axes, we have
                                    du du      du     du
                           X1 + X =    +    u+     v+    w,                       (2)
                                    dt   dx     dy    dz
                                    dv   dv    dv     dv
                           Y1 + Y =    +    u + v + w,                            (3)
                                    dt   dx    dy     dz
                                    dw dw        dw    dw
                           Z1 + Z =    +     u+     v+    w,                      (4)
                                    dt    dx     dy    dz
from the mere geometry of the question.                                                 p. 30

  14
       p. 36, below. Ed.



                                           33
  Finally, Gauss’s principle teaches us that
                     ZZZ
                              dx dy dz{X1 ∆X1 + Y1 ∆Y1 + Z1 ∆Z1 } = 0.                            (β)

Now
                              d(X + X1 ) d(Y + Y1 ) d(Z + Z1 )
                                        +          +
                                 dx         dy         dz
                     2                2                2
                du                dv                dw           dv dw dw du du dv
                                                                                       
        =                 +                 +                 +2       +       +       ,
                dx                dy                dz           dz dy   dx dz   dy dx
as appears from the equations (1), (2), (3), (4); and hence
                                       d∆X1 d∆Y1 d∆Z1
                                           +    +     = 0,
                                        dx   dy   dz
the complete solution of which, free from the sign of integration, is
                                                      dψ dϕ
                                                ∆X1 =    −    ,
                                                      dy   dz
                                                      dω dψ
                                                ∆Y1 =    −    ,
                                                      dz   dx
                                                      dϕ dω
                                                ∆Z1 =    −    ,
                                                      dx   dy
ω, ϕ, ψ being any three independent functions of x, y, z.
   On substituting these values in equation (β) we obtain

                    dψ      dψ                                               dω      dω
  ZZZ                                                     ZZZ                       
        dx dy dz X1    − Y1                             +       dx dy dz Y1     − Z1
                    dy      dx                                               dz      dy
                                                                                 dϕ       dϕ
                                                                ZZZ                         
                                                              +      dx dy dz Z1     − X1      = 0.
                                                                                 dx       dz
This may be put under the form
                                           d             d
                          Z        ZZ                                        
                              dz             (ψX1 ) −
                                            dx dy           (ψY1 )
                                          dy            dx
                                             d             d
                            Z   ZZ                                 
                          + dx      dy dz       (ωY1 ) − (ωZ1 )
                                             dz           dy
                                              d            d
                            Z   ZZ                                 
                          + dy      dz dx        (ϕZ1 ) − (ϕX1 )
                                             dx            dz
                                                dX1 dY1
                            ZZZ                             
                          −     dx dy dz · ψ        −
                                                 dy      dx
                                                dY1 dZ1
                            ZZZ                             
                          −     dx dy dz · ω        −
                                                dz      dy
                                               dZ1 dX1
                            ZZZ                             
                          −     dx dy dz · ϕ        −          = 0.
                                                dx      dz

                                                              34
                                                                                       p. 31
   Here it must be remembered that ω, ϕ, ψ are perfectly independent of each
other. Also the values of the three first written quantities depend upon the values
of X1 , Y1 , Z1 at the bounding surface; the values of the three last-written depend
upon the general values of X1 , Y1 , Z1 . It is clear therefore that each system of
three equations and each member of each system must be separately zero.
   The three latter equations give
                                  dX1 dY1
                                                   
                                      −    = 0,
                                               
                                   dy   dx
                                               
                                               
                                               
                                               
                                               
                                  dY1 dZ1      
                                      −    = 0,                                 (γ)
                                   dz   dy     
                                               
                                  dZ1 dX1
                                               
                                               
                                           = 0,
                                               
                                      −        
                                   dx   dz
The three former require that for each section of the surface parallel to the plane
xy                         Z
                                  ψ(X1 dx + Y1 dy) = 0,

for each section parallel to yz
                            Z                          
                                ω(Y1 dy + Z1 dz) = 0,
                                                     
                                                     
                                                           ∗
                            Z                                                    (δ)
                                ϕ(Z1 dz + X1 dx) = 0
                                                    
                                                    


for each section parallel to zx and these equations are to hold good whatever
ψ, ϕ, ω may be. From the equations (γ) we derive

                           X1 dx + Y1 dy + Z1 dz = df,                           (5)

from equations (δ) we obtain

      f = constant for all points in any section of the bounding surface
      parallel to the plane of xy,
      f = constant for all points in any section of the bounding surface
      parallel to the plane of yz,
      f = constant for all points in any section of the bounding surface
      parallel to the plane of zx.

Now by drawing through all the points in a plane parallel to xy, planes parallel
to yz, we may cover the whole surface; hence f is constant all over the surface
bounding the fluid.

                            * See remark at introduction.


                                          35
                                                                                      p. 32
  Therefore
                           X1 dx + Y1 dy + Z1 dz = 0,                           (6)
for all variations of dx, dy, dz taken upon the surface.
   The equations (1, 2, 3, 4, 5, 6) are coincident with those obtained by the usual
method; with this difference, that X1 , Y1 , Z1 here take the place of
                                  dp         dp         dp
                              −      ,   −      ,   −      .
                                  dx         dy         dz
Thus then we have obtained all the conditions requisite for determining the
motion of fluids from the universal principle of least constraint conjoined with
the specific character of the system in question.

                                General Remarks.

   In the case of equilibrium, that is in the case where no particle moves, we have
X1 + X = 0, Y1 + Y = 0, Z1 + Z = 0. Hence X dx + Y dy + Z dz is a complete
differential always and zero for the surface.
   The above results have been obtained upon the principles of the differential
calculus, and the continuity of the forces has been tacitly assumed. If now we
were to suppose forces of finite magnitude (as compared with the whole sum
acting upon the entire system) to be applied to a layer of single particles or to
a layer of a thickness of the same order of magnitude as the distances between
the particles themselves, (which has been treated as an infinitesimal) it would
appear that our results would be no longer applicable, just in the same manner
as it would be erroneous to apply the principle of vis-viva (for example) without
modification, to the case of impulsive forces, because we had deduced it by the
calculus in the case of the motion being continuous. Hence the above equations
ought not strictly to apply to the motion or rest of a fluid contained between
physical surfaces; for the pressure afforded by these surfaces, whatever its actual
value may be, we know à priori is commensurable with the whole amount of
force acting on the fluid; but the immediate application of this pressure (alias
repulsive force) is confined to the bounding layer of fluid particles, or at most
extends to a distance bearing a low ratio to the distances between the particles
themselves.
   Accordingly, to the non-applicability of the equations for free fluids to the
case of fluids confined at the boundaries, and to an independent investigation
upon the minimum principle for this class of problems, it is, that I look for the
true explanation of the phenomena of capillary attraction (vulgarly so called).




                                         36
                                           3.
              On the Motion and Rest of Rigid Bodies
               [Philosophical Magazine, XIV. (1839), pp. 188–190]
                                                                                            p. 33
In the subjoined investigation, which, as far as I know, is my own, I apply the
same method to rigid as in the preceding paper I applied to fluid systems.
   Let x, y, z be the coordinates of any particle in a rigid body; x′ , y ′ , z ′ the
coordinates of some other particle, and let

                       x′ = x + h,     y ′ = y + k,     z ′ = z + l.

Call ∆x, ∆y, ∆z the increments which x, y, z receive after the lapse of a small
interval of time; so that terms in which they enter in two or more dimensions
may be neglected.
   Then
                                   d∆x    d∆x    d∆x
                    ∆(x′ ) = ∆x +      h+     k+      l + P,
                                    dx     dy     dz
                                   d∆y    d∆y    d∆y
                    ∆(y ′ ) = ∆y +     h+     k+     l + Q,
                                    dx     dy     dz
                                   d∆z    d∆z    d∆z
                    ∆(z ′ ) = ∆z +     h+     k+     l + R,
                                    dx     dy     dz
P, Q, R containing binary and higher combinations of h, k, l, which we shall have
no occasion to express.
   At the commencement of the interval the squared distance of the two particles
was (x′ − x)2 + (y ′ − y)2 + (z ′ − z)2 ; at the end of the interval the distance squared
is

  (x′ − x + ∆(x′ ) − ∆x)2 + (y ′ − y + ∆(y ′ ) − ∆y)2 + (z ′ − z + ∆(z ′ ) − ∆z)2 ,

and these two expressions must be the same by the conditions of rigidity whatever
h, k, and l may be; that is

                                                    2
                         d∆x     d∆x     d∆x
                                                        
  h2 + k 2 + l2 = h +        h+      k+      l+P
                          dx      dy      dz
                                                        2
                              d∆y     d∆y     d∆y
                         
                        + k+      h+       k+      l+Q
                               dx      dy       dz
                                                                     2
                                              d∆z     d∆z     d∆z
                                         
                                       + l+        h+      k+     l+R ,
                                               dx      dy      dz
for all values of h, k, and l.                                                              p. 34


                                           37
   Hence rejecting infinitesimals of the second order and equating to zero sepa-
rately the coefficients of h2 , k 2 , l2 , and of kl, lh, hk, we have

                  d∆x                        d∆y d∆z
                      = 0,        (a)            +    = 0,     (d)
                   dx                         dz   dy
                  d∆y                        d∆z d∆x
                      = 0,               (b)     +         = 0,(e)
                   dy                         dx   dz
                  d∆z                        d∆x d∆y
                      = 0,               (c)     +         = 0.(f )
                   dz                         dy   dx

By differentiating (d), (e), (f ) with respect to z, x, y respectively, and substituting
from (a), (b), (c), we obtain

                     d2 ∆y              d2 ∆z          d2 ∆x
                           = 0,               = 0,           = 0.
                      dz 2               dx2            dy 2
By differentiating the same with respect to y, z, x respectively, and proceeding
as before, we have

                     d2 ∆z              d2 ∆x          d2 ∆y
                           = 0,               = 0,           = 0.
                      dy 2               dz 2           dx2
Thus, then, we have

                        d∆x             d2 ∆x        d2 ∆x
                            = 0,              = 0,         = 0,
                         dx              dy 2         dz 2
                        d∆y             d2 ∆y        d2 ∆y
                            = 0,              = 0,         = 0,
                         dy              dz 2         dx2
                        d∆z             d2 ∆z        d2 ∆z
                            = 0,              = 0,         = 0,
                         dz              dx2          dy 2
therefore

                                ∆x = A + By + Cz,                                    (o)
                                ∆y = D + Ez + F x,                                   (p)
                                ∆z = G + Hx + Ky,                                    (q)

A, B, C, D, E, F being constant for a given instant of time; between which by
virtue of the equations (d), (e), (f ), we have the relations

                   E + K = 0,           H + C = 0,      B + F = 0.

If we call u, v, w the three component velocities of the particles at x, y, z parallel
to the three axes, and X1 , Y1 , Z1 , the three internal forces, it is at once seen that



                                            38
u, v, w, as also ∆X1 , ∆Y1 , ∆Z1 must be subject to the same equations as limit
∆x, ∆y, ∆z; so that

                                    u = a + γy − βz,                                  (1)
                                    v = b + αz − γx,                                  (2)
                                    w = c + βx − αy,                                  (3)
                                ∆X1 = a1 + γ1 y − β1 z,                              (h)
                                ∆Y1 = b1 + α1 z − γ1 x,                               (j)
                                ∆Z1 = c1 + β1 x − α1 y.                              (k)
                                                                                            p. 35
   Also if X, Y, Z be the impressed forces, we have
                                              du
                                     X1 + X =    ,                                    (4)
                                              dt
                                              dv
                                     Y1 + Y =    ,                                    (5)
                                              dt
                                              dw
                                     Z1 + Z =     .                                   (6)
                                              dt
And by Gauss’s principle, calling m the mass of the particle at x, y, z,

                            ∆       m(X12 + Y12 + Z12 ) = 0.
                                X


Hence equating separately to zero the coefficients of a1 , b1 , c1 and of α1 , β1 , γ1 in
the quantity m(X1 ∆X1 + Y1 ∆Y1 + Z1 ∆Z1 ) we have
            P


                                                mX1 = 0,
                                            X           
                                                        
                                                        
                                                      
                                                 mY1 = 0,
                                            X         
                                                      
                                                         
                                                         
                                                      
                                                      
                                              mZ1 = 0,
                                            X         
                                                      
                                                                                  (7–12)
                                    m(Z1 y − Y1 z) = 0,
                                X
                                                       
                                                       
                                                      
                                    m(X1 z − Z1 x) = 0,
                                X                  
                                                   
                                                       
                                                       
                                                   
                                                   
                                  m(Y x − X y) = 0.
                                X                  
                                        1         1

Lastly, we have the equations
                                           dx
                                         u=   ,                                     (13)
                                           dt
                                           dy
                                        v=    ,                                     (14)
                                           dt
                                           dz
                                        w=    .                                     (15)
                                           dt
From the fifteen equations marked (1) to (15), the motion may be determined
by assigning the position of each particle at the end of the time t in terms of its

                                            39
three initial coordinates, its three initial velocities, and the initial values of the
nine quantities
                                                    mx2 ,
                             P        P          P
                                mx,      myz,
                                                    my 2 ,
                             P        P          P
                                my,      mzx,
                                                    mz 2 .
                             P        P          P
                                mz,      mxy,
In the case of rest X1 = −X, Y1 = −Y , Z1 = −Z, and the equations (7) to (12)
inclusively taken, express the conditions of equilibrium.
   The equations (o), (p), (q), which have been obtained from conditions purely
geometrical, establish the well-known but interesting and not obvious fact, that
any small motion of a rigid body may be conceived as made up of a motion of
translation and a motion about one axis.




                                         40
                                           4.
On Definite Double Integration, Supplementary to a Former
          Paper on the Motion and Rest of Fluids
              [Philosophical Magazine, XIV. (1839), pp. 298–300]
                                                                                      p. 36
In a paper on Fluids which appeared in the December Number of this Magazine,
I had occasion to remark, that the mass of an area having at the point (x, y) a
density du
        dx + dy could be expressed by the simple formula
             dv

                               Z 0
                                         dy  dx
                                                        
                                        u −v    ds;
                               l         ds  ds
l being the length, and ds an element of the bounding curve: this may be thought
to require some explanation.
   (1) Let AP Bq represent any oval; P pL, QqM any two contiguous ordinates
cutting the curve in P p, Qq respectively, AC, BD the two extreme RR     tangents
parallel to Oy, and ρ the density at any point (x, y). The expression       ρdxdy
will serve to denote the mass of the oval area AP Bq, and the limits may be twice
taken, that is, (i) the two values of y corresponding to any one of x; and (ii) the
two values of x corresponding to C and D. This method is in fact tantamount
to taking the sum of the columns P p, qQ; but this is not necessary, for AP Bq
may be considered as the algebraical sum of the mixtilinear area AP QBDC,
and the mixtilinear area BDCApq, or (if any line O′ C ′ D′ be drawn parallel to
OCLM D) of AP QBD′ C ′ and BD′ C ′ Apq.

                      y


                                             P Q
                                                             B

                               A
                                                    q
                                                p

                     O             C′           L′M ′       D′   x
                     O′            C            LM          D
                                         Fig. 1.

   Thus then the mass = dx ( ρdy), ρdy being left indeterminate, and the
                           R       R        R

extremity of x travelled round from C to D, and back again from D to C.         p. 37
   This will be better expressed by transforming the variable, and summing
with respect to some quantity, such as the arc of the curve, which continuously

                                          41
increases, or if we please, with respect to θ, the angle subtending any point taken
within the curve.
   The mass is then              Z 0
                                                    dx
                                         Z          
                            =±        dθ      ρdy        ;
                                   2π               dθ
always remembering that no constant need be added to ρdy, and that the
                                                                                              R

doubtful sign arises from the choice of ways in which θ may be measured round.
If the area be not included by one line; but by several, as for example, by a curve
and a right line, the above integral, if broken up into as many parts as there are
breaches of continuity, will still apply.
    (2) Let us suppose that we have two areas exactly coinciding with, and
overlapping one another; but the density of the one at (x, y) to be ρ, and of the
other ρ′ .
    Let the mass of the first be treated as the sum of columns parallel to Oy, and
that of the second as the sum of columns parallel to Ox.
    The one will be represented by
                                          Z 0
                                                                      dx
                                                     Z          
                                      ±         dθ         ρdy           ,
                                           2π                         dθ
the other will be represented by
                                          Z 0
                                                                      dy
                                                     Z           
                                                            ′
                                      ±         dθ         ρ dx          ,
                                           2π                         dθ
and the sum of the two, or the joint mass, by
                      Z 0                                  Z 0
                                           dx                                                 dy
                                 Z                                  Z                  
                                                                                  ′
                  ±         dθ         ρdy    ±                  dθ              ρ dx            .
                       2π                  dθ               2π                                dθ
So long as these two operations are performed separately, the doubtful signs
may be preserved in each term, because s need not be travelled round in the
same direction for the two summations; but if we perform the second integration
conjointly for the two masses, their sum
                        Z 0
                                                    dx                        dy
                                      Z                        Z                         
                                                                             ′
                  =±             dθ             ρdy    ±?                ρ dx    ,
                            2π                      dθ                        dθ
the mark of interrogation denoting that one or the other, but not either of the
signs ± must be used, and the question is, which?
   This will be answered by taking different points in the bounding line which may
be continuous or not. Now every line returning into itself, whether continuous or
not, will naturally divide with respect of any given                               p. 38
   system of axes, into at most four parts, or sets of parts; two in which dx and
dy both increase or both decrease, and two in which one increases and the other
decreases.

                                                      42
  Take P1 , P2 , P3 , P4 , any points in the four quadrants respectively, it will be
observed that,

                        y

                                                             P2
                                             P1
                                                                  P3
                                        P4


                                                                           x
                                                  Fig. 2.

  At P1 the ρ column enters additively, and the ρ′ column subtractively.
  At P2 both columns are additive.
  At P3 the ρ′ column is additive and the ρ column subtractive.
  At P4 both columns enter subtractively.
  Again, reckoning round in the direction of the arrows,

     At P1 , x and y are both increasing.
     At P2 , x is increasing and y decreasing.
     At P3 , x and y both decrease.
     At P4 , x is decreasing and y increasing.

Thus when ρdy and ρ′ dx       are affected with the same signs, dx and dy are of
            R               R

opposite signs; and when ρdy, ρ dx are of opposite signs, dx and dy are of
                           R      R ′

the same sign.
   Hence it appears that the mass of the area, whose density at (x, y) is ρ + ρ′ ,
is capable of being represented by
                        Z 0
                                                dx                    dy
                                      Z                  Z            
                                                                  ′
                    ±            dθ         ρdy    −             ρ dx    .
                            2π                  dθ                    dθ




                                                      43
                                                5.
       On an Extension of Sir John Wilson’s Theorem to All
                       Numbers Whatever
                   [Philosophical Magazine, XIII. (1838), p. 454]
                                                                                                    p. 39
The annexed original theorem in numbers will serve as a pendant to the elegant
discovery announced by the ever-to-be-lamented and commemorated Horner15
with his dying voice, in your valued pages16 .

                                          Theorem.

   If N be any number whatever and

                                       p1 , p2 , p3 . . . pc

be all the numbers less than N and prime to it, then either

                                    p1 · p2 · p3 . . . pc + 1,

or else
                                    p1 · p2 · p3 . . . pc − 1,
is a multiple of N .




  15
     Horner’s proof is highly valuable as a novel and highly ingenious form of reasoning, but his
theorem may be deduced with infinitely more ease and brevity from Fermat’s than he seems to
have been aware of.
  16
    Phil. Mag. Vol. XI. p. 456. Ed.


                                               44
                                          6.
                              Note to the Foregoing
                   [Philosophical Magazine, XIV. (1839), pp. 47, 48]
                                                                                     p. 39
I have to apologize for calling “original” (in the last Number of the Magazine)
the theorem of numbers which I termed “a pendant to Horner’s theorem.” This
Mr Ivory has done me the honour to inform me may be found in Gauss’s
Disquisitiones Arithmeticæ, p. 76. As Horner’s extension of Fermat’s theorem
suggested this extension of Sir John Wilson’s to me, so I concluded that had this
extension of Wilson’s been known to the world it would naturally have suggested
his to Horner. No acknowledgment of this kind having been made, I took it for
granted that the theorem I gave was new. Undoubtedly had Mr Horner been
aware of Gauss’s theorem he would have made mention of it.
   I take this opportunity of adding that my acquaintance with Gauss’s principle17
has not been derived from the study of his works, but from a casual statement
of it in an English work, Dynamics, by Mr Earnshaw, of St John’s College,
Cambridge.




  17
       See p. 28 above. Ed.


                                          45
                                               7.
On Rational Derivation from Equations of Coexistence, that
is to say, a New and Extended Theory of Elimination. Part I
                 [Philosophical Magazine, XV. (1839), pp. 428–435]
                                                                                                     p. 40
  Any number of equations existing at the same time and having the same
quantities repeated, may be termed equations of coexistence: in the present
paper we consider only the case of two algebraical equations:

                        xm + a1 xm−1 + a2 xm−2 + · · · + am = 0,

                          xn + b1 xn−1 + b2 xn−2 + · · · + bn = 0.
The above being “equations of coexistence,” x is called “the repeating term.”
  If we suppose the equation

                         c0 xr + c1 xr−1 + c2 xr−2 + · · · + cr = 0

to be capable of being deduced from the two above, and, therefore, necessarily
implied by them, this will be called “a Particular Derivative” from the equations
of coexistence, of the rth degree, (r being supposed less than m and n18 ), and
the coefficients being rational functions of the coefficients of the equations of
coexistence.
   There will be an indefinite number in general of such derivatives, and the
form involving arbitrary quantities which includes them all is called “the general
derivative of the rth degree.”
   Any “Particular Derivative,” in which the terms are all integral, numerically
as well as literally speaking, is called an “Integral Derivative.”
   That “Integral Derivative” of any given degree in which the literal parts of
the coefficients are of the lowest possible dimensions19 , and the numerical parts
as low as they can be made, is called the “Prime Derivative” 20                    p. 41
   of that degree. So that there is nothing left ambiguous in the prime derivative
save the sign.
   The “Derivative by succession” is that particular derivative which is obtained
by performing upon the equations of coexistence the process commonly employed
  18
      This restriction upon the value of r is not essentially requisite, and is only introduced to
keep the attention fixed upon the particular objects of this first Part.
   19
      Of course the dimensions of the coefficients in the equations of coexistence are to be
understood as denoted by the indices subscribed.
   20
      The results of this and some following papers were repeated, with demonstrations, in the
paper “On a Theory of the Syzygetic Relations of two rational integral functions comprising an
application to the Theory of Sturm’s Functions, and that of the greatest Algebraical Common
Measure,” Phil. Trans. Royal Soc. Vol. CXLIII., Part I. pp. 407–548, 1853. See below Section
II. Art. (16) of that paper. Ed.


                                               46
for the discovery of the greatest common measure, and equating the successive
remainders to zero.
   To express the product of the sums formed by adding each of one row of
quantities to each of another row, we simply write the one row above the other;
a notation clearly capable of extension to any number of rows, which would not
be the case if we spoke of differences instead of sums21 .


                                        Theorem 1.

   Let h1 , h2 , . . . , hm , be the roots of one equation of coexistence, k1 , k2 , . . . , kn ,
the roots of the other. The general derivative of the rth degree is represented by
                                                                    (                      )!
                                                                   hr+1 , hr+2 . . . hm
Σ SR(h1 , h2 , h3 . . . hr ){(x − h1 )(x − h2 ) · · · (x − hr )} ×                              = 0,
                                                                   −k1 , −k2 . . . − kn

SR(h1 , h2 , h3 . . . hr ) denoting any symmetrical rational (integral or fractional)
function of h1 , h2 . . . hr ;     (                      )
                                     hr+1 , hr+2 . . . hm
                                     −k1 , −k2 . . . − kn
being to be interpreted as above explained, and Σ of course including as many
terms as there are ways of putting n things r and r together22 .
   A form tantamount to the above, and which may be substituted for it, is its
analogue,
                                                                (                      )!
                                                               kr+1 , kr+2 . . . kn
Σ SR(k1 , k2 . . . kr ){(x − k1 )(x − k2 ) · · · (x − kr )} ×                               = 0.
                                                              −h1 , −h2 . . . − hm

When r = 0 the theorem gives simply
                                (                        )
                                    h1 , h2 . . . hm
                                                             = 0,
                                  −k1 , −k2 . . . − kn

and is coincident with that given by Bezout in his Theory of Elimination.                           p. 42


                                 Subsidiary Theorem (A).
  21
      The wider views which I have attained since writing the above, and which will be developed
in a future paper, lead me to request that this notation may be considered only as temporary.
It would have been more in accordance with these views to have used the two rows to denote
products of differences than of sums. But a change now in the text would be very apt to cause
errors in printing.
   22
      The general derivative may clearly be expressed also by the sum of any two particular
derivatives affected respectively with arbitrary rational coefficients. The equivalency of an
arbitrary function to two arbitrary multipliers is very remarkable, and analogous to what occurs
in the solution of certain differential equations.


                                              47
   If h1 , h2 . . . hm be the roots of the equation
                      xm + a1 xm−1 + a2 xm−2 + · · · + am = 0,
and if
                    em + a1 em−1 + a2 em−2 + · · · + am − u = 0,
then
                             hr1                       1 d
              Σ                                    =          Σ(er+1 ),
             (h1 − h2 )(h1 − h3 ) · · · (h1 − hm )   r + 1 du
u being made zero after differentiation.
  Cor. If R(h1 ) denote any integral rational function of h1 , then
                                            R(h1 )
                          Σ
                              (h1 − h2 )(h1 − h3 ) · · · (h1 − hm )
is always integral and is zero when the dimensions of R(h1 ) fall short of (m − 1).

                                 Subsidiary Theorem (B).

                                       SR(h1 , h2 . . . hr )
                               Σ(                              )
                                        h1 , h2 . . . hr
                                    −hr+1 , −hr+2 . . . − hm
can be expressed by the sum of terms, each of which is the product of series of
the form
                                         R(h1 )
                         Σ                                       ,
                           (h1 − h2 )(h1 − h3 ) · · · (h1 − hm )
it is always integral, and when the dimensions of the numerator fall short of
(m − r)r it vanishes23 .

                                 Subsidiary Theorem (C).

   The only modes of satisfying the equation
                     Σ{f (h1 , h2 . . . hr ) × SR(h1 , h2 . . . hr )} = 0,
for all forms of the latter factors short of (m − r)(n − r) dimensions, are to put
f (h1 , h2 . . . hr ) = 0, or else
                                                      constant
                   f (h1 , h2 . . . hr ) = (                              ).
                                                   h1 , h2 . . . hr
                                               −hr+1 , −hr+2 . . . − hm
                                                                                              p. 43

  23
     It may be remarked also in passing, that any term in the numerator which contains any
one power not greater than m − 2r may be neglected and thrown out of calculation. Moreover,
an analogous proposition may be stated of fractions in the denominators of which any number
of rows are written one under the other; see the first note, page 41.


                                                48
                                     Theorem 2.

   By virtue of the subsidiary theorem (B), the two equations
                                                (                         )
                                                     hr+1 , hr+2 . . . hm
           
                                                     −k1 , −k2 . . . − kn 
           (x − h1 )(x − h2 ) · · · (x − hr ) × (                       = 0,
                                                                         
        ±Σ                                                            )
                                                  hr+1 , hr+2 . . . hm 
                                                    −h1 , −h2 . . . − hr
                                               (                          )
                                                     kr+1 , kr+2 . . . kn
           
                                                    −h1 , −h2 . . . − hm 
           (x − k1 )(x − k2 ) · · · (x − kr ) × (                       = 0,
                                                                         
        ±Σ                                                            )
                                                  kr+1 , kr+2 . . . kn 
                                                    −k1 , −k2 . . . − kr
are each integer derivatives of the rth degree.

                                     Theorem 3.

   And by virtue of the subsidiary theorem (C), the two above equations are the
“Prime Integer Derivatives,” and are exactly identical with each other.
   Cor. 1.. The leading coefficient of the “prime derivative” of the rth degree
is always of (m − r)(n − r) dimensions.
   Cor. 2.. If Pr be the prime derivative of the rth degree and if (X = 0, Y = 0)
be the two equations of coexistence, and λr , µr the two “prime constituents of
multiplication” to the said derivative, that is if λr and µr satisfy the equation
λr X + µr Y = Pr , then the coefficient of the leading terms in λr and in µr is of
(m − r − 1)(n − r − 1) dimensions.

                                     Theorem 4.

   The “Prime Derivative” of any given degree is an exact factor of the “derivative
by succession,” of the same degree. The quotient resulting from striking out this
factor is called “the quotient of succession.”

                                     Theorem 5.

   If L1 , L2 , L3 , &c., be the leading coefficients of the derivatives occurring first,
second, third, &c., in order after the equations of coexistence, and if Q1 , Q2 , Q3 ,
&c., represent the first, second, third, “quotients of succession” reckoned in the
same order, then
                                   1                 L41              L42
              Q1 = 1,       Q2 =       ,    Q3 =         ,    Q4 =           ,
                                   L21               L22             L41 L23

                                           49
 and in general                                                                                          p. 44
                                     L2 L4 · · · L42n−4 L42n−2
                                Q2n = 24 44                    ,
                                     L1 L3 · · · L42n−3 L22n−1
                                           L41 L43 · · · L42n−3 L42n−1
                               Q2n+1 =                                 .
                                            L42 L44 · · · L42n−2 L22n
   Cor.     Hence, in place of Sturm’s auxiliary functions, we may substitute
                                                                           df x
                                                                                  
the functions derived from the equations of coexistence f x = 0,                =0
                                                                            dx
according to Theorem 2, due regard being had to the sign.
   Scholium. Hitherto it has been supposed that the values of the coefficients
in the equations of coexistence are independent of one another, but particular
relations may be supposed to exist which shall cause the leading terms given by
Theorem 2 to vanish, giving rise to anormal or singular primes, as they may be
called, of the degree r of fewer than (m−r)(n−r) dimensions. The theory of this,
the failing case (so to say), is highly interesting, and I have already discovered
the law of formation for the quotients of succession on the supposition of any
number of primes vanishing consecutively; but I forbear to vex the patience of
my reader further, the more so, as I hope soon to be able to present a complete
memoir, with all the steps here indicated filled up, and numerous important
additions, (the perfect image of which this is but a rough mould), as homage
to the learned and illustrious society which has lately done me the honour of
admitting me into its ranks.
   Why this has not already been done must be excused, by the fact of the theory
having suggested itself abroad in the intervals of sickness24 . Yet thus much will
I add in general terms, namely, that as many primes as vanish consecutively, so
many units must be added to the index 2 of the accessions 25 26                      p. 45
   received in the numerator and denominator of the subsequent quotient; and in
the quotient after that, it is not the square of the leading term of the penultimate
  24
      The reflections which Sturm’s memorable theorem had originally excited, were revived by
happening to be present at a sitting of the French Institute, where a letter was read from the
Minister of Public Instruction, requesting an opinion upon the expediency of forming tables of
elimination between two equations as high as the 5th or 6th degree containing one repeating
term. The offer was rejected, on the ground of the excessive labour that would be required. I
think that this has been very much overrated; and probably many will be of the same opinion
who have dwelt upon the fact that no numerical quantity will occur in the result higher than
the highest index of the repeating term. Would it not redound to the honour of British science
that some painstaking ingenious person should gird himself to the task? and would not this be
a proper object to meet with encouragement from the Scientific Association of Great Britain?
   25
      That the appearance of the index 4 may not startle, let my reader bear in mind that there
are what may be termed secondary derivatives of succession for every degree appearing in the
process of successive division.
   26
      The prime derivatives must be capable of yielding an internal evidence of the truth of
Sturm’s theorem. In fact, for the case of all the roots being possible, a little consideration    will
serve to show that the leading term of each prime derivative of the equation f x df           = 0 will
                                                                                          x
                                                                                            
                                                                                         dx
consist of a series of fractions, each of which fractions is, numerically speaking, of the same sign.


                                                 50
prime,–but the product of this term by the leading term of that anormal prime
of the same degree which has the lowest dimensions,–that finds its way into the
numerator. The rest of the formation remaining undisturbed, unless and until a
new failure have taken place.

                           Note on Sturm’s Theorem.

   When one of the equations of coexistence is the differential coefficient with
respect to the repeated term of the other, the prime derivatives given in Theorem
2 which coincide in this case with Sturm’s auxiliary functions reduced to their
lowest terms, may be exhibited under an integral aspect.
   Let SPD intimate that the squared product of the differences is to be taken
of the quantities which follow it.
   Let S1 indicate the sum of the quantities to which it is prefixed.
        S2 the sum of the binary products.
        S3 the sum of the ternary products, and so on
   Let h1 , h2 . . . hn be the roots of any equation.
   Then Sturm’s last auxiliary function may be replaced by

                                  SPD(h1 , h2 . . . hn ).

The last but one may be replaced by

      ΣSPD(h1 , h2 . . . hn−1 )x + ΣS1 (h2 , h3 . . . hn−1 )SPD(h1 , h2 . . . hn−1 ).

The one preceding by

  ΣSPD(h1 , h2 . . . hn−2 )x2 + ΣS1 (h1 , h2 . . . hn−2 )SPD(h1 , h2 . . . hn−2 )x
                                       + ΣS2 (h1 , h2 . . . hn−2 )SPD(h1 , h2 . . . hn−2 ),

and so on.
  Thus then Sturm’s rule for determining the absolute number of real roots in
an equation is based wholly and solely upon the following

                            Algebraical Proposition.

  If there be n quantities, real and imaginary, the imaginary ones entering in
pairs, as many changes of sign as there are in the terms
                                     ΣSPD(h1 , h2 ),
                                   ΣSPD(h1 , h2 , h3 ),
                                     ···············
                                ΣSPD(h1 , h2 . . . hn−1 ),
                                 ΣSPD(h1 , h2 . . . hn ),

                                            51
so many in number are these pairs.                                                  p. 46
Query (1). Is there no proposition applicable to any n quantities whatever?
   Query (2). Is there no faintly analogous proposition applicable to higher
powers than the squares?
   Query (3). Seeing that in forming the coefficients in the equation of the
squares of the differences, we pass from n functions of the roots to n n−1  2 and
not n functions, of their squared differences, does not a natural passage to the
former lie through n functions of the squared differences?
   In other words, may not the quantities ΣSPD(h1 , h2 . . . hn ), &c., serve as
natural and valuable intermediaries between the coefficients of an equation
involving simple quantities and the coefficients of the equation involving the
squares of their differences?
   P.S. In the next part I trust to be able to present the readers of this Magazine
with a direct and symmetrical method of eliminating any number of unknown
quantities between any number of equations of any degree, by a newly invented
process of symbolical multiplication, and the use of compound symbols of nota-
tion.
   I must not omit to state that the constituents of multiplication λr and µr
explained in Cor. 2 to Theorem 3 are equal to the expression
                                                 (                             )
                                                    k1 , k2 . . . kn−r−1
                                                   −h1 , −h2 . . . − hm
            Σ(x − k1 )(x − k2 ) · · · (x − kn−r−1 ) (                          ) ,
                                                        k1 , k2 . . . kn−r−1
                                                        −kn−r , . . . − kn

and its analogue respectively.




                                         52
                                                8.
On Derivation of Coexistence. Part II. Being the Theory of
       Simultaneous Simple Homogeneous Equations
                 [Philosophical Magazine, XVI. (1840), pp. 37–43]
                                                                                                   p. 47
Art. (1).     We shall have constant occasion in this paper to denote different
quantities by the same letter affected with different subscribed numerical indices.
   Such a letter is to be termed a “Base.”
   Every character consisting of a base and an inferior index, this index is called
an argument of the base, namely, the first, second, or nth argument, according
as 1, 2, or in general n, be the number subscribed.
Art. (2). I use the symbol PD to denote the product of the differences of the
quantities to which it is prefixed (each being to be subtracted from each that
follows); thus
                  PD(a, b, c) indicates (b − a)(c − a)(c − b).
                   PD(0, a, b, c) indicates abc(b − a)(c − a)(c − b).
                PD(0, a, b, c . . . l) indicates abc . . . l × PD(a, b, c . . . l).

Art. (3). For want of a better symbol I use the Greek letter ζ to denote that
the product of factors to which it is prefixed is to be effected after a certain
symbolical manner. This I shall distinguish as the zeta-ic product.
   The symbol ζ will never be prefixed except to factors, each of which is made
up of one or more terms, consisting solely of linear arguments of different bases,
that is, characters bearing indices below but none above.
   I am thereby enabled to give this short rule for zeta-ic multiplication: “Imagine
all the inferior indices to become superior, so that each argument is transformed
into a power of its base; multiply according to the rules of ordinary algebra;
after the multiplication has been done fully out depress all the indices into their
original position; the result is the zeta-ic product27 .”                                p. 48
   Thus for example ζ(ar , bs ) is the same as simply ar bs , but ζ(ar , as ) represents
not ar as but ar+s .
   So in like manner

                ζ{(ah − bk )(al − bm )} = ah+l − ah bm − bk al + bm+k ,

                              ζ{(a1 − b1 )(a1 − c1 )(b1 − c1 )}
  27
    It is scarcely necessary to add that an analogous interpretation may be extended to any
                                                                                    a2    a4
zeta-ic function whatever. Thus ζ(a1 + b1 )2 = a2 + 2a1 b1 + b2 , ζ cos(a1 ) = 1 −     +       ,
                                                                                   1·2 1·2·3·4
&c.


                                                53
                 = the depressed product of (a − b)(a − c)(b − c)
            = the depressed value of a2 (b − c) + b2 (c − a) + c2 (a − b),
that is,
                     = a2 b1 − a2 c1 + b2 c1 − b2 a1 + c2 a1 − c2 b1 .

Art. (4).   We shall have occasion in this part to combine the two symbols
ζ, PD: thus we shall use

                            ζPD(a1 b1 ) to denote ζ(b1 − a1 ),

              ζPD(a1 b1 c1 ) to denote ζ{(b1 − a1 )(c1 − a1 )(c1 − b1 )}.

Art. (5). For the sake of elegance of diction I shall in future sometimes omit
to insert the inferior index when it is unity; but the reader must always bear in
mind that it is to be understood though not expressed.
   I shall thus be able to speak of the zeta-ic product of such and such bases
mentioned by name.
Art. (6). We are not yet come to the limit of the powers of our notation. The
zeta-ic product of the sum of arguments will consist of the sum of products of
arguments, each argument being (as I have defined) made up of a base and an
inferior index. Now we may imagine each index of every term of the zeta-ic
product after it is fully expanded to be increased or diminished by unity, or each
at the same time to be increased or diminished by 2, or each in general to be
increased or diminished by r. I shall denote this alteration by affixing an r with
the positive or negative sign to the ζ. Thus

            ζ(a1 − b1 )(a1 − c1 ) being equal to a2 − a1 c1 + b1 c1 − b1 a1 ,

            ζ+1 (a1 − b1 )(a1 − c1 ) is equal to a3 − a2 c2 + b2 c2 − b2 a2 ,
            ζ−1 (a1 − b1 )(a1 − c1 ) is equal to a1 − a0 c0 + b0 c0 − b0 a0 .
In like manner ζPD(a, b, c) indicating

                      b2 a1 − b2 c1 + c2 b1 − c2 a1 + a2 c1 − a2 b1 ,

ζ±r PD(a, b, c) indicates

       b2±r a1±r − b2±r c1±r + c2±r b1±r − c2±r a1±r + a2±r c1±r − a2±r b1±r .

I shall in general denote ζ+r PD(a, b, c . . . l) actually expanded as the zeta-ic
product of a, b, c, . . . l in its rth phase.                                      p. 49

Art. (7).    General Properties of Zeta-ic Products of Differences.



                                            54
   If there be made one interchange in the order of the bases to which ζ is
prefixed, the zeta-ic product, in whatever phase it be taken, remains unaltered
in magnitude, but changes its sign.
   If in any phase of a zeta-ic product two of the bases be made to coincide, the
expansion vanishes.
   Let f1 be used, agreeably to the ordinary notation, to denote the sum of the
quantities to which it is prefixed, f2 to denote the sum of the binary products,
f3 of the ternary ones, and so on.
   Thus let f1 (a1 b1 c1 ) or f1 (a, b, c) indicate a1 + b1 + c1 ,

                 and f2 (a1 b1 c1 ) or f2 (a, b, c) indicate a1 b1 + a1 c1 + b1 c1 ,

                        and f3 (a1 b1 c1 ) or f3 (a, b, c) indicate a1 b1 c1 ,
we shall be able now to state the following remarkable proposition connecting
the several phases of certain the same zeta-ic products.
Art. (8). Let a, b, c, . . . l, denote any number of independent bases, say (n − 1);
but let the arguments of each base be periodic, and the number of terms in each
period the same for every base, namely n, so that

                           ar = ar+n = ar−n ,             an = a0 = a−n ,
                            br = br+n = br−n ,            bn = b0 = b−n ,
                            cr = cr+n = cr−n ,            cn = c0 = c−n ,
                                    ·········           ········· ,
                             lr = lr+n = lr−n ,           ln = l0 = l−n ,

r being any number whatever. Then

              ζ−1 PD(0, a, b, c . . . l) = ζ{f1 (a, b, c . . . l) ζPD(0, a, b, c . . . l)},

              ζ−2 PD(0, a, b, c . . . l) = ζ{f2 (a, b, c . . . l) ζPD(0, a, b, c . . . l)},
                              ·································
              ζ−r PD(0, a, b, c . . . l) = ζ{fr (a, b, c . . . l) ζPD(0, a, b, c . . . l)}.
This proposition admits of a great generalization28 , but we have now all that
is requisite for enabling us to arrive at a proposition exhibiting under one coup
d’oeil every combination and every effect of every combination that can possibly
be made with any number of coexisting equations of the first degree, containing
any number of repeated, or to use the ordinary language of analysts, (variable
or) unknown quantities.                                                           p. 50
   For the sake of symmetry I make every equation homogeneous; so that to
eliminate n repeated terms, no more than n equations will be required.
  28
       See the Postscript to this paper for one specimen.


                                                   55
   In like manner the problem of determining n quantities from n equations will
be here represented by the case in which we have to determine the ratios of
(n + 1) quantities from n equations.
Art. (9). Statement of the Equations of Coexistence.
   Let there be any number of bases (a, b, c . . . l), and as many repeated terms
(x, y, z . . . t), and let the number of equations be any whatever, say n. The system
may be represented by the type equation

                            ar x + br y + cr z + · · · + lr t = 0,

in which r can take up all integer values from −∞ to +∞. The specific number
of equations given will be represented by making the arguments of each base
periodic, so that

             ar = aµn+r ,     br = bµn+r ,     cr = cµn+r , . . .    lr = lµn+r ,

µ being any integer whatever.
Art. (10). Combination of the given Equations.–Leading Theorem.
   Take f, g, . . . k as the arbitrary bases of new and absolutely independent but
periodic arguments, having the same index of periodicity (n) as a, b, c . . . l, and
being in number (n − 1), that is, one fewer than there are units in that index.
   The number of differing arbitrary constants thus manufactured is n(n − 1).
   Let Ax + By + Cz + · · · + Lt = 0 be the general prime derivative from the
given equations, then we may make

                                A = ζPD(0, a, f, g . . . k),
                                B = ζPD(0, b, f, g . . . k),
                                C = ζPD(0, c, f, g . . . k),
                                     ···············
                                L = ζPD(0, l, f, g . . . k).

Art. (11). Cor. 1.. Inferences from the Leading Theorem.
   Let the number of equations, or, which is the same thing, the index of
periodicity (n), be the same as the number of repeated terms (x, y, z . . . t), then
one relation exists between the coefficients: this is found by making the (n − 1)
new bases coincide with (n − 1) out of the old bases. We get accordingly, as the
result of elimination,
                             ζPD(0, a, b, c . . . l) = 0.
                                                                                        p. 51
Art. (11).   Cor. 2.. Let the number of equations be one more than that
of the given bases, there will then be two equations of condition. These are


                                             56
represented by preserving one new arbitrary base, as λ. The result of elimination
being in this case
                           ζPD(0, a, b, c . . . l, λ) = 0.
  Example.     The result of eliminating between

                                    a1 x + b1 y = 0,

                                    a2 x + b2 y = 0,
                                    a3 x + b3 y = 0,
is ζPD(0, a, b, λ) = 0, that is

           λ3 b2 a1 − λ3 b1 a2 + λ1 b3 a2 − λ1 b2 a3 + λ2 b1 a3 − λ2 b3 a1 = 0,

from which we infer, seeing that λ3 , λ2 , λ1 are independent,

                                   b2 a1 − b1 a2 = 0,

                                   b3 a2 − b2 a3 = 0,
                                   b1 a3 − b3 a1 = 0,
any two of which imply the third.
   In like manner, in general, if the number of equations exceed in any manner
the number of bases or repeated terms, the rule is to introduce so many new
and arbitrary bases as together with the old bases shall make up the number of
equations, and then equate the zeta-ic product of the differences of zero, the old
bases and the new bases, to nothing.
Art. (12).    Cor. 3.. Let the number of equations be one fewer than the
number (n) of bases or repeated terms; the number of introduced bases in the
general theorem is here (n − 2). Make these (n − 2) bases equal severally to the
bases which in the type equation are affixed to z, u . . . t, then

                      C = 0,       D = 0,        ··· ,      L = 0,

and we have left simply

                ζPD(0, a, c, d . . . kl)x + ζPD(0, b, c, d . . . kl)y = 0.

In like manner we may make to vanish all but A and C, and thus get

                ζPD(0, a, b, d . . . kl)x + ζPD(0, c, b, d . . . kl)z = 0,
                                                                                     p. 52
  and similarly

                    ζPD(0, a, b . . . k)x + ζPD(0, b, c . . . l)t = 0.

                                           57
Hence                                                                  
                     x                            ζPD(0, b, c . . . l) 
                                                     ζPD(a, 0, c . . . l) 
                                                                       
                      y 
                    
                                                 
                                                                         
                    
                       
                                                  
                                                                         
                                                                          
                    
                     z 
                                                   ζPD(a, b, 0 . . . l) 
                                                                         
                                are severally as                                .
                    
                     · 
                                                  
                                                        ············      
                                                                         
                    
                    
                     · 
                        
                        
                                                   
                                                   
                                                        ············      
                                                                            
                                                       ζPD(a, b, c . . . 0)
                                                                         
                        t
                                                                         

This is the symbolical representation as a formula of the remarkable method
discovered by Cramer, perfected by Bezout and demonstrated by Laplace for the
solution of simultaneous simple equations.
Art. (13). Cor. 4.. In like manner if the number of repeated terms be two
greater than the number of equations, we have for the relation between any three
of them, taken at pleasure, for instance, x, y, z,

            ζPD(0, a, d . . . l)x + ζPD(0, b, d . . . l)y + ζPD(0, c, d . . . l)z = 0.

And in like manner we may proceed, however much in excess the number of
repeated terms (unknown quantities) is over the number of equations.

                            Art. (14).   Subcorollary to Corollary 3.

  If there be any number of bases (a, b, c . . . l), and any other two fewer in
number (f, g . . . k)
                                                                   
                       ζPD(a, f, g . . . k) × ζPD(b, c . . . l)    
                                                                   
                                                                   
                        + ζPD(b, f, g . . . k) × ζPD(a, c . . . l) 
                                                                   
                                                                   
                                                                   
                                                                   
                        + ζPD(c, f, g . . . k) × ζPD(b, c . . . l)       = 0,
                                                                     
                             ························
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                        + ζPD(l, f, g . . . k) × ζPD(a, b, c . . .)
                                                                     
                                                                     

a formula that from its very nature suggests and proves a wide extension of
itself29 .
   In conclusion I feel myself bound to state that the principal substance of
Corollaries (1), (2) and (3) may be found in Garnier’s Analyse Algébrique, in the
chapter headed “Développement de la Théorie donnée par M. Laplace, &c.” But
I am not aware of having been anticipated either in the fertile notation which
serves to express them nor in the general theorems to which it has given birth.
   P.S. I shall content myself for the present with barely enunciating a theorem,
one of a class destined it seems to the author to play no secondary part in the
development of some of the most curious and interesting points of analysis.       p. 53

 29
      The cross is used to denote ordinary algebraical multiplication.



                                               58
   Let there be (n − 1) bases a, b, c . . . l, and let the arguments of each be
“recurrents of the nth order,”30 that is to say let
           2πt                     2πt                         2πt                       2πt
                                                                                           
at = ϕ cos     ,        bt = ψ cos     ,            ct = χ cos     ,...       lt = ω cos     .
            n                       n                           n                         n
Let Rr denote that any symmetrical function of the rth degree is to be taken
of the quantities in a parenthesis which come after it, and let S indicate any
function whatever. Then the zeta-ic product

                      ζ{Rr (a, b, c . . . l) × Sρ ζPD(0, a, b, c . . . l)}

is equal to the product of the number

             2π √        2π             4π √         4π
                                                    
  Rr        cos + −1 sin      , cos        + −1 sin       ,
              n           n              n            n
             6π √        6π                 2(n − 1)π √       2(n − 1)π
                                                                     
         cos    + −1 sin      , . . . , cos          + −1 sin              ,
              n           n                     n                 n

multiplied by the zeta-ic phase

                                ζρ−r SPD(0, a, b, c . . . l)!!




  30
     I am indebted for this term to Professor De Morgan, whose pupil I may boast to have been.
I have the sanction also of his authority, and that of another profound analyst, my colleague
Mr Graves, for the use of the arbitrary terms zeta-ic, zeta-ically. I take this opportunity of
retracting the symbol SPD used in my last paper, the letter S having no meaning except for
English readers. I substitute for it QDP, where Q represents the Latin word Quadratus. On
some future occasion I shall enlarge upon a new method of notation, whereby the language of
analysis may be rendered much more expressive, depending essentially upon the use of similar
figures inserted within one another, and containing numbers or letters, according as quantities
or operations are to be denoted. This system to be carried out would require special but very
simple printing types to be founded for the purpose. In the next part of this paper an easy
and symmetrical mode will be given of representing any polynomial either in its developable or
expanded form.


                                               59
                                           9.
A Method of Determining by Mere Inspection the Derivatives
           from Two Equations of any Degree
               [Philosophical Magazine, XVI. (1840), pp. 132–135]
                                                                                            p. 54
   Let there be two equations, one of the nth, the other of the mth degree in x;
let the coefficients of the first equation be an , an−1 , an−2 . . . a0 , each power of x
having a coefficient attached to it, an belonging to xn and a0 to the constant
term.
   In like manner let bm , bm−1 . . . b0 be the coefficients of the second equation.
   I begin with

                        A Rule for absolutely eliminating x.

   Form out of the (a) progression of coefficients m lines, and in like manner out
of the (b) progression of coefficients form n lines in the following manner:
   1. (a). Attach (m − 1) zeros all to the right of the terms in the (a) progression;
next attach (m − 2) zeros to the right and carry over to the left; next attach
(m − 3) zeros to the right and carry over 2 to the left. Proceed in like manner
until all the (m − 1) zeros are carried over to the left and none remain on the
right.
   The m lines thus formed are to be written under one another.
   1. (b) Proceed in like manner to form n lines out of the (b) progression by
scattering (n − 1) zeros between the right and left.
   2. If we write these n lines under the m lines last obtained, we shall have a
solid square (m + n) terms deep and (m + n) terms broad.
   3. Denote the lines of this square by arbitrary characters, which write down in
vertical order and permute in every possible way, but separate the permutations
that can be derived from one another by an even number of interchanges (effected
between contiguous terms) from the rest; there will thus be half of one kind and
half of another.                                                                      p. 55
   4. Now arrange the (m + n) lines accordingly, so as to obtain

                           2 {(m + n)(m + n − 1) · · · 2 · 1}
                           1


squares of one kind which shall be called positive squares, and an equal number
of the opposite kind which shall be called negative.
   Draw diagonals in the same direction in all the squares; multiply the coefficients
that stand in any diagonal line together: take the sum of the diagonal products
of the positive squares, and the sum of the diagonal products of the negative
squares; the difference between these two sums is the prime derivative of the

                                           60
zero degree, that is, is the result of elimination between the two given equations
reduced to its ultimate state of simplicity, there will be no irrelevant factors to
reject, and no terms which mutually destroy.
   Example. To eliminate between

                    ax2 + bx + c = 0,          lx2 + mx + n = 0,

I write down
                              a, b, c,         0,       (1)
                              0, a, b,         c,       (2)
                              l, m, n,         0,       (3)
                              0, l, m,         n.       (4)
I permute the four characters (1), (2), (3), (4), distinguishing them into positive
and negative; thus I write together

                              Positive Permutations.

                      1   2   3   1   2   3   2     1   3   4   4   4
                      2   3   1   4   4   4   1     3   2   2   1   3
                      3   1   2   2   3   1   4     4   4   1   3   2
                      4   4   4   3   1   2   3     2   1   3   2   1
and again

                              Negative Permutations.

                      1   2   3   4   4   4   2     1   3   2   1   3
                      2   3   1   1   2   3   4     4   4   1   3   2
                      4   4   4   2   3   1   1     3   2   3   2   1
                      3   1   2   3   1   2   3     2   1   4   4   4
                                                                                      p. 56
   I reject from the permutations of each species all those where 1 or 3 appear in
the fourth place, and also those where 2 or 4 appear in the first place, for these
will be presently seen to give rise to diagonal products which are zero.
   The permutations remaining are

                          Positive effectual permutations.

                                      1   3   3     1
                                      2   1   4     3
                                      3   2   1     4
                                      4   4   2     2

                          Negative effectual permutations.


                                          61
                                      3    1   1     3
                                      1    4   3     2
                                      4    3   2     1
                                      2    2   4     4
   I now accordingly form four positive squares, which are
           a b c       0    l m n         0        l m n 0         a b c 0
           0 a b       c    a b c         0        0 l m n         l m n 0
           l m n       0    0 a b         c        a b c 0         0 l m n
           0 l m       n    0 l m         n        0 a b c         0 a b c
Drawing diagonal lines from left to right, and taking the sum of the diagonal
products, I obtain a2 n2 + lb2 n + l2 c2 + am2 c. Again, the four negative squares
           l m n 0          a b c 0                a b c     0     l m n    0
           a b c 0          0 l m n                l m n     0     0 a b    c
           0 l m n          l m n 0                0 a b     c     a b c    0
           0 a b c          0 a b c                0 l m     n     0 l m    n
give as the sum of the diagonal products
                              lbmc + alnc + ambn + lacn,
that is,
                                 lbmc + ambn + 2acln.
Thus the result of eliminating between
                       ax2 + bx + c = 0,           lx2 + mx + n = 0,
ought to be, and is
               a2 n2 + l2 c2 − 2acln + lb2 n + am2 c − lbmc − ambn = 0.
                                                                                     p. 57

           Rule for finding the prime derivative of the first degree, which is
                                  of the form Ax − B.

   Begin as before, only attach one zero less to each progression; we shall thus
obtain not a square, but an oblong broader than it is deep, containing (m + n − 2)
rows, and (m + n − 1) terms in each row: in a word, (m + n − 2) rows, and
(m + n − 1) columns.
   To find A reject the column at the extreme right, we thus recover a square
arrangement (m + n − 2) terms broad and deep.
   Proceed with this new square as with the former one; the difference between
the sums of the positive and negative diagonal products will give A.
   To find B, do just the same thing, with the exception of striking off not the
last column, but the last but one.

                                              62
   Rule for finding the prime derivative of any degree, say the rth, namely,
                         Ar xr − Ar−1 xr−1 + · · · ± A0 .

   Begin with adding zeros as before, but the number to be added to the (a)
progression is (m − r) and to the (b) progression (n − r).
   There will thus be formed an oblong containing (m + n − 2r) rows, and
(m + n − r) terms in each row, and therefore the same number of columns.
   To find any coefficient as As , strike off all the last (r + 1) columns except
that which is (s) places distant from the extreme right, and proceed with the
resulting squares as before.




   Through the well-known ingenuity and kindly proferred help of a distinguished
friend, I trust to be able to get a machine made for working Sturm’s theorem,
and indeed all problems of derivation, after the method here expounded; on
which subject I have a great deal more yet to say, than can be inferred from this
or my preceding papers.




                                       63
                                             10.
                               Note on Elimination
                 [Philosophical Magazine, XVII. (1840), pp. 379, 380]
                                                                                                   p. 58
   The object of this brief note is to generalise Theorem 2 in my paper on
Elimination31 which appeared in the last December number of this Magazine.
The theorem so generalised presents a symmetry which before was wanting. Here,
as in so many other instances, the whole occupies in the memory a less space
than the part.
   To avoid the ill-looking and slippery negative symbols, I warn my reader that I
now use two rows of quantities written one over the other, to denote the product
of the terms resulting from taking away each quantity in the under from each in
the upper row.
   Let h1 , h2 . . . hm be the roots of one equation of coexistence,

                                k1 , k2 . . . kn of the other,

and let the prime derivative of the degree r be required. Take any two integers p
and q, such that p + q = r. The derivative in question may be written
                                                                   !                          !       
                                                         h1 h2 · · · hp   hp+1 hp+2 · · · hm
                                                                        ·
  
                                                                                                      
                                                                                                       
                                                                                                      
  
                                                        k1 k2 · · · kq   kq+1 kq+2 · · · kn           
                                                                                                       
Σ (x − h1 ) · · · (x − hp )(x − k1 ) · · · (x − kq )                    !                          !       .
                                                       h1 h2 · · · hp       k1 k2 · · · kq            
                                                                          ·
  
                                                                                                      
                                                                                                       
                                                                                                      
                                                      hp+1 hp+2 · · · hm   kq+1 kq+2 · · · kn         


   N.B. Whatever p and q be taken, so long only as p+q = r, the above expression
changes nothing but its sign; which, therefore, upon transcendental grounds, it
is easy to see is of one name or another, according as p is odd or even.
   In the original paper, I asserted this theorem only for the case of p = 0, or
q = 0.




  31
       p. 43 above. Ed.


                                              64
                                             11.
   On the Relation of Sturm’s Auxiliary Functions to the
             Roots of an Algebraic Equation
        [Plymouth British Association Report 1841, (Pt. II.), pp. 23, 24]
                                                                                           p. 59
   The author availed himself of the present meeting of the British Association
to bring under the more general notice of mathematicians his discovery, made
in the year 1839, of the real nature and constitution of the auxiliary functions
(so-called) which Sturm makes use of in locating the roots of an equation: these
are obtained by proceeding with the left-hand side of the equation and its first
differential coefficient as if it were our object to obtain their greatest common
factor; the successive remainders, with their signs alternately changed and
preserved, constitute the functions in question. Each of these may be put under
the form of a fraction, the denominator of which is a perfect square, or in fact
the product of many: likewise the numerator contains a huge heap of factors of
a similar form.
   These therefore, as well as the denominator, since they cannot influence the
series of signs, may be rejected; and furthermore we may, if we please, again
make every other function, beginning from the last but one, change its sign, if
we consent to use changes wherever Sturm speaks of continuations of sign, and
vice versâ.
   The functions of Sturm, thus modified and purged of irrelevancy, the author,
by way of distinction, and still to attribute honour where it is really most due,
proposes to call “Sturm’s Determinators”; and he proceeds to lay bare the internal
anatomy of these remarkable forms.
   He uses the Greek letter “ζ” to indicate that the squared product of the
differences of the letters before which it is prefixed is to be taken.
   Let the roots of the equation be called respectively a, b, c, e . . . l, the determi-
nators taken in the inverse order are as follows:

                                     ζ(a, b, c, e . . . l).

                        Σ{ζ(b, c, e . . . l)x − Σaζ(b, c, e . . . l)}.
           Σ{ζ(c, e . . . l)x2 − Σ(a + b)ζ(c, e . . . l)x + Σab · ζ(c, e . . . l)}.
                Σ{ζ(k, l)(x − a)(x − b)(x − c)(x − e) · · · (x − h)}.
                                                                                           p. 60
   It may be here remarked, that the work of assigning the total number of real
and of imaginary roots falls exclusively upon the coefficients of the leading terms,
which the author proposes to call “Sturm’s Superiors”; these superiors are only
partial symmetric functions of the squared differences, but complete symmetric
functions of the roots themselves, differing in the former respect from those other

                                              65
(at first sight similar-looking) functions of the squared differences of the roots,
in which, from the time of Waring downwards, the conditions of reality have
been sought for. It seems to have escaped observation, that the series of terms
constituting any one of the coefficients in the equation of the squares of the
differences (with the exception of the first and last) each admit of being separated
and classified into various subordinate groups in such a way, that instead of being
treated as a single symmetric function of the roots, they ought to be viewed as
aggregates of many. In fact, Sturm’s superior No. 1 is identical with Waring’s
coefficient No. 1; Sturm’s superior No. 2 is a part of Waring’s coefficient No. 3;
Sturm’s superior No. 3 is a part of Waring’s coefficient No. 6; and so forth till we
come to Sturm’s final superior, which is again coextensive and identical with the
last coefficient in the equation of the squares of the differences. The theory of
symmetric functions of forms which are themselves symmetric functions of simple
letters, or even of other forms, the author states his belief is here for the first
time shadowed forth, but would be beside his present object to enter further into.
He would conclude by calling attention to the importance to the general interests
of algebraical and arithmetical science that a searching investigation should be
instituted for showing, a priori, how, when a set of quantities is known to be
made up partly of possible and partly of pairs of impossible values, symmetrical
functions of these, one less in number than the quantities themselves, may be
formed, from the signs of the ratios of which to unity and to one another the
respective amounts of possible and impossible quantities may at once be inferred;
in short, we ought not to rest satisfied, until, from the very form of Sturm’s
Determinators, without caring to know how they may have been obtained, we
are able to pronounce upon the uses to which they may be applied.




                                        66
                                         12.
Examples of the Dialytic Method of Elimination as Applied
            to Ternary Systems of Equations
          [Cambridge Mathematical Journal, II. (1841), pp. 232–236]
                                                                                           p. 61
   This method is of universal application, and at once enables us to reduce
any case of elimination to the form of a problem, where that operation is to
be effected between quantities linearly involved in the equations which contain
them.
   As applied to a binary system, f x = 0, ϕx = 0, the method furnishes a rule
by which we may unfailingly arrive at the determinant, free from every species
of irrelevancy, whether of a linear, factorial, or numerical kind.
   The rule itself is given in the Philosophical Magazine (London and Edin-
burgh, Dec. 1840). The principle of the rule will be found correctly stated by
Professor Richelot, of Königsberg, in a late number of Crelle’s Journal, at the
commencement of a memoir in Latin bordering on the same subject (“Nota ad
Eliminationem pertinens”).
   My object at present is to supply a few instances of its application to ternary
systems of equations.
Ex. 1.    To eliminate x, y, z, between the three homogeneous equations
                              Ay 2 − 2C ′ xy + Bx2 = 0,                             (1)
                                         ′
                              Bz − 2A yz + Cy = 0,
                                  2               2
                                                                                    (2)
                                         ′
                              Cx − 2B zx + Az = 0.
                                  2               2
                                                                                    (3)
Multiply the equations in order by −z 2 , x2 , y 2 , add together, and divide out by
2xy; we obtain
                       C ′ z 2 + Cxy − A′ xz − B ′ yz = 0.                        (4)
By similar processes we obtain
                          A′ x2 + Ayz − B ′ yx − C ′ zx = 0,                        (5)
                            ′ 2               ′       ′
                          B y + Bzx − C zy − A xy = 0.                              (6)
                                                                                           p. 62
  Between these six, treated as simple equations, the six functions of x, y, z,
namely, x2 , y 2 , z 2 , xy, xz, yz, treated as independent of each other, may be elimi-
nated; the results may be seen, by mere inspection, to come out
               ABC(ABC − AB ′2 − BC ′2 − CA′2 + 2A′ B ′ C ′ ) = 0,
or rejecting the special (N.B. not irrelevant) factor ABC, we obtain
                  ABC − AB ′2 − BC ′2 − CA′2 + 2A′ B ′ C ′ = 0.

                                             67
  I may remark, that the equations (1), (2), (3), or (4), (5), (6), express the
condition of
               Ax2 + By 2 + Cz 2 + 2A′ yz + 2B ′ zx + 2C ′ xy,
having a factor λx + µy + νz; a general symbolical formula of which I am in
possession for determining in general the condition of any polynomial of any
degree having a factor, furnishes me at once with either of the two systems
indifferently. The aversion I felt to reject either, led me to employ both, and
thus was the occasion of the Dialytic Principle of Solution manifesting itself.
Ex. 2.

                              Ax2 + ayz + bzx + cxy = 0,                              (1)
                            M y + lyz + mzx + nxy = 0,
                                2
                                                                                      (2)
                             Rz + pyz + qzx + rxy = 0.
                                 2
                                                                                      (3)

Multiply equation (1) by βy +γz, equations (2) and (3) by νz and κy respectively,
and add the products together, we obtain terms of which y 2 z and yz 2 are the
only two into which x does not enter.
  Make now the coefficients of each of these zero, and we have

                    aγ + lν + Rκ = 0,           aβ + M ν + pκ = 0.

Let ν = a, κ = a, then γ = −(l + R), β = −(M + p).
  Hence, multiplying as directed, and then dividing out by x, we obtain

   (mν + bγ)z 2 + (rκ + cβ)y 2 + (bβ + cγ + nν + qκ)yz + Aβxy + Aγxz = 0,

or by substitution,

  {ra − c(M + p)}y 2 + {ma − b(l + R)}z 2 + {an + aq − b(M + p) − c(l + R)}yz
                                                − A(M + p)xy − A(l + R)xz = 0. (4)

Similarly, by preparing the equations so as to admit in turn of y and z as a
divisor, we obtain

  {ma − l(R + b)}z 2 + {mr − n(A + q)}x2 + {mc + mp − n(R + b) − l(A + γ)}xz
                                                − M (R + b)yz − A(A + q)xy = 0, (5)


  {rm − q(A + n)}x2 + {ra − p(M + c)}y 2 + {rl + rb − p(A + n) − q(M + c)}xy
                                              − R(A + n)xz − R(M + c)yz = 0. (6)
                                                                                             p. 63
  Between the six equations (1), (2), (3), (4), (5), (6), x2 , y 2 , z 2 , xy, xz, yz, may
be eliminated; the result will be a function of nine letters {three out of each

                                           68
equation (1), (2), (3)} equated to zero. Perhaps the determinant may be found
to contain a special factor of three letters; and if so, may be replaced by a simpler
function of six letters only.
Ex. 3.     To eliminate between the three general equations

                    Ax2 + By 2 + Cz 2 + 2Dyz + 2Ezx + 2F xy = 0,
                  Lx2 + M y 2 + N z 2 + 2P yz + 2Qzx + 2Rxy = 0,
                                                       f x + gy + hz = 0.

By virtue of one of the two canons which limit the forms in which the letters
can appear combined in the determinant of a general system of equations, we
know that the determinant in this case (freed of irrelevant factors) ought to be
made up in every term of eight letters (powers being counted as repetitions),
namely, (A, B, C, D, E, F ) must enter in binary combinations, (L, M, N, P, Q, R)
the same, whereas f, g, h must enter in quaternary combinations.
   To obtain the determinant, write

                        Ax2 + By 2 + Cz 2 + Dyz + Ezx + F xy = 0,                     (1)
                     Lx + M y + N z + P yz + Qzx + Rxy = 0,
                          2        2      2
                                                                                      (2)
                                                f x + gyx + hzx = 0,
                                                   2
                                                                                      (3)
                                                f xy + gy + hzy = 0,
                                                          2
                                                                                      (4)
                                                f xz + gyz + hz 2 = 0.                (5)

We want one equation more of three letters between x2 , y 2 , z 2 , xy, xz, yz. To
obtain this, write

         (Ax + Ez + F y)x1 + (By + F x + Dz)y1 + (Cz + Dy + Ex)z1 = 0,
         (Lx + Qz + Ry)x1 + (M y + Rx + P z)y1 + (N z + P y + Qx)z1 = 0,
                                                              f x1 + gy1 + hz1 = 0.

Forget that x1 = x, y1 = y, z1 = z, and eliminate x1 , y1 , z1 , we obtain

                        (Ax + Ez + F y)(M y + Rx + P z)
                                                               
                h       −(By + F x + Dz)(Lx + Qz + Ry)
                              (Cz + Dy + Ex)(Lx + Qz + Ry)
                                                                   
                 +g           −(N z + P y + Qx)(Ax + Ez + F y)
                              (N z + P y + Qx)(By + F x + Dz)
                                                                   
                 +f           −(Cz + Dy + Ex)(M y + Rx + P z)           = 0.
                                                                                            p. 64
  This may be put under the form

                        αx2 + βy 2 + γz 2 + α′ yz + β ′ zx + γ ′ xy = 0,              (6)

                                              69
where the coefficients are of the first order in respect to f, g, h, L, M, N, P, Q, R, A, B, C, D, E
in all of the third order.
   Between the equations marked from (1) to (6), the process of linear elimination
being gone through, we obtain as equated to zero a function of 5 + 3, or of eight
letters, two belonging to the first equation, two to the second, and four to the
third; so that the determinant is clear of all factorial irrelevancy.
Ex. 4.      To eliminate x, y, z between the three equations

                        Ax2 + By 2 + Cz 2 + 2A′ yz + 2B ′ zx + 2C ′ xy = 0,
                  Lx2 + M y 2 + N z 2 + 2L′ yz + 2M ′ zx + 2N ′ xy = 0,
                        P x2 + Qy 2 + Rz 2 + 2P ′ yz + 2Q′ zx + 2R′ xy = 0.

Call these three equations U = 0, V = 0, W = 0, respectively. Write

                                   xU = 0, yU = 0, zU = 0,
                                   xV = 0, yV = 0, zV = 0,                                            (1–9)
                                   xW = 0, yW = 0, zW = 0.

We have here nine unilateral equations: one more is wanted to enable us to
eliminate linearly the ten quantities

                           x3 , y 3 , z 3 , x2 y, x2 z, xy 2 , xz 2 , xyz, y 2 z, yz 2 .

This tenth may be found by eliminating x, y, z between the three equations

         x(Ax + B ′ z + C ′ y) + y(By + C ′ x + A′ z) + z(Cz + A′ y + B ′ x) = 0,
      x(Lx + M ′ z + N ′ y) + y(M y + N ′ x + L′ z) + z(N z + L′ y + M ′ x) = 0,
         x(P x + Q′ z + R′ y) + y(Qy + R′ x + P ′ z) + z(Rz + P ′ y + Q′ x) = 0;

for, by forgetting the relations between the bracketed and unbracketed letters,
we obtain
                              (M y + N ′ x + L′ z)(Rz + P ′ y + Q′ x)
                                                                                      
           ′        ′
(Ax + B z + C y)              −(Qy + R′ x + P ′ z)(N z + L′ y + M ′ x)                     + &c. + &c. = 0,

which may be put under the form

                               αx3 + βy 3 + γz 3 + δx2 y + · · · = 0∗ .                                (10)

                                                                                                              p. 65
    31
       We might dispense with a 10th equation, using the nine above given, to determine the
ratios of the ten quantities involved to one another; and then by means of any such relations as
x3 y × xy 3 = x2 y 2 × x2 y 2 , or x3 × y 3 = x2 y × xy 2 , &c. obtain a determinant. But it is easy
to see that this would be made up of terms, each containing literal combinations of the 18th
order. Again, we might use five out of the nine equations to obtain a new equation free from
y 3 , y 2 z, yz 2 , z 3 ; that is, containing x in every term: which being divided by x, and multiplied


                                                       70
   By eliminating linearly between the equations marked from (1) to (10), we
obtain as zero a quantity of the twelfth order in all, being of the fourth order in
respect to the coefficients of each of the three equations, which is therefore the
determinant in its simplest form.
   I have purposely, in this brief paper, avoided discussing any theoretical question.
I may take some other opportunity of enlarging upon several points which have
hitherto been little considered in the theory of elimination, such as the Canons
of Form, the Doctrine of Special Factors, the Method of Multipliers as extended
to a system of any order, the Connexion between the Method of Multipliers and
the Dialytic Process, the Idea of Derivations and of Prime Derivatives extended
to ultra-binary Systems. For the present I conclude with the expression of my
best wishes for the continued success of this valuable Journal.




   31
      by y, or by z, would furnish a 10th equation no longer linearly involved in the 9 already
found. The determinant, however, found in this way, would consist of 14-ary combinations of
letters. Finally, we might, instead of a system of ten equations, employ a system of 15, obtained
by multiplying each of the given three by any 5 out of the 6 quantities x2 , y 2 , z 2 , xy, xz, yz; but
the determinant, besides being not totally symmetrical, would contain combinations of the 15th
order. I may take this opportunity of just adverting to the fact, that the method in the text
does in fact contain a solution of the equation

                                        xr λ + y s µ + z t ν = 0,

where r + s + t = 1, and λ, µ, ν are functions of the second degree in regard to x, y, z to be
determined.


                                                  71
                                           13.
Introduction to an Essay on the Amount and Distribution of
  the Multiplicity of the Roots of an Algebraic Equation
              [Philosophical Magazine, XVIII. (1841), pp. 136–139]
                                                                                               p. 66
   I use the word multiplicity to denote a number, and distinguish between the
total and partial multiplicities of the roots of an algebraic equation.
   There may be r different roots repeated respectively h1 , h2 . . . hr times.
   r is the index of distribution.
   h1 , h2 . . . hr are the partial multiplicities, and if h = h1 + h2 + · · · + hr , h is
the total multiplicity.
   The total multiplicity it is clear may be defined as the difference between
the index of the equation and the number of its roots distinguishable from one
another.
   In this Introduction, I propose merely to consider how existing methods may
be applied to determine the amount and distribution of multiplicity in a given
equation, and conversely, how equations of condition can be formed which shall
imply a given distribution and amount.
   Let the greatest common factor between f x (the argument of the proposed
                    df x
equation) and            be called f1 x.
                     dx
                                                                             df1 x
   And in like manner, let the greatest common factor of f1 x and                  be called
                                                                              dx
f2 x, and so on, till in the end we come to fr x, which has no common factor with
dfr x
      .
 dx
   Let k1 , k2 . . . kr denote the degrees in x of f x, f1 x . . . fr x respectively.
   It is easy to see that

   k1 − k2 , partial multiplicities, are less than 2, that is, are each units,
   k2 − k3 , partial multiplicities, will be less than 3, and therefore either 1 or 2,
             in value respectively, and so on till we come to
 kr−1 − kr , which will severally be between zero and r − 1, and
    kr − 0, of values intermediate between zero and r.
                                                                                               p. 67
   Hence there will be
                 k1 − 2k2 + k3       multiplicities each of the value      1,
                 k2 − 2k3 + k4                                             2,
         ·····················
                    kr−1 − 2kr                   of the value              r − 1,
                            kr                   of the value              r.


                                            72
                               df x                   df x        d2 f x
   In place of f x with             we might employ         with         and so on for the
                                dx                     dx          dx2
rest; the values of k2 , k3 . . . kr will remain unaffected by this change; but the
former method would be more expeditious in practice.
   The total multiplicity is, of course, = k1 .
   Suppose now that we propose to ourselves the converse problem to determine
the conditions that an algebraic equation may have a given amount of multiplicity
distributed in a given manner.
   If h1 , h2 , h3 . . . hr be used to denote the given number of partial multiplicities
which are respectively of the values 1, 2, 3 . . . r, it is easy to see that the quantities
derived above by k1 , k2 . . . kr are respectively equal to

                                 h1 + 2h2 + · · · + rhr ,
                                h2 + 2h3 + · · · + rhr−1 ,
                                h3 + 2h4 + · · · + rhr−2 ,
                                    ···············
                                          hr .

              df x
   Now from        having a factor of the degree k1 common with f x we obtain
               dx
                     df1 x
k1 conditions, from        having a factor of the degree k2 common with f1 x we
                      dx
obtain k2 more, and so on. So that altogether we obtain in this way

                            k1 + k2 + · · · + kr conditions.

But it may easily be seen that the total multiplicity being k1 , the number of
conditions need never to exceed k1 in number, no matter what its distribution
may be. Hence, besides the enormous labour of the process, and the extreme
complexity of the results, we obtain by this method more equations by far than
are necessary, and it requires some caution to know which to reject.
   In my forthcoming paper (to appear in Philosophical Magazine of next month)
I shall show, by a most simple means, how without the use of derived or
other subsidiary functions, to obtain the simplest equations of condition which
correspond to a given distribution of a given amount of multiplicity.
   The total multiplicity, say m, being given in as many ways as that number can
be broken into parts, so many different systems of m equations can be formed
differing each from the other in the dimensions of the terms.                        p. 68
   These systems may be arranged in order so that each in the series shall imply
all those that follow it, and be implied in all those that go before, without the
converse being satisfied.
   The subject of the unreciprocal implication of systems of equations is a very
curious one, upon which the limits assigned to me prevent me from enlarging at
present. It is closely connected with a part of the theory of elimination, which, as

                                            73
far as I am aware, has either been overlooked, or has not met with the attention
which it deserves; I mean the theory of Special Factors.
   An example may make what I mean by these clear.
   Let C be a function (if my reader please) void of x, which equivalent to zero
implies two given equations in x having a common root.
   Let C be rid of all irrelevant factors, that is, let C be the simplest form of the
determinant, when the coefficients of the two equations are perfectly independent
qualities. Now suppose, as is quite possible in a variety of ways, that such
relations are instituted between the coefficients alluded to as make C split up
into factors, so that C = L × M × N = 0.
   Only one of the factors L, M, N will satisfy the condition of the coexistence
of the two given equations: the others are clearly, however, not to be confounded
with factors of solution, or irrelevant factors, as they are termed, but are of quite
a different nature, and enjoy remarkable properties, which point to an enlarged
theory of elimination, and constitute what I call special or singular factors.
   I shall feel much obliged to any of the readers of your widely circulated
Journal, interested in the subject of this paper, who would do me the honour
of communicating with me upon it, and especially if they would (between now
and the next coming out of the Magazine) inform me whether anything, and if
so how much, different from what is here stated has been done in the matter of
determining the relations between the coefficients of an equation corresponding
to a given amount and distribution of multiplicity in its roots.
   I ought to add, that my method enables me not merely to determine the
conditions of multiplicity, but also to decompose the equations containing multiple
roots into others free of multiplicity, that is, to find, à priori, the values of the
several quantities
                    f x f2 x     f1 x f3 x             fr−1 x
                             ,             ,    ...,           ,   fr x.
                    (f1 x)2      (f2 x)2               (fr x)2

Moreover, other decompositions, not necessary to be enlarged upon in this place,
may be obtained with equal facility.




                                               74
                                                 14.
     A New and More General Theory of Multiple Roots
               [Philosophical Magazine, XVIII. (1841), pp. 249–254]
                                                                                                  p. 69
   I shall begin with developing the theory of polynomials containing perfect
square factors, one or more.
   First, let us proceed to determine the relations which must exist between the
coefficients of such polynomials, and afterwards show how they may be broken
up into others of an inferior degree.
   A parallelogram filled with letters standing in one row is intended to express
the product of the squared difference of the quantities contained. Thus ab
indicates (a − b)2 , abc is used to indicate (a − b)2 (a − c)2 (b − c)2 , and so forth.
   Suppose now that two of the roots e1 , e2 . . . en belonging to the equation
f x = 0 are equal to one another, it is clear that e1 , e2 . . . en = 0; and moreover
is a symmetric function, and can be calculated in terms of the coefficients of f x.
   Next let us suppose that we have two couples of equals (as for instance a and
b, two of the roots equal, as also c and d two others), it is clear, that on leaving
any one of the roots out, the (n − 1) that are left will still contain one equality,
and therefore we have
              e2 , e3 . . . en = 0,     e1 , e3 . . . en = 0 . . . e1 , e2 . . . en−1 = 0.
None of the parallelogrammatic functions above taken singly, are symmetric
functions of the coefficients, but their sum is; so also is the sum of the product
of each into the quantity left out.
    Now in general, suppose that the polynomial f x contains r perfect square
factors, so that we have r couples of equal roots belonging to the equation f x = 0,
it is clear that er , er+1 . . . en and all the other
                                      n(n − 1) · · · (n − r + 2)
                                          1 · 2 · · · (r − 1)
functions of which it is the type are severally zero. Moreover, the sum of         p. 70
   these or the sum of the products of each by any symmetrical function of the
(r − 1) letters left out will be a symmetrical function of the coefficients of the
powers of x in f x. To express now the affirmative 32 conditions corresponding to
the case of there being r pairs of equal roots, we might employ the r equations,
                                          e1 , e2 . . . en = 0,
                                        Σe2 , e3 . . . en = 0,
                                             ·········
                                      Σer , er+1 . . . en = 0.
  32
     The importance of the restriction hinted at by the use of the word affirmative will appear
hereafter.


                                                  75
But these, except the last, are not the simplest that can be employed; that is to
say, we can write down r others, the terms of which shall be of lower dimensions
in respect to the roots.
   Let fµ denote that any rational symmetrical function of the µth degree is to
be taken of the quantities which it precedes.
   Then the r equations in question are all contained in the general equation

                       Σ{fµ (e1 , e2 . . . er−1 ) × er , er+1 . . . en } = 0.

µ being taken from 0 up to (r − 1) we obtain r equations, which in respect to
the roots are respectively of all degrees between

          n(n − 1) · · · (n − r + 2)             n(n − 1) · · · (n − r + 2)
                                         and                                + (r − 1)
              1 · 2 · · · (r − 1)                    1 · 2 · · · (r − 1)

reckoned inclusively.
   Now at this stage it is important to remark that the above r equations,
although necessary, are not sufficient; and indeed, no mere affirmations of
equality can be sufficient to ensure there being r pairs of equal roots.
   To make this manifest, suppose r = 2. Then in order that an equation may
have two pairs of equal roots, we must have by the above formula

                     Σe2 , e3 . . . en = 0,       Σ{e1 e2 , e3 . . . en } = 0.

But if instead of there being two perfect square factors there be one perfect cube
factor in f x, it may be shown by the same reasoning as above, that the very
same two equations apply. In fact, it may be shown in general that no such
equations as those given above can be affirmed in consequence of there being an
amount r of multiplicity consisting of unit parts which may not be affirmed with
equal truth as necessary consequences of the same                                  p. 71
   amount distributed in any other manner whatever. How to obtain affirmative
equations sufficient as well as necessary (under certain limitations) will appear
at the close of this present paper.
   It is worthy of being remarked, that if we make fµ denote the sum of the
products of the quantities to which it is prefixed, taken µ and µ together, the
equations of affirmation become identical with those obtained by eliminating
                  df x 33
between f x and           .
                   dx
   It can scarcely be doubted that the illustrious Lagrange, had he chosen to
perfect the incomplete theory of equal roots given in the Résolution Numérique,
by applying to it his own favourite engine of symmetric functions, could scarcely
have failed of stumbling by a back passage upon Sturm’s memorable theorem.
 33
      See my note on Sturm’s Theorem, Phil. Mag., December, 1839 [p. 45 above. Ed.].



                                                76
   Let us now proceed to show how a polynomial known to contain one or more
perfect square factors may be decomposed.
   Let us begin with supposing that it contains but one such factor; so that
f x = ϕx(x − a)2 .
   I shall show how to obtain the equations

     C(x − a) = 0,          Dϕx(x − a) = 0,             E(x − a)2 = 0,            F (ϕx) = 0,

each in its lowest terms.
   1. To form the equation Lx + M = 0, where x = a, it is easy to see that if we
write down in general the expression (x − e1 )e2 , e3 . . . en this will become zero
whenever the root e1 left out is not one of the equal roots (a); so that in fact
(calling the two equal roots e1 , e2 respectively)

       Σ{(x − e1 )e2 , e3 . . . en } = (x − e1 )e2 , e3 . . . en + (x − e2 )e1 , e3 . . . en ,

or simply
                                    = 2(x − a)e2 , e3 . . . en .
Hence by making

                         xΣe2 , e3 . . . en − Σ{e1 e2 , e3 . . . en } = 0,

we have an equation for finding the equal roots e1 , e2 .
  Again, it is easily seen upon the same hypothesis, that

Σ{(x−e2 )(x−e3 ) · · · (x−en )e2 , e3 . . . en } = 2(x−e2 )(x−e3 ) · · · (x−en )e2 , e3 . . . en .
                                                                                                     p. 72
   Hence, to form the equation having the same roots as (x − a)ϕx, we have only
to make

         xn−1 Σe2 , e3 . . . en − xn−2 Σ{(e2 + e3 + · · · + en )e2 , e3 . . . en } + · · ·

                             ±Σ{(e2 e3 · · · en )e2 , e3 . . . en } = 0.

   Suppose now in general that we have r perfect square factors, so that

                        f x = ϕx(x − a1 )2 (x − a2 )2 · · · (x − ar )2 .

To form the equation G(x − a1 )(x − a2 ) · · · (x − ar ) = 0, we have only to make

                 Σ{(x − e1 )(x − e2 ) · · · (x − er )er+1 , er+2 . . . en } = 0.

And to obtain
                        Dϕx × (x − a1 )(x − a2 ) · · · (x − ar ) = 0,



                                                 77
we must make

              Σ{(x − er+1 )(x − er+2 ) · · · (x − en )er+1 , er+2 . . . en } = 0.

    The theory of perfect square factors is not yet complete until it has been shown
how to obtain constructively ϕx, and, as analogy suggests, the complementary
part D(x − a1 )2 (x − a2 )2 · · · (x − ar )2 , each in its lowest terms. To effect the
latter it might be said that it is only necessary to take the square of C(x −
a1 )(x − a2 ) · · · (x − ar ). It is true the polynomial so formed would contain every
pair of equal factors, but not in the lowest terms as regards the coefficients (as
we shall presently show).
    To solve this last part of the problem, let it be agreed that two rows of
letters inclosed in a parenthesis shall indicate the product of the squares of the
differences got by subtracting each in the row from each in the other, so that
   !                       !                                       !
 a                       a                                     a, b
       = (a−b) ,2
                                 = (a−b) (a−c) ,
                                         2         2
                                                                       = (a−c)2 (a−d)2 (b−c)2 (b−d)2 .
 b                      b, c                                   c, d

Let us begin with supposing that f x has one pair only of equal roots; to form
the simplest quadratic equation containing this pair, write down
                      (                                                              !)
                                                                      e1 , e2
                    Σ (x − e1 )(x − e2 )e3 , e4 . . . en                                  .
                                                                  e3 , e4 . . . en

Now if e1 and e2 are the two equal roots in question neither of the multipliers of
(x − e1 )(x − e2 ) vanishes.
   If e1 and e2 are neither of them equal roots e3 , e4 . . . en = 0.
   If one of the two only belong to the pair of equal roots
                                                           !
                                            e1 , e2
                                                               = 0.
                                        e3 , e4 . . . en
                                                                                                           p. 73
   Hence it is clear that
                    (                                                              !)
                                                                    e1 , e2
                 Σ (x − e1 )(x − e2 )e3 , e4 . . . en                                   =0
                                                                e3 , e4 . . . en

is the equation desired.
   In like manner if there be r pairs of equal roots the equation of the (2r)th
degree which contains them all may be written
          (                                                                                    !)
                                                                       e1 , e2 , . . . , e2r
        Σ (x − e1 )(x − e2 ) · · · (x − e2r )e2r+1 . . . en                                         = 0.
                                                                       e2r+1 , . . . , en

The coefficient of x2r in this equation is clearly of

                               (n − 2r)(n − 2r − 1) + 4r(n − 2r),

                                                 78
that is, of (n + 2r − 1)(n − 2r) dimensions. The coefficient of xr in the equation
which contains the r equal roots unyoked together is of (n − r)(n − r − 1)
dimensions, and consequently the coefficient of x2r in the square of this equation
would be of 2(n − r)(n − r − 1) dimensions, that is, would be n2 + 6r2 − (4r + 1)n
dimensions higher than needful.
   Finally, to obtain an equation clear of simple as well as double appearances of
the equal roots, we have only to write the complementary form
      (                                                                                  !)
                                                                 e1 , e2 , . . . , e2r
    Σ (x − e2r+1 )(x − e2r+2 ) · · · (x − en )e2r+1 . . . en                                  = 0.
                                                                  er+1 , . . . , en

   Let us, now that we are more familiarized with the notation essential to this
method, revert to the question with which we set out, and endeavour to obtain
r such equations as shall imply unambiguously the existence of r pairs of equal
roots.
   The existence of r such pairs enables us to assert the following disjunctive
proposition, which cannot be asserted when the same amount of multiplicity is
distributed in any other way.
   To wit, on selecting any r roots out of the entire number, either these r will
all be found again in those that are left, or those that are left will contain inter
se, one repetition at least; so that except on the latter supposition any (r − 1)
may be absolutely sunk out of those that are left, and there will still be one root
common to the (n − 2r + 1) remaining, and to the r originally selected to be left
out.
   Wherefore calling the roots e1 , e2 . . . en , and giving µ any value whatever, we
have
           (                                                                      !)
                                                             e1 , e2 . . . er
       Σ fµ (e1 , e2 . . . er ) × er+1 , er+2 . . . en × Σ                               = 0.
                                                           e2r , er+1 . . . en
                                                                                                     p. 74
    Hence the simplest distinctive equations indicative of the existence of r pairs
of equal roots are to be found by putting µ equal in succession to all values from
0 up to (r − 1).
    For instance, if we require that an equation of the seventh degree shall have
three pairs of equal roots, we need only to call the seven roots respectively
a, b, c, d, e, f, g, and then our type equation becomes
                                             !             !                !
                                           d, e      d, f      d, g
     Σ fµ (a, b, c) × d, e, f, g ×                +         +
                                          a, b, c   a, b, c   a, b, c
                                                   !             !                !
                                            e, f      e, g      f, g
                                         +         +         +                            = 0.
                                           a, b, c   a, b, c   a, b, c

   From this it appears that the r distinctive equations for r pairs of equal roots
are of different dimensions from the r general or overlying ones corresponding to

                                              79
the multiples r, anyhow distributed; the lowest of the latter being of (n − r +
1)(n − r), the lowest of the former of

                            (n − r)(n − r − 1) + 2r(n − 2r + 1),

that is, of n(n − 1) − 3r(n − 1) dimensions. In general we shall find that the
more unequally distributed the multiplicity may be the lower are the dimensions
of the distinctive equations, and are accordingly lowest when the multiplicity is
absolutely undistributed34 .




  34
     It must not, however, be overlooked, that the equations above given, although decisive
as to the existence of r pairs of equal roots when the multiplicity is known to be not greater
than r, do not enable us to affirm with certainty their existence when this limitation is absent;
for should the multiplicity exceed r, then inevitably (no matter how it may be distributed)
er+1 , er+2 . . . en is always zero, and consequently nullifies each term of every one of the equations
in question. In fact (repugnant as it may appear to be to the ordinary assumptions of analytical
reasoning), it is not possible to express with absolute unambiguity the conditions of there being
a multiplicity (r) distributed in any assigned manner by means of r affirmative equations alone.


                                                 80
                                           15.
On a Linear Method of Eliminating between Double, Treble,
        and other Systems of Algebraic Equations
              [Philosophical Magazine, XVIII. (1841), pp. 425–435]
                                                                                              p. 75

                            Part I.      Binary Systems.

   Let U and V be two integer complete homogeneous functions of x and y, one
of the mth, the other of the nth degree; and let it be required to express the
condition of the coexistence of the two equations U = 0, V = 0 by means of the
equation C = 0, where C is free from all appearances of x or y.
   This equation, according to the system of notation developed in a preceding
paper, and which has been since adopted and sanctioned by the high authority
of M. Cauchy, I call the final derivative: the quantity C is designated the final
derivee: and it is our present object to show how this may be obtained in a prime
form, that is to say, divested of irrelevant factors: in this state it must consist of
terms, each containing m + n letters, of which n belong to the coefficients of U ,
and m to those of V .
   Of course in applying this rule it is to be understood that every combination
of powers in U or V has a single letter prefixed for its coefficient, and that in
the final derivee powers are represented by repetitions of the same character.
   Every term in U or V being of the form Cxp y q , xp y q is called an argument,
C its prefix.
   Assume two integer positive numbers r and r′ , and also two others s and s′ ,
such that r + r′ = n − 1, s + s′ = m − 1, and form from U = 0, V = 0 two new
equations,
                                  ′                 ′
                            xr y r U = 0,     xs y s V = 0.
Such equations are termed the augmentatives of the two given ones respectively;
           ′
also xr y r U and its fellow are termed the augmentees of U and V .               p. 76
   r and r′ are termed the indices of augmentation belonging to U , s and s′ the
same belonging to V .
   Finally, it will be useful hereafter to call the given polynomials U and V
themselves the proposees, and the given equations which assert their nullity, the
propositive equations, or, briefly, the propositives.
   Now as many augmentees of either proposee can be formed as there are ways
of stowing away between two lockers (vacancies admissible) a number of things
equal to the index of the other35 ; hence we shall have n augmentees of U , and m
  35
    “Tot augmenta utriusvis ex æquationibus propositis formari possunt quot modi sint inter
duo receptacula (utriusvis vel ambobus omnino vacare licet) rerum, quarum numerus indicem
alterius æquat, distributionem faciendi.”


                                            81
of V : thus there will be m + n augmentatives each of the degree m + n − 1, and
the number of arguments is clearly m + n also, so that they can be eliminated
linearly, and the final derivee thus found, containing m + n letters (properly
aggregated) in each term, will be in its prime form, that is, incapable of further
reduction, and void of irrelevant factors.
   It is worthy of remark, that the final derivee obtained by arranging in square
battalion the prefixes of the augmentees, permuting the rows or columns, and
reading off diagonal products, affected each with the proper sign (according to
the well known rule of Duality), will not only be free from factorial irrelevancy,
but also of linear redundancy, which latter term I use to signify the reappearance
of the same combination of prefixes, sometimes with positive and sometimes with
negative signs: furthermore, it follows obviously from the nature of the process
that no numerical quantity in the final derivee will be greater than the higher of
the indices of the two given polynomials.

                               Part II.       Ternary Systems.

                               Case A.        Indices all equal.

                                             Method 1.

   Let there be now three proposees, U , V , W , integer complete homogeneous
functions of x, y, z, each of the degree n: let

         r + r′ + r′′ = n − 1,           s + s′ + s′′ = n − 1,       t + t′ + t′′ = n − 1,
                               ′   ′′             ′    ′′        ′    ′′
                         xr y r z r U,       xs y s z s V,   xt y t z t W,
will, as above, be called the augmentees of U , V , W , and every other part of the
notation previously described is to be preserved.                                   p. 77
   Suppose now
                          U = 0,     V = 0,       W = 0,
we shall have as many augmentative equations formed from each proposee as there
are ways of stowing away n things between three lockers (vacancies admissible)36 ,
         n(n + 1)                                    n(n + 1)
that is,          of each kind; in all, therefore, 3          , and every one of these
             2                                          2
will be of the degree 2n − 1, so that the number of arguments to be eliminated is
equal to the number of ways of stowing away 2n − 1 things between three lockers
(empty ones counting), that is

                                            2n(2n + 1)
                                                       .
                                                2
  36
       See for Latin translation the preceding note.



                                                 82
As yet, then, we have not enough equations for eliminating these linearly.
  Make, however,
                             α + β + γ = n + 1,
and write
                            U = xα F + y β F ′ + z γ F ′′ = 0,
                            V = xα G + y β G′ + z γ G′′ = 0,
                           W = xα H + y β H ′ + z γ H ′′ = 0,
it will always be possible to make the multipliers of xα , y β , z γ integer functions:
for if we look to any argument in U , V , or W , it is of the form xa y b z c , and one
of the letters a, b, c must be not less than its correspondent α, β, γ, for otherwise
a + b + c would be not greater than α + β + γ − 3, that is, n would be not greater
than (n + 1) − 3, or n − 2, which is absurd: if now any one, as a, be equal to or
greater than α, it may be made to supply an integer part to the multiplier of xα .
   Here it may be asked what is to be done with such terms as Kxa y b z c , when
two letters a, b are each not less than their correspondents α, β: the answer is,
such terms may be made to enter under the multiplier of xα , or of y β , or to
supply a part to both in any proportion at pleasure37 .
   From the equations above we get, by linear elimination,
         F G′ H ′′ + GH ′ F ′′ + HF ′ G′′ − GF ′ H ′′ − HG′ F ′′ − F H ′ G′′ = 0.
This may be denoted thus: Π(α, β, γ) = 0, which equation I call a secondary
derivative, and the left side of it a secondary derivee; α, β, γ may likewise be
termed the indices of derivation (as r, s, t, &c. are of augmentation).
   Now since α + β + γ = n + 1, it is clear that the index of Π(α, β, γ) is always
n + n + n − (n + 1); that is, 2n − 1.                                              p. 78
   1st. Let any two of the indices of derivation be taken zero, then it is easily
seen that all the terms in Π(α, β, γ) vanish, and consequently the secondary
derivative equations obtained upon this hypothesis become mere identities, and
are of no use.
   2nd. Let any one of them become zero.
   It is manifest, from the doctrine of simple equations, that Π(α, β, γ) may be
made equal to
                                                   1
                              {λU + µV + νW } α ,
                                                  x
or
                              ′                    1
                              λ U + µ′ V + ν ′ W β ,
                                                   y
or
                             ′′                     1
                             λ U + µ′′ V + ν ′′ W γ ,
                                                    z
  37
     The prefixes of any such terms (say K) may be conceived as made up of two parts, an
arbitrary constant, as e and (K − e); e will disappear spontaneously from the final derivee.


                                            83
upon the understanding that
             λ = G′ H ′′ − G′′ H ′ , µ = H ′ F ′′ − H ′′ F ′ , ν = F ′ G′′ − F ′′ G′ ,
             λ′ = G′′ H − GH ′′ , µ′ = H ′′ F − HF ′′ , ν ′ = F ′′ G − F G′′ ,
             λ′′ = GH ′ − G′ H, µ′′ = HF ′ − H ′ F, ν ′′ = F G′ − F ′ G.
The three rows of coefficients will be respectively of the degrees
          (n − β) + (n − γ),          (n − γ) + (n − α),          (n − α) + (n − β).
Thus if any one of the indices α, β, γ be zero, Π(α, β, γ) becomes identical with
λ′ U + µ′ V + ν ′ W , where the multipliers of U, V, W are of 2n − (α + β + γ)
dimensions, that is of (n − 1) dimensions, and may accordingly be put under the
form
                          ′   ′′           ′   ′′           ′   ′′
                  ΣAxr y r z r U + ΣBxs y s z s V + ΣCxt y t z t W,
that is to say, becomes a linear function of the augmentatives, and therefore if
combined with them in the process of linear elimination would give rise to the
identity 0 = 0.
   Hence we must reject all such secondary derivatives as have zero for one
of the indices of derivation. But all others, it may be shown, will be linearly
independent of one another, and of the augmentees previously found. Hence,
          n(n + 1)
besides 3           equations of augment of the degree 2n − 1, we shall have of
              2
the same degree so many equations of derivation as there are ways of stowing
away between three lockers (n + 1) things, under the condition that no locker
                                  n(n − 1) 38
shall ever be left empty, that is             .
                                     2
   Thus, then, in all we have
                       n(n − 1)      n(n + 1)    2n(2n + 1)
                                 +3            =
                           2             2            2
equations, which is exactly equal to the number of arguments to be eliminated.
Hence                                                                               p. 79
   the final derivee can be obtained by the usual explicit rule of permutation,
                                                                          n(n + 1)
and moreover will be its lowest form, for it will contain in each term
                                                                              2
prefixes belonging to the augmentatives of U , and a like number pertaining to
                                  n(n − 1)
those of V and of W , as well as            belonging to the secondary derivatives,
                                      2
each prefix in any one of which is triliteral, containing a prefix drawn out of
those belonging to each of the proposees.
                                        n+1      n−1
   Thus every member containing n            +n        , that is n2 of the original
                                          2         2
prefixes belonging to U, V, W , singly and respectively, the final derivee evolved
by this process will be in its lowest terms; as was to be proved.
 38
      Vide page 76 for the Latin version.


                                                84
                            Case A.       Indices all equal.

                                         Method 2.

   It is remarkable that we may vary the method just given by making

       r + r′ + r′′ = n − 2,        s + s′ + s′′ = n − 2,         t + t′ + t′′ = n − 2.

The augmentatives will thus be of the degree 2n − 2.
   Furthermore, we must make α + β + γ = n + 2. It will still be possible to
satisfy by integer multipliers the equations

                                U = xα F + y β F ′ + z γ F ′′ ,
                                V = xα G + y β G′ + z γ G′′ ,
                               W = xα H + y β H ′ + z γ H ′′ ,

[these it will be useful in future to term the equations, xα , y β , z γ being the
arguments, and F, G, H, &c. the factors of decomposition] for otherwise calling
the indices of x, y, z in any original argument a, b, c, their sum or n would be not
greater than (n + 2) − 3, that is (n − 1), which is absurd.
   For the same reasons as in the last case no index of augmentation must
be made zero: the degree of each will be (n − α) + (n − β) + (n − γ), that
                                   (n + 1)n
is (2n − 2), and their number               ; the number of augmentatives will be
                                       2
  (n − 1)n
3           linearly uninvolved, each of the degree 2n − 2, and therefore containing
      2
(2n − 1)2n
              arguments.
      2
   Now
                         (n + 1)n     (n − 1)n      (2n − 1)2n
                                  +3             =              .
                             2            2              2
                                                                                     p. 80
   Hence the final derivee may be found, and it will be in its lowest terms,
                                   3(n − 1)n
for every member will contain                  letters due to the augmentative, and
                                        2
3(n + 1)n
            due to the partial derivative equations; in all then there will be 3n2
     2
letters in each term.
   This second method being applied to three quadratic equations of the most
general form, leads to the problem of eliminating between six simple equations
which lies within the limits of practical feasibility, and it is my intention to
register the final derivee upon the pages of some one of our scientific Transactions
as a standing monument for the guidance of hereafter coming explorers39 .
  39
    Elimination between two quadratics leads to a final derivee made up of seven terms only;
the final derivee of three quadratics is made up of at least several thousand; nay, I believe I
may safely say, several myriads of terms!


                                              85
                             Scholium to Case A.

  If we attempt to carry forward these processes to quaternary systems, it
becomes necessary to make

                          α + β + γ + δ = (r − 2)n + 1

or else
                          α + β + γ + δ = (r − 2)n + 2,
where r is the number of proposees.
   Now if the factors in the equations of decomposition are all integer, one of the
indices of derivation must be not greater than the corresponding index in any of
the original arguments, which may easily be shown to be always impossible for
a system of equations, complete in all their terms, whenever their number r is
greater than three, if α+β +γ +δ = (r −2)n+2; but if α+β +γ +δ = (r −2)n+1
only possible for the case of n = 2.

          Particular method applicable to four Quadratics.

  Let U = 0, V = 0, W = 0, Z = 0, be four quadratic equations existing
between x, y, z, t.
  Make
                    xU = 0, xV = 0, xW = 0, xZ = 0,
                    yU = 0, yV = 0, yW = 0, yZ = 0,
                    zU = 0, zV = 0, zW = 0, zZ = 0,
                    tU = 0, tV = 0, tW = 0, tZ = 0.
                                                                                      p. 81
  Also write
                       U = x2 F + yF ′ + zF ′′ + tF ′′′ = 0,
                       V = x2 G + yG′ + zG′′ + tG′′′ = 0,
                      W = x2 H + yH ′ + zH ′′ + tH ′′′ = 0,
                       Z = x2 K + yK ′ + zK ′′ + tK ′′′ = 0.
By eliminating linearly we get

                        Σ{F ΣG′ (H ′′ K ′′′ − H ′′′ K ′′ )} = 0,

which will be of the third degree, since the factors represented by the unmarked
letters F, G, H, K are of zero, and all the rest of unit dimensions.
   Similarly we may obtain other equations, so that besides the sixteen augmen-
tatives already written down, we have four secondary derivatives, namely,

      Π(2 111) = 0,     Π(12 11) = 0,          Π(11 21) = 0,       Π(111 2) = 0.


                                          86
Thus we have twenty equations and as many arguments to eliminate, since a
perfect cubic function of four letters contains twenty terms.
   The final derivee will contain 16 + 4 · 4 letters, that is 32, 8 or 23 belonging
to each system of original prefixes in each member, and will therefore be in
its lowest terms: for one of the canons of form teaches us, à priori, that every
member of the derivee deduced from any number of assumed equations must
contain in each member as many prefixes belonging to one equation of the system
as there are units in the product of the indices of all the rest taken together.

                                 Corollary to Case A.

  Either of the two methods given as applicable to this case enables us to
determine integer values of X, Y, Z, which shall satisfy the equation

                            XU + Y V + ZW = F xp y q z r ,

where F is the final derivee and p + q + r = 3n − 2. For by the doctrine of
simple equations we know how to express F in terms of the linear functions, out
of which it is obtained by permutation, that is we are able to assign values of
A, B, C, and their antitypes, as also of L and its antitype, which shall satisfy
the equation
                        ′   ′′               ′   ′′              ′   ′′
               Σ(Axr y r z r U ) + Σ(Bxs y s z s V ) + Σ(Cxt y t z t W )
                                                                                 (1)
                             + Σ{LΠ(α, β, γ)} = F xf y g z h ,
where A, B, C, as well as L and all the quantities formed after them, are made
up of integer combinations of the original prefixes.
   Now the functions Π(α, β, γ) may be expressed in three ways in terms of
U, V, W , as has been already shown.                                           p. 82
   We may therefore suppose these functions to be divided into three groups,
and make
                 QU + Q′ V + Q′′ W      RU + R′ V + R′′ W     SU + S ′ V + S ′′ W
ΣLΠ(αβγ) = Σ               α
                                    +Σ             β
                                                          +Σ                       .
                         x                       y                     zγ
                                                                               (2)
And it is evident that the equations (1) and (2) lead immediately to the equation

                       XU + Y V + ZW = F xa+f y b+g z c+h ,

if we call a, b, c the greatest values attributed respectively to α, β, γ.
   Now if we suppose the first method to be followed,

                                   f + g + h = 2n − 1.

And it will always be possible to make a, b, c of what values we please subject to
the condition of a + b + c = n − 1; for one at least of the indices of derivation in

                                           87
Π(α, β, γ) must be not greater than its correspondent among a, b, c; otherwise
α + β + γ would be not less than (a + b + c) + 3; but

                       α + β + γ = n + 1,              a + b + c = n − 1,

which is absurd.
    Hence we can satisfy XU + Y V + ZW = F xp y q z r , p, q, r being subject to the
condition of p + q + r = 3n − 2, but otherwise arbitrary.
    Moreover, we can not do so if p+q+r be less than 3n−2, for that would require
a + b + c to be less than n − 1. Now if two of the indices of derivation, as α and
β, be made equal to a + 1, b + 1 respectively, the third γ = (n + 1) − (a + b + 2) =
(n − 1) − (a + b), and is therefore greater than c: so that α + β + γ for this case
becomes greater than a + b + c, and the method falls to the ground.
    In fact, I have discovered a theorem which lets me know this, à priori, a
law which serves as a staff to guide my feet from falling into error in devising
linear methods of solution, and the importance of which all candid judges who
have studied the general theory of elimination cannot fail to recognize. To
wit, if X1 , X2 , X3 . . . Xn be n integer complete polynomial functions of n letters
x1 , x2 . . . xn , and severally of the degree b1 , b2 , b3 . . . bn ; then it is always possible
to satisfy the identity

             P1 X1 + P2 X2 + P3 X3 + · · · + Pn Xn = F xa11 xa22 xa33 · · · xann ,
                                                                                                    p. 83
   if a1 + a2 + a3 + · · · + an be equal to or greater than b1 + b2 + b3 + · · · + bn − n +1,
but otherwise not 40 .
   This again is founded immediately upon a simple proposition, of which I have
obtained a very interesting and instructive demonstration, shortly to appear, and
which may be enumerated thus: “The number of augmentees of the same degree
that can be formed, linearly independent of one another, out of any number of
polynomial functions of as many variables, may be either equal to or less than the
number of distinct arguments contained in such augmentees, but never greater.
The latter will be the case when the index of the augmentees diminished by unity
  40
     Hence it is apparent, that in applying the method of multipliers, a curious and important
distinction exists between the cases of there being two equations, and there being a greater
number to eliminate from: for in the first case the element of arbitrariness needs never to
appear; in the latter it cannot possibly be excluded from appearing in the multipliers.
   This will explain how it comes to pass that the method of the text may be employed to give
various solutions of the XU + Y V + ZW = F xp y q z r ; thus not only can p, q and r be variously
made up of (f + a), (g + b), (h + c), but also Π(α, β, γ) when two of the indices (α, β suppose)
are each not greater than the assigned greatest values a, b may be made to figure indifferently
either under the form
                       λU + µV + νW                      λ′ U + µ′ V + ν ′ W
                                          or that of                         .
                            xα                                   yβ




                                               88
is less than the sum of the indices of the original unaugmented polynomials each
so diminished; the former, when the aforesaid index is equal to or greater than
the aforesaid sum.”
    To return to the particular case of finding X, Y, Z to satisfy

                          XU + Y V + ZW = F xp y q z r .

This has been already done according to the first method; if we employ the
second method of elimination we shall have

                                  f + g + h = 2n − 2.

But, now since α + β + γ = n + 2, we shall easily see by the same method
as above, that the least value of a + b + c {where a, b, c denote respectively
the greatest values of α, β, γ, appearing in the denominator of the fractional
forms used to express Π(α, β, γ)}, will be one greater than before, or n; so that
f + g + h + a + b + c will still be equal to 3n − 2, as we might, à priori, by virtue
of our rule, have been assured.

                                  Ternary Systems.

    Case B.     Two of the indices equal; the third less by a unit.

   Let U = 0, V = 0, W = 0, be the three given equations severally of the degree
n, n, (n − 1).                                                                   p. 84
   Make

      r + r′ + r′′ = n − 2,       s + s′ + s′′ = n − 2,    t + t′ + t′′ = n − 1,
                              ′    ′′             ′   ′′             ′   ′′
by multiplying U into xr y r z r , V into xs y s z s , W into xt y t z t , we obtain
augmentees each of the same, namely, the (2n − 2)th degree.
  The number of these is
                        (n − 1)n (n − 1)n n(n + 1)
                                +        +         .
                           2         2       2
Again, make
                                  α + β + γ = n + 1.
It will still be possible, as before, to form equations of decomposition in which
xα , y β , z γ are the arguments, and affected with integer factors. For if we look
to W even, all its arguments are of the form xa y b z c , where a + b + c = (n − 1),
and each of these cannot be less than its correspondent, for that would be to say
that (n − 1) is not greater (n + 1) − 3, à fortiori, U and V can be decomposed in
the manner described. Thus, then, we shall obtain as many secondary derivees


                                          89
                                       n(n − 1)
as in the last case (Method 1), that is,          {since α + β + γ is still equal to
                                           2
(n+1)}, as before. Moreover, each of these will be of (n−α)+(n−β)+(n−1−γ),
that is of 2n − 2 dimensions.
  Altogether, therefore, we have

                      (n − 1)n (n − 1)n n(n + 1)             (n − 1)n
                                                    
                              +        +                 +
                          2       2        2                    2
linear independent equations of the degree 2n − 2, and the number of arguments
                 (2n − 1)2n
to eliminate is             . Now these two numbers are equal.
                      2
                                                                         (n − 1)n
   Thus we obtain a final derivee containing of U ’s coefficients                 +
                                                                            2
(n − 1)n                                         n(n + 1) (n − 1)n
          , an equal number of V ’s, but of W ’s          +          ; now n(n−1),
    2                                               2           2
n(n − 1) and n exactly express the number that ought to appear of each of
                 2

these respectively: hence the final derivee is clear of irrelevant factors.

                                Ternary Systems.

Case C.     Two of the indices equal; the third one greater by a unit.

  Here, calling n the highest index, the augmentees must each be made of the
degree (2n − 3), their number will evidently be

                        (n − 2)(n − 1) (n − 1)n (n − 1)n
                                      +        +         ,
                              2           2         2
                                                                                       p. 85
   making the sum of the indices of derivation now, as before, equal to (n + 1); it
will be still possible to form integer equations of decomposition, which will give
rise to augmentatives of the degree (n − α) + (n − 1) − β + (n − 1) − γ, that is, of
(2n − 3) dimensions. The total number of equations, what with augmentatives
and secondary derivatives, will be

                  (n − 2)(n − 1) (n − 1)n (n − 1)n          n(n − 1)
                                                        
                                +          +              +
                         2            2          2             2
                        4n − 4n + 2
                           2           (2n − 2)(2n − 1)
                     =              =                   ,
                             2                2
that is, is equal to the exact number of distinct arguments contained between
them.
   Also the final derivative will contain in each member
                             (n − 2)(n − 1) n(n − 1)
                                           +         ,
                                   2           2


                                           90
that is, (n − 1)(n − 1), letters belonging to the first equation, and

                              (n − 1)n n(n − 1)
                                      +         ,
                                 2        2
that is, n(n − 1) belonging to those of the second and of the third, and will
therefore be in its lowest terms.

                       Corollary to Cases B and C.

  It is not necessary, after all that has been already said, to do more than just
point out that the processes applicable to these cases enable us to determine
X, Y, Z, which satisfy the equation

                         XU + Y V + ZW = F xf y g z h ,

where
                        f + g + h = 3n − 3 for Case B,
and
                        f + g + h = 3n − 4 for Case C.




                                        91
                                             16.
    Memoir on the Dialytic Method of Elimination. Part I
[Philosophical Magazine, XXI. (1842), pp. 534–539; reprinted from Proc. Roy.
                  Irish Acad., Vol. II. (1840–1844), p. 130]
                                                                                                   p. 86
   The author confines himself in this part to the treatment of two equations,
the final and other derivees of which form the subject of investigation.
   The author was led to reconsider his former labours in this department of the
general theory by finding certain results announced by M. Cauchy in L’Institut,
March Number of the present year, which flow as obvious and immediate conse-
quences from Mr Sylvester’s own previously published principles and method.
   Let there be two equations in x,

                    U = axn + bxn−1 + cxn−2 + exn−3 + &c. = 0,
                    V = αxm + βxm−1 + γxm−2 + &c. = 0,

and let n = m + ι, where ι is zero or any positive value (as may be).
  Let any such quantities as xr U , xs V , be termed augmentatives of U or V .
  To obtain the derivee of a degree s units lower than V , we must join s
augmentatives of U with s + ι of V . Then out of 2s + ι equations

                 x0 U = 0,     x1 U = 0,     x2 U = 0, . . . . . . xs−1 U = 0,
                x0 V = 0,     x1 V = 0,     x2 V = 0, . . . . . . xι+s−1 V = 0,

we may eliminate linearly 2s + ι − 1 quantities.
   Now these equations contain no power of x higher than m+ι+s−1; accordingly,
all powers of x, superior to m − s, may be eliminated, and the derivee of the
degree (m − s) obtained in its prime form.
   Thus to obtain the final derivee (which is the derivee of the degree zero), we
take m augmentatives of U with n of V , and eliminate (m + n − 1) quantities,
namely,
                       x, x2 , x3 , . . . . . . up to xm+n−1 .
                                                                                                   p. 87
  This process, founded upon the dialytic principle, admits of a very simple
modification. Let us begin with the case where ι = 0, or m = n. Let the
augmentatives of U be termed U0 , U1 , U2 , U3 , . . . and of V , V0 , V1 , V2 , V3 , . . ., the
equations themselves being written

                            U = axn + bxn−1 + cxn−2 + &c.
                            V = a′ xn + b′ xn−1 + c′ xn−2 + &c.



                                              92
It will readily be seen that

                                      a′ U0 − aV0 ,
                             (b′ U0 − bV0 ) + (a′ U1 − aV1 ),
                 (c′ U0 − cV0 ) + (b′ U1 − bV1 ) + (a′ U2 − aV2 ), &c.

will be each linearly independent functions of x, x2 , . . . xm−1 , no higher power of
x remaining. Whence it follows, that to obtain a derivee of the degree (m − s) in
its prime form, we have only to employ the s of those which occur first in order,
and amongst them eliminate xm−1 , xm−2 , . . . xm−s+1 . Thus, to obtain the final
derivee, we must make use of n, that is, the entire number of them.
   Now, let us suppose that ι is not zero, but m = n − ι. The equation V may
be conceived to be of n instead of m dimensions, if we write it under the form

        0xn + 0xn−1 + 0xn−2 + · · · + 0xm+1 + αxm + βxm−1 + &c. = 0,

and we are able to apply the same method as above; but as the first ι of the
coefficients in the equation above written are zero, the first ι of the quantities

              (a′ U0 − aV0 ),       (b′ U0 − bV0 ) + (a′ U1 − aV1 ),      &c.

may be read simply

              −aV0 ,      −bV0 − aV1 ,           −cV0 − bV1 − aV2 ,       &c.

and evidently their office can be supplied by the simple augmentatives themselves,

                   V0 = 0,        V1 = 0,        V2 = 0 . . . Vι−1 = 0;

and thus ι letters, which otherwise would be irrelevant, fall out of the several
derivees.
   The author then proceeds with remarks upon the general theory of simple
equations, and shows how by virtue of that theory his method contains a solution
of the identity
                              Xr U + Yr V = Dr ,
where Dr is a derivee of the rth degree of U and V , and accordingly, Xr of the
form
                        λ + µx + νx2 + · · · + θxm−r−1 ,
and Yr of the form
                                l + mx + · · · + txn−r−1 ,
                                                                                         p. 88
   and accounts à priori for the fact of not more than (n − r) simple equations
being required for the determination of the (m + n − 2r) quantities λ, µ, ν, &c.
l, m, n, &c., by exhibiting these latter as known linear functions of no more than
(n − r) unknown quantities left to be determined.

                                            93
   Upon this remarkable relation may be constructed a method well adapted for
the expeditious computation of numerical values of the different derivees.
   He next, as a point of curiosity, exhibits the values of the secondary functions,

                                       a′ U0 − aV0 ,
                                b′ U0 − bV0 + a′ U1 − aV1 ,
                      c′ U0 − cV0 + b′ U1 − bV1 + a′ U2 − aV2 ,   &c.

under the form of symmetric functions of the roots of the equations U = 0, V = 0,
by aid of the theorems developed in the London and Edinburgh Philosophical
Magazine, December 183941 , and afterwards proceeds to a more close examination
of the final derivee resulting from two equations each of the same (any given)
degree.
    He conceives a number of cubic blocks each of which has two numbers, termed
its characteristics, inscribed upon one of its faces, upon which the value of such
a block (itself called an element) depends.
    For instance, the value of the element, whose characteristics are r, s, is the
difference between two products: the one of the coefficient rth in order occurring
in the polynomial U , by that which comes sth in order in V ; the other product
is that of the coefficient sth in order of the polynomial U , by that rth in order of
V ; so that if the degree of each equation be n, there will be altogether 12 n(n + 1)
such elements.
    The blocks are formed into squares or flats (plafonds) of which the number is
n      n+1
   or        , according as n is even or odd. The first of these contains n blanks
 2        2
in a side, the next (n − 2), the next (n − 4), till finally we reach a square of four
blocks or of one, according as n is even or odd. These flats are laid upon one
another so as to form a regularly ascending pyramid, of which the two diagonal
planes are termed the planes of separation and symmetry respectively. The
former divides the pyramid into two halves, such that no element on the one side
of it is the same as that of any block in the other. The plane of symmetry, as
the name denotes, divides the pyramid into two exactly similar parts; it being a
rule, that all elements lying in any given line of a square (plafond) parallel to the
plane of separation are identical; moreover, the sum of the characteristics is the
same, for all elements lying anywhere in a plane parallel to that of separation. p. 89
    All the terms in the final derivee are made up by multiplying n elements of
the pile together, under the sole restriction, that no two or more terms of the
said product shall lie in any one plane out of the two sets of planes perpendicular
to the sides of the squares. The sign of any such product is determined by the
places of either set of planes parallel to a side of the squares and to one another,
in which the elements composing it may be conceived to lie.
  41
       p. 40 above. ED.



                                            94
   The author then enters into a disquisition relating to the number of terms which
will appear in the final derivee, and concludes this first part with the statement of
two general canons, each of which affords as many tests for determining whether
a prepared combination of coefficients can enter into the final derivee of any
number of equations as there are units in that number, but so connected as
together only to afford double that number, less one, of independent conditions.
   The first of these canons refers simply to the number of letters drawn out
of each of the given equations (supposed homogeneous); the second to what he
proposes to call the weight of every term in the derivee in respect to each of the
variables which are to be eliminated.
   The author subjoins, for the purpose of conveying a more accurate conception
of his Pyramid of derivation, examples of the mode in which it is constructed.

                                                       When n = 2 there is one flat, viz.
When n = 1 there is one flat, viz.                                           2, 3      2, 4
                  1, 2                                                       2, 4      3, 4
                                                     Let n = 4, there will still be two flats
Let n = 3, there will be two flats:
                                                                     only:
                  2, 3
                                                                             2, 3      2, 4
        1, 2      1, 3     1, 4                                              2, 4      3, 4
        1, 3      1, 4     2, 4                                     1, 2      1, 3     1, 4    1, 5
        1, 4      2, 4     3, 4                                     1, 3      1, 4     1, 5    2, 5
                                                                    1, 4      1, 5     2, 5    3, 5
                                                                    1, 5      2, 5     3, 5    4, 5
                                                                                                      p. 90
   Let n = 5, there will be three flats:

                                                     1, 2    1, 3      1, 4          1, 5     1, 6
                         2, 3 2, 4 2, 5              1, 3    1, 4      1, 5          1, 6     2, 6
           3, 4          2, 4 2, 5 3, 5              1, 4    1, 5      1, 6          2, 6     3, 6
                         2, 5 3, 5 4, 5              1, 5    1, 6      2, 6          3, 6     4, 6
                                                     1, 6    2, 6      3, 6          4, 6     5, 6

   Let n = 6, there will be three flats:
                                              2, 3    2, 4    2, 5         2, 6
                             3, 4    3, 5     2, 4    2, 5    2, 6         3, 6
                             3, 5    4, 5     2, 5    2, 6    3, 6         4, 6
                                              2, 6    3, 6    4, 6         5, 6

                              1, 2    1, 3   1, 4    1, 5    1, 6     1, 7
                              1, 3    1, 4   1, 5    1, 6    1, 7     2, 7
                              1, 4    1, 5   1, 6    1, 7    2, 7     3, 7
                              1, 5    1, 6   1, 7    2, 7    3, 7     4, 7
                              1, 6    1, 7   2, 7    3, 7    4, 7     5, 7
                              1, 7    2, 7   3, 7    4, 7    5, 7     6, 7


                                                95
                                                                          n+1
  Thus the work of computation reduces itself merely to calculating n
                                                                            2
elements, or the n(n + 1) cross-products out of which they are constituted, and
combining them factorially after that law of the pyramid, to which allusion has
been already made.




                                      96
                                            17.
   Elementary Researches in the Analysis of Combinatorial
                       Aggregation
               [Philosophical Magazine, XXIV. (1844), pp. 285–296]
                                                                                                 p. 91
   The ensuing inquiries will be found to relate to combination-systems, that is,
to combinations viewed in an aggregative capacity, whose species being given,
we shall have to discover rules for ranging or evolving them in classes amenable
to certain prescribed conditions. The question of numerical amount will only
appear incidentally, and never be made the primary object of investigation42 .
   The number of things combined will be termed the modulus of the system
to which they belong. The elements taken singly, or combined in twos, threes,
&c., will be denominated accordingly the monadic, duadic, triadic elements, or
simply the monads, duads, or triads of the system.
   Let us agree to denote by the word syntheme43 any aggregate of combinations
in which all the monads of a given system appear once, and once only.
   It is manifest that many such synthemes totally diverse in every term may be
obtained for a given system to any modulus, and for any order of combination.
   Let us begin with considering the case of duad synthemes. Take the modulus
4 and call the elements a, b, c, d.

                           (ab, cd),      (ac, bd),       (ad, cb)
constitute three perfectly independent synthemes, and these three synthemes
include between them all the duad elements, so that no more independent
synthemes can be obtained from them.                                             p. 92
   Again, let a, b, c, d, e, f be the monads; we can write down five independent
synthemes, to wit,                              
                                    ab, cd, ef 
                                    ad, cf, eb 
                                                
                                                
                                                
                                    ac, de, f b .
                                                
                                    af, bd, ce 
                                                
                                                
                                    ae, df, bc
                                                

We can write no more than these without repeating duads which have already
appeared44 .
  42
     The present theory may be considered as belonging to a part of mathematics which bears
to the combinatorial analysis much the same relation as the geometry of position to that of
measure, or the theory of numbers to computative arithmetic; number, place, and combination
(as it seems to the author of this paper) being the three intersecting but distinct spheres of
thought to which all mathematical ideas admit of being referred.
  43
     From συν and τίθηµι.
  44
     Such an aggregate of synthemes may be therefore termed a Total.


                                             97
    We propose to ourselves this problem:—A system to any even 45 modulus being
given, to arrange the whole of its duads 46 in the form of synthemes; or in other
words, to evolve a Total of duad synthemes to any given even modulus 47 .
    When the modulus is odd, as before remarked, the formation of a duad
syntheme is of course impossible, for any number of duads must necessarily
contain an even number of monadic elements; but there is nothing to prevent us
from forming in all cases what may be termed a bisyntheme or diplotheme, that
is, an aggregate of combinations, where each element occurs twice and no more.
    For instance, if the elements be called after the letters of the alphabet, we
have                                               !
                                ab, bc, cd, de, ea
                                                     ,
                                ac, ce, eb, bd, da
the bisynthematic total to modulus 5; and in like manner
                                                          
            ab, bc, cd, de, ef, f g, ga 
                                        
            ac, ce, eg, gb, bd, df, f a   the total to modulus 7.
                                        
            ad, dg, gc, cf, f b, be, ea 
                                                                                                      p. 93
                                                            n−1
  In general, if n be the modulus, the number of duads is n     ; n being even,
                                                             2
n
  duads go to each syntheme, and therefore the total contains (n − 1) of these.
2
  45
     The modulus must be even, as otherwise it is manifest no single syntheme can be formed.
We shall before long extend the scope of our inquiry so as to take in the case of odd moduli.
  46
     Triadic systems will be treated of hereafter.
  47
     It is scarcely necessary to advert here to the fact of the problem being in general indetermi-
nate and admitting of a great variety of solutions.
  When the modulus is four there is only one synthematic arrangement possible, and there is no
indeterminateness of any kind; from this we can infer, a priori, the reducibility of a biquadratic
equation; for using Φ, ϕ, F to denote rational symmetrical forms of function, it follows that

                                           f {ϕ(a, b), ϕ(c, d)}
                                       (                          )
                                   F       f {ϕ(a, c), ϕ(b, d)}
                                           f {ϕ(a, d), ϕ(b, c)}

is itself a rational symmetric function of a, b, c, d. Whence it follows that if a, b, c, d be the
roots of a biquadratic equation, f {ϕ(a, b), ϕ(c, d)} can be found by the solution of a cubic: for
instance, (a + b) × (c + d) can be thus determined, whence immediately the sum of any two of
the roots comes out from a quadratic equation.
   To the modulus 6 there are fifteen different synthemes capable of being constructed; at first
sight it might be supposed that these could be classed in natural families of three or of five
each, on which supposition the equation of the sixth degree could be depressed; but on inquiry
this hope will prove to be futile, not but what natural affinities do exist between the totals; but
in order to separate them into families each will have to be taken twice over, or in other words,
the fifteen synthemes to modulus 6 being reduplicated subdivide into six natural families of five
each. Again, it is true that the triads to modulus 6 (just like the duads to modulus 4) admit
of being thrown into but one synthematic total, but then this will contain ten synthemes, a
number greater than the modulus itself.


                                                  98
If n be odd, then, since always n duads go to a bisyntheme, the number of such
                n−1
in the total is       .
                  2
    Before proceeding to the solution of the problem first proposed, let us investi-
gate the theory of diplothematic arrangement. Here we shall find another term
convenient to employ. By a cyclotheme, I designate a fixed arrangement of the
elements in one or more circles, in which, although for typographical purposes
they are written out in a straight line, the last term is to be viewed as contiguous
and antecedent to the first; the recurrence may be denoted by laying a dot upon
the two opened ends of the circle; ȧ.b.c.d.ė will thus denote a cyclotheme to
                                                                          ˙
modulus 5; ȧ.b.c.d.e.f.g.h.k̇ the same to modulus 9; so also is ȧ.b.c, d.e.f, ġ.h.k̇
a cyclotheme of another species to the same modulus. In general the number of
terms will be alike in each division of a cyclotheme.
    Now it is evident that every cyclotheme, on taking together the elements that
lie in conjunction, may be developed into a diplotheme. Thus
                                  1̇.2.3̇ = 12, 23, 31,
                               1̇.2.3.4̇ = 12, 23, 34, 41,
                                                    12, 23, 31
                                                                      

                    (1̇.2.3̇; 4̇.5.6̇; 7̇.8.9̇) =  45, 56, 64  .
                                                              
                                                    78, 89, 97
Hence we shall derive a rule for throwing the duads of any system into bisyn-
themes.
   Let m = 3, we have simply ȧbċ;
                           m = 5, we write            ȧ.b.c.d.ė,
                                                                ˙
                                                      ȧ.c.e.b.d,
the second being derived from the first by omitting every alternate term; similarly
below, the lines are derived each from its antecedent.
                         m = 7, we have             ȧ.b.c.d.e.f.ġ,
                                                    ȧ.c.e.g.b.d.f˙,
                                                                  ˙
                                                    ȧ.e.b.f.c.g.d.
                                                                                          p. 94
   A very little consideration will serve to prove that in this way, m being a
                 m−1
prime number,           cyclothemes may be formed, such that no element will
                   2
ever be found more than once in contact on either side with any other; whence
the rule for obtaining the diplothematic total to any prime-number modulus is
apparent.
   For example, to modulus 7 the total reads thus:—
                      1st. ab, bc, cd, de, ef, f g, ga
                     2nd. ac, ce, eg, gb, bd, df, f a
                     3rd. ae, eb, bf, f c, cg, gd, da

                                           99
and no more remains to be said on this special case.
   Let us now return to the theory of even moduli, and show how to apply what
has been just done to constructing a synthematic total to a modulus which is
the double of a prime number.
   Suppose the modulus to be six, the number of synthemes is five. Let the six
elements, a, b, c, d, e, f , be taken in three parts, so that each part contains two of
them; let these parts be called A, B, C, where A denotes ab, B, cd, and C, ef .
   Now the duads will evidently admit of a distinction into two classes, those
that lie in one part, and those that lie between two; thus ab, cd, ef will be each
unipartite duads, the rest will be bipartite.
   The unipartite duads may be conveniently formed into a syntheme by them-
selves; it only remains to form the four remaining bipartite duad synthemes.
   Write the parts in cyclothematic order, as below:

                                        ȦB Ċ.

It will be observed that each part may be written in two positions; thus

                                                  a             b
                    A    may be expressed by            or by     ,
                                                  b             a
                                                  c             d
                   B                                              ,
                                                  d             c
                                                  e             f
                    C                                             .
                                                  f             e

Now we may form a cyclic table of positions as below:

                                      Ȧ B Ċ
                                      1 1 1
                                      1 2 2
                                      2 1 2
                                      2 2 1
                                                                                          p. 95
   Here the numbers in each horizontal line denote the synchronic positions of
the parts.
   On inspection it will be discovered that A will be found in each of its two
positions, with B in each of its two; similarly B with C, and C with A. In fact
the four permutations, 11, 12, 21, 22, occur, though in different orders, in any
two assigned vertical columns.
   Now develope the preceding table, and we have

                                ace adf     bcf bde,
                                bdf bce     ade acf ;


                                          100
and these being read off (the superior of each antecedent with the inferior of each
consequent48 ) must manifestly give the four independent bipartite synthemes
which we were in quest of, videlicet
              (ad, cf, eb),           (ac, de, f b),            (bd, ce, f a),            (bc, df, ea) :
these four, together with the syntheme first described (ab, cd, ef ), constitute a
duad synthematic total to modulus 6.
   Before proceeding further let us take occasion to remark that the foregoing
table of positions may evidently be extended to any odd number of terms by
repetition of the second and third places, as seen in the annexed tables of position.
                              1̇     1   1    1̇       1̇   1    1   1   1       1   1̇
                              1̇     2   2    2̇       1̇   2    2   2   2       2   2̇
                              2̇     1   2    1̇       2̇   1    2   1   2       1   2̇
                              2̇     2   1    1̇       2̇   2    1   2   1       2   1̇
Now let 10 be the modulus.
   As before divide the elements into five parts, which call A, B, C, D, E.
   The unipartite duads fall into a single syntheme; the eight remaining bipartite
synthemes may be found as follows:—
                             n−1
                                              
   Arrange in cyclothemes           in number the odd modulus system A, B, C, D, E.
                               2
We have thus
                            ȦBCDĖ,         ȦCEB Ḋ.
                                                                                                           p. 96
  Let each cyclotheme be taken in the four positions given in the table above,
we have thus 2 × 4, that is, eight arguments.
                                   ȧbcdė,   ȧβγδ ϵ̇,     ȧbγdϵ̇,     ȧβcδ ė,
                                   αβγδϵ,     abcde,        aβcδe,       abγde,
                                         ˙
                                   ȧcebd,    ȧγeβ δ̇,     ȧcebδ̇,     ȧγeβ d,˙
                                   αγϵβδ,     acebd,        aγeβd,       acebδ.
And each of these arguments will furnish one bipartite syntheme, by reading off,
as before, the superior of each antecedent with the inferior of each consequent;
and the least reflection will serve to show that the same duad can never appear
in two distinct arguments.
   In like manner, if the modulus be 14 and seven parts be taken, the bipartite
synthemes, twelve in number, may be expressed symbolically thus:
                       1̇.1.1.1.1.1.1̇
                                                  
                                                              
                      +1̇.2.2.2.2.2.2̇      Ȧ.B.C.D.E.F.Ġ
                                         ×  +Ȧ.C.E.G.B.D.Ḟ  .
                                                            
                      +2̇.1.2.1.2.1.2̇ 
                     
                                              +Ȧ.E.B.F.C.G.Ḋ
                       +2̇.2.1.2.1.2.1̇
  48
       Any other fixed order of successive conjunction would answer equally well.


                                                        101
Nay more, from the above table, if we agree to name the elements

                                      A1 B 1
                                             ,      &c.,
                                      A2 B 2

we can at once proceed to calculate each of the twelve synthemes in question by
an easy algorithm. For instance,

(1̇.2.2.2.2.2.2̇)×(Ȧ.C.E.G.B.D.Ḟ ) = (A1 C1 , C2 E1 , E2 G1 , G2 B1 , B2 D1 , D2 F1 , F2 A2 ).

And again

(2̇.1.2.1.2.1.2̇)×(Ȧ.E.B.F.C.G.Ḋ) = A2 E2 , E1 B1 , B2 F2 , F1 C1 , C2 G2 , G1 D1 , D2 A1 ;

each figure occurring once unchanged as an antecedent and once changed as a
consequent.
   If it were thought worth while it would not be difficult, by using numbers
instead of letters, to obtain a general analytical formula, from which all similarly
constituted synthemes to any modulus might be evolved.
   But the rule of proceeding must be now sufficiently obvious; the modulus
                                                                              p−1
being 2p, we divide the elements into p classes; these may be arranged into
                                                                                 2
distinct forms of cyclothematic arrangement, and each of the cyclothemes taken
                                    p−1
in four positions, thus giving 4 ×        , that is, 2p − 2 bipartite synthemes, the
                                      2
whole number that can be formed to the given modulus 2p.                             p. 97
   I shall now proceed to the theory of bipartite synthemes to the modulus
2m × p, by which it is to be understood that we have p parts each containing 2m
terms, and p is at present supposed to be a prime number; the total number of
synthemes to the modulus 2mp being 2mp−1, and 2m−1 of these evidently being
capable of being made unipartite; the remainder, 2mp − 2m, that is, (p − 1)2m,
will be the number of bipartites to be obtained49 :
                                               p−1
                               2m(p − 1) =         × 4m;
                                                2
p−1
       denotes the total number of cyclothemes to modulus p; 4m, as will be
  2
presently shown, the number of lines or syzygies in the Table of position.
   To fix our ideas let the modulus be 4 × 3, and let A, B, C be three parts:
                                     
                 a1 a2 a3 a4 , 
                               
                 b1 b2 b3 b4 ,   their constituents respectively.
                               
                 c1 c2 c3 c4 
  49
    In general, if there be π parts of µ terms each, and µπ be even, the number of bipartite
synthemes is (π − 1)µ, as is easily shown from dividing the whole number of bipartite duads by
the semi-modulus.


                                            102
Give a fixed order to the constituents of each part, then each of them may be
taken in four positions; thus A may be written

                                    a1    a2        a3      a4 ,
                                    a2    a3        a4      a1 ,
                                    a3    a4        a1      a2 ,
                                    a4    a1        a2      a3 .

Assume some particular position for each, as, for instance,

                                         a1    b1        c1 ,
                                         a2    b2        c2 ,
                                         a3    b3        c3 ,
                                         a4    b4        c4 ,

and read off by coupling the first and third vertical places of each antecedent
with the second and fourth respectively of each consequent; we have accordingly,

                     a1 b2 , b1 c2 , c1 a2 ,         a3 b4 , b3 c4 , c3 a4 .

It is apparent that the same combinations will recur if any two contiguous
parts revolve simultaneously through two steps; or in other words, that Ar Bs =
Ar+2 Bs+2 , where µ is any number, odd or even.                                 p. 98
   Symbolically speaking, therefore, as regards our table of position,

                                 r : s = r + 2 : s + 2,

or more generally,
                              = r + 2 ± 4i : s + 2 ± 4i.
So that
                          1 : 1 = 3 : 3,       2 : 1 = 4 : 3,
                          1 : 2 = 3 : 4,       2 : 2 = 4 : 4,
                          1 : 3 = 3 : 1,       2 : 3 = 4 : 1,
                          1 : 4 = 3 : 2,       2 : 4 = 4 : 2.
There are therefore no more than eight independent unequivalent permutations
to every pair of parts. Now inspect the following table of position:—

                                      1.1.1,        2.1.2,
                                      1.2.3,        2.2.4,
                                      1.3.2,        2.3.1,
                                      1.4.4,        2.4.3.

It will be seen that in the first and second, second and third, third and first
places, all the eight independent permutations occur under different names; the
law of formation of such and similar tables will be explained in due time; enough

                                               103
for our present object to see how, by means of this table, we are able to obtain
the bipartite synthemes to the given modulus 4 × 3; the number according to our
                   3−1
formula is 2 × 4 ×       = 8, and they may be denoted symbolically as follows:—
                    2
                                                                   !
                               1.1.1 + 1.2.3 + 1.3.2 + 1.4.4
                  (Ȧ.B.Ċ)                                            .
                               +2.1.2 + 2.2.4 + 2.3.1 + 2.4.3

Each of the eight terms connected by the sign of + gives a distinct syntheme;
for example, let us operate on

                                  Ȧ.B.Ċ × (2.3.1).

2.3.1 denotes 2.3, 3.1, 1.2.

             2.3 gives rise to 2(3 + 1) + (2 + 2).(3 + 3) = 2.4 + 4.2.
             3.1 gives rise to 3(1 + 1) + (3 + 2).(1 + 3) = 3.2 + 1.4.
             1.2 gives rise to 1(2 + 1) + (1 + 2).(2 + 3) = 1.3 + 3.1.

The syntheme in question is therefore

                   A2 B4 , A4 B2 , B3 C2 , B1 C4 , C1 A3 , C3 A1 ,

and so on for all the rest, the rule being that

                         r : s = r(s + 1) + (r + 2)(s + 3).
                                                                                       p. 99
   Now, as before, it is evident that if we look only to contiguous terms, the
above table of position may be extended to any number of odd terms, simply by
repetition of the second and third figures in each syzygy; and hence the rule for
obtaining the bipartite synthemes to the modulus 4 × p is apparent. For instance,
                             7−1
let p = 7, there will be 8 ×     , that is, 8 × 3 of them denoted as follows:—
                              2
                                            1.1.1.1.1.1.1 + 2.1.2.1.2.1.2 
                                                                          
                                 
          Ȧ.B.C.D.E.F.Ġ
                                          
                                                                          
                                 
                                          +1.2.3.2.3.2.3 + 2.2.4.2.4.2.4 
                                                                          
            +Ȧ.C.E.G.B.D.Ḟ          ×                                            .
          +Ȧ.E.B.F.C.G.Ḋ 
                                        
                                           +1.3.2.3.2.3.2 + 2.3.1.3.1.3.1 
                                                                           
                                            +1.4.4.4.4.4.4 + 2.4.3.4.3.4.3
                                          
                                                                          
                                                                           

As an example of the mode of development, let us take the term

                         Ȧ.E.B.F.C.G.Ḋ × 2̇.4.3.4.3.4.3̇,

           2̇.4.3.4.3.4.3̇ = (2 : 4, 4 : 3, 3 : 4, 4 : 3, 3 : 4, 4 : 3, 3 : 2)
                                                                           !
                   2.1  4.4  3.1  4.4  3.1  4.4  3.3
             =                                                                 ,
                  +4.3 +2.2 +1.3 +2.2 +1.3 +2.2 +1.1

                                            104
            Ȧ.E.B.F.C.G.Ḋ = AE, EB, BF, F C, CG, GD, DA,
and the product
                                                                       !
             A2 E1 , E4 B4 , B3 F1 , F4 C4 , C3 G1 , G4 D4 , D3 A3
        =                                                                  .
             A4 E3 , E2 B2 , B1 F3 , F2 C2 , C1 G3 , G2 D2 , D1 A1
Let the modulus be 6 × 3; as before, give a fixed cyclic order to the constituents
of each part, and each will admit of being exhibited in six positions.
   Write similarly as before,
                                  a1 b1 c1 ,
                                  a2 b2 c2 ,
                                  a3 b3 c3 ,
                                  a4 b4 c4 ,
                                  a5 b5 c5 ,
                                  a6 b6 c6 ,
and take the odd places of each antecedent with the even places of each consequent;
it will now be seen that
                       r : s = r + 2 : s + 2 = r + 4 : s + 4,
                                                6.6
and the number of independent permutations is       = 2.6; and so in general, if
                                                 3
there be 2m constituents in a part, the number of independent permutations is
                                  2m.2m
                                        = 4m.
                                    m
                                                                                      p. 100
   The rule for the formation of the table will be apparent on inspection. I
suppose only three parts, as the rule may always be extended to any number
by reiteration of the second and third terms. The table will be found to resolve
itself naturally into four parts, each containing m lines.
   Let m = 1, we have
                                    1.1.1 2.1.2
                                    1.2.2 2.2.1
m = 2, we have
                                   1.1.1   2.1.2
                                   1.2.3   2.2.4
                                   1.3.2   2.3.1
                                   1.4.4   2.4.3
m = 3, we have
                                   1.1.1   2.1.2
                                   1.2.3   2.2.4
                                   1.3.5   2.3.6
                                   1.4.2   2.4.1
                                   1.5.4   2.5.3
                                   1.6.6   2.6.5

                                        105
m = 4, we have
                                    1.1.1   2.1.2
                                    1.2.3   2.2.4
                                    1.3.5   2.3.6
                                    1.4.7   2.4.8
                                    1.5.2   2.5.1
                                    1.6.4   2.6.3
                                    1.7.6   2.7.5
                                    1.8.8   2.8.7
So that x, going through all its values from 1 to m, the general expression for
the four parts is      (                               )
                         1.x(2x − 1) + 1(m + x)2x
                     Σ                                   .
                         +2.x.2x + 2(m + x)(2x − 1)
To show the use of this formula, let us suppose that we have seven parts, each
containing ten terms, the general expression for the bipartite duad synthemes is

                                 1.x(2x − 1)x(2x − 1)x(2x − 1)
                                                                                  
                     
 Ȧ.B.C.D.E.F.Ġ
                               
                                                                                  
                                                                                   
                     
                               +2.x.2x.x.2x.x.2x
                                                                                  
                                                                                   
   +Ȧ.C.E.G.B.D.Ḟ       ×Σ                                                           .
 +Ȧ.E.B.F.C.G.Ḋ 
                             
                                +1(5 + x)2x(5 + x)2x(5 + x)2x                     
                                                                                   
                                 +2(5 + x)(2x − 1)(5 + x)(2x − 1)(5 + x)(2x − 1)
                               
                                                                                  
                                                                                   
                                                                                   p. 101
   Make, for example, x = 3, one of the synthemes in question out of the twelve
corresponding to this value will be

                        Ȧ.C.E.G.B.D.Ḟ × 2̇.3.6.3.6.3.6̇.

Here
           Ȧ.C.E.G.B.D.Ḟ = AC, CE, EG, GB, BD, DF, F A,
                         2.4     3.7   6.4  3.7   6.4  3.7   6.3
                                                                        
                      +4.6     +5.9 +8.6 +5.9 +8.6 +5.9 +8.5 
                                                                
   2̇.3.6.3.6.3.6̇ =  +6.8     +7.1 +10.8 +7.1 +10.8 +7.1 +10.7  ,
                                                                
                      +8.10    +9.3 +2.10 +9.3 +2.10 +9.3 +2.9 
                                                                

                       +10.2    +1.5 +4.2 +1.5 +4.2 +1.5 +4.1
and the product

           = A2 C4 , C3 E7 , E6 G4 , G3 B7 , B6 D4 , D3 F7 , F6 A3
             A4 C6 , C5 E9 , E8 G6 , G5 B9 , B8 D6 , D5 F9 , F8 A5 ,
                     &c.             &c.             &c.

To prove the rule for the table of formation, it will be sufficient to show that
no two contiguous duads ever contain the same or equivalent permutations; the
equation of equivalence it will be remembered is

                       r : s = r + 2i ± 2m : s + 2i ± 2m.

                                        106
Now, as regards the first and second terms, it is manifest that 1 : x cannot be
equivalent, either to 1 : x′ nor to 2 : x, nor to 2 : x′ , where x′ is any number
differing from x.
   Similarly, as regards the last and first terms, x : 1 cannot be equivalent to
x′ : 1, nor to x : 2, nor to x′ : 2; therefore there is no danger as far as the first
term is concerned, either as antecedent or consequent.
   Again, it is clear that x : (2x−1) cannot interfere with x′ : 2x′ , nor (m+x) : 2x
with (m + x′ ) : (2x′ − 1); neither can (2x − 1) : x with 2x′ : x′ , nor 2x : (m + x)
with (2x′ − 1) : (m + x′ ).
   Again, if possible, let

                        x : (2x − 1) = (m + x′ ) : (2x′ − 1);

then
                                 m + x′ − x = 2i,
and
                                  2x′ − 2x = 2i,
therefore
                                     2m = 2i,
or
                                      m = i,
which is impossible, since ±i is the difference between two indices, each less than
m.                                                                                  p. 102
   Similarly,
                          m + x : 2x cannot = x′ : 2x′ ,
and vice versâ with the terms changed

                         2x : (m + x) cannot = 2x′ : x′ ,

and
                   (2x − 1) : x cannot = (2x′ − 1) : (m + x′ ),
which proves the rule for the table of formation.
    So much for the bipartite duad synthemes. As regards the unipartite synthemes
little need be said, for every part may be treated as a separate system, and as
each will produce an equal number of synthemes, these being taken one with
another, will furnish just as many unipartite synthemes of the whole system as
there are synthemes due to each part. Thus then the synthematic resolution of
the modulus 2m × p may be made to depend on the synthematization of 2m and
the cyclothematization of p. This has been already shown (whatever m may be)
for the case of p being a prime number; but I proceed now to extend the rule to
the more general case of p being any number whatever.

                                        107
                                        18.
On the Existence of Absolute Criteria for Determining the
              Roots of Numerical Equations
              [Philosophical Magazine, XXV. (1844), pp. 442–445]
                                                                                        p. 103
   I wish to indicate in this brief notice a fact which I believe has escaped
observation hitherto, that there exist, certainly in some cases, and probably in
all, infallible criteria for determining whether a given equation has all its roots
rational or not.
   In the equation of the second degree it is enough, in order that this may be
the case, that the expression for the square of the difference of the roots shall be
a perfect square; in other words, if x2 − px + q = 0 have its roots rational, p2 − 4q
must be not only a positive number (the condition of the roots being real), but
that number must also be a complete square. In this case it is further evident
that p must be either prime to q, or if not, the greatest common measure of p2
and q must be a perfect square; but this condition is contained in the former,
which is a sufficient criterion in itself.
   If we now consider the equation of the third degree,

                              x3 − px2 + qx − r = 0,

one condition is, that the product of the squared differences shall be a perfect
square; in other words, the equation cannot have all its roots rational unless

                        p2 q 2 − 4q 3 − 18pqr − 4p3 r − 27r2

be a positive square number.
   This remark is made at the end of the second supplement of Legendre’s Theory
of Numbers, and is indeed self-evident; and in like manner one condition may be
obtained for an equation of any degree which is to have all its roots rational; but
this is far from being the sole condition required.                                 p. 104
   In the equation of the third degree, however, one other condition, conjoined
with that above expressed, will serve to determine infallibly whether all the roots
are rational or not.
   To obtain this condition, let us suppose that by making 3x = y + p we obtain
the equation
                                 y 3 − Qy − R = 0.
Calling the three roots of this new equation α, β, γ (all of which it is evident
must be rational if those of the first equation are so), we have

                                  α + β + γ = 0,

                                        108
                     Q = −(αβ + αγ + βγ) = α2 + αβ + β 2 ,
                                       R = αβγ.
From the last two equations it is easily seen that if k be any prime factor common
to Q and R, k 2 will be contained in Q, and k 3 in R; or, in other words, k will be
a common measure of α, β, γ.
   We have therefore a second condition, that 9q −3p2 shall be a negative quantity,
which is either prime to 2p3 − 9qp + 27r, or else so related to it, that the greatest
common measure of the cube of the first and the square of the second is a perfect
sixth power.
   I now proceed to show the converse, that if these two conditions be both
satisfied (and it will appear in the course of the inquiry that the first does not
involve the second), the roots cannot help being all rational.
   It is evident that the two conditions in question are tantamount to supposing
that the roots of the proposed equation are linearly connected with those of
another z 3 − Qz − R = 0 (by virtue of the assumption 3x = kz + p), where Q
may be considered as prime to R; and where 4Q3 − 27R2 is a perfect square.
   Let now 4Q3 − 27R2 = D2 , then D2 + 27R2 = 4Q3 , or D2 + 3(3R)2 = 4Q3 .
   Here, as Q is prime to R, D can have no common measure but 3, with 3R.
   Firstly, let Q be prime to 3R.
   Then putting f 2 +3g 2 = Q3 , the complete solution of the equation immediately
preceding is contained in the two systems:
                        1st. D = 2f, 3R = 2g.
                        2nd. D = (f ± 3g), 3R = f ∓ g,
and for both systems,
                                √            √
                           f ± g −3 = {h ± 3k −3}3 .
                                                                                        p. 105
   The second system must therefore be rejected, for g evidently contains 3, and
therefore f = 3R ± g will contain 3, and therefore D and therefore Q will do the
same, contrary to supposition.
   Hence
          v                           v(
          u       s                            s
                                                      1
                         3     2
                                        u               )
              R        Q     R             R   D
          u                           u
                                      =
          3                            3
                ±   −      −                 ±      −
          u                             t
              2            27      4              2    2       27
          t                         

                                             v(      s
                                                         1
                                             u             )
                                             u  g
                                           =
                                             3
                                             t    ±f   −
                                                  3          27
                                               1               √
                                                      q
                                           =∓ √
                                                       3
                                                         {f ± g −3}
                                             3 −3
                                                   h         √
                                           = −K ± √   = λ ± µ −3;
                                                 3 −3

                                         109
and the three roots of the equation being
                        √              √
              {λ + µ −3} + {λ − µ −3},
             
             
                     √                       √
                 1 ± −3          √        1 ∓ −3       √
                         {λ + µ −3} +           {λ − µ −3},
                     2                       2
             

will evidently be all rational, which of course includes the necessity of their being
also integer.
   Again, secondly, if we suppose that Q does contain 3, D2 will contain 27, and
consequently D will contain 9; and we shall have
                                                 2                3
                                             D                 Q
                                                          
                                R2 + 3                =4                .
                                             9                 3
                     D
Here R being prime to , it may be shown, as in the last case, that the complete
                     9
solution is
                       R    D√               √
                          ±     −3 = {h ± k −3}3 ;
                       2    18
consequently       v
                        u     s           
                        u
                           R      R 2   Q3 
                                                     √
                        3
                             ±        −       = h ± k −3;
                        u
                          2      4     27 
                        t


and the three roots of the equation are

                               2h,       h − 3k,           h + 3k

respectively, and are therefore all rational.
   Here it may be observed that the condition of R being an even number, which
we know, à priori, is the case when all the roots are rational, is                   p. 106
   involved in the two more general conditions already expressed. It will now be
evident that the first condition by no means involves the second, as it is perfectly
easy to satisfy the equation f 2 + 3g 2 = Q3 without supposing anything relative
to k, the common measure of f, g, Q, except that it be itself of the form λ2 + 3µ2 ,
which will give
                        2         2
                        f            g
                              +3             = (λ2 + 3µ2 )(r2 + 3s2 )3 ,
                         k           k
an equation which can be solved in rational terms for all values of λ, µ, r, s; and
consequently the product of the squares of the differences of the roots may be a
square, and at the same time the roots themselves may be irrational50 .
  50
    Thus then it appears that the total rationality of the roots of the equation x3 − qx − r = 0
may be determined by a direct method without having recourse to the method of divisors to
determine the roots themselves; the two conditions being that 4q 3 − 27r2 shall be a perfect
square, and the greatest common measure of q 3 and r2 a perfect sixth power.


                                                 110
   I believe it will be found on inquiry that the equation xn − qx + r = 0 will
always have two rational roots if

                            (n − 1)n−1 · q n − nn · rn−1

be a complete square, provided that q be prime to r.
   Furthermore, viewing the striking analogy of the general nature of the con-
ditions of rationality already obtained, to those which serve to determine the
reality of the roots of equations, I am strongly of opinion that a theorem remains
to be discovered, which will enable us to pronounce on the existence of integer, as
Sturm’s theorem on that of possible roots of a complete equation of any degree:
the analogy of the two cases fails however in this respect, that while imaginary
roots enter an equation in pairs, irrational roots are limited to entering in groups,
each containing two or more.




                                        111
                                         19.
    An Account of a Discovery in the Theory of Numbers
       Relative to the Equation Ax3 + By 3 + Cz 3 = Dxyz
             [Philosophical Magazine, XXXI. (1847), pp. 189–191]


              First General Theorem of Transformation.
                                                                                          p. 107
  If in the equation
                           Ax3 + By 3 + Cz 3 = Dxyz,                                (1)
A and B are equal, or in the ratio of two cube numbers to one another, and if
27ABC − D3 (which I shall call the Determinant) is free from all single or square
prime positive factors of the form 6n + 1, but without exclusion of cubic factors
of such form, and if A and B are each odd, and C the double or quadruple of
an odd number, or if A and B are each even and C odd, then, I say, the given
equation may be made to depend upon another of the form

                          A′ u3 + B ′ v 3 + C ′ w3 = D′ uvw;

where
           A′ B ′ C ′ = ABC,      D′ = D,       uvw = some factor of z.
The following are some of the consequences which I deduce from the above
theorem. In stating them it will be convenient to use the term Pure Factorial to
designate any number into the composition of which no single or square prime
positive factor of the form 6n + 1 enters.
   The equations

x3 + y 3 + 2z 3 = Dxyz,        x3 + y 3 + 4z 3 = Dxyz,         2x3 + 2y 3 + z 3 = Dxyz,

are insoluble in integer numbers, provided that the Determinant in each case is
a Pure Factorial.                                                               p. 108
   The equation
                            x3 + y 3 + Az 3 = 9Bxyz
is insoluble in integer numbers, provided that the Determinant, for which in this
case we may substitute A − 27B 3 , is a pure factorial whenever A is of the form
9n ± 1, and equal to 2p3i±1 or 4p3i±1 , p being any prime number whatever.
    I wish however to limit my assertion as to the insolubility of the equations
above given. The theorem from which this conclusion is deduced does not
preclude the possibility of two of the three quantities x, y, z being taken positive
or negative units, either in the given equation itself or in one or the other of
those into which it may admit of being transformed. Should such values of two

                                         112
of the variables afford a particular solution, then instead of affirming that the
equations are insoluble, I should affirm that the general solution can be obtained
by equations in finite differences51 .

               Second General Theorem of Transformation.

   The equation
                               f 3 x3 + g 3 y 3 + h3 z 3 = Kxyz                              (2)
may always be made to depend upon an equation of the form

                               Au3 + Bv 3 + Cw3 = Duvw,

where
                             ABC = R3 − S 3 ,           D = 3R;
and uvw = some factor of f x + gy + hz.

           R    representing K + 6f gh,             S                   K − 3f gh.
                                                                                                    p. 109
   I have not leisure to show the consequences of this theorem of transformation
in connexion with the one first given, but shall content myself with a single
numerical example of its applications:

                                   x3 + y 3 + z 3 = −6xyz

may be made to depend on the equation

                                     u3 + v 3 + w3 = 0,

and is therefore insoluble.
   It is moreover apparent that the Determinant of equation (2) transformed is
in general −27R3 , and is therefore always a Pure Factorial, and consequently
the equation
                           f 3 x3 + g 3 y 3 + h3 z 3 = Kxyz
will be itself insoluble, being convertible into an insoluble form, provided that
K + 6f gh is divisible by 9, and provided further that (K + 6f gh)3 − (K − 3f gh)3
  51
     Take for instance the equation x3 + y 3 + 2z 3 = 3xyz. The Determinant 27.25 is a Pure
Factorial: consequently if the solution be possible, since in this case the transformed must be
identical with the given equation, this latter must be capable of being satisfied by making x
and y positive or negative units. Upon trial we find that x = 1, y = 1, z = 2 will satisfy the
equation. I believe, but have not fully gone through the work of verification, that these are the
only possible values (prime to one another) which will satisfy the equation. Should they not be
so, my method will infallibly enable me to discover and to give the law for the formation of all
the others.
  Here, then, under any circumstances, is an example, the first on record, of the complete
resolution of a numerical equation of the third degree between three variables.


                                              113
belongs to the form m3 Q, where Q is of the form 9n ± 1, and also of one or the
other of the two forms 2p3i±1 , 4p3i±1 , p being any prime number whatever.
    Pressing avocations prevent me from entering into further developments or
simplifications at this present time.
    It remains for me to state my reasons for putting forward these discoveries in
so imperfect a shape. They occurred to me in the course of a rapid tour on the
continent, and the results were communicated by me to my illustrious friend M.
Sturm in Paris, who kindly undertook to make them known on my part to the
Institute.
    Unfortunately, in the heat of invention I got confused about the law of oddness
and evenness, to which the coefficients of the given equation are in the first
theorem generally (in order for the successful application of my method as far as
it is yet developed) required to be subject. I stated this law erroneously, and
consequently drew erroneous conclusions from my Theorems of Transformation,
which I am very anxious to seize the earliest opportunity of correcting. I venture
to flatter myself that as opening out a new field in connexion with Fermat’s
renowned Last Theorem, and as breaking ground in the solution of equations of
the third degree, these results will be generally allowed to constitute an important
and substantial accession to our knowledge of the Theory of Numbers.




                                       114
                                        20.
 On the Equation in Numbers Ax3 + By 3 + Cz 3 = Dxyz, and its
              Associate System of Equations
             [Philosophical Magazine, XXXI. (1847), pp. 293–296]
                                                                                        p. 110
   In the last Number of this Magazine I gave an account of a remarkable
transformation to which the equation
                            Ax3 + By 3 + Cz 3 = Dxyz
is subject when certain conditions between the coefficients A, B, C, D are satisfied;
which conditions I shall begin by expressing with more generality and precision
than I was enabled to do in my former communication.
    1. Two of the quantities A, B, C are to be to one another in the ratio of two
cubes.
    2. 27ABC − D3 must contain no positive prime factor whatever of the form
6n + 1. I erred in my former communication in not excluding cubic factors of
this form.
    3. If 2m is the highest power of 2 which enters into ABC, and 2n the highest
power of 2 which enters into D, then either m must be of the form 3n ± 1, or if
not, then m must be greater than 3n.
    These three conditions being satisfied, the given equation can always be
transformed into another,
                          A′ u3 + B ′ v 3 + C ′ w3 = D′ uvw,
where
             A′ B ′ C ′ = ABC,      D′ = D,       uvw = a factor of z.
The consequence of this is, as stated in my former paper, that wherever A, B, C, D,
besides satisfying the conditions above stated, are taken so as likewise to satisfy
the condition,–firstly, of ABC being equal to 23m±1 , or secondly, of ABC being
equal to 23m±1 · p3n±1 , provided in the second case that ABC is of the form
9m ± 1, and that D is divisible by 9, p being in both cases a prime, then the
given equation will be generally insoluble.                                         p. 111
   And I am now enabled to add that the only solution of which it will in any
case admit, is the solitary one found by making two of the terms Ax3 , By 3 , Cz 3
equal to one another; so that, for instance, if the given equation should be of the
form
                            x3 + y 3 + ABCz 3 = Dxyz,
then the above conditions being satisfied, the one solitary solution of which the
equation can possibly admit, is x = 1, y = 1,
                                 Az 3 − Dz + 2 = 0,

                                         115
which may or may not have possible roots. I call this a solitary or singular
solution, because it exists alone and no other solution can be deduced from it;
whereas in general I shall show that any one solution of the equation

                                Ax3 + By 3 + Cz 3 = Dxyz

can be made to furnish an infinity of other solutions independent of the one
supposed given, that is, not reducible thereto by expelling a common factor from
the new system of values of x, y, z deduced from the given system.
   The following is the Theorem of Derivation in question:
   Let
                           Aα3 + Bβ 3 + Cγ 3 = Dαβγ.
Then if we write

                        F = Aα3 ,         G = Bβ 3 ,        H = Cγ 3 ,

and make

     x = F 2 G + G2 H + H 2 F − 3F GH,              y = F G2 + GH 2 + HF 2 − 3F GH,
                                 1
                           z=      {F 3 + G3 + H 3 − 3F GH},
                                 D
or
                     = αβγ{F 2 + G2 + H 2 − F G − F H − GH},
we shall have
                               x3 + y 3 + ABCz 3 = Dxyz.
I am hence enabled to show that whenever x3 + y 3 + Az 3 = Dxyz is insoluble,
there will be a whole family of allied equations equally insoluble. For instance,
because x3 + y 3 + z 3 = 0 is insoluble in integer numbers, I know likewise that

                           x6 + y 6 + z 6 = x3 y 3 + x3 z 3 + y 3 z 3

                          x6 + y 6 + z 6 = x3 y 3 + x3 z 3 − 2y 3 z 3
are each equally insoluble.                                                           p. 112
   In fact
                (x3 + y 3 + z 3 ) × (x6 + y 6 + z 6 − x3 y 3 − x3 z 3 − y 3 z 3 )
                   × (x6 + y 6 + z 6 − x3 y 3 − x3 z 3 + 2y 3 z 3 )
                   × (x6 + y 6 + z 6 − y 3 z 3 − y 3 x3 + 2x3 z 3 )
                   × (x6 + y 6 + z 6 − x3 z 3 − z 3 y 3 + 2y 3 x3 )
                = u3 + v 3 + w3 ,

where u, v, w are rational integral functions of x, y, z.

                                              116
   Hence each of the factors must be incapable of becoming zero52 .
   As a particular instance of my general theory of transformation and elevation,
take the equation
                             x3 + y 3 + 2z 3 = M xyz.
Then, with the exception of the singular or solitary solution x = 1, y = 1, of
which I take no account, I am able to affirm that for all values of M between 7
and −6, both inclusive, with the exception of M = −2, the equation is insoluble
in integer numbers.
   Take now the equation where M = −2, namely

                                  x3 + y 3 + 2z 3 + 2xyz = 0.

One particular solution of this is

                               x = 1,        y = −1,          z = 1.

Another, which I shall call the second53 , is

                               x = 1,        y = 3,         z = −2.

From the first solution I can deduce in succession the following:

                        x = 11,          y = 5,          z = −7,
                x = −793269121, y = 1179490001, z = −1189735855,
                            &c.             &c.              &c.

From the second,

                          x = −10085, y = 8921, z = −8442,
                              x = &c.   y = &c.    z = &c.

As another example, take the equation

                                    x3 + y 3 + 6z 3 = 6xyz.
                                                                                               p. 113
   One solution of the transformed equation

                                   u3 + 2v 3 + 3w3 = 6uvw

is evidently
                                u = 1,        v = 1,        w = 1.
  52
       It is however sufficiently evident from their intrinsic form, which may be reduced to
1
4
  (M 2 + 3N 2 ), that this impossibility exists for all the factors except the first.
   53
       See Postscript.




                                                117
Hence I can deduce an infinite series of solutions of the given equation, of which
the first in order of ascent will be

                            x = 5,        y = 7,      z = 3.

Again, the lowest possible solution in integers of the equation

                                    x3 + y 3 + 6z 3 = 0

will be
                          x = 17,       y = 37,      z = −21.
The equation
                                    x3 + y 3 + 9z 3 = 0
admits of the solutions
                              x = 1,  y = 2,   z = −1,
                          x = −271, y = 919, z = −438.

I trust that my readers will do me the justice to believe that I am in possession of
a strict demonstration of all that has been here advanced without proof. Certain
of the writer’s friends on the continent have, in their comments upon one of his
former papers which appeared in this Magazine, complimented his powers of
divination at the expense of his judgment, in rather gratuitously assuming that
the author of the Theory of Elimination was unprovided with the demonstrations,
which he was too inert or too beset with worldly cares and distractions to present
to the public in a sufficiently digested form. The proof of whatever has been
here advanced exists not merely as a conception of the author’s mind, but fairly
drawn out in writing, and in a form fit for publication.
   P.S. It must not be supposed that the two primary or basic solutions above
given of the equation
                            x3 + y 3 + 2z 3 + 2xyz = 0,
namely,
                             x = 1, y = −1,    z = 1,
                             x = 1,   y = 3, z = −2,
are independent of one another. The second may be derived from the first, as
I shall show in a future communication. In fact there exist three independent
processes, by combining which together, one particular solution may be made
to give rise to an infinite series of infinite series of infinite series of correlated
solutions, which it may possibly be discovered contain between them the general
complete solution of the equation

                              x3 + y 3 + Az 3 = Dxyz.


                                           118
                                         21.
On the General Solution (in Certain Cases) of the Equation
                 x3 + y 3 + Az 3 = M xyz, &c.
             [Philosophical Magazine, XXXI. (1847), pp. 467–471]
                                                                                         p. 114
   I shall restrict the enunciation of the proposition I am about to advance to
much narrower limits than I believe are necessary to the truth, with a view to
avoid making any statement which I may hereafter have occasion to modify. Let
us then suppose in the equation

                              x3 + y 3 + Az 3 = M xyz

that A is a prime number, and that 27A − M 3 is positive, but exempt from
positive prime factors of the form 6i + 1. Then I say, and have succeeded in
demonstrating, that all the possible solutions in integer numbers of the given
equation may be obtained by explicit processes from one particular solution or
system of values of x, y, z, which may be called the Primitive system.
   This system of roots or of values of x, y, z is that system in which the value of
                                        1
the greatest of the three terms x, y, A 3 · z (which may be called the Dominant)
is the least possible of all such dominants. I believe that in general the system
of the least Dominant is identical with the system of the least Content, meaning
by the latter term the product of the three terms out of which the Dominant is
elected. I proceed to show the law of derivation.
   To express this simply, I must premise that I shall have to employ such an
expression as S ′ = ϕ(S) to indicate, not that a certain quantity, S ′ , is a function
of S, but that a certain system of quantities disconnected from one another,
denoted by S ′ , are severally functions of a certain other system of quantities
denoted by S; and, as usual, I shall denote ϕϕS by ϕ2 S, ϕϕ2 S by ϕ3 S, and so
forth.
   Let now P be the Primitive system of solution of the equation

                              x3 + y 3 + Az 3 = M xyz,

P denoting a certain system of values of and written in the order of the letters
x, y, z, which may always be found by a limited number of trials (provided that
the equation admits of any solution).                                                  p. 115
   That this is the case is obvious, since we have only to give the Dominant every
                                                         1
possible value from the integer next greatest to A 3 upwards, and combine the
values of x3 , y 3 , Az 3 so that none shall ever exceed at each step the cube of such
dominant, and we must at last, if there exist any solution, arrive at the System
of the Least Dominant.


                                         119
   Now, every system of solution is of one or the other of two characters. Either
x and y must be odd and z even, or x and y must be one odd and the other even
and z odd. That all three should be odd is inconsistent with the given conditions
as to A being odd and M even; and if all three were even, by driving out the
common factor we should revert to one or the other of the foregoing cases.
   The systems of solution where z is even may be termed Reducible, those where
z is odd Irreducible. Let ϕ denote a certain symbol of transformation hereafter
to be explained.
   Then the Reducible systems of the first order may be expressed by

                          ϕP, ϕ2 P, ϕ3 P, ad inf initum;

or in general by ϕn1 P , n1 being absolutely arbitrary. I will anticipate by stating
that the function ϕ involves no variable constants; that is to say, ϕ(S) may be
found explicitly from S without any reference to the particular equation to which
S belongs. Let now ψ denote another symbol of transformation, also hereafter to
be defined, and differing from ϕ insofar as it does involve as constants the three
values of x, y, z contained in P : then the general representations of Irreducible
systems of the first order will be denoted by ψϕn1 P .
   It is proper to state here that the symbol ψ is ambiguous; and ψϕn1 P , when
P and n1 are given, will have two values, according to the way in which the
terms represented by P are compared with x, y, z in the given equation

                              x3 + y 3 + Az 3 = M xyz;

for it is obvious that if x = a, y = b, z = c satisfies the equation, so likewise will

                           x = b,      y = a,        z = c.

Each however of these values of ψϕn1 P gives a solution of the kind above
designated.
   Proceeding in like manner as before, the Reducible system of the second order
may be designated by ϕn2 · ψϕn1 · P , the Irreducible by ψϕn2 · ψϕn1 · P ; and in
general every possible system of values of x, y, z satisfying the proposed equation,
in which z is even, is comprised under the form

                           ϕnr · ψϕnr−1 · ψ . . . ψϕn1 · P ;
                                                                                         p. 116
   and every possible system of such values, in which z is odd, is comprised under
the form
                          ψϕnr · ψϕnr−1 · ψ . . . ψϕn1 · P :
the quantities n1 , n2 . . . nr being of course all independent of one another, and
unlimited in number and value.


                                         120
    Thus then we may be said to have the general solution of the given equation
in the same sense as an arbitrary sum of terms, each of a certain form, is in
certain cases accepted as the complete solution of a partial differential equation.
    As regards the value of the symbols ψ and ϕ, ϕ indicates the process by which
a, b, c becomes transformed into α, β, γ, the relations between the two sets of
elements being contained in the following equations:

                        a′ = a3 ,        b′ = b3 ,            c′ = Ac3 ,

                        α = a′2 b′ + b′2 c′ + c′2 a′ − 3a′ b′ c′ ,
                         β = a′ b′2 + b′ c′2 + c′ a′2 − 3a′ b′ c′ ,
                   γ = abc{a′2 + b′2 + c′2 − a′ b′ − a′ c′ − b′ c′ }.
Next, as to the effect of the Duplex symbol ψ. Let e, g, ι be the elements of the
Primitive system P : ι being the value of z and e, g of x and y taken in either
mode of combination, each with each, which satisfy the proposed equation

                              x3 + y 3 + Az 3 = M xyz.

Let l, m, n represent any system S,

                       λ, µ, ν represent any system ψ(S),

ψS has two values, which we may denote by ψ ′ S, ′ψS respectively, and accentu-
ating the elements λ, µ, ν accordingly to correspond, we shall have

             λ′ = 3gm(gl − em) + 3Aιn(ιl − en) − M (gιl2 − e2 lm),

            µ′ = 3Aιn(ιm − gl) + 3el(em − gl) − M (eιm2 − g 2 lm),
             ν ′ = 3el(en − ιl) + 3gm(gn − ιm) − M (egn2 − ι2 lm) :
we have then
                                    ψ ′ S = λ′ , µ′ , ν ′ ,
and in like manner
                                     ′
                                     ψS = ′λ, ′µ, ′ν,
′ψS being derived from ψ ′ S by the mere interchange of e and g one with the

other.                                                                              p. 117
   I have stated that every possible solution of the proposed equation comes
under one or the other of the orders, infinite in number and infinite to the power
of infinity in variety of degree, above given: this is not strictly true, unless we
understand that all systems of solution are considered to be equivalent which
differ only in a multiplier common to all three terms of each; that is to say,
which may be rendered identical by the expulsion of a common factor. So that
mα, mβ, mγ as a system is treated as identical with α, β, γ, which of course

                                            121
substantially it is; and it should be remarked that there is nothing to prevent the
operations denoted by ϕ and ψ introducing a common factor into the systems
which they serve to generate, and the latter in particular will have a strong
tendency so to do.
   I believe that this theorem may be extended with scarcely any modification
to the case where A, instead of being a prime, is any power of the same, and to
suppositions still more general. I believe also that, subject to certain very limited
restrictions, the theorem may prove to apply to the case where the determinant
27A − M 3 becomes negative.
   The peculiarity of this case which distinguishes it from the former, is that it
admits of all the three variables x, y, z in the equation

                                 x3 + y 3 + Az 3 = M xyz

having the same sign, which is impossible when the determinant is positive; or
in other words, the curve of the third degree represented by the equation
                                                    M
                                 Y 3 + X3 + 1 =       1   XY
                                                    A3
(in which I call the coefficient of XY the characteristic), which, as long as the
quantity last named is less than 3, is a single continuous curve extending on
both sides to infinity, as soon as the characteristic becomes equal to 3 assumes
to itself an isolated point, the germ of an oval or closed branch, which continues
to swell out (always lying apart from the infinite branch) as the characteristic
continues indefinitely to increase.
   I ought not to omit to call attention to the fact that the theorem above
detailed is always applicable to the case of the equation

                                   x3 + y 3 + Az 3 = 0,

when A is any power of a prime number not of the form 6i + 1; in other words,
the above always belongs to the class of equations having Monogenous solutions,
which for the sake of brevity may be termed themselves Monogenous Equations54 . p. 118
   On the probable existence of such a class of equations I hazarded a conjecture
at the conclusion of my last communication to this Magazine. As I hope shortly
to bring out a paper on this subject in a more complete form, I shall content
myself at this time with merely stating a theorem of much importance to the
completion of the theory of insoluble and of Monogenous equations of the third
degree; to wit, that the equation in integers

       a(x3 + y 3 + z 3 ) + c(x2 y + y 2 z + z 2 x + xy 2 + yz 2 + zx2 ) + exyz = 0
  54
    Thus the equation x3 + y 3 + 9z 3 = 0 alluded to by Legendre is Monogenous, and the
Primitive system of solution is x = 1, y = 2, z = −1, from which every other possible solution
in Integers may be deduced.


                                            122
may always be transformed so as to depend upon the equation

                      f u3 + gv 3 + hw3 = (6a − e)uvw,

wherein
                 f gh = ae2 − (c2 + 3a2 )e + 9a2 − 3ac2 − 2c3 .
By means of the above theorem, among other and more remarkable consequences,
we are enabled to give a theory of the irresoluble and monogenous cases of the
equation
                           x3 + y 3 + m3 z 3 = M xyz,
when m is some power of 2, or of certain other numbers.




                                      123
                                         22.
On the Intersections, Contacts, and Other Correlations of
   Two Conics Expressed by Indeterminate Coordinates
    [Cambridge and Dublin Mathematical Journal, v. (1850), pp. 262–282]
                                                                                       p. 119
   Let U = 0, V = 0 be two homogeneous equations of the second degree with
real coefficients, between the same three variables ξ, η, ζ.
   The direct and most general mode of determining the intersections of the
conics expressed by these equations would be to make

                    aξ + bη + cζ = t,       a′ ξ + b′ η + c′ ζ = u :

eliminating ξ, η, ζ between the four equations in which they appear, there results
a biquadratic equation between t and u. The nature of the intersections will
depend upon the nature of the roots of this biquadratic; and thus the conditions
may be expressed analytically, which will represent the several cases of all the
intersections being real or all imaginary, or one pair real and the other imaginary.
These analytical conditions will depend upon the signs of certain functions of
the coefficients of the given and the assumed equations being of an assigned
character; my endeavour has been to obtain conditions of a character perfectly
symmetrical and free from the coefficients arbitrarily introduced.
   In this research I have only partially succeeded, but the method employed,
and some of the collateral results, will, I think, be found of sufficient interest to
justify their appearance in the pages of this Journal.
   Adopting Mr Cayley’s excellent designation, let the four points of intersection
of the two conics be called a quadrangle. This quadrangle will have three pairs of
sides; the intersections of each pair, from principles of analogy, I call the vertices
of the quadrangle. Then, inasmuch as the four                                          p. 120
   sets of ratios ξ : η : ζ, corresponding with the four sets of the ratio t : u, must
be so related that we may always make
                        ξ1        √          η1        √
                           = a + b −1,          = c + d −1,
                        ζ1                   ζ1
                        ξ2        √          η2        √
                           = a − b −1,          = c − d −1,
                        ζ2                   ζ2
                        ξ3        √          η3        √
                           = α + β −1,          = γ + δ −1,
                        ζ3                   ζ3
                        ξ4        √          η4        √
                           = α − β −1,          = γ − δ −1,
                        ζ4                   ζ4
we may easily draw the following conclusions.


                                         124
    If all the four points of the quadrangle of intersection are real, the three vertices
and the three pairs of sides are all real. If only two points of the quadrangle
are real, one vertex and one of the three pairs of sides will be real; the other
two vertices and two pairs of sides being imaginary. If all four points of the
quadrangle are unreal, one pair of sides will be real and the other two pairs
imaginary, as in the last case; but all the three vertices will remain real, as in the
first case. Hence we have a direct and simple criterion for distinguishing the case
of mixed intersection from intersection wholly real or wholly imaginary; namely,
that the cubic equation of the roots of which the coordinates of the vertices are
real linear functions shall have a pair of imaginary roots. This is the sole and
unequivocal condition required.
    The equation in question is, or ought to be, well known to be the determinant
in respect to ξ, η, ζ of λU + µV . In fact, if we write

                     U = aξ 2 + bη 2 + cζ 2 + 2a′ ηζ + 2b′ ζξ + 2c′ ξη,
                     V = αξ 2 + βη 2 + γζ 2 + 2α′ ηζ + 2β ′ ζξ + 2γ ′ ξη,
            λU + µV = (aλ + αµ)ξ 2 + &c.
                       = Aξ 2 + Bη 2 + Cζ 2 + 2A′ ηζ + 2B ′ ζξ + 2C ′ ξη,

the ratios of the coordinates ξ, η, ζ of the vertex of λU + µV may easily be shown
to be identical with

                       AB − C ′2 : C ′ A′ − B ′ B : B ′ C ′ − A′ A,

and will be real or imaginary as λ : µ is one or the other.
  If then the cubic equation in λ : µ, namely, □ξηζ (λU + µV ) = 0, has a pair of
imaginary roots, that is, if

                                 □λµ □ξηζ (λU + µV )

is a positive quantity, the intersections of U and V are of a mixed kind, that is,
the two conics have two real points in common.                                     p. 121
   I may remark here, en passant, that if we form the biquadratic equation in t
and u, ϕ(t, u) = 0 from the equations

        U = 0,       V = 0,       aξ + bη + cζ = t,             a′ ξ + b′ η + c′ ζ = u,

and if any reducing cubic of this equation be P (θ, ω) = 0, the determinant of
P (θ, ω) must, from what has been shown above, be identical with

                                 □λµ □ξηζ (λU + µV )

multiplied by some squared function of the extraneous coefficients

                                   a, b, c;    a′ , b′ , c′ .

                                              125
   If □□(λU + µV ) is a negative quantity, it remains to distinguish between the
cases of the conics intersecting really in four points or not at all.
   The most obvious mode of proceeding to distinguish between purely real and
purely imaginary intersections would be as follows. Let λ1 , µ1 ; λ2 , µ2 ; λ3 , µ3 be
the three sets of values of λ, µ which satisfy the equation

                                    □(λU + µV ) = 0

and make
          A1 = aλ1 + αµ1 ,                 A2 = aλ2 + αµ2 ,                A3 = aλ3 + αµ3 ,
          C1 = cλ1 + γµ1 ,                 C2 = cλ2 + γµ2 ,                C3 = cλ3 + γµ3 ,
        B1′ = b′ λ1 + β ′ µ1 ,           B2′ = b′ λ2 + β ′ µ2 ,            B3′ = b′ λ3 + β ′ µ3 ,
A1 C1 − B1′2 = e1 ,              A2 C2 − B2′2 = e2 ,              A3 C3 − B3′2 = e3 .

Now if the equation

                  Aξ 2 + Bη 2 + Cζ 2 + 2A′ ηζ + 2B ′ ζξ + 2C ′ ξη = 0

represent a pair of straight lines, it may be thrown into the form

                                         AC − B ′2 2
                                 Au2 +            v = 0,
                                            A
where u and v are linear functions of ξ, η, ζ, and the straight lines will be real
or imaginary, according as B ′2 − AC is positive or negative; hence one or else
all of the quantities e1 , e2 , e3 will be necessarily negative, and the intersections
will be all real or all imaginary, according as all three are negative or only one
is so. A cubic equation in e may be formed containing e1 , e2 , e3 as its roots by
eliminating between the equations

                        e = AC − B ′2 ,       □(λU + µV ) = 0,

and the conditions for the reality of the intersections will be that all four
coefficients of this cubic shall be of the same sign, which in reality amount only
to two, since the first and last must in all cases have the same sign.              p. 122
   The same objection however of want of symmetry and consequent irrelevancy
and complexity attaches to this as much as to the method originally proposed.
The following treatment of the question relieves the objection of want of symmetry
as far as the coefficients of the same equation are concerned, but in its practical
application necessitates an arbitrary and therefore unsymmetrical election to be
made between the two sets of coefficients appertaining to the two equations. It
is however, I think, too curious and suggestive to be suppressed.
   I observe that if the four intersections are all real, an imaginary conic cannot
be drawn through them; for the equation to an imaginary conic may always

                                           126
be reduced to the form Ax2 + By 2 + Cz 2 = 0, where A, B, C are all positive
and can therefore have at utmost one real point. Consequently the case of
total non-intersection is distinguishable from that of complete intersection by
the peculiarity that in the one case µ may be so taken that U + µV = 0 shall
represent an imaginary conic, that is, U + µV will be a function whose sign never
changes for real values of ξ, η, ζ, whereas in the latter case no value of µ will make
U + µV = 0 the equation to an imaginary conic, and therefore U + µV will have
values on both sides of zero. On the other hand, it is obvious that an infinite
number of real as well as unreal conics may be drawn through four imaginary
points of intersection. Consequently if we make U + µV = 0 (supposing the
intersections of U and V to be imaginary), there will be a range or ranges of
values of µ consistent, and another range or ranges of values of µ inconsistent
with real values of ξ, η, ζ; in other words, U + µV = 0 treated as an equation
between the four variables ξ, η, ζ, µ, will give one or more maxima or minima
values of µ, in the case supposed, but no such values when the intersections are
two or all of them real.
   To determine these values of µ, let dµ = 0; then we have
          d                          d                        d
             (U − µV ) = 0,            (U − µV ) = 0,            (U − µV ) = 0,
          dξ                        dη                        dζ
that is
                                  □ξηζ (U − µV ) = 0.
In order that any value of µ found from this equation may be a maximum or
minimum, Lagrange’s condition requires that
                                                     2
                                    d      d    d
                              
                                h      +k    +l           µ
                                    dξ    dη    dζ
may be a function of unchangeable sign.                                                  p. 123
   Now
                              dU        dV       dµ
                                   =µ        +V     ;
                              dξ         dξ      dξ
therefore since dµ = 0,
                            d2 U        d2 V     d2 µ
                                   =  µ      + V       .
                             dξ 2       dξ 2      dξ 2
Hence
                         d2 µ      1 d 2
                                      
                              =              {U − µV };
                         dξ 2     V dξ
similarly
                          d d        1 d d
                                  =          {U − µV },
                         dξ dη      V dξ dη
and so on. Making now as before
                 U = aξ 2 + bη 2 + &c.,         V = αξ 2 + βη 2 + &c.,

                                          127
                           a − µα = A,          b − µβ = B,           &c.,
the condition for µ, a root of □{U − µV } = 0, giving µ a maximum or minimum,
may be expressed by saying that

                         Ah2 + Bk 2 + Cl2 + 2A′ kl + 2B ′ hl + 2C ′ hk

shall be unchangeable in sign for all real values of h, k, l.
   The above quantity, by virtue of the equation □ = 0, is always the product
of two linear functions. Hence we see, as above indicated, that if all these pairs
are real, that is, if all the points of intersection of U and V are real, there is no
maximum or minimum value of µ; but if only one pair be real and the other two
pairs be imaginary, that is, if all the four intersections are imaginary, then two
of the values of µ, namely those corresponding to the imaginary pairs, are real
maxima or minima values of µ, but the third is illusory.
   Now I shall show that if V = 0 is a real conic, but the intersections of U and
V are all unreal, the value of µ which makes U + µV the product of real linear
functions of ξ, η, ζ, is always one or the other extreme of the three values of µ
which satisfy the equation
                                   □(U − µV ) = 0.
Assume as the three axes of coordinates the three lines joining the vertices of
the quadrangle each with each, the two non-intersecting conics may evidently be
written under the form

       U = c(x2 + y 2 ) − e(y 2 + z 2 ) = 0,         V = −γ(x2 + y 2 ) + ϵ(y 2 + z 2 ) = 0;
                                                                                              p. 124
   these equations being only other modes of writing

                  U = Ax2 + By 2 + Cz 2 ,            V = A′ x 2 + B ′ y 2 + C ′ z 2 ,

in which A, B, C; A′ , B ′ , C ′ will be real, because by hypothesis □(U + µV ) = 0
has all its roots real.
   Hence x, y, z are linear functions of ξ, η, ζ and consequently, by a simple
inference from a theorem of Prof. Boole55 , the roots of □ξηζ {U + µV } are
identical with those of
                                   □xyz {U + µV } = 0.
These latter are evidently
                                    c          e        c−e
                                      ,          ,          ,
                                    γ          ϵ        γ−ϵ
the third of which is the one which makes U + µV the product of two real linears,
for we have
                          γU + cV = (cϵ − γe)(y 2 + z 2 ),
  55
       See Postscript.


                                                128
                              ϵU + eV = (ϵc − eγ)(x2 + y 2 ),
                      (γ − ϵ)U + (c − e)V = (cϵ − eγ)(z 2 − x2 )56 .
Now
                  c   c−e   eγ − cϵ                 e c−e   eγ − cϵ
                    −     =          ,               −    =          ;
                  γ γ−ϵ     γ(γ − ϵ)                ϵ γ−ϵ   ϵ(γ − ϵ)
and ϵ, γ are supposed to have the same sign, as otherwise V would be an unreal
conic; hence the ascending or descending order of magnitudes of the three values
of λ follows the scale
                                c    e    c−e
                                  ,    ,       ,
                                γ    ϵ    γ−ϵ
as was to be shown.
   Imagine now lengths reckoned on a line corresponding to all values of µ from
−∞ to +∞, and mark off upon this line by the letters A, B, C, the lengths
corresponding with the three roots of □(U + µV ) = 0. Then observing that when
µ = ±∞, U + µV is of the same nature as V , and is therefore a possible conic by
hypothesis, and agreeing to understand by a possible and impossible region of
µ, a range of values for which U + µV corresponds to a possible and impossible
conic respectively, one or the other of the annexed schemes will represent the
circumstances of the case supposed:

−∞ Poss. Reg. A              Imposs. Reg.       B    Poss. Reg. C         Poss. Reg.       + ∞,

−∞ Poss. Reg. A              Poss. Reg. B         Imposs. Reg.       C    Poss. Reg.       + ∞.
But in either scheme it is essential to observe that the middle root of □(U +µV ) =
0 divides a possible from an impossible region; and therefore                       p. 125
   if we can find u, v, any two values lying between the first and second and
second and third roots of the above equation arranged in order of their magnitude,
one of the two equations U + vV = 0, U + uV = 0, will represent a possible and
the other an impossible conic: one such couple of values may always be found by
taking the roots of the quadratic equation
                                      d
                                        □{U + µV } = 0.
                                     dµ

Hence calling the two roots thereof m and M , we see (which is in itself a theorem)
that one at least of the conics U + mV = 0, U + M V = 0, must be a possible
conic, provided only that V = 0 be a possible conic: if both U +mV and U +M V
are possible conics, the intersections of U and V are all real, and if not, not57 .
    z − x2 = 0 of course represents a real pair of lines.
  56 2
  57
    It must be well observed however that the possibility of the conics U + mV and of U + M V
does not imply the reality of the intersections unless the conic V is known to be possible. For if
V be impossible ϵ and γ have opposite signs, and therefore γ−ϵ c−e
                                                                   is intermediate between cϵ and



                                              129
The criteria for distinguishing possible from impossible conics being well known
need not be stated in this place.
  We may of course proceed analogously by forming the two conics lU + V ,
LU + V , where l and L are roots of
                                     d
                                       □{λU + V } = 0
                                    dλ
upon the supposition of U = 0 being a possible conic.
   If either of the two U and V be not possible, their intersections are of course
impossible, and the question is already decided.
   It will be seen as pre-indicated that this method only fails in symmetry because
of the choice between the couples m, M , and l, L. But moreover a perfect method
for the discrimination of the two cases of unmixed intersection one from the
other should (perhaps?) require the application of only a single test (in lieu
of the two conditions which the above method supposes), over and above the
condition which expresses the fact of the intersections being so unmixed. Such
more perfect method I have not yet been able to achieve.
   Another interesting question of intersections remains to be discussed, namely,
supposing the two conics are known to be non-intersecting, how are we to
ascertain if they are external to one another, or if one contains the other? In
order to settle this point we must first establish a criterion for determining
whether a given point is internal or external to a given conic; the point being in
general said to be external when two real tangents can be drawn from it to the
curve, and internal when this cannot be done.                                       p. 126
   Let now

             ϕ(x, y, z) = ax2 + by 2 + cz 2 + 2a′ yz + 2b′ zx + 2c′ xy = 0,

be the equation to any conic: l, m, n the coordinates of any point. Let

                  A = bc − a′2 ,       B = ca − b′2 ,            C = ab − c′2 ,
                 A′ = aa′ − b′ c′ , B ′ = bb′ − c′ a′ , C ′ = cc′ − a′ b′ .

Then the reciprocal equation to the conic is

                  Aξ 2 + Bη 2 + Cζ 2 + 2A′ ηζ + 2B ′ ζξ + 2C ′ ξη = 0,

and in making lξ + mη + nζ = 0, the ratios of ξ, η, ζ must be real if the tangents
γ
ϵ
  , and the scheme for µ will be as here annexed:

          −∞    Impossible.   A    Possible.   B     Possible.    C   Impossible.   + ∞,

so that U + mV and U + M V will both represent possible conics.




                                               130
drawn from l, m, n are real: this will be found to imply that the determinant

                                     A C ′ B′ l
                                     C ′ B A′ m
                                     B ′ A′ C n
                                      l m n 0

shall be negative58 . This determinant may be shown59 to be equal to the product
of the determinant
                                    a c′ b′
                                    c′ b a′
                                    b′ a′ c
by the quantity

                     al2 + bm2 + cn2 − 2a′ mn − 2b′ ln − 2c′ lm,

that is, equal to ϕ(l, m, n) × □.
   Hence l, m, n is internal or external to ϕ(x, y, z) according as ϕ(l, m, n) and
□ϕ have the same or contrary sign.
   If ϕ(l, m, n) = 0, the point lies on the conic, and the point is neither internal
nor external; if □ϕ = 0, the conic becomes a pair of straight lines, and no point
can be said either to be within or without such a system. Hence our criterion
fails, as it ought to do, just in the very two cases where the distinction vanishes.
I believe that this criterion is here given for the first time.                      p. 127
   To return to the two non-intersecting conics. Let us again throw them under
the form

        U = (x2 + y 2 ) − e2 (z 2 + y 2 ),     V = k(x2 + y 2 ) − kϵ2 (z 2 + y 2 ),

e and ϵ being real, that is, U and V being both functions corresponding to
possible conics. Suppose U external to V ; then any point in U is an external
point to V .
   Take in U either of the two points represented by the equations y = 0,
x = e2 z 2 ; substituting these values of y and x, V becomes k(e2 − ϵ2 )z 2 , and
 2

□V becomes −k 3 ϵ2 (1 − ϵ2 ); therefore (1 − ϵ2 )(e2 − ϵ2 ) must be positive, that is,
ϵ2 must be one of the extremes of the three values 1, e2 , ϵ2 . In like manner, if V
  58
    See theorem of the “Diminished Determinant” in Postscript to this paper.
  59
    As we know à priori by virtue of a theorem given by M. Cauchy, and which is included
as a particular case in a theorem of my own, relating to Compound Determinants, that is,
Determinants of Determinants, which will take its place as an immediate consequence of
my fundamental Theorem given in a Memoir about to appear. The well-known rule for the
multiplication of Determinants is also a direct and simple consequence from my theorem on
Compound Determinants, which indeed comprises, I believe, in one glance, all the heretofore
existing Doctrine of Determinants.



                                             131
is external to U , e will be also one of the extremes of the same three quantities;
and hence, if the two conics are mutually external, unity will be the middle
magnitude of the group e2 , 1, ϵ2 .
   Now the three roots of □(V + λU ) = 0, are

                                       ϵ2                  1 − ϵ2
                   λ = −k,       λ = −k 2 ,       λ = −k          .
                                       e                   1 − e2
Hence if U and V be without one another, or, as it may be termed, are extra-
spatial, the third value of λ will be of a different sign from the first two; but if
the two conics be co-spatial, that is, if one includes the other, all the three values
of λ will have the same sign. Hence we have the following elegant criterion of
co-spatiality of two possible conics expressed by the equations U = 0, V = 0,
between indeterminate coordinates ξ, η, ζ; the coefficients of the cubic function
□ξηζ (λU + µV ) must give only changes or only continuations of sign.
   If this test be not satisfied, it will remain to determine which of the two conics
contains, and which is contained by the other. Let U contain V , then the order
                                                1−ϵ2
of magnitudes will be 1, e2 , ϵ2 ; therefore k 1−e 2 is greater than k, and therefore
    2
      2 , which is that root of the equation □(V + λU ) = 0 which is always one or
  1−ϵ
k 1−e
the other of the extremes, is the greatest of the three. Hence the scheme for the
impossible and possible regions of λ will be as below:

            −∞ Poss.       A   Imposs. B      Poss.   C    Poss.      + ∞.

Hence if the two roots of dλ d
                               {V + λU } = 0 be l and L, and of the two conics
V + lU = 0, V + LU = 0, the former be the possible, and the latter the impossible
one, U contains V or is contained in it according as l is greater or less than L. p. 128
   Observe that if U and V be non-cospatial, so that the three values of µ in
□(U + µV ) = 0 have not all the same sign and consequently zero lies between the
greatest and least of them, it will not be necessary to make trial of the characters
of the two curves U + mV = 0, and U + M V = 0, in order to ascertain whether U
and V intersect or not; for it will be sufficient to find which of the two quantities
m and M substituted for µ in □(U + µV ) causes it to have the opposite sign to
□(U + 0V ), that is, □U , and this one of the two it is, if either, which will make
U + µV an impossible conic, and will thus alone serve to determine whether the
intersections of U and V are unreal, or the contrary.
   It might be a curious question to consider whether, in a certain sense, conics
not both possible may not be said to lie one within or without the other. Upon
general logical grounds, I think it not improbable that two impossible conics
might be discovered each to contain the other; but this is an inquiry which I
have not had leisure to enter upon.
   I have thus far supposed the roots of □(λU + V ) = 0 to be all distinct from
one another. I now approach the discussion of the contact of two conics, in which


                                        132
event two or more of the roots will be equal. The condition for simple contact is
evidently
                           □λµ □ξηζ (λU + µV ) = 0.

   The unpaired value of λ in □(λU + V ) makes λU + V an impossible pair
of lines, and therefore, in the scheme for λ drawn as above, will separate the
possible from the impossible region.
   Whether the conics intersect in two real or two unreal points, besides the point
of contact, will be known at once by ascertaining whether U + µV = 0 represents
two real or two imaginary lines. If the latter, the two curves lie dos-à-dos or one
within the other, according as the successions of sign in □(λU + V ) are all of
the same kind or not; if they be all of the same kind, one will include the other,
namely, U will include V if the equal roots are greater, and be included in it if
they be less than the unequal one. This last conclusion however, it should be
observed, is inferred upon the principle of continuity, by making two values of λ
approach indefinitely near to one another, but cannot be strictly deduced from
the equations given for U and V applicable to the general case, in which the
axes of coordinates are the three axes joining the vertices; since these latter, in
the case supposed, reduce to two only, and consequently such representation of
U and V becomes illusory.
   If all three values of λ are equal, the three vertices come together, and hence
the two conics will have three consecutive points in common, that is, will have
the same circle of curvature. On this supposition the two curves cut at the point
of contact, and all four points of intersection are of course real.                 p. 129
   The classification of contacts between two conics may be stated as follows:
   Simple contact = one case.
   Second degree contact = two cases, namely, common curvature or double
contact.
   Third degree contact = one case, namely, contact in four consecutive points.
   These four cases of course correspond to the several suppositions of there
being two equal roots, three equal roots, two pairs of equal roots, or four equal
roots in the biquadratic equation obtained between two variables by elimination
performed in any manner between the given equations in the two conics.
   The first species and the first case of the second species have been already
disposed of. I proceed to assign the conditions appertaining to the second case
of the second species, when U and V have a double contact.
   Let A, A′ , B, B ′ be the two pairs of coincident points in which the conics
are supposed to meet; either pair of lines AB, A′ B ′ , and AB ′ , A′ B, becomes a
coincident pair. Hence such a value of µ can be found as will make U + µV the
square of a linear function of ξ, η, ζ. If therefore we make U + µV = W , and




                                        133
form the determinant
                                d2 W     d2 W     d2 W
                                                           p
                                 dξ 2    dξdη     dξdζ
                                d2 W     d2 W     d2 W
                                                           q
                                dηdξ      dη 2    dηdζ
                                d2 W     d2 W     d2 W
                                                           r
                                dζdξ     dζdη      dζ 2
                                  p        q        r      0
                   = Ap2 + Bq 2 + Cr2 + 2F qr + 2Grp + 2Hpq,
where all the coefficients are quadratic functions of µ, and make

              A = 0,     B = 0,     C = 0,       F = 0,    G = 0,      H = 0,

each of these six equations in µ will have one and the same root in common.
   It is, however, enough to select any three; if these vanish together for any
value of µ, the remaining three must also vanish. This is a simple application of
a general law60 which will appear in a forthcoming memoir on “Determinants
and Quadratic Forms,” of which this paper is to be considered as an accidental
episode.                                                                          p. 130
   Take now any three of the six equations which for the sake of generality call
P = 0, Q = 0, R = 0. The hypothesis of double contact requires that P and Q,
Q and R, R and P shall have a factor in common; but these conditions are not
sufficiently explicit for our present object, since P, Q, R might be of the form

           κ(λ − a)(λ − b),         κ′ (λ − b)(λ − c),         κ′′ (λ − c)(λ − a),

and would thus satisfy the conditions above stated, without P, Q, R having a
common factor. A sufficient criterion is that f Q+gR and P shall have a common
factor for all values of f and g.
   Let then the resultant of f Q + gR and P be

                                  Lf 2 + M f g + N g 2 ;

we must have
                           L = 0,        M = 0,           N = 0,
where L is the resultant of P and Q, N is the resultant of R and Q, and M is a
new function, which if we call Q = ϕ(λ), R = ψ(λ), and suppose a and b to be
the two roots of P = 0, is easily seen to be equal to

                                    ϕa · ψb + ϕb · ψa.
   For statement of this law called the Homaloidal Law, see Philosophical Magazine of this
  60

month “On Certain Additions, &c.” [p. 150 below. Ed.]


                                           134
This I call the connective of P · Q and P · R.
  L, M, N may conveniently be denoted by the forms

                           P · Q,       P · R,      Q · P · R.

   We may now take more generally

                        aP + bQ + cR,            αP + βQ + γR,

which will have a factor in common for all values of a, b, c, α, β, γ.
   I am indebted to Mr Cayley for the remark that the resultant of these two
functions is a new quadratic function, which, according to my notation just given,
may be put under the form
            P Q(aβ − bα)2 + QR(bγ − cβ)2 + RP (cα − aγ)2
               + P RQ(bγ − cβ)(cα − aγ) + QP R(cα − aγ)(aβ − bα)
               + RQP (aβ − bα)(bγ − cβ).
Ternary systems of the six coefficients formed upon the type of (P Q, QR, RP ),
I call complete systems, because the three functions included in such a system
equated severally to zero, imply that the remaining three coefficients are all zero.
Such a system as (P Q, QR, RP ) I term an incomplete ternary system as not
drawing with it the like implication. Probably (?) we should find on investigation
that P RQ, QP R, RQP , would also be an                                              p. 131
   incomplete system, but that systems formed after the type of P RQ, RQ, RQP
are complete. This however is only matter of conjecture, as I have been too
much occupied with other things to enter upon the inquiry. The distinct types
of ternary systems are altogether six in number, namely, four of a symmetrical
species,
                                P Q, QR, RP,
                               P RQ,     QP R,      RQP,
                                 P Q,    P QR,      QR,
                                P RQ,     RQ,      RQP ;
and two of an unsymmetrical species, namely,
                                 P Q,    P QR,      P R,
                               P RQ,     RQ,      QP R.61

   If instead of confining ourselves to three out of the six original quantities,
A, B, C; F, G, H, we take them all into account, and write down the resultant of

                          aA + bB + cC + f F + gG + hH,
  61
     P Q, QR, RP , may be compared in a general way with the angles, and P RQ, QP R, RQP ,
with the sides of a triangle.


                                          135
                              αA + βB + γC + ϕF + χG + ηH;
we shall obtain a quadratic function of 15 variables (not however all independent)
having 120 coefficients, all of which must be zero. It would be extremely
interesting to determine how many complete ternary groups can be formed out
of these 120 terms.
   It will be recollected that we have assigned as the condition of contact in
three consecutive points, that a certain cubic equation shall have all its roots
real. Now, as well remarked by Mr Cayley, we cannot express this fact by less
than three equations in integral terms of the coefficients. Thus if the cubic be
written
                            aλ3 + 3bλ2 + 3cλ + d = 0,
we have as one of such ternary systems,

             U = ac − b2 = 0,          V = bd − c2 = 0,           W = bc − ad = 0.

The significant parts of these equations are of course, however, capable of being
connected by integral multipliers U ′ , V ′ , W ′ , such that

                                   U ′ U + V ′ V + W ′ W = 0.
                                                                                       p. 132
     Any number of functions U, V, W so related, I call syzygetic functions, and
U ′ , V ′ , W ′ I term the syzygetic multipliers 62 . These in the case supposed are
c, a, b, respectively.
     In like manner it is evident that the members of any group of functions, more
than two in number, whose nullity is implied in the relation of double contact,
whether such group form a complete system or not, must be in syzygy.
     Thus P Q, P QR, QR, must form a syzygy; nor is there any difficulty in assign-
ing a system of multipliers to exhibit such syzygy. Calling P = ϕ(λ), R = ψ(λ),
a and b the two roots of Q = 0, I have found that

   {(ψa)2 + (ψb)2 }P Q − (ϕa · ψa + ϕb · ψb)P QR + {(ϕa)2 + (ϕb)2 }QR = 0.

Again, if we take the incomplete system

                                 (P Q),       (QR),         (RP ),

it will be found that

                              L(QR) + M (RP ) + N (P Q) = 0,
  62
       There will be in general various such systems of multipliers.




                                                136
provided that, calling a, b; c, d; e, f , the roots of P = 0, Q = 0, R = 0, respectively,
we make
                                            (a − c)(a − d)(a − e)(a − f )
   L = (k0 + k1 a + k2 a2 + k3 a3 + k4 a4 )
                                                         a−b
                                              (b  − c)(b − d)(b − e)(b − f )
       + (k0 + k1 b + k2 b2 + k3 b3 + k4 b4 )                                ,
                                                           b−a
                                           (c − a)(c − b)(c − d)(c − e)(c − f )
  M = (k0 + k1 c + k2 c2 + k3 c3 + k4 c4 )
                                                           c−d
                                               (d  − a)(d − b)(d − c)(d − e)(d − f )
       + (k0 + k1 d + k2 d2 + k3 d3 + k4 d4 )                                        ,
                                                                d−c
                                           (e − a)(e − b)(e − c)(e − d)
  N = (k0 + k1 e + k2 e2 + k3 e3 + k4 e4 )
                                                        e−f
                                                 (f − a)(f − b)(f − c)(f − d)
       + (k0 + k1 f + k2 f 2 + k3 f 3 + k4 f 4 )                               ;
                                                             f −e
k0 , k1 , k2 , k3 , k4 being quite arbitrary, and L, M, N , although presented in a
fractional form, being essentially integral.
    This fact of L, M, N constituting a system of multipliers to the syzygy
QR, RP, P Q, is easily demonstrated; for

                           QR = (c − e)(c − f )(d − e)(d − f ),

                           RP = (e − a)(e − b)(f − a)(f − b),
                           P Q = (a − c)(a − d)(b − c)(b − d).
                                                                                                    p. 133
   Hence
L(QR) + M (RP ) + N (P Q)
= (a − c)(a − d)(a − e)(a − f )(b − c)(b − d)(b − e)(b − f )(c − e)(c − f )(d − e)(d − f )
                 k0 + k1 a + k2 a2 + k3 a3 + k4 a4
                                                    = 0.
           X
       ×
               (a − b)(a − c)(a − d)(a − e)(a − f )
My theory of elimination enables me to explain exactly the nature of L, M, N ,
and the reason of their appearance as syzygetic factors.
   Let Lr , Mr , Nr signify what L, M, N become, when all the k’s except kr are
taken zero. Then the theory given by me in the Philosophical Magazine for the
year 1838, or thereabouts63 , shows that L0 λ + L1 is the prime derivee of the first
  63
    This cannot be obtained directly from what is stated in the paper referred to, although
contained in the general theory of derivation there given. The arbitrary functions which enter
into the expression for the general derivees have been in that paper evaluated only for the prime
derivees, which however are only particular phenomena, with reference to the general results
of Dialytic Elimination. Hereafter I may give a more general exposition of this remarkable,
although ignored or neglected theory. The prime derivees of f x and f ′ x are Sturm’s Functions,
cleared of quadratic factors, and are expressed by virtue of the general theorems there laid
down as functions of x and of symmetrical functions of the roots of f x. [p. 40 above. Ed.]


                                              137
degree between the two equations P and Q × R, or, in other words, will be the
remainder integralized of
                                     QR
                                         .
                                     P
In like manner M0 λ + M1 , N0 λ + N1 are the integralized remainders of
                                  RP              PQ
                                        and of
                                  Q                R
respectively.
   If now the resultant of P, Q and of Q, R are each zero, but the resultant of P
and R is not zero, it will be evident that P, Q, R must be of the form

            f (λ + a)(λ + c),      g(λ + c)(λ + d),        h(λ + d)(λ + b);

and therefore P × R will contain Q, and consequently we must have

                                M0 = 0,         M1 = 0.

More generally, if we write

                Q = 0,       λQ = 0,        λ2 Q = 0,      P × R = 0,

and eliminate dialytically, that is, treating λ4 , λ3 , λ2 , λ as distinct quantities, we
shall obtain
                  λ4 : λ3 : λ2 : λ : 1 :: M4 : M3 : M2 : M1 : M0 ;
and therefore when P × R contains Q,

               M0 = 0,      M1 = 0,    M2 = 0,      M3 = 0,     M4 = 0.
                                                                                            p. 134
   In like manner, when Q × P contains R,

                N0 = 0,     N1 = 0,     N2 = 0,    N3 = 0,     N4 = 0;

and when R × Q contains P ,

                 L0 = 0,     L1 = 0,    L2 = 0,    L3 = 0,    L4 = 0.

Accordingly, we see from the equation

                           L(QR) + M (RP ) + N (P Q) = 0,

that if QR = 0, RP = 0; but P Q not = 0, then N = 0; and therefore

                N0 = 0,     N1 = 0,     N2 = 0,    N3 = 0,     N4 = 0,



                                          138
and so in like manner for the remaining corresponding two suppositions64 .
   Before proceeding to consider the remaining case of the highest species of
contact, I must observe that besides the equations involved in the condition that
A, B, C; F, G, H, or, which is the same thing, that any three of them shall all
have a factor in common, we must have □(U + λV ) containing the square of
such common factor. In the memoir before adverted to a general theorem will be
given and proved, which shows that this latter condition is involved in the former
one; in fact, more generally (but still only as a particular case) that when U and
V are quadratic functions of n letters, but U + eV admits of being represented
as a complete function of (n − 2) quantities only, which are themselves linear
functions of the n letters, then □(U + λV ), which is of course a function of λ of
the nth degree, will contain the factor (λ − e)2 .
   When the two conics have four consecutive points in common, the characters
of double-point contact and of contact in three consecutive points must exist
simultaneously; and consequently the factor common to A, B, C; F, G, H, will
enter not as a binary but as a ternary factor into □(U + λV ). This gives the
extra condition required. As an example take the two conics,

                                                y2
                                         U=        + x2 − z 2 = 0,
                                               1−k

                              V = y 2 + x2 − 2kxz + (2k − 1)z 2 = 0,
                             1
                                         
           U + λV =             + λ y 2 + (1 + λ)x2 − {1 + λ(1 − 2k)}z 2 − 2kλxz.
                            1−k
                                                                                                p. 135
     The complete determinant of U + λV is then
     1                                                       1
−       {1 + (1 − k)λ}{(1 + λ)2 − 2kλ(1 + λ) + k 2 λ2 } = −     {1 + (1 − k)λ}3 .
    1−k                                                     1−k
A, B, C are the determinants of U + λV , when x = 0, y = 0, z = 0, respectively.
Thus
                                  1
                                         
                          A=          + λ (1 + λ),
                                 1−k
                              1
                                    
                      B=          + λ {1 + λ(1 − 2k)},
                            1−k
    64
         Since we are able to assign the values of the syzygetic multipliers in the equations

                                           L(P Q) + M (QR) + N (RP ) = 0,
                                        L′ (P Q) + M ′ (P QR) + N ′ (QR) = 0,
                                  L′′ (QR) + M ′′ (QRP ) + N ′′ (RP ) = 0,
                                L (RP ) + M ′′′ (RP Q) + N ′′′ (P Q) = 0,
                                  ′′′


it follows that we may eliminate between these four equations any three of the six quantities
(P Q), (P RQ), &c., and thus express any one of them in terms of any two others: this method,
however, is not practically convenient. I may probably hereafter return to this subject.


                                                      139
        C = k 2 λ2 − (1 + λ){1 + λ(1 − 2k)} = λ2 (1 − k)2 − 2λ(1 − k) − 1;
                                               1
                                      λ=−
                                             1−k
makes A = 0, B = 0, C = 0, and the factor λ + 1−k    1
                                                        enters cubed into □(U + λV ).
   Hence the two conics have a contact of the third order.
   This is easily verified; for if we pass from general to Cartesian and rectangular
coordinates, and make √    z unity; U = 0 will represent an ellipse with centre at
the origin, eccentricity k, and mean focal distance 1, and V = 0 the circle of
curvature at the extremity of the axis major65 .
   I had intended to have added some other remarks connected with the present
discussion, and also to have appended an à posteriori proof of the propositions
relative to the reality and otherwise of the vertices and chordal pairs of intersection
which I have, at the commencement of this paper, deduced quite legitimately,
but in a manner not at first sight perhaps easily intelligible, from the general
principles of conjugate forms; but this discussion has run on already to a length
so much greater than I had anticipated and than the importance of the inquiry
may seem to justify, that I must reserve for a future number of the Journal what
further matter I may have to communicate concerning it.

                                           Postscript

  As I have alluded to Professor Boole’s theorem relative to Linear Transfor-
mations, it may be proper to mention my theorem on the subject, which is of
a much more general character, and includes Mr Boole’s (so far as it refers to
Quadratic Functions) as a corollary to a particular case. The demonstration will
be given in the forthcoming memoir above alluded to.
  Let U be a quadratic function of any number of letters x1 , x2 . . . xn , and let
any number r of linear equations of the general form
                           1
                               ar x1 + 2 ar x2 + · · · + n ar xn = 0
                                                                                                 p. 136
  be instituted between them: and by means of these equations let U be
expressed as a function of any (n − r) of the given letters, say of xr+1 , xr+2 . . . xn ,
and let U , so expressed, be called M . Let
                                1
                                    ar x1 + 2 ar x2 + · · · + n ar xn
be called Lr . Then the determinant of M in respect to the (n − r) letters above
given is equal to the determinant of
                        U + L1 xn+1 + L2 xn+2 + · · · + Lr xn+r ,
  65
    We have thus discussed all the four cases of biconical contact: for an exactly parallel
discussion of the theory of contact of a plane with the curve of double curvature in which two
surfaces of the second order intersect, see the paper in the Philosophical Magazine for this
month, before referred to. [p. 148 below. Ed.]


                                                 140
considered as a function of the (n + r) letters

                                     x1 x2 . . . xn+r ,

divided by the square of the determinant
                               1a      2a      ···   ra
                                 1       1             1
                               1a      2a      ···   ra
                                 2       2             2
                                                            .
                               ···      ···    ···   ···
                               1a       2a     ···   ra
                                 r         r            r

This I call the theorem of Diminished Determinants.
   If now we have U a function of r letters, and V of r other letters, and V is
derived from U by linear transformations, that is, by r equations connecting the
2r letters; then, since U may be considered as a function of all the 2r letters with
abortive coefficients for all the terms where any of the second set of r letters
enter, we may apply our theorem of diminished determinants to the question so
considered, and the result may be found to represent Mr Boole’s theorem in a
form rather more general and symmetrical, but substantially identical with that
given by Mr Boole.
   Thus suppose 12 ax2 + bxy + 12 cy 2 say P , and 12 αu2 + βuv + 12 γv 2 say Q, are
mutually transformable by virtue of the linear equations

                 lx + my = λu + µv,            l′ x + m′ y = λ′ u + µ′ v,

P may be considered as a function of x, y, u, v, and Q as the value of P , when
we eliminate x and y by virtue of the two linear equations

       L1 = lx + my − λu − µv = 0,             L2 = l′ x + m′ y − λ′ u − µ′ v = 0;

we have therefore by our theorem the determinant of Q equal to the squared
reciprocal of the determinant
                                  l m
                                  l ′ m′
multiplied by the determinant

                          a b    0   0  l  l′
                          b c    0   0  m m′
                          0 0    0   0 −λ −λ′
                                              .
                          0 0    0   0 −µ −µ′
                          l m −λ −µ 0      0
                          l′ m′ −λ′ −µ′ 0  0
                                                                                       p. 137




                                           141
  which last determinant is evidently equal to the determinant of P multiplied
by the square of the determinant

                                           λ µ
                                                  .
                                           λ′ µ ′

Whence we see that the determinant of Q divided by the square of

                                           λ µ
                                           λ ′ µ′

is equal to the determinant of P divided by the square of

                                           l m
                                                 .
                                           l′ m′

There is also another way more simple, but less direct, by means of which the
theorem of diminished determinants may be made to yield Mr Boole’s theorem
of transformation66 . Some unavowed use has been made in the foregoing pages
of this former theorem, one of the highest importance in the analytical and
geometrical theory of quadratic functions. It has been nearly a year in my
possession, and I trust and believe that I am committing no act of involuntary
misappropriation in announcing it as a result of my own researches.




  66
     Namely, by considering P and Q as each derived from some common function of x, y, u, v, w,
by means of the equations L1 = 0, L2 = 0; the law of Diminished Determinants will then indicate
the determinants of P and Q, each under the form of fractions having the same numerator, but
                              λ µ           l m
whose denominators will be ′         and ′          respectively.
                             λ µ′          l m′


                                             142
                                       23.
An Instantaneous Demonstration of Pascal’s Theorem by the
           Method of Indeterminate Coordinates
               [Philosophical Magazine, XXXVII. (1850), p. 212]
                                                                                       p. 138
   The new analytical geometry consists essentially of two parts—the one deter-
minate, the other indeterminate.
   The determinate analysis comprehends that class of questions in which it is
necessary to assume independent linear coordinates, or else to take cognizance
of the equations by which they are connected if they are not independent. The
indeterminate analysis assumes at will any number of coordinates, and leaves
the relations which connect them more or less indefinite, and reasons chiefly
through the medium of the general properties of algebraic forms, and their
correspondencies with the objects of geometrical speculation. Pascal’s theorem of
the mystic hexagon, and the annexed demonstration of its fundamental property,
belong to this branch of the subject, and afford an instructive and striking
example of the application of the pure method of indeterminate coordinates.
   Let x, y, z, t, u, v be the sides of a hexagon inscribed in the conic U . Let the
hexagon be divided by a new line ϕ in any manner into two quadrilaterals, say
xyzϕ, tuvϕ. Then
                             ayϕ + bvx = U = auϕ + βtv;
therefore
                            (ay − αu)ϕ = βtv − bvx;
therefore ay − αu and ϕ are the diagonals of the quadrilateral txvz.
   By construction, ϕ is the diagonal joining x, v (that is, the intersection of x
and v) with z, t; and thus we see that ay − αu is the line joining t, x with v, z;
but this line passes through y, u. Therefore x, t; y, u; z, v lie in one and the
same right line. Q.E.D.




                                       143
                                            24.
       On a New Class of Theorems in Elimination Between
                     Quadratic Functions
             [Philosophical Magazine, XXXVII. (1850), pp. 213–218]
                                                                                                p. 139
   In a forthcoming memoir on determinants and quadratic functions, I have
demonstrated the following remarkable theorem as a particular case of one much
more general, also there given and demonstrated.
   Let U and V be respectively quadratic functions of the same 2n letters, and
let it be supposed possible to institute n such linear equations between these
letters as shall make U and V both simultaneously become identically zero67 .
Then the determinant of λU + µV , which is of course a function of λ and of µ of
the 2nth degree, will become a perfect square of a function of λ and µ of the nth
degree; and conversely, if this determinant be a perfect square, U and V may be
made to vanish simultaneously by the institution of n linear equations between
the 2n letters68 .
   Let now P and Q be respectively quadratic functions of three letters only, say
x, y, z; and let
                            U = P + (lx + my + nz)t,
                               V = Q + k(lx + my + nz)t.
The determinant of λU + µV in respect to x, y, z, t is easily seen to be (λ + kµ)2
times the determinant of

                               λP + µQ + (lx + my + nz)t

in respect to x, y, z, t. Hence if we call

                          λP + µQ + (lx + my + nz)t = W,

and make □xyzt W a squared function of λ, µ or which is the same thing, if

                                   □λµ □xyzt {W } = 0,
                                                                                                p. 140
  U and V will vanish simultaneously when two linear relations are instituted
between the quantities (all or some of them) x, y, z, t.
  In order that this may be the case, it will be seen to be sufficient that

                      P = 0,       Q = 0,         lx + my + nz = 0,
  67
     In the more general theorem above alluded to, the number of letters is any number m, the
number of linear equations being any number not exceeding m 2
                                                              .
  68
     When n = 1, we obtain a theorem of elimination between two quadratics, which has been
already given by Professor Boole.


                                            144
shall coexist; for then two equations between x, y, z of which lx + my + nz = 0
will be one, will suffice to make U and V each identically zero. Hence we have
the following theorem:

                     □λµ □xyzt {λU + µV + (lx + my + nz)t}

is a factor of the resultant of

                    P = 0,         Q = 0,         lx + my + nz = 0.

A comparison of the orders of the resultant and the determinant shows that
they must be identical, à-ci-près, of a numerical factor, which, if the resultant be
taken in its general lowest terms, may no doubt be easily shown to be unity.
   As an illustration of our theorem, let

                  P = xy + yz + zx,           Q = cxy + ayz + bzx.

Then
                                             0    λ + cµ λ + bµ l
                                           λ + cµ    0   λ + aµ m
       □xyzt {λP + µQ + (lx + my + nz)t} =
                                           λ + bµ λ + aµ    0   n
                                              l     m      n    0

        = n2 (λ + cµ)2 + m2 (λ + bµ)2 + l2 (λ + aµ)2
           − 2lm(λ + bµ)(λ + aµ) − 2mn(λ + cµ)(λ + bµ)
           − 2nl(λ + aµ)(λ + cµ)
        = λ2 {n2 + m2 + l2 − 2lm − 2mn − 2nl}
           + 2λµ{cn2 + bm2 + al2 − lm(a + b) − mn(b + c) − nl(c + a)}
           + µ2 {c2 n2 + b2 m2 + a2 l2 − 2ablm − 2bcmn − 2canl}.
And we thus obtain, finally,

           □λµ □xyzt {λP + µQ + (lx + my + nz)t}
         = (n2 + m2 + l2 − 2lm − 2mn − 2nl)
              × (c2 n2 + b2 m2 + a2 l2 − 2ablm − 2bcmn − 2canl)
           − {cn2 + bm2 + al2 − lm(a + b) − mn(b + c) − nl(c + a)}2
         = − 4lmn{(a − b)(a − c)l + (b − a)(b − c)m + (c − a)(c − b)n}.
                                                                                       p. 141
  Now to obtain the resultant of

       xy + yz + zx = 0,          cxy + azy + bxz = 0,       lx + my + nz = 0,

we need only find the four systems in their lowest terms of x : y : z, which
satisfy the first two equations, and multiply the four linear functions obtained

                                            145
by substituting these values of x, y, z in the fourth: the product will contain the
resultant of the system affected with some numerical factor. In the present case,
the four systems of x, y, z are

                             x = 0,   y = 0,     z = 1,

                             y = 0,   z = 0,     x = 1,
                             z = 0,   x = 0,     y = 1,
          x = (a − b)(a − c),   y = (b − a)(b − c),       z = (c − a)(c − b),
and accordingly the product of

                                 lx1 + my1 + nz1 ,
                                 lx2 + my2 + nz2 ,
                                 lx3 + my3 + nz3 ,
                                 lx4 + my4 + nz4 ,

becomes

           lmn{(a − b)(a − c)l + (b − a)(b − c)m + (c − a)(c − b)n},

agreeing with the result obtained by my theorem,—a special numerical factor
4, arising from the peculiar form of the equations, having disappeared from the
resultant.
   A geometrical demonstration may be given of the theorem which is instructive
in itself, and will suggest a remarkable extension of it to functions containing
more than three letters; the equation

                    □xyzt {λU + µV + (lx + my + nz)t} = 0,

which is a quadratic equation in λ : µ, may easily be shown to imply that the
conic λU + µV is touched by the straight line

                                lx + my + nz = 0.

And we thus see that in general two conics,

                                  λU + µV = 0,

passing through the intersections of two given conics,

                                U = 0,         V = 0,
                                                                                      p. 142




                                         146
   may be drawn to touch a given line. If, however, the given line passes through
any of the four points of intersection, in such case only one conic can be drawn
to touch it; accordingly

                       □□{λU + µV + (lx + my + nz)t}

must be zero when l, m, n are so taken as to satisfy this condition, that is, if

                              lx1 + my1 + nz1 = 0,

or
                              lx2 + my2 + nz2 = 0,
or
                              lx3 + my3 + nz3 = 0,
or
                              lx4 + my4 + nz4 = 0,
whence the theorem.
  Now suppose U and V to be each functions of four letters, x, y, z, t; when

                □xyztu {λU + µV + (lx + my + nz + pt)u} = 0,

the conoid λU + µV touches the plane

                             lx + my + nz + pt = 0;

and □ = 0 being a cubic equation, in general three such conoids can be drawn.
   Considerations of analogy make it obvious to the intuition, that in the partic-
ular case of two of these becoming coincident, the given plane

                               lx + my + nz + pt

must be a tangent plane to those two coincident conoids at one of the points
where it meets the intersections of U = 0, V = 0; that is

                             lx + my + nz + pt = 0

will pass through a tangent line to, or in other words, may be termed a tangent
plane to the intersections. Hence the following analytical theorem, derived from
supposing q, r, s, t to be proportional to the areas of the triangular faces of the
pyramid cut out of space by the four coordinate planes to which x, y, z, t refer.
As these planes are left indefinite, q, r, s, t are perfectly arbitrary.
   Theorem. The resultant of
             1. U = 0
                           where U and V are functions of x, y, z, t;
             2. V = 0

                                       147
and
                                   3.      lx + my + nz + pt = 0;
together with
                                          dU   dU      dU     dU
                                          dx   dy      dz     dt
                                          dV   dV      dV     dV
                              4.                                 = 0;
                                          dx   dy      dz     dt
                                           l   m        n      p
                                           q    r       s      t
                                                                                          p. 143
  which system, it will be observed, consists of three quadratic functions, and
one linear function of x, y, z, t, contains the factor

                       □λµ □xyzt {λU + µV + (lx + my + nz + pt)u}.

This last quantity is of the 4 × 3th, that is, the 12th order in respect of the
coefficients in U and V combined; of the 4 × 2th, that is, the 8th order in respect
of l, m, n, p; and of the zero order in respect of q, r, s, t.
   The resultant which contains it is of the (4 + 4 + 2 · 4)th, that is, 16th order
in respect to the coefficients in U and V ; of the (4 + 8)th, that is, the 12th, in
respect of l, m, n, p; and of the 4th in respect of q, r, s, t. Hence the special (and,
as far as the geometry of the question is concerned, the unnecessary, I may not
say extraneous or irrelevant) factor which enters into the resultant is of the 4th
order in respect to the combined coefficients of U and V 69 ; and of the same order
in respect to l, m, n, p, and in respect to q, r, s, t.
   I have not yet succeeded in divining its general value.
   In the very particular example, of the system,

                                           αx2 + βy 2 = 0,

                                            cz 2 + dt2 = 0,
                                        lx + my + nz + pt = 0,
                                         αx βy 0 0
                                          0  0 cz dt
                                                     = 0,
                                          l m n p
                                          q  0 0 0
I find that the double determinant is

                             c2 d2 α2 β 2 (cp2 + dn2 )2 (m2 α + l2 β)2 ,

and the resultant is

                            q 4 c2 d2 α2 β 4 (cp2 + dn2 )4 (m2 α + l2 β)2 ,
  69
       And consequently of the second in respect to the separate coefficients of each.


                                                 148
giving as the special factor

                                q 4 β 2 (cp2 + dn2 )2 .

I believe that the theorem which I have here given for determining the condition
that lx + my + nz + pt shall be a tangent plane to the intersection of two conoids
U and V , namely, that the determinant of

                        λU + µV + (lx + my + nz + pt)u

shall have two equal roots, is altogether novel.                                  p. 144
   What is the meaning of all three roots of this determinant becoming equal,
that is, of only one conoid being capable of being drawn through the intersection
of U and V to touch the plane

                               lx + my + nz + pt?

Evidently (ex vi analogiae) that this plane shall pass through three consecutive
points of the curve of intersection, that is, that it shall be the osculating plane
to the curve.
   If we return to the intersection of two co-planar conics, and if we suppose a
line to be drawn through two of the points of intersection, the conics capable of
being drawn through the four points of intersection to touch the line, besides
becoming coincident, evidently degenerate each into a pair of right lines. It would
seem, therefore, by analogy, that if a plane be drawn including any two tangent
lines to the curve of intersection of two surfaces of the second degree, this should
be touched by two coincident cones drawn through the curve of intersection,
and consequently every such double tangent plane to the intersection of two
conoids (and it is evident that one or more of these can be taken at every point
of the curve) must pass through one of the vertices of the four cones in which
the intersection may also be considered to lie; and it would appear from this,
that in general four double tangent planes admit of being drawn to the curve,
which is the intersection of two conoids, at each point thereof. At particular
points a tangent plane may be drawn passing through more than one of the
vertices, and then of course the number of double tangent planes that can be
drawn will be lessened. These results, indicated by analogy, become immediately
apparent on considering the curve in question as traced upon any one of the four
containing cones. For the plane drawn through a tangent at any point, and the
vertex of the cone being a tangent plane to the cone, must evidently touch the
curve again where it meets it. We thus have an additional confirmation of the
analogy between a point of intersection of two curves and the tangent at any
point of the intersection of two surfaces.
   I might extend the analytical theorems which have been established for
functions of three and four to functions of a greater number of variables; but

                                         149
enough has been done to point out the path to a new and interesting class of
theorems at once in elimination and in geometry, which is all that I have at
present leisure or the disposition to undertake.




                                    150
                                         25.
  Additions to the Articles “On a New Class of Theorems,”
                 and “On Pascal’s Theorem”
            [Philosophical Magazine, XXXVII. (1850), pp. 363–370]
                                                                                       p. 145
   First addition.—I have alluded in the second of the above articles to a more
general theorem, comprising, as a particular case, the theorem there given for
the simultaneous evanescence of two quadratic functions of 2n letters, on n linear
equations becoming instituted between the letters.
   In order to make this generalization intelligible, I must premise a few words on
the Theory of Orders, a term which I have invented with particular reference to
quadratic functions, although obviously admitting of a more extended application.
A linear function of all the letters entering into a function or system of functions
under consideration I call an order of the letters, or simply an order. Now it
is clear that we may always consider a function of any number of letters as a
function of as many orders as there are letters; but in certain cases a function
may be expressed in terms of a fewer number of orders than it has letters, as
when the general characteristic function of a conic becomes that of a pair of
crossing lines or a pair of coincident lines, in which event it loses respectively one
and two orders, and so for the characteristic of a conoid becoming that of a cone,
a pair of planes or two coincident planes, in which several events, a function
of four letters becomes that of only three orders, or two orders, or one order,
respectively. When a function may be expressed by means of r orders less than
it contains letters, I call it a function minus r orders. I now proceed to state my
theorem.
   Let U and V be functions each of the same m letters, and suppose that the
determinant in respect of those letters of U + µV contains i pairs of                  p. 146
   equal linear factors of µ; then it is possible, by means of i linear equations
instituted between the letters, to make U and V each become functions of the
same m − 2i orders; and conversely, if by i equations between the letters U and
V may be made functions of the same m − 2i orders, the determinant of U + µV
considered as a function of µ will contain i square factors.
   Thus when m = 2n and i = n, U and V will each become functions of zero
orders, that is, will both disappear, provided that on the institution of a certain
system of n linear equations, among the letters of which U and V are functions,
the determinant of (U + µV ) is a perfect square,—which is the theorem given in
the article referred to.
   So for example if U and V be quadratic functions of four letters, and therefore
the characteristics of two conoids, □(U + µV ) being a perfect square, expresses
that these conoids have a straight line in common lying upon each of their
surfaces.

                                         151
   If U and V be quadratic functions of three letters only, and admit therefore
of being considered as the characteristics of two conics, □(U + µV ) containing a
square factor, is indicative of these conics having a common tangent at a common
point, that is, of their touching each other at some point; for it is easily shown
that the disappearance of two orders from any quadratic function by virtue of
one linear function of its letters being zero, indicates that the line, plane, &c. of
which the linear function is the characteristic is a tangent to the curve, surface,
&c. of which the quadratic function is the characteristic.
   I pass now to a generalization of the theorem which shows how to express,
under the form of a double determinant, the resultant of one linear and two
quadratic homogeneous functions of three letters (which I should have given
in the original paper, had I not there been more intent upon developing an
ascending scale than of expatiating upon a superficial ramification of analogies),
and which constitutes my Second addition to that paper, to wit—
   If U and V be homogeneous quadratic, and L1 , L2 . . . Ln homogeneous linear
functions of (n + 2) letters x1 , x2 . . . xn+2 , the determinant of the entire system
of n + 2 functions is equal to

           □λ,µ □x1 ,x2 ,...,xn+2 ,t1 ,t2 ,...,tn {λU + µV + L1 t1 + L2 t2 + · · · + Ln tn };

the demonstration is precisely similar to the analytical one given in the September
Number70 for the particular case of n = 1.
   When n = 0, we revert to Mr Boole’s theorem of elimination between U and
V already adverted to. The proof, it will be easily recognized, does not require
the application of the more general theorem relative to the simultaneous            p. 147
   depression of orders of two quadratic functions, but only the limited one before
given, which supplies the conditions of their simultaneous disparition. I now
proceed to develop more particularly certain analogies between the theory of the
mutual contacts of two conics, and that of the tangencies to the intersection of
two conoids.
   But here again I must anticipate some of the results which will be given in my
forthcoming memoir on Determinants and Quadratic Functions, by explaining
what is to be understood by minor determinants, and the relation in which they
stand to the complete determinant in which they are included. This preliminary
explanation, and the statement of the analogies above alluded to, will constitute
my Third and last addition.
   Imagine any determinant set out under the form of a square array of terms.
This square may be considered as divisible into lines and columns. Now conceive
any one line and any one column to be struck out, we get in this way a square,
one term less in breadth and depth than the original square; and by varying in
every possible manner the selection of the line and column excluded, we obtain,
  70
       p. 140 above.


                                                 152
supposing the original square to consist of n lines and n columns, n2 such minor
squares, each of which will represent what I term a First Minor Determinant
relative to the principal or complete determinant. Now suppose two lines and
two columns struck out from the original square, we shall obtain a system of
                                                      2
                                           n(n − 1)
                                       

                                              2
squares, each two terms lower than the principal square, and representing a
determinant of one lower order than those above referred to. These constitute
what I term a system of Second Minor Determinants; and so in general we
can form a system of rth minor determinants by the exclusion of r lines and r
columns, and such system in general will contain
                                                              2
                                 n(n − 1) · · · (n − r + 1)
                             

                                        1 · 2···r
distinct determinants.
   I say “in general”; because if the principal determinant be totally or partially
symmetrical in respect to either or each of its diagonals, the number of distinct
determinants appertaining to each system of minors will undergo a material
diminution, which is easily calculable.
   Now I have established the following law:—
   The whole of a system of rth minors being zero, implies only (r +1)2 equations,
that is, by making (r + 1)2 of these minors zero, all will become zero; and this is
true, no matter what may be the dimensions or form of the complete determinant.
But furthermore, if the complete determinant be formed from a quadratic function,
so as to be symmetrical about one of its diagonals, then 12 (r + 1)(r + 2) only of
the rth minors being zero, will serve                                                    p. 148
   to imply that all these minors are zero. Of course, in applying these theorems,
care must be taken that the (r + 1)2 or 12 (r + 1)(r + 2) selected equations must
be mutually non-implicative, and shall constitute independent conditions.
   In the application I am about to make of these principles, we shall have only
to deal with a system of first minors and of a symmetrical determinant. If three
of these properly selected be zero, from the foregoing it appears that all must be
zero.
   Now let U and V be characteristics of two conics, that is, let each be a function
of only three letters, it may be shown (see my paper71 in the Cambridge and
Dublin Mathematical Journal for November, 1850) that the different species of
contacts between these two conics will correspond to peculiar properties of the
compound characteristic U + µV .
   If the determinant of this function have two equal roots, the conics simply
touch; if it have three equal roots, the conics have a single contact of a higher order,
  71
       p. 119 above.


                                             153
that is, the same curvature; if its six first minors become zero simultaneously for
the same value of µ, the conics have a double contact. If the same value of µ,
which makes all these first minors zero, be at the same time not merely a double
root (as of analytical necessity it always must be) but a treble root of
                                       □(U + µV ) = 0,
then the conics have a single contact of the highest possible order short of
absolute coincidence, that is, they meet in four consecutive points.
    The parallelism between this theory and that of two quadratic functions
P, Q, and one linear function L72 of four letters, say x, y, z, t, is exact73 . For let
P + Lu + µQ be now taken as our compound characteristic (a function, it will
be observed, of five letters, x, y, z, t, u); if its determinant have two equal roots,
L has two consecutive points in common with the intersection of P and Q, that
is, passes through a tangent to that intersection; if it have three equal roots, L
has three consecutive points in common with the said intersection, that is, is
an osculating plane thereto; if its fifteen first minors admit of all being made
simultaneously zero, L has a double contact with the intersection of P and Q,
that is, it is a tangent plane to some one of the four cones of the second order
containing this intersection;                                                           p. 149
    if the same linear function of µ which enters into all these first minors be
contained cubically in the complete determinant, then the plane L passes through
four consecutive points of the intersection of P and Q, and the points where it
meets the curve will be points of contrary plane flexure; and, as it seems to me,
at such points the tangential direction of the curve must point to the summit of
one or other of the four cones above alluded to74 . In assigning the conditions for
L being a double tangent plane to the intersection of P and Q, we may take any
three independent minors at pleasure equal to zero. One of these may be selected
so as to be clear of the coefficients of L; in fact, the determinant of P + µQ will
be a first minor of P + µQ + Lu; µ may thus be determined by a biquadratic
equation; and then, by properly selecting the two other minors, we may obtain
two equations in which only the first powers of the coefficients of x, y, z, t in L
appear, and may consequently obtain L under the form of
                     (ae + α)x + (be + β)y + (ce + γ)z + (de + δ)t,
  72
      Observe that P = 0, Q = 0, L = 0 now express the equations to two conoids and a plane
respectively.
   73
      This parallelism may be easily shown analytically to imply, and be implied, in the geometrical
fact, that the contact of the plane L with the intersection of the two surfaces P and Q, is of
exactly the same kind as the contact (which must exist) between the two conics which are the
intersections of P and Q respectively with the plane L.
   74
      If this be so, then we have the following geometrical theorem:—“The summit of one of the
four cones of the second degree which contain the intersections of two surfaces of the second
order drawn in any manner respectively through two given conics lying in the same plane, and
having with one another a contact of the third degree, will always be found in the same right
line, namely, in the tangent line to the two given conics at the point of contact.”


                                               154
where a, α; b, β; c, γ; d, δ will be known functions of any one of the four values
of µ. The point of contact being given will then serve to determine e, and we
shall thus have the equation to each of the four double tangent planes at any
given point fully determined.
   In the foregoing discussions I have freely employed the word characteristic
without previously defining its meaning, trusting to that being apparent from
the mode of its use. It is a term of exceeding value for its significance and brevity.
The characteristic of a geometrical figure75 is the function which, equated to zero,
constitutes the equation to such figure. Pluecker, I think, somewhere calls it the
line or surface function, as the case may be. Geometry, analytically considered,
resolves itself into a system of rules for the construction and interpretation
of characteristics. One more remark, and I have done. A very comprehensive
theorem has been given at the commencement of this commentary, for interpreting
the effect of a complete determinant of a linear function of two quadratic functions
(U + µV ), having                                                                      p. 150
   one or more pairs of equal factors (e + ϵµ). But here a far wider theory
presents itself, of which the aim should be to determine the effect and meaning of
this determinant, having any amount and distribution of multiplicity whatsoever
among its roots. Nor must our investigations end at that point; but we must
be able to determine the meaning and effect of common factors, one or more
entering into the successive systems of minor determinants derived from the
complete determinant of U + µV .
   Nor are we necessarily confined to two, but may take several quadratic
functions simultaneously into account.
   Aspiring to these wide generalizations, the analysis of quadratic functions
soars to a pitch from whence it may look proudly down on the feeble and vain
attempts of geometry proper to rise to its level or to emulate it in its flights.
   The law which I have stated for assigning the number of independent, or
to speak more accurately, non-coevanescent determinants belonging to a given
system of minors, I call the Homaloidal law, because it is a corollary to a
proposition which represents analytically the indefinite extension of a property
common to lines and surfaces to all loci (whether in ordinary or transcendental
space) of the first order, all of which loci may, by an abstraction derived from the
idea of levelness common to straight lines and planes, be called Homaloids. The
property in question is, that neither two straight lines nor two planes can have
  75
     More generally, the characteristic of any fact or existence is the function which, equated
to zero, expresses the condition of the actuality of such fact or existence. Perhaps the most
important pervading principle of modern analysis, but which has never hitherto been articulately
expressed, is that, according to which we infer, that when one fact of whatever kind is implied in
another, the characteristic of the first must contain as a factor the characteristic of the second;
and that when two facts are mutually involved, their characteristics will be powers of the same
integral function. The doctrine of characteristics, applied to dependent systems of facts, admits
of a wide development, logical and analytical.


                                               155
a common segment; in other words, if n independent relations of rectilinearity
or of coplanarity, as the case may be, exist between triadic groups of a series of
n + 2, or between tetradic groups of a series of n + 3 points respectively, then
every triad or tetrad of the series, according to the respective suppositions made,
will be in rectilinear or in plane order. So, too, if n independent relations of
coincidence exist between the duads formed out of n + 1 points, every duad will
constitute a coincidence.
   This homaloidal law has not been stated in the above commentary in its
form of greatest generality. For this purpose we must commence, not with a
square, but with an oblong arrangement of terms consisting, suppose, of m lines
and n columns. This will not in itself represent a determinant, but is, as it
were, a Matrix out of which we may form various systems of determinants by
fixing upon a number p, and selecting at will p lines and p columns, the squares
corresponding to which may be termed determinants of the pth order. We have,
then, the following proposition. The number of uncoevanescent determinants
constituting a system of the pth order derived from a given matrix, n terms
broad and m terms deep, may equal, but can never exceed the number

                              (n − p + 1)(m − p + 1).

             Remark on Pascal’s and Brianchon’s Theorems
                                                                                           p. 151
   I omitted to state, in the September Number of the Journal 76 , that the
demonstration there given by me for Pascal’s, applied equally to Brianchon’s
theorem. This remark is of the more importance, because the fault of the
analytical demonstrations hitherto given of these theorems has been, that they
make Brianchon’s consequence of Pascal’s, instead of causing the two to flow
simultaneously from the application of the same principles. No demonstration
can be held valid in method, or as touching the essence of the subject-matter,
in which the indifference of the duadic law is departed from. Until these recent
times, the analytic method of geometry, as given by Descartes, had been suffered
to go on halting as it were on one foot. To Pluecker was reserved the honour
of setting it firmly on its two equal supports by supplying the complementary
system of coordinates. This invention, however, had become inevitable, after the
profound views promulgated by Steiner, in the introduction to his Geometry, had
once taken hold of the minds of mathematicians. To make the demonstration in
the article referred to apply, totidem literis, to Brianchon’s theorem (recourse
being had to the correlative system of coordinates), it is only needful to consider
U as the characteristic of the tangential envelope of the conic, x, y, z, t, u, v as the
characteristics of the six points of the circumscribed hexagon, ϕ the characteristic
of the point in which the line x, v meets the line z, t; ay −αu will then be shown to
characterize the point in which t, x meets v, z; and thus we see that y, u; t, x; v, z,
  76
       p. 138 above.


                                         156
the three pairs of opposite sides of the hexagon, will meet in one and the same
point, which is Brianchon’s theorem.




                                     157
                                        26.
 On the Solution of a System of Equations in which Three
  Homogeneous Quadratic Functions of Three Unknown
Quantities are Respectively Equated to Numerical Multiples
   of a Fourth Non-Homogeneous Function of the Same
             [Philosophical Magazine, xxxvii. (1850), pp. 370–373]
                                                                                      p. 152
   Let U, V, W be three homogeneous quadratic functions of x, y, z, and let ω
be any function of x, y, z of the nth degree, and suppose that there is given for
solution the system of equations

                      U = Aω,        V = Bω,         W = Cω.

  Theorem. The above system can be solved by the solution of a cubic equation,
and an equation of the nth degree.
  For let D be the determinant in respect to x, y, z of

                                   f U + gV + hW,

then D is a cubic function of f, g, h. Now make

                         D = 0,       Af + Bg + Ch = 0;

the ratios of f : g : h which satisfy the last two equations can be determined by
the solution of a cubic equation, and there will accordingly be three systems of
f, g, h which satisfy the same, as

                                    f1 , g1 , h1 ,
                                    f2 , g2 , h2 ,
                                    f3 , g3 , h3 .

Now D = 0 implies that f U + gV + hW breaks up into two linear factors;
accordingly we shall find

                   (l1 x + m1 y + n1 z)(λ1 x + µ1 y + ν1 z) = 0,
                   (l2 x + m2 y + n2 z)(λ2 x + µ2 y + ν2 z) = 0,
                   (l3 x + m3 y + n3 z)(λ3 x + µ3 y + ν3 z) = 0,

 in which the several sets of l, √
                                 m, n; λ, µ, ν can be expressed without difficulty in p. 153
                                     √ √
terms of the several values of f , g, h.
   Let the above equations be written under the form

                      P P ′ = 0,      QQ′ = 0,       RR′ = 0.


                                         158
Since the given equations are perfectly general, it is readily seen that the equations

               (P = 0, P ′ = 0),     (Q = 0, Q′ = 0),          (R = 0, R′ = 0),

will severally represent pairs of opposite sides of a quadrangle expressed by
general coordinates x, y, z; so that one of the two functions R, R′ will be a linear
function of P and Q and also of P ′ and Q′ , and the other will be a linear function
of P and Q′ and also of P ′ and Q.
   In order to solve the equations, we need only consider two such pairs: as
P P ′ = 0, QQ′ = 0; we then make
                               P     =     0,         Q    =   0,
                               P     =     0,         Q′   =   0,
                               P′    =     0,         Q    =   0,
                               P′    =     0,         Q′   =   0.
Any one of these four systems will give the ratios of x : y : z; and then, by
substitution in any one of the given equations, we obtain the values of x, y, z by
the solution of an ordinary equation of the nth degree. The number of systems
x, y, z is therefore always 4n.
   The equations connected with the solution of Malfatti’s celebrated problem,
“In a given triangle to inscribe three circles such that each circle touches the
remaining two circles and also two sides of the triangle,” given by Mr Cayley
in the November Number for 1849 of the Cambridge and Dublin Mathematical
Journal, to wit,
                        by 2 + cz 2 + 2f yz = θ2 a(bc − f 2 ) = A,
                        cz 2 + ax2 + 2gzx = θ2 b(ca − g 2 ) = B,
                       ax2 + by 2 + 2hxy = θ2 c(ab − h2 ) = C,
come under the general form which has just been solved. It so happens, however,
that in this particular case                 
                                f1 , g1 , h1 
                                             
                                f2 , g2 , h2
                                             
                                f3 , g3 , h3 
77                                                                                             p. 154
      become respectively
                                            1     1 
                                     0,       , −  
                                            B     C 
                                                    
                                     1           1 
                                                    
                                    −  ,    0,        ,
                                     B           C 
                                     1      1
                                                    
                                                    
                                                 0 
                                                    
                                    − ,       ,
                                                    
                                     C      B
    Were it not for this being the case, the number of solutions would be n times the number
     77

of ways of obtaining duads out of three sets of two things, excluding the duads forming the
sets, that is, the number of solutions would be 12n in place of 4n, the true number.


                                                159
and the cubic equation is resolved without extraction of roots.
   It follows from my theorem that the eight intersections of three concentric
surfaces of the second order can be found by the solution of one cubic and
one quadratic equation; and in general, if we have ϕ, ψ, θ any three quadratic
functions of x, y, z, and ϕ = 0, ψ = 0, θ = 0 be the system of equations to be
solved, provided that we can by linear transformations express ϕ, ψ, θ under the
form of
                        U − aw,     V − bw,     W − cw,
U, V, W being homogeneous functions, and w a non-homogeneous function of
three new variables, x′ , y ′ , z ′ , we can find the eight points of intersection of the
three surfaces, of which U, V, W are the characteristics, by the solution of one
cubic and one quadratic. But (as I am indebted to Mr Cayley for remarking
to me) that this may be possible, implies the coincidence of the vertices of one
cone of each of the systems of four cones in which the intersections of the three
surfaces taken two and two are contained.
   I may perhaps enter further hereafter into the discussion of this elegant little
theory. At present I shall only remark, that a somewhat analogous mode of
solution is applicable to two equations,

                               U = aP 2 ,         V = bP 2 ,

in which U, V are homogeneous quadratic functions, and P some non-homogeneous
function of x, y.
   We have only to make the determinant of f U + gV equal to zero, and we shall
obtain two systems of values of f, g, wherefrom we derive

         l1 x + m1 y = ± af1 + bg1 P,             l2 x + m2 y = ± af2 + bg2 P,
                          p                                     p


from which x and y may be determined.




                                            160
                                             27.
 On a Porismatic Property of Two Conics Having with One
         Another a Contact of the Third Order
               [Philosophical Magazine, xxxvii. (1850), pp. 438, 439]
                                                                                                  p. 155
   If two conics have with one another a contact of the third order, that is, if they
intersect in four consecutive points, it will easily be seen that their characteristics
referred to coordinate axes in the plane containing them must be of the relative
forms x2 +yz, k(y 2 +x2 +yz) respectively, y characterizing their common tangent
at the point of contact78 .
   Hence if we take planes of reference in space, and call t the characteristic of
the plane of the conics, the equations to any two conoids drawn through them
respectively will be of the relative forms

               U = x2 + yz + tu = 0,           V = y 2 + x2 + yz + tv = 0.

Using W to denote V − U , and (W ) to denote what W becomes when ey is
substituted for t, we see that W and (W ) are of the respective forms y 2 + tw
and yθ; showing that the former is the characteristic of a cone which will be
cut by any plane t − ey drawn through the line (t, y) in a pair of right lines;
or, in other words, that one of the cones containing the intersection of the two
variable conoids (V and U ) will have its vertex in the invariable line which is
the common tangent to the two fixed conics: this proves the theorem stated by
me hypothetically in a foot-note in one of my papers in the last number of the
Magazine 79 . The steps of the geometrical proof there hinted at are as follows. p. 156
    The four consecutive points in which the two conics intersect will be consecutive
points in the curve of intersection of the two variable conoids. This curve lies
in each of four cones of the second degree. Every double tangent plane to it
passes through the vertex of one amongst these. The plane containing four, that
is, two (consecutive) pairs of consecutive points, is a double tangent plane, and
will therefore pass through a vertex; but four consecutive points of a curve of
the fourth order described upon a cone, and lying in one tangent       plane thereto,
can only be conceived generally as disposed in the form of an , of which the
                                                                    R

belly part will point to the vertex; or, in other words, at any point where two
consecutive osculating planes coincide so that the spherical curvature vanishes,
  78
     These relative or conjugate forms are taken from a table which I shall publish in a future
number of this Magazine, exhibiting the conjugate characteristics in their simplest forms,
correspondent to all the various species of contacts possible between lines and surfaces of the
second degree. This table is as important to the geometer as the fundamental trigonometrical
formula to the analyst, or the multiplication table to the arithmetician; and it is surprising
that no one has hitherto thought of constructing such.
  79
     P. 149 above.


                                             161
the linear curvature will also vanish, that is, there will be a point of inflexion
at which, of course, the tangent line must pass through the vertex of the cone.
This is the assumption felt to be true, but stated by me hypothetically in the
paper referred to, because a ready demonstration did not at the moment occur
to me. The legitimacy of this inference is now vindicated by the above analytical
demonstration.
   The methods of general and correlative coordinates and of determinants
combined possess a perfectly irresistible force (to which I can only compare that
of the steam-hammer in the physical world) for bringing under the grasp of
intuitive perception the most complicated and refractory forms of geometrical
truth.




                                       162
                                                   28.
        On the Rotation of a Rigid Body about a Fixed Point
                  [Philosophical Magazine, xxxvii. (1850), pp. 440–444]
                                                                                    p. 157
   In the Cambridge and Dublin Mathematical Journal for March 1848, an article
by Professor Stokes, of the University of Cambridge, is ushered in with the words
following:—

            “The most general instantaneous80 motion of a rigid body move-
         able in all directions about a fixed point consists in a motion of
         rotation about an axis passing through that point. This elementary
         proposition is sometimes assumed as self-evident, and sometimes
         deduced as the result of an analytical process. It ought hardly per-
         haps to be assumed, but it does not seem desirable to refer to a long
         algebraical process for the demonstration of a theorem so simple. Yet
         I am not aware of a geometrical proof anywhere published which
         might be referred to.”

   The learned and ingenious professor is indubitably right, and might have
trusted himself to assert less hesitatingly the necessity of demonstrating this
proposition, which possesses none of the characters of a self-evident truth; but it
is to be regretted that he should have stated it in such a form as naturally to
lead the incautious reader to mistake the nature and grounds of its existence,
which consist in this fact—that any kind of displacement of a body moveable
about a fixed axis, whether instantaneous and infinitesimal, or secular and finite,
is capable of being effected by a single rotation about a single axis.
   The annexed simple proof of this capital law has the advantage of affording a
rule for compounding into one any two (and therefore any number of) rotations
given in direction, magnitude and order of succession.                              p. 158
   It will somewhat conduce to simplicity if we fix our attention upon a spherical
surface rigidly connected with the rotating body, and having its centre at the
fixed point thereof. When the positions of two points in this are given, the
position of the body is completely determined.
   Now evidently two points A, B may be brought respectively to A′ B ′ (if
AB = A′ B ′ ) by two rotations; the first taking place about a pole situated
anywhere in the great circle bisecting AA′ at right angles, the second about A′ ,
the position into which it is brought by the first rotation. This view leads us to
consider the effect of two rotations taking place successively about two axes fixed
in the rotating body. Or again, we may make the plane A′ B ′ revolve into the
position AB round a pole taken at the node in which the two planes intersect,
  80
       The italics do not exist in the original.


                                                   163
and then the points A, B swing into their new positions A′ , B ′ by means of a
rotation about the pole of the great circle, of which A′ B ′ forms a part. This
mode of effecting the displacement naturally suggests the consideration of the
effect of rotations taking place successively about two axes fixed in space.
   First, then, let us study the effect of the combination of a rotation (α) having
P for its pole, followed by another (β), of which Q is the pole, P and Q being
points in the surface of the revolving sphere.

                                               r


                                                                         P
                              Q
                                                          R
                                                                   R′

                                          Q′


   In drawing the annexed figure, I have supposed that the two rotations are of
the same kind, each tending, when a spectator is standing with his head to the
respective poles and his feet to the centre, to make a point at his right-hand
pass in front of his face towards his left-hand. Let now P Q revolve through α2
positively into the position of P R, and through β2 negatively into that of QR.
   Then I say that the two impressed rotations α and β about P and Q will be
equivalent to a single rotation about R, equal to twice the acute angle between
QR, RP .
   Let the first rotation about P bring Q to Q′ and R to R′ ; it is clear that
QP R, Q′ P R, Q′ P R′ are all equal triangles. Therefore R′ Q′ R = 2P QR = β.
Consequently the positive rotation β about Q′ (the new position of Q) will carry
R′ back again to R, its original position. Hence the actual motion which results
from the successive rotations combined being consistent with R remaining at
rest, must be equivalent to a single rotation about R.
   To find its magnitude, let the second rotation carry P to P ′81 ; then the angular
displacement P RP ′ (which is the required rotation of the whole body) is equal p. 159
to twice the acute angle between Q′ R, RP , which is the same as that between
QR, RP , as was to be shown. Thus we see that the semi-rotations about three
poles (considered as the angular points of a spherical triangle), which, taken in
order, would bring the sphere back to its first, undisturbed position, are equal to
the included angles at such poles respectively.
   If in our figure the order of the rotations had been reversed, P Qr, QP r would
 81
      The reader is requested to fill in the point P ′ and join P ′ R.


                                                   164
have been taken respectively equal to P QR, QP R, but on the opposite side of
P Q, and r would have been the resultant pole, the resultant rotation remaining
in amount the same as before.
   If either of the rotations had been negative, the resultant pole would be found
in QR produced, namely, at the intersection of rQ or rP with P Q.
   Calling the resultant rotation γ, we have always
                         α      β     γ
                   sin     : sin : sin :: sin QR : sin RP : sin P Q.
                         2      2     2
When the component rotations are infinitesimal in amount, R and r will come
together in QP ; the order of succession of the rotations will be indifferent, and
we shall have
                                α      β     γ
             α : β : γ :: sin     : sin : sin :: sin QR : sin RP : sin P Q,
                                2      2     2
which gives the rule for the parallelogrammatic composition of two simultaneously
impressed rotations82 .
   If, next, we consider the effect of rotations about two poles, P and Q, fixed in
space (supposing, as above, that they take place first about P and then about
Q), we must take QP r equal to half the contrary of the rotation about P , and
P Qr to half the direct rotation about Q (the angle being now taken positive
which was on the first supposition negative, and vice versâ); so that, retaining
the original figure, the first rotation will bring r to R, and the second carry R
back to r; showing that r is the resultant pole, and that83 P ′ rP , the resultant
rotation, will be double the acute angle between Qr, rP , as in the former case.
   To popular apprehension the important doctrine of uniaxial rotation may be
made intelligible by the following mode of statement. Take a pocket-globe, open
the case and roll about the sphere within it in any manner whatever; then closing
the case, there will unavoidably remain two points on the terrestrial surface
touching the same two points on the celestial surface as they were in apposition
with before the sphere was so turned about in its case.                             p. 160
   It is right to bear in mind that the whole of this doctrine is comprised in, and
convertible with, the following easy geometrical proposition relative to arcs of
great circles on any spherical surface, including the plane as an extreme case.
   “The arcs joining the extremities (each with each in either order) of two other
equal arcs, subtend equal angles at either of the points of intersection of two
great circles bisecting at right angles the first-named connecting arcs84 .”
  82
      Compare Mr Airy’s Tracts, Art. “On Precession and Nutation.”
  83
      P ′ is not expressed in the figure given.
   84
      This proposition will be seen to be immediately demonstrable, by the comparison of equal
triangles, when viewed as the converse of this other. “The arcs (or right lines) joining the
correspondent extremities of the bases of two similar isosceles spherical (or plane) triangles
having a common vertex, are equal to each other.”


                                            165
   The spherico-triangular mode of compounding rotations given in the above
simple disquisition may easily be made the parent of a whole brood of geometrical
consequences, which, however, I must leave to the ingenuity and care of those
who have a turn for this kind of invention.
   But I ought not to omit to invite attention to a remarkable form, which may
be imparted to the theorems above stated for the composition of finite rotations,
or rather to a theorem which may be derived from them by an obvious process
of inference.
   Let P, Q, R . . . X, Z be any number of points on a sphere capable of moving
about its centre, joined together by arcs of great circles so as to form a spherical
polygon. Imagine any number of rotations to take place about these points
in succession as poles. It matters not which is considered the first pole of
rotation, but the order of the circulation must be supposed given, as, for instance,
P QR . . . XZ, or QR . . . XZP , or R . . . XZP Q, &c. This will be one order; the
reverse order would be P ZX . . . RQ, or QP ZX . . . R, &c.
   I shall suppose the circulation to be of the kind first above written. Now we
may make two hypotheses:—

                  1. That the poles are fixed in space.
                  2. That they are fixed in the rotating body.

In the first case, let the rotations about the given poles P, Q, R, S . . . X, Z be
double the amounts which would serve to transport P Q to QR, QR to RS . . . XZ
to ZP respectively.
    In the second case, let the rotations be double the amounts which would carry
P Z to ZX . . . SR to RQ, RQ to QP respectively. Then, on either supposition,
the sum of the combined rotations is zero; or, to use a more convenient and
suggestive form of expression, if the poles of rotation form a closed spherical
polygon whose angles are respectively equal to the semi-rotations about the
poles, the resultant rotation is zero.                                               p. 161
    This proposition is immediately derivable from the fundamental one relative
to three poles, given above, by dividing the polygon into triangles by arcs, joining
any one of the poles with all the rest, or (as pointed out to me by my eminent
friend Prof. W. Thomson) it becomes apparent as a particular case of a more
general proposition, on representing the motion about the successive axes as
effected by two equal pyramids having a common vertex at the centre of motion,
of which the one is fixed in space, and the other is fixed in the revolving body
and rolls over the first, so that the corresponding equal faces are successively
brought into coincident apposition.
    P.S. To find the pole of rotation whereby P Q may be brought into the position
P ′ Q′ , we may use the following simple construction.
    Measure off from O the node of the great circles (or right lines) containing
P Q and P ′ Q′ , two distances in the proper direction upon each (four distinct

                                        166
assumptions may be made), say OR and OS equal to one another and to the
difference between OP and OP ′ , then the pole of rotation required, say E, is
the centre of the circle described about ROS, and the amount of rotation is the
angle subtended by OR or OS at E. The writer of this paper suggests that axis
of displacement would be a convenient term for designating the line whereby any
finite change in the position of a body moveable about a fixed centre may be
brought about; a geometrical theory of rotation leading to the investigation of a
very curious species of correlation, now opens upon the view, the general object
of which may be stated as follows:
   “Given upon a sphere or plane any curve considered as the locus of successive
poles of instantaneous rotation, and the ratio of the rotation about each pole to
its distance from the one that follows85 , to construct the curve of the poles of
displacement, and to determine the amount of rotation corresponding to each
such pole.”
   The discussion of this question offers a fine field for the exercise of geometrical
taste and skill.




  85
       Which by analogy may be termed the “density of rotation.”


                                             167
                                           29.
                      On the Intersections of Two Conics
       [Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 18–20]
                                                                                    p. 162
  Let the two conics be written
                  U = ax2 + by 2 + cz 2 + 2a′ yz + 2b′ zx + 2c′ xy = 0,
                  V = αx2 + βy 2 + γz 2 + 2α′ yz + 2β ′ zx + 2γ ′ xy = 0,

and make

               U + λV = Ax2 + By 2 + Cz 2 + 2A′ yz + 2B ′ zx + 2C ′ xy.

In my paper in the last number of the Journal 86 , I showed that the case of
intersection of the two conics in two points was distinguishable from all other
cases by the equation □(U + λV ) = 0 having two imaginary roots. When all the
roots are real, the curves either intersect in four points or not at all.
   On the former supposition,

                      −C ′2 + AB,      −A′2 + BC,        −B ′2 + CA,

which are quadratic functions of λ, will be negative for all three values of λ.
On the contrary supposition, one value of λ will make all these three quantities
negative, but the other two values with each make them all three positive.
   Hence we obtain a symmetrical criterion (which I strangely omitted to state
in my former paper) by forming the quantity

                           A2 + B 2 + C 2 − AB − AC − BC.

A cubic equation
                               Ly 3 + M y 2 + N y + P = 0
may be then constructed, of which the three values of the above function
corresponding to three values of λ will be the roots.
   The condition for real intersection is that L, M, N, P should be all of the same
sign. The conics being supposed real, L and P are necessarily in both cases of
the same sign. The condition is therefore satisfied if either L, M , N , or M, N, P p. 163
be of the same sign, and is consequently equivalent to the condition that M    L and
 L shall be both positive, or P and P both positive. It does not appear to be
N                              N       M

possible in the nature of the question to find a criterion for distinguishing between
the two cases, dependent on the sign of one single function of the coefficients.
 86
      P. 119 above.



                                           168
   The case of double contact, abstraction being made of binary intersection, is a
sort of intermediary state between intersection in four points and non-intersection;
and accordingly, as shown in my former paper for this case, the two equal values
of λ will make the three quantities

                    AB − C ′2 ,           BC − A′2 ,       CA − B ′2

all real; so that two of the values of y corresponding to the equal values of λ are
zero, and the criterion becomes nugatory as it ought to do.
   Again, when the two conics do not intersect, I distinguished two cases according
as they lie each without, or one within the other, that is, according as they have
four common tangents or none.
   But, as Mr Cayley has well remarked to me, a similar distinction exists when
the conics intersect in four points; in that case also they may have four common
tangents or not any: when they intersect in two points they have necessarily two
and only two common tangents. There is no difficulty in separating these four
cases.
   Let the conics be written

               (U ) = ξ 2 + η 2 − ζ 2 ,       (V ) = Aξ 2 + Bη 2 − Cζ 2 ,

(U ) and (V ) being what U and V become when the coordinates are changed
from x, y, z to ξ, η, ζ.
   A, B, C are the three values of λ in the equation

                                   □(V − λU ) = 0.

If the curves intersect A − C, B − C must have different signs, that is, C must
be an intermediary quantity between A and B.
   Again, the tangential equations to the conics expressed by the correlative
system of coordinates will be

                                                   ξ12 η12 ζ12
                    ξ12 + η12 − ζ12 = 0,              +   −    = 0;
                                                   A    B   C
and that these may have four real systems of roots,
                                 1  1              1   1
                                   − ,               −
                                 A C               C   B
must have the same sign; and consequently, as A − C and C − B are supposed p. 164
to have the same sign, A and B, and therefore all three A, B, C, have the same
sign. We have therefore the following rule:
   Let the equation in λ, namely, □(U + λV ) = 0, be called θ = 0, and the
equation in y, above given, ω = 0. By an equation being congruent or incongruent,
understand that its roots have all the same sign or not all the same sign.

                                             169
   Then ω congruent, θ congruent, implies that the intersections and common
tangents are both real; ω congruent, θ incongruent, implies that the intersections
are real, but the common tangents imaginary; ω incongruent, θ congruent, implies
that the intersections and common tangents are both imaginary; ω incongruent,
θ incongruent, implies that the intersections are imaginary, but the common
tangents real.
   In like manner, as the cases of contact of lines are limiting cases to those
which relate to the relative configurations of their points of intersection, so the
cases of contact of surfaces are limiting cases in which the characters which
usually separate the different forms of their curve of intersection exist blended
and indistinguishable. The first step therefore to the study of the particular
species of the curve of the fourth degree87 in which two surfaces of the second
degree intersect, is to obtain the analytical and geometrical characters of their
various species of contact. Accordingly I have made an enumeration of these
different species, no less than 12 in number, many of them highly curious and I
believe unsuspected, which the reader may consult in the Philosophical Magazine
for February, 185188 .
   By the aid of these landmarks, I have little doubt, should time and leisure
permit, of mapping out a natural arrangement of the principal distinctions of
form between that class at least of lines in space of the fourth order which admit
of being considered the complete intersection of two surfaces.




  87
      I have found that the 16 points of spherical flexure in this curve are the four sets of four
points in which it meets the four faces of the pyramid whose summits are the vertices of the
four cones of the second degree in which the curve is completely contained, which 16 points
reduce to 4 when the two surfaces have an ordinary contact, and to 1 when they have a cuspidal
contact: of course in the case of contact the pyramid above described in a manner folds up and
vanishes, as there are no longer 4 distinct vertices. I have found also that when the factors of
D(U + λV ), (U and V being the characteristics of the two surfaces) are all unreal, the points
of flexure are all unreal. When two factors are real and two imaginary, two of the faces of the
pyramid (namely, its two real faces) will each contain one (and only one) pair of real points of
flexure, and the other two planes none; and lastly, when the factors of □(U + λV ) are all real,
then either all the points of flexure are imaginary, or else all the eight contained in a certain
two of the pyramidal faces are real: and these two cases admit of being distinguished by a
method analogous in its general features to that whereby I have shown in the text above how
to distinguish between the cases of 4 real and 4 imaginary points of intersection of two conics.
Where the two surfaces have an ordinary contact, the curve of intersection, it is well known,
has a double point; and where the surfaces have a higher contact, the curve has a cusp. Thus
in the fact of the 16 flexures reducing to 4 and to 1 in these respective cases, we see a beautiful
analogy to what takes place with the 9 flexures of a plane curve of the third degree, which
contract to 3 and 1, according as the curve has a double point or a cusp.
   88
      P. 219 below.


                                               170
                                          30.
 On Certain General Properties of Homogeneous Functions
     [Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 1–17]
                                                                                        p. 165
  Let χ denote the operation
                               d        d                d
                         x1       + x2     + · · · + xn     ,
                              da1      da2              dan
and A the operation
                               d        d                d
                         a1       + a2     + · · · + an     ;
                              dx1      dx2              dxn
and now suppose that ω, a homogeneous function of ι dimensions of a1 , a2 . . . an ,
and not of any of the quantities x1 , x2 . . . xn , is subjected to the successive
operations indicated by As χr .
  We have
                               As χr ω = As−1 Aχr ω,
              d         d                 d          d        d                d r
                                                                               
Aχ ω = a1
   r
                  + a2      + · · · + an         x1     + x2     + · · · + xn       ω
             dx1       dx2               dxn        da1      da2              dan
                d         d                 d
                                              
       = r a1      + a2      + · · · + an        χr−1 ω
               da1      da2               dan
       = r(ι − r + 1)χr−1 ω,
for χr−1 ω is of (r − 1) dimensions, lower than ω (which is of ι dimensions) in
a1 , a2 . . . an .
    Hence
As χr ω = r(ι − r + 1)As−1 χr−1 ω
        = &c.
      = {r(r − 1) · · · (r − s + 1)}{(ι − r + 1)(ι − r + 2) · · · (ι − r + s)}χr−s ω.
                                                                                   (1) p. 166
  Now in the expression
                                χr ω(a1 , a2 . . . an ),
suppose that we write
                                  x1 = u1 + a1 ε,
                                  x2 = u2 + a2 ε,
                                     ·········
                                  xn = un + an ε,
we have, by Taylor’s theorem,
                                              ε2                          εr
        χr ω = U r ω + AU r ω ε + A2 U r ω       + · · · + Ar U r ω               ,
                                             1·2                    1 · 2 · 3···r

                                          171
where U r ω denotes what χr ω becomes, on substituting u’s for x’s, and A now
represents
                           d        d                d
                       u1     + u2     + · · · + un     .
                          da1      da2              dan
This expansion stops spontaneously at the (r + 1)th term, because χr ω is only
of r dimensions in x1 , x2 . . . xn .
   Applying now theorem (1), we obtain

 χr ω = U r ω + r(ι − r + 1)U r−1 ωε + 12 r(r − 1){(ι − r + 1)(ι − r + 2)}U r−2 ωε2
          + · · · + {(ι − r + 1)(ι − r + 2) · · · ι}ωεr .       (2)

   In using this theorem in the course of the ensuing pages, it will be found
                                                                            xn
convenient to assign to ε a specific value, and I shall suppose it equal to    ; this
                                                                            an
gives
                     a1                        a2                              an
        u1 = x1 −       xn ,       u2 = x2 −      xn ,   ···   un = xn −          xn = 0.
                     an                        an                              an
And inasmuch as the U symbol now contains a1 , a2 , . . . , an , so that U U r no
longer equals U r+1 , I shall write Ur for U r . Theorem (2) will thus assume the
form
                                                                                                2
                                        xn 1                                        xn
                                                                                            
χr ω = Ur ω + r(ι − r + 1)Ur−1 ω           + 2 r(r − 1)(ι − r + 1)(ι − r + 2)Ur−2 ω
                                        an                                          an
                                           r
                                           xn
         + · · · + {(ι − r + 1) · · · ι}ω         .    (3)
                                           an
                                                                                                   p. 167
   where Ur for all values of r denotes what
                                                                      r
                                 d        d                   d
                       
                           x1       + x2     + · · · + xn−1                ω
                                da1      da2                dan−1
becomes, on substituting u1 , u2 , . . . , un−1 for x1 , x2 , . . . , xn−1 , after the processes
of derivation have been completed: this it is essential to observe, because
u1 , u2 , . . . , un−1 now involve a1 , a2 , . . . , an−1 , an . The term xn dadn is omitted
from the symbol of linear derivation, because in the substitutions xn will be
replaced by zero.
    As an example of this last theorem, take

                                    ω = a3 + b3 + c3 + kabc;

then
                 χω = 3a2 x + 3b2 y + 3c2 z + kbcx + kcay + kabz,
                χ2 ω = 6ax2 + 6by 2 + 6cz 2 + 2kcxy + 2kayz + 2kbzx,
                χ3 ω = 6x3 + 6y 3 + 6z 3 + 6kxyz.

                                               172
                        az                 bz                  az               bz
                                                                            
    U1 ω = 3a2 x −            + 3b2 y −          + kbc x −          + kca y −      ,
                          c                  c                  c                c
                       az 2               bz 2                 az        bz
                                                                     
    U2 ω = 6a x −             + 6b y −           + 2kc x −           y−       ,
                        c                  c                    c         c
                     az 3              bz 3
                                       
    U3 ω = 6 x −             +6 y−             ,
                      c                 c
and it will be found that the equations given by theorem (3) are satisfied, namely
                                 z
                 χω = U ω + 3 ω,
                                 c
                                     z             z2
                χ2 ω = U2 ω + 2 · 2 U ω + 2 · 3 2 ω,
                                     c             c
                                  z                 z2                z3
                χ3 ω = U3 ω + 3 U2 ω + 3 · 1 · 2 2 U ω + 1 · 2 · 3 3 ω.
                                  c                 c                  c
Probably, as this theorem is of rather a novel character, the annexed sketch of
a somewhat different course of demonstration may be not unacceptable to my
readers.
   We have
                                   d           d               d
                                                                 
                     χω = x1          + x2       + · · · + xn       ω;
                                 da1         da2              dan
and by the well-known law for homogeneous functions,
                                  d        d                d
                                                               
                         ιω = a1     + a2     + · · · + an     ω.
                                 da1      da2              dan
                                                                                              p. 168
   Hence
                     xn         d        d                   d
                                                                  
               χ−ι      ω = u1     + u2     + · · · + un−1       ω = U ω.
                     an        da1      da2                dan−1
Hence
                           xn
                                 
                χω = U + ι      ω,
                           an
                                  xn         xn
                                             
               χ2 ω = U + (ι − 1)       U +ι      ω,
                                  an         an
                                  xn                xn        xn
                                                            
               χ3 ω = U + (ι − 2)       U + (ι − 1)      U +ι      ω,
                                  an                an        an
               &c. = &c.
But in performing the process indicated by the several factors it must be carefully
borne in mind that U Ur is not = Ur+1 ; this would be the case were it not for
the terms − aan1 xn , − aan2 xn , &c., which enter into u1 , u2 . . . un−1 . But on account
of these terms, we have
                                                                                                  r
                d        d                   d            d         d                    d
                                                   
U Ur ω =       u1 + u2      + · · · + un−1            u1      + u2      + · · · + un−1                 ω
              da1       da2                dan−1         da1       da2                 dan−1
                                                                 r−1
                     xn      d          d                    d
                        
        = Ur+1 ω − r     u1       + u2     + · · · + un−1              ω,
                     an     da1        da2                dan−1

                                           173
for
                    d        d                 d           xn
                       u1 =     u2 = · · · =       un−1 = − .
                   da1      da2              dan−1         an
Hence
                                                          xn
                              U Ur ω = Ur+1 ω − r            Ur ω.
                                                          an
Let xann be called ε; we find

          χ = U + ιε,
          χ2 = {U + (ι − 1)ε}(U + ιε)
            = U U + (2ι − 1)εU + (ι − 1)ιε2
            = U2 + 2(ι − 1)εU + (ι − 1)ιε2 ;
          χ3 = {U + (ι − 2)ε}χ2
            = U U2 + 2(ι − 1)εU U + (ι − 1)ιε2 U
              + (ι − 2)εU2 + 2(ι − 2)(ι − 1)ε2 U + (ι − 2)(ι − 1)ιε3
            = U3 + 3(ι − 2)εU2 + 3(ι − 2)(ι − 1)ε2 U + (ι − 2)(ι − 1)ιε3 .
                                                                                      p. 169
   The same process being continued will lead to results identical with those
previously obtained and expressed in theorem (3).
   The expansion of χr , treated according to this second method, appears to
require the solution of the partial equation in differences

                           ar+1,s+1 = ar,s+1 + (ι − 2r)ar,s ,

a0,s being given as unity for s = 1 and as zero for all other values of s.
   It is probable however that the solution of this equation might be evaded by
some artifice peculiar to the particular case to be dealt with. I do not propose
to dwell upon this inquiry, which would be foreign to the object of my present
research. It may however not be out of place to make the passing remark, that
the equations expressing χr in terms of powers of U admit easily of being reverted,
as indeed may the more general form
                                                  1
                    χr = ur + εr ur−1 +              εr εr−1 ur−2 + &c.,
                                                 1·2
which becomes the equation of formula (3), on making
                                     xn
               εr = r(ι + 1 − r)        ,         χr = χr ω,           ur = Ur ω;
                                     an
for let
                     ur = ε1 ε2 · · · εr vr ,         χr = ε1 ε2 · · · εr yr ,
then
                                                vr−2    vr−3
                     yr = vr + vr−1 +                +       + &c.;
                                                1·2    1·2·3

                                                174
whence
                           d                        yr−2    yr−3
                vr = e− dr yr = yr − yr−1 +              −       + &c.;
                                                    1·2    1·2·3
and therefore
                        ur = χr − εr χr−1 + 21 εr εr−1 χr−2 + &c.
Thus we obtain, from equation (3),
                                                           xn
                    Ur ω = χr ω − r(ι − r + 1)χr−1 ω          + &c.
                                                           an
   As a first application of theorem (3), I shall proceed to show how Joachim-
sthal’s equation to the surface drawn from a given point (α, β, γ, δ) through the
intersection of two surfaces ϕ(x, y, z, t) = 0, θ(x, y, z, t) = 0, may be expressed
under the explicit form of the equation to a cone.
   The equation in question is obtained by eliminating λ between
                                             1 2 m−2
                   ϕλm + χϕλm−1 +               χ ϕλ + &c. = 0,
                                            1·2
                                  1 2 n−2     1
          θλn + χθλn−1 +             χ θλ +       χ3 θλn−3 + &c. = 0,
                                 1·2        1·2·3
where                                                                                 p. 170

                                                               d     d     d   d
   ϕ = ϕ(α, β, γ, δ),          θ = θ(α, β, γ, δ),      χ=x       +y    +z    +t .
                                                              dα    dβ    dγ   dδ
By theorem (3), these two equations, on writing xann = ε, become

                                                                          λm−2
ϕλm + {U ϕ + mϕε}λm−1 + {U 2 ϕ + 2(m − 1)U ϕε + (m − 1)mϕε2 }                  + &c. = 0,
                                                                           1·2
                                              λn−2
θλn + {U θ + nθε}λn−1 + {U 2 θ + &c.}
                                              1·2
                                                                               λn−3
  + {U 3 θ + 3(n − 2)U 2 θε + 3(n − 2)(n − 1)U θε2 + (n − 2)(n − 1)nε3 }            + &c. = 0.
                                                                              1·2·3
Now on writing λ = µ − ε, these equations take the forms
                                                    µm−2
                        ϕµm + U ϕµm−1 + U 2 ϕ            + &c. = 0,
                                                     1·2
                                              µn−2
                        θµn + U θµn−1 + U 2 θ        + &c. = 0,
                                              1·2
as is easily seen by substituting back λ + ε in place of µ. Consequently ε no
longer appears in the coefficients of the terms of the equations between which
the elimination is to be performed, and the resultant will accordingly come out
as a function only of ϕ, U ϕ, U 2 ϕ, &c., that is, of α, β, γ, δ, and of
                                  α             β             γ
                           x−       t,     y−     t,     z−     t,
                                  δ             δ             δ

                                             175
showing that the equation in x, y, z, t, is of the form of that to a cone, as we
know à priori it ought to be. Precisely a similar method may be applied to the
elucidation of the corresponding theorem for a system of rays drawn from a given
point through the locus of the intersection of two curves.
   Before entering upon some further and more interesting applications of theorem
(3), it will be convenient to explain a nomenclature which has been employed by
me on another occasion, and which is almost indispensable in inquiries of the
nature we are now engaged upon. Homogeneous functions may be characterized
by their degree, by the number of letters which enter into them, and lastly, by
the lowest number of linear functions of the letters which may be introduced
in place of the letters to represent such functions. Any such linear function I
designate as an order, and am now able to discriminate between the number of
letters and the number of orders which enter into a given function. The latter
number, generally speaking, is the same as the former; it can never exceed it,
but may be any number of units less than it.                                            p. 171
   I need scarcely observe that a pair of points becoming coincident, a conic
becoming a pair of lines, a conoid becoming a cone, and so forth, for the higher
realms of space, will be expressed by the homogeneous function of the second
order which characterizes such loci89 , losing one order, that is, having an order
less than the number of letters entering therein. Calling such characteristic
ϕ(x, y, z . . . t), it is well known that the condition of such loss of an order is the
vanishing of the determinant
                                 d2 ϕ     d2 ϕ           d2 ϕ
                                                   ···
                                 dx2     dxdy            dxdt
                                 d2 ϕ     d2 ϕ           d2 ϕ
                                                   ···
                                dydx      dy 2           dydt   .
                                 ···      ···      ···    ···
                                d2 ϕ      d2 ϕ           d2 ϕ
                                                   ···
                                dtdx      dtdy           dt2
   A conoid becoming a pair of planes, a cone becoming a pair of coincident
lines, a pair of points becoming indeterminate, will, in like manner, be denoted
by their characteristic losing two orders, and so forth, for the higher degrees
of degradation. In like manner, in general, a homogeneous function of three
letters of any degree losing an order, typifies that the locus to which it is the
characteristic will break up into a system of right lines.
   Now let ω be a homogeneous function of α, β, γ . . . δ, and suppose that we
have the equations ω = 0, χω = 0, χ2 ω = 0, where χ as above
                                 d     d     d          d
                           =x      +y    +z    + ··· + t .
                                dα    dβ    dγ          dδ
  89
     If U = 0 is the equation to any locus, U may be said to characterize the same, or to be its
characteristic.


                                             176
I say that on eliminating any of the variables x, y, z . . . t between the second and
third of the above equations, the resulting equation will be of one order less than
the number of letters, that is, the expulsion of one letter will be attended by the
expulsion of two orders.
   For we have, by theorem (3),
                                                   xn
                                 χω = U ω + 2         ω = 0,
                                                   an
                                                            2
                                          xn        xn
                                                       
                        χ2 ω = U2 ω + 2      Uω + 2              ω = 0,
                                          an        an
and by hypothesis
                                           ω = 0.
Hence we have also
                                  U ω = 0,         U2 ω = 0;
and since U ω, U2 ω contain one order less than the number of letters in ω, the p. 172
resultant of the elimination between them will contain two orders less than the
number of letters in ω; and consequently, whichever of the letters x, y, z . . . t
we eliminate between χω = 0 and χ2 ω = 0, provided that ω = 0, the resultant
equation will contain one order less than the number of letters remaining.
   Thus we see how it is that the tangent line to a conic meets it in two coincident
points, the tangent plane to a conoid in two intersecting lines, and so forth,
for the higher regions of space90 . For instance, if we take ω(x, y, z, t) = 0, the
equation to a conoid, and α, β, γ, δ, the coordinates to any point therein, we
shall have ω(α, β, γ, δ) = 0,

                         d     d     d    d
                                                  
                      x    +y    +z    +t    ω,            that is, χω = 0,
                        dα    dβ    dγ    dδ

and ω(x, y, z, t), that is, χ2 ω = 0, x, y, z, t representing the coordinates of any
point in the intersection of the conoid by the tangent plane.
   Consequently, by what has been shown above, on eliminating any one of the
four letters x, y, z, t, the resultant function of three letters will contain only two
orders, and will thus represent a pair of lines, real or imaginary, intersecting one
another at α, β, γ, δ.
   The fact which has just been demonstrated (that the resultant of χω = 0,
χ2 ω = 0, loses an order if ω = 0), indicates that on expressing one of the
quantities x, y, z . . . t in terms of the others, by means of the first equation, and
then substituting this value in the second, the determinant of the equation so
obtained must be zero.
  90
     Thus a tangential section of a hyperlocus of the second degree at any point cuts it in two
cones.



                                             177
   Now by virtue of a theorem which was given by me in a note91 to my paper
in the preceding number of this Journal, this determinant will be equal to
the squared reciprocal of the coefficient in the equation χω = 0 of the letter
eliminated multiplied by the determinant in respect to x, y, z . . . t, λ of

                                                    χ2 ω + λχω.

This latter determinant is therefore zero; but this determinant is the resultant of
the equations
                             2
               d     d  d           d     d  d
                                                   
                   x + y + &c. ω +      x + y + · · · ω = 0,
            
            
            
            
            
            
             dx    da  db         dx    da  db
                             2
               d     d  d           d    d   d
                                                  
                                +y        + &c.      ω+           x + y + ···                  ω = 0,
            
                       x
            
              dy           da        db                       dy    da   db
                                          &c.       &c.         &c.    &c.,
            
            
            
            
            
            
                                                               d  d
            
                                                                                    
                       χω = 0,            that is,           x + y + ···                  ω = 0.
            
            
            
                                                              da  db
                                                                                                        p. 173
   Thus we obtain the singular law, that the symmetrical determinant
                                   d d           d d                d d           d
                                  da da ω       da db ω      ···   da dl ω       da ω
                                  d d           d d                d d           d
                                  db da ω       db db ω      ···   db dl ω       db ω
                                  d d           d d                d d           d
                                  dc da ω       dc db ω      ···   dc dl ω       dc ω

                                  ................................
                                  d d     d d           d d     d
                                  dl da ω dl db ω · · · dl dl ω dl ω
                                      d
                                     da ω
                                                  d
                                                  db ω       ···    d
                                                                    dl ω          0

is zero when ω is zero.
   This is easily shown independently by means of a remarkable and I believe
novel theorem, relative to homogeneous functions.
   If ω be any homogeneous function of ι dimensions of a, b, c . . . l, we have (by
Euler’s theorem already repeatedly applied), remembering that dω  da , db . . . dl are
                                                                        dω      dω

all homogeneous,
                                        d  d            d
                                                                            
                                −ιω + a + b + · · · + l    ω = 0,
                                       da  db           dl
                          dω      d d      d d             d d
                                                                                          
                 −(ι − 1)    + a       +b       + ··· + l       ω = 0,
                          da     da da    da db           da dl
                                    dω     d d             d d
                                                                                 
                           −(ι − 1)    + a       + ··· + l       ω = 0,
                                    db     db da           db dl
  91
       P. 135 above.


                                                          178
                                         &c.&c.    &c.
                          dω         d d               d d
                                                            
                −(ι − 1)       + a          + ··· + l          ω = 0.
                           dl        dl da            dl dl
Between these equations we may eliminate all the letters, a, b, c . . . l, and we
obtain
                   d d        d d            d d         d
                  da da ω da db ω · · · da dl ω         da ω
                   d d        d d            d d         d
                   db da ω    db db ω · · · db dl ω     db ω
                   d d        d d            d d         d
                   dc da ω    dc db ω · · · dc dl ω     dc ω    = 0.
                   .................................
                   d d        d d            d d         d
                   dl da ω    dl db ω · · ·  dl dl ω    dl ω
                     d          d              d         ι
                    da ω       db ω    ···    dl ω     ι−1 ω
                                                                                                 p. 174
   As a corollary to this theorem, we see that if ω = 0 the determinant obtained
in the previous investigation becomes zero, agreeing with what has been already
shown; in fact the last-named determinant is always equal to
                                                      d d                    d d
                                                     da da ω          ···   da dl ω
                                ι−1
                                    ω×                ···             ···        ···         .
                                 ι                   d d                        d d
                                                     dl da ω          ···       dl dl ω

This remarkable theorem, which I have communicated to friends nearly a twelve-
month back, is here, I believe, published for the first time92
   Suppose next that ω(x, y, z) is the characteristic of a line of any degree, to
which a tangent is drawn at the point α, β, γ, using U in a manner correspondent
to its previous signification to denote
                                            αz        d      βz                        d
                                                                               
                                     x−                 + y−                             ,
                                            γ        dα      γ                        dβ
and understanding ω(α, β, γ) by ω, we have for determining the point of inter-
section, ω = 0, χω = 0, χ2 ω = 0; and consequently, by aid of our theorem (3),
we shall obtain
                                              z
              ω = 0,     U ω = 0,     Un ω + n Un−1 ω + · · · = 0.
                                              γ
  92
       Thus let z be a homogeneous function in x and y of ι dimensions, and let
                                     dz       dz         d2 z          d2 z        d2 z
                                        ,        ,            ,             ,
                                     dx       dy         dx2          dxdy         dy 2
be called p, q, r, s, t; we shall find

                                               r     s        p
                                               s     t        q          = 0,
                                                             ι
                                               p     q      ι−1
                                                                ω

that is,
                                               ι − 1 rq 2 − 2pqs + tp2
                                         ω=                            .
                                                 ι         rt − s2



                                                          179
By means of the two latter equations, we obtain
                                       2
                                  αz             αz
                                                      
                               x−           F x−            = 0,
                                  γ              γ
                                       2
                                  βz             βz
                                                      
                               y−           G y−            = 0,
                                  γ              γ
 where F and G are each of only (n − 2) dimensions, and serve to determine p. 175
the intersections of the tangent with the curve, extraneous to the two coincident
ones at the point of contact.
   Again, suppose that ω is a function of any degree of any number of letters
α, β, γ, &c., and that we have given ω = 0, χω = 0, χ2 ω = 0, . . . χm ω = 0; it is
evident from our fundamental theorem that these equations may be replaced by

                 ω = 0,    U1 ω = 0,        U2 ω = 0,   ...    Um ω = 0;

and consequently that the expulsion of (m − 1) letters, by aid of the last m of
the given equations, will be attended by the disappearance of m orders, or, in
other words, the resultant will be minus an order, that is, will have one order
less than the number of letters remaining in it.
   In applying to space conceptions the preceding theorem, it will be convenient
to use a general nomenclature for geometrical species of various dimensions.
   Thus we may call a line a monotheme, a surface a ditheme, the species beyond
a tritheme, and so on, ad infinitum.
   A system of points according to the same system of nomenclature would be
called a kenotheme.
   An n-theme has for its characteristic a homogeneous function of (n + 2) letters.
   Again, it will be convenient to give a general name to all themes expressed
by equations of the first degree. Right lines and planes agree in conveying an
idea of levelness and uniformity; they may both be said to be homalous. I shall
therefore employ the word homaloid to signify in general any theme of the first
degree.
   Now let ω(x, y, z . . . t) be the characteristic to an n-theme of the nth degree.
   The number of letters x, y, z . . . t is (n + 2).
   As usual, let ω represent ω(α, β, γ . . . δ), and suppose

                    ω = 0,      χω = 0,       χ2 ω = 0 · · · χn ω = 0,

and consequently
                          U1 ω = 0,    U2 ω = 0 · · · Un ω = 0.
On eliminating (n − 1) letters between the n last equations, the resulting function
will be of three letters but of only two orders, and of the 1 · 2 · 3 · · · n degree.


                                             180
Hence we see that at every point of an n-theme of the nth degree, and lying p. 176
in the tangent homaloid thereto, 1 · 2 · · · n right lines may be drawn coinciding
throughout with the n-theme.
   Thus one right line can be drawn at each point of a line of the first order lying
on the line; two right lines at each point of a surface of the second order lying
on the surface; six right lines at each point of a hyperlocus of the third degree,
and so forth.
   It is obvious that a surface may be treated as the homaloidal section of a
tritheme, just as a plane curve may be regarded as a section of a surface. I shall
proceed to show upon this view, how we may obtain a theorem given by Mr
Salmon for surfaces of the third degree of a particular character from the law
just laid down, according to which a tritheme of the third degree admits of six
right lines being drawn upon it at every point93 .
   Let ω(x, y, z, t, u) be the characteristic of any tritheme of the third degree;
α, β, γ, δ, ϵ, coordinates to any point in the same. Then ω(α, β, γ, δ, ϵ) = 0,
and the equation to the tangent homaloid will be χω(α, β, γ, δ, ϵ) = 0, and the
equation to the polar of the second degree to the given tritheme in relation to
the assumed point as origin, (that is, the infinite system of homaloids that may
be drawn from the point to touch the tritheme), will be χ2 ω(α, β, γ, δ, ϵ) = 0.
   But the section of any polar through its origin is the polar of the section to
the same origin; hence the polar to the intersection of ω(x, y, z, t, u) = 0, with
χω(α, β, γ, δ, ϵ) = 0, is the intersection of χω = 0 with χ2 ω = 0.
   The projections of these intersections upon the space x, y, z, t will be found
by eliminating u, and getting the correspondent two equations between x, y, z, t.
Hence we see that the projection of the latter intersection upon any space x, y, z, t
is a cone; or, in other words, this intersection itself, that is, the polar to the
intersection of the tritheme with its tangent homaloid, is a cone; that is to say,
the surface of the third degree formed by cutting a tritheme of the third degree
by any tangent homaloid has a conical point at the point of contact; so that
every surface of the third degree with a conical point may be considered as
the intersection of a tritheme of the third degree with any tangent homaloid
thereto94 .                                                                           p. 177
  93
     The reduction of any equation of the sixth degree to depend upon one of the fifth may be
shown by Mr Jerrard’s method to be equivalent to drawing a straight line upon a tritheme of
the third degree, just as the reduction of the equation of the fifth degree to a trinomial form
may be shown to be dependent upon our being able to draw a straight line upon a ditheme
of the second degree. Now at every point of a tritheme straight lines may be drawn, but as
they keep together in groups of sixes they cannot be found in general at a given point without
solving an equation of the sixth degree.
  94
     So in like manner a surface of the third degree with more than one conical point may be
generated by the intersection of the tritheme with a pluri-tangent plane; and so too we may get
other varieties by taking homaloidal sections of trithemes whose characteristics are minus one
or more orders.




                                             181
   Hence then we see, as an instantaneous deduction from our general theorem,
that at any conical point (when one exists) of a surface of the third degree six
right lines may be drawn lying completely upon it. This theorem is thus brought
into an immediate and natural connexion with the well-known one, that at every
point in a surface of the second degree, two right lines can be drawn lying wholly
upon the surface95 .
   The last geometrical application of the theorem (3) which I shall make, refers
to the equations employed by Mr Salmon in No. xxi. (New Series) of this Journal,
to obtain the locus of the points on any surface at which four consecutive tangent
lines can be drawn passing through points. I remark in passing that these
equations may be obtained by rather simpler considerations than Mr Salmon has
employed so to do, and without any reference to Joachimsthal’s theorem; for if
we take ξ, η, ζ, θ, as the coordinates of any point in one of the tangent lines above
described, and if we take the first polar to the surface with this point as origin,
three out of the four original points will be found in such polar consecutive but
distinct; and consequently in the second polar, referred to the same origin, two
will continue consecutive but distinct, and consequently one will remain over in
the third polar.
   Hence writing the equation to the surface ω(x, y, z, t) = 0, and using D to
denote ξ dx
          d
            + η dyd
                     + ζ dz
                         d
                            + θ dt
                                d
                                   , we shall evidently have

                  ω = 0,        Dω = 0,         D2 ω = 0,         D3 ω = 0,
as obtained by Mr Salmon. And the same kind of reasoning precisely applies
to the theory of points of inflexion in curves; three consecutive points in a right
line in this case corresponding to four such in the case above considered.
   If now we make
                       x                y                z
                  ξ − θ = u,       η − θ = v,        ζ − θ = w,
                       t                t                t
 the equations (2), (3), (4), by our theorem, may be expressed in terms of p. 178
u, v, w, which being eliminated we obtain an equation between x, y, z, t, which
will express the surface whose intersection with the given surface ω = 0 serves to
determine the locus of the points in question.
   Hence if we proceed in the ordinary manner to eliminate two of the four
letters, as ξ and η, between the equations (2), (3), (4), the resultant will be
  95
     If we have an indeterminate system of algebraical equations consisting of one quadratic
and another nc function of three variables, this may be completely resolved by considering
the first as an equation to a surface of the second degree, finding at any point thereof the two
lines which lie upon the surface, and determining their respective intersections with the surface
represented by the second equation. This will require therefore the solution only of a quadratic
and an nc equation. In like manner an indeterminate system of two equations of four variables,
one of the third and the other of the nth degree, may be completely resolved (with the aid of
the theorem in the text) by means of two equations, one of the sixth and the other of the nth
degree.


                                              182
of the form M × ϕ(ζ, θ), where M does not contain ξ, η, ζ or θ, and where by
the general laws of elimination ϕ(ζ, θ) will be an integral function of the sixth
degree in respect to ζ, θ: and it is manifest that M × ϕ(ζ, θ) will be identical
with the resultant of (2), (3), (4) expressed in terms of u, v, w, when u and
v are eliminated cy-près of an integralizing factor, showing that ϕ(ζ, θ) is w6
integralized, that is, is equal to (tζ − zθ)6 . Consequently as M ϕ is of the order
(n − 1)2 · 3 + (n − 2)1 · 3 + (n − 3)1 · 2, that is, 11n − 18 in respect to x, y, z, t, it
follows that M = 0, the equation to the second surface spoken of above, will be
of the order 11n − 24, agreeable to Mr Salmon’s showing.
    I shall conclude this paper by showing the application of our theorem to
the subject propounded by Mr Jerrard and Sir William Hamilton, of systems
of equations containing a sufficient number of variable letters for effecting the
solution without elevation of degree.
    If we have n homogeneous equations containing a sufficient number of letters
a1 , a2 . . . am to enable us to express the solution of (n − 1) of the equations under
the form

           a1 = α1 + λβ1 ,        a2 = α2 + λβ2 ,      ···    am = αm + λβm ,

where α1 , α2 . . . αm , β1 , β2 . . . βm are supposed known, and λ is indeterminate, it
is evident that by substituting these values in the nth equation, λ may be found
by solving an equation of the same degree as that equation contains dimensions
of a1 , a2 . . . am .
   Let us then propose this question: how many letters a1 , a2 . . . ar are needed
to obtain a linear solution of a system of n equations

                          ϕ1 = 0,     ϕ2 = 0,    ...    ϕn = 0,

of the several degrees ι1 , ι2 . . . ιn , without elevation of degree; by a linear solution
being understood a solution under the form

            a1 = α1 + λβ1 ,         a2 = α2 + λβ2 ,     ···   ar = αr + λβr ,

where λ is left indeterminate.                                                            p. 179
   Let us suppose that α1 , α2 . . . αr , substituted respectively for a1 , a2 . . . ar ,
satisfy the given system of equations. The determination of these values without
elevation of degree will, from what has been said before, depend upon the linear
solution of a system of equations differing from the given system by the omission
of any one of them at pleasure.
   Now make
                                d          d                d
                       D = a1       + a2      + · · · + ar     ,
                               da1        da2              dar




                                           183
and then write
                   Dϕ1 = 0,
                                                                  
                                                     
                                                     
                   D ϕ1 = 0 . . . D ϕ1 = 0,
                     2               ι1
                                                     
                                                     
                                                     
                                                     
                   Dϕ2 = 0, D ϕ2 = 0 . . . D ϕ2 = 0,
                                  2         ι2                              (θ)
                   ············
                                                     
                                                     
                                                     
                                                     
                   Dϕn = 0, D ϕn = 0 . . . D ϕn = 0
                                   2         ι
                                                     
                                              n      

The values of a1 , a2 . . . ar derived from this system, say (a)1 , (a)2 . . . (a)r , give

             a1 = α1 + λ(a)1 ,      a2 = α2 + λ(a)2 , . . . ar = αr + λ(a)r ,

a solution under the required form, where λ is left indeterminate.
   The solution of this new system without elevation of degree depends on the
linear solution of all but one of them; this excepted one may be taken the one
whose dimensions ιr are the highest or as high as any of the quantities ι1 , ι2 . . . ιn .
   Consequently, if we use the symbol (k1 , k2 . . . kr ) to denote the number of
letters required for the linear solution (without elevation of degree) of k1 equations
of the first degree, k2 of the second, k3 of the third, . . ., kr of the rth, it would
at first sight appear from the preceding reduction that we must have

                       (k1 , k2 . . . kr ) = {K1 , K2 . . . Kr−1 , Kr′ },

where
                             K1 = k1 + k2 + · · · + kr−1 + kr ,
                             K2 = k2 + · · · + kr−1 + kr ,
                                 ············ ,
                          Kr−1 = kr−1 + kr ,
                             Kr′ = kr − 1.
But now steps in our theorem (3), and shows that the system (θ) may be
superseded by another, in which the variables, instead of being a1 , a2 . . . an , will
be
                      α1            α2                   αn−1
                a1 −     an , a2 −     an , . . . an−1 −      an ;
                      an            an                    an
consequently the number of really independent variables is only (n − 1); we must
therefore have
                    (k1 , k2 . . . kr ) = 1 + {K1 , K2 . . . Kr′ }.
                                                                                             p. 180
   Since the introduction of a new simple equation is equivalent to the requirement
of one more disposable letter, we may write the above more symmetrically under
the form
                      (k1 , k2 . . . kr ) = (K1 , K2 . . . Kr−1 , Kr′ ),
where
                   K1 = 1 + k1 + k2 + · · · + kr ,           Kr′ = kr − 1.

                                             184
By means of this formula of reduction (k1 , k2 . . . kr ) may be finally brought down
to the form (L), and the value of (L) being the number of letters required for
the linear solution of a system of L linear equations is evidently L + 2.
   Thus, to determine the number of letters required for the linear solution of a
single quadratic, we write
                                (0, 1) = (2) = 4.
For two quadratics, we write

                             (0, 2) = (3, 1) = (5) = 7;

for a quadratic and a cubic,

                       (0, 1, 1) = (3, 2) = (6, 1) = (8) = 10;

for two cubics,

           (0, 0, 2) = (3, 2, 1) = (7, 3) = (11, 2) = (14, 1) = (16) = 18.

These results coincide with those obtained by Sir William Hamilton in his
Report on Mr Jerrard’s Transformation of the Equation of the Fifth Degree in
the Transactions of the British Association. I have much more to say on the
subject of the linear solution of a system of indeterminate equations, and am, I
believe, able to present the subject in a more general light than has hitherto been
done; but my observations on this matter must be deferred until a subsequent
communication.




                                        185
                                         31.
   Reply to Professor Boole’s Observations on a Theorem
   contained in the last November Number of the Journal
       [Cambridge and Dublin Mathematical Journal, VI. (1851), pp. 171–174]
                                                                                     p. 181
   The restricted space that can be spared for discussion in these pages, necessi-
tates me to compress within the narrowest limit the remarks which I feel bound
to make on Mr Boole’s extraordinary observations96 in the February number
of this Journal, on my theorem contained in the antecedent number thereof97 ,
which statements I cannot, in the interests of truth and honesty, suffer to pass
unchallenged. The object of that theorem was to show how the determinant of
the quadratic function resulting from the elimination of any set of the variables
between a given quadratic function and a number of linear functions of the
same variables, could be represented without performing the actual elimination
by a fraction, of which the numerator would be constant whichever set of the
variables might be selected for elimination, and the denominator the square of the
determinant corresponding to the coefficients of the variables so eliminated. The
numerator itself is a determinant, obtained by forming the square corresponding
to the determinant of the given quadratic function, and bordering it horizontally
and vertically with the lines and columns corresponding to the coefficients of all
the variables in the given linear equations. An immediate corollary from this
theorem leads to Mr Boole’s. Conversely upon the principle that “tout est dans
tout” Mr Boole devotes a page and a half of close print merely to indicate the
steps of a method by which from his theorem mine is capable of being deduced,
ending with the announcement, that the numerator in question is equal to the
quantity
                                  ϕ1 ϕ2 · · · ϕr θ(Q),
(the symbols above employed being Mr Boole’s own), and concludes with assuring
his readers that “he has ascertained that Mr Sylvester’s result is reducible to the
above form.” Mr Sylvester would be very sorry to put his                            p. 182
   result under any such form. Mr Boole could scarcely have reflected upon the
effect of his words when he indulged in the remark which follows—“there cannot
be a doubt that for the discovery of the actual relation in question, the above
theorem is far more convenient than Mr Sylvester’s.” Of the value to be attached
to this assertion the annexed comparison of results is submitted as a specimen.
   Let the quadratic function be

         ax2 + by 2 + cz 2 + dt2 + 2exy + 2εzt + 2gxz + 2γyt + 2hyz + 2ηxt,
   Cambr. and Dublin Math. Jour. VI. (1851), pp. 90, 284.
  96

   p. 135 above.
  97




                                         186
and the linear functions (taken two in number)

                      lx + my + nz + pt,       l′ x + m′ y + n′ z + p′ t.

My numerator will be the determinant (hereinafter cited as the extended deter-
minant),
                          a e g η l l′
                          e b h γ m m′
                          g h c ε n n′
                                                   .
                          η γ ε d p p′
                          l m n p 0 0
                          l ′ m ′ n′ p ′ 0 0
To find the numerator of Mr Boole’s fraction, we must form the symbolical
operator
               d        d         d        d         d         d      d
        
            l2   + m2 + n2 + p2              + 2lm + 2np + 2ln
              da        db       dc       dd        de        dε      dg
                       d          d           d
                                                
                +2mp       + 2lp      + 2mn
                      dγ         dh          dη
                    d          d         d        d          d         d        d
               
             × l′2     + m′2 + n′2 + p′2             + 2l′ m′ + 2n′ p′ + 2l′ n′
                   da         db         dc      dd          de       dε        dg
                         d            d           d
                                                    
                +2m′ p′     + 2l′ p′    + 2m′ n′      ,
                        dγ           dh          dη
and after expanding the determinant hereunder written

                                       a   e   g   η
                                       e   b   h   γ
                                                     ,
                                       g   h   c   ε
                                       η   γ   ε   d

perform the operations above indicated upon the result so obtained.
   These are the operations and processes which, on Professor Boole’s authority,
we are to accept “as without doubt far more convenient” than the one simple
process of forming, and when necessary, calculating the                           p. 183
   extended determinant above given. Here for the present I leave the case
between Mr Boole and myself to the judgment of the readers of this Journal.
   In the April Number of the Philosophical Magazine 98 , I have shown that the
extended determinant serves, not only to represent the full and complete determi-
nant of the reduced quadratic function, but likewise all the minor determinants
thereof; the last set of which will be evidently no other than the coefficients
themselves. For instance, in the example above given, if we wish to find the
 98
      p. 241 below.


                                            187
coefficient of x2 after z and t have been eliminated, we have only to strike out
the line and column e b h γ m m′ from the extended determinant; if we wish to
find the coefficient of y 2 , we must strike out the line and column a e g η l l′ ; to
find the coefficient of xy, we must strike out the line a e g η l l′ and the column
e b h γ m m′ , or vice versâ.
   In each of these cases the determinant so obtained is the numerator of the
equivalent fraction; the denominator remaining always the same function of the
coefficients of transformation as in the original theorem.
   Again, if there be taken only one linear equation, and by aid of it x is supposed
to be eliminated; and if the reduced quadratic function be called

                       Ly 2 + M z 2 + N t2 + 2P zt + 2Qyt + 2Rzy,

the same extended determinant as before given will serve, when stripped of its
outer border, consisting of the line and column l′ m′ n′ p′ , to produce the various
equivalent fractions: thus form the square

                                         L R     Q
                                         R M     P .
                                         Q P     N

The numerator of the fraction equivalent to

                            L R
                                     ,   that is, to LM − R2 ,
                            R M

may be found by striking out from the form of the extended determinant the
line and column η γ ε d p; that corresponding to

                              L Q
                                  ,       that is, LP − RQ,
                              R P

will be found by striking out the line g h c ε n and the column η γ ε d p, or vice
versâ; and so forth for all the first minor determinants; and similarly the second
minors, that is, L, M, N, P, Q, R, may be obtained by striking out in each case a
correspondent pair of lines and pair of columns. Thus, to find the numerator
of L the same pair of lines and columns, namely, (g h c ε n), (η γ ε d p), must
be elided. To find the numerator of R, the pair of lines (g h c ε n), (η γ ε d p),
and the pair of columns (e b h γ m), (η γ ε d p), or vice versâ, will have to
be elided; and so forth for the remaining second minors. I may conclude with
observing, that the theorem contested by Mr Boole is an immediate corollary
from the general Theory of Relative Determinants alluded99 to in the “Sketch”
inserted in the present number of the Journal.
  99
       p. 188 below.


                                           188
                                                 32.
  Sketch of a Memoir on Elimination, Transformation, and
                    Canonical Forms
   [Cambridge and Dublin Mathematical Journal, VI. (1851), pp. 186–200]
                                                                                           p. 184
   There exists a peculiar system of analytical logic, founded upon the properties
of zero, whereby, from dependencies of equations, transition may be made to the
relations of functional forms, and vice versâ: this I call the logic of characteristics.
   The resultant of a given system of homogeneous equations of as many variables,
is the function whose nullity implies and is implied by the possibility of their
coexistence, that is, is the characteristic of such possibility; but inasmuch as
any numerical product of any power of a characteristic is itself an equivalent
characteristic, in order to give definiteness to the notion of a resultant, it must
further be restricted to signify the characteristic taken in the lowest form of
which it in general admits.
   The following very general and important proposition for the change of the
independent variables in the process of elimination, is an immediate consequence
of the doctrine of characteristics.
   Let there be two sets of homogeneous forms of function;
   the 1st,
                                  ϕ1 , ϕ2 . . . ϕn ,
the 2nd,
                                      ψ1 ,       ψ2 . . . ψn .
Let the results of applying these forms to any sets of n variables be called

                                   (ϕ1 ),       (ϕ2 ) . . . (ϕn ),

                                  (ψ1 ),        (ψ2 ) . . . (ψn );
then will the resultant (in respect to those variables) of

                                 ϕ1 {(ψ1 ), (ψ2 ) . . . (ψn )},
                                 ϕ2 {(ψ1 ), (ψ2 ) . . . (ψn )},
                                      ···············
                                 ϕn {(ψ1 ), (ψ2 ) . . . (ψn )},
                                                                                           p. 185
   be the product of powers (assignable by the law of homogeneity) of the separate
resultants of the two systems,

                  {(ϕ1 ), (ϕ2 ) . . . (ϕn )},          {(ψ1 ), (ψ2 ) . . . (ψn )}.


                                                 189
   By means of the doctrine of characteristics the following general problem may
be resolved.
   Given any number of functions of as many letters, and an inferior number of
functions of the same inferior number of letters, obtained by combining, inter
se, in a known manner, the given functions, to determine the factor by which,
the resultant of the reduced system being divided, the resultant of the original
system may be obtained.
   If in the theorem for the change of the independent variables both sets of forms
of functions be taken linear, we obtain the common rule for the multiplication of
determinants: if we take one set linear and the other not, we deduce two rules,
namely, That the resultant of a given set of functional forms of a given set of
variables, enters as a factor into the resultant,
   1st, of linear functions of the given functions of the given variables;
   2nd, of the given functions of linear functions of the given variables:
   the extraneous factor in each case being a power of what may be conveniently
termed the modulus of transformation, that is, the resultant of the imported
linear forms of functions.
   From the second of these rules we obtain the law first stated I believe for
functions beyond the second degree by Mr Boole, to wit, that the determinant of
any homogeneous algebraical function (meaning thereby the resultant of its first
partial differential coefficients) is unaltered by any linear transformations of the
variables, except so far as regards the introduction of a power of the modulus of
transformation. This is also abundantly apparent from the fact, that the nullity
of such determinant implies an immutable, that is, a fixed and inherent, property
of a certain corresponding geometrical locus.
   There exist (as is now well known) other functions besides the determinant,
called by their discoverer (Mr Cayley) hyperdeterminants, gifted with a similar
property of immutability. I have discovered a process for finding hyperdetermi-
nants of functions of any degree of any number of letters, by means of a process
of Compound Permutation. All Mr Cayley’s forms for functions of two letters
may be obtained in this manner by the aid of one of the two processes (to wit,
that one which will hereafter be called the derivational process), for passing from
immutable constants to immutable forms. Such constants and forms, derived
from given forms, may be best                                                                 p. 186
   termed adjunctive; a term slightly varied from that employed by M. Hermite
in a more restricted sense.
   The two processes alluded to may be termed respectively appositional and
derivational. The appositional is founded upon the properties of the binary
function xξ + yη + zζ + · · · ; in which, whether we substitute linear functions of
x, y, z, &c., or linear functions of ξ, η, ζ, &c., in place of x, y, z, &c., or ξ, η, ζ, &c.,
the result is the same.


                                            190
   Consequently, if we apply the form ϕ to ξ, η . . . ζ, and take any constant (in
respect to ξ, η . . . ζ) adjunctive to

                      ϕ(ξ, η . . . ζ) + (xξ + yη + · · · + zζ + kt)n−1 ,

calling this quantity ψ(x, y . . . z, t), the form ψ is evidently adjunctive to the
form ϕ: and if we expand so as to obtain

               ψ(x, y . . . z, t) = ψ1 (x, y . . . z)ta + ψ2 (x, y . . . z)tβ + &c.,

it is evident ψ1 , ψ2 , &c. will be each separately adjunctive to ϕ. These forms,
when ψ is obtained by finding the determinant in respect to ξ, η . . . ζ of S, are,
in fact, identical with Hermite’s “formes adjointes.”
   The derivational mode of generating forms from constants depends upon the
property of the operative symbol
                                             d     d          d
                                    χ=ξ        +η    + ··· + ζ ,
                                            dx    dy          dz
applied to ϕ a function of x, y . . . z; namely, that if in ϕ, in place of these letters,
we write linear functions thereof, to wit x′ , y ′ . . . z ′ , we may write
                                            d        d               d
                                  χ = ξ′      ′
                                                + η′ ′ + · · · + ζ ′ ′ ,
                                           dx       dy              dz

where ξ ′ , η ′ . . . ζ ′ will be the same functions of ξ, η . . . ζ that x′ , y ′ . . . z ′ are of
x, y . . . z.
   Suppose now, in the first place, that in regard to ξ, η . . . ζ, ψ(x, y . . . z) is
adjunctive to χr ϕ(x, y . . . z); then is the form ψ adjunctive to the form ϕ, for on
changing x, y . . . z to x′ , y ′ . . . z ′ ,
                                                          r
                                  d     d           d
                          
                             ξ      +η    + ··· + ζ            ϕ(x′ , y ′ . . . z ′ )
                                 dx    dy           dz
becomes
                               d        d               d
                                                            
                        ξ′        + η ′ ′ + · · · + ζ ′ ′ ϕ(x′ , y ′ . . . z ′ );
                              dx′      dy              dz
and consequently ψ(x, y . . . z) becomes ψ(x′ , y ′ . . . z ′ ), multiplied by a power of
the modulus of transformation, the modulus of that transformation, be it well
observed, whereby x′ , y ′ . . . z ′ would be replaced by x, y . . . z, and not as in the
appositional mode of that converse transformation according to which                       p. 187
   x, y . . . z would be replaced by x , y . . . z . It is on account of this converseness
                                        ′  ′      ′

of the modes of transformation that the appositional and derivational modes of
generating forms cannot except for a certain class of restricted linear transfor-
mations be combined in a single process. More generally, if instead of a single
function χr ϕ(x, y . . . z), we take as many such with different indices to χ as there

                                                   191
are variables, and form either the resultant in respect to ξ, η . . . ζ, or any other
immutable constant in regard to those variables, (presuming in extension of the
hyperdeterminant theory and as no doubt is the case, that such exist), every such
resultant or other constant will give a form of function of x, y . . . z adjunctive to
the given form ϕ.
    It may be shown that every such resultant so formed will contain ϕ as a factor.
    Again, in the former more available determinant mode of generation, if we take
the determinant in respect to ξ, η . . . ζ, it may be shown that all the adjunctive
functions so obtained will be algebraical derivees of the partial differential
coefficients of ϕ in respect to x, y . . . z; that is to say, if these be respectively zero,
all such adjunctive functions so derived, as last aforesaid, will be zero, or in other
words, each such adjunctive is a syzygetic function of the partial differential
coefficients of the primitive function.
    To Mr Boole is due the high praise of discovering and announcing, under a
somewhat different and more qualified form and mode of statement, this marvel-
working process of derivational generation of adjunctive forms. I was led back to
it, in ignorance of what Mr Boole had done, by the necessity which I felt to exist
of combining Hesse’s so-called functional determinant, under a common point of
view with the common constant determinant of a function; under pressure of
which sense of necessity, it was not long before I perceived that they formed the
two ends of a chain of which Hesse’s end exists for all homogeneous functions,
but the other only when such functions are algebraical.
    In fact, if we give to r every value from 2 upwards, the successive determinants
in respect to ξ, η . . . ζ of
                                                      r
                               d     d           d
                        
                          ξ      +η    + ··· + ζ           ϕ(x, y, z),
                              dx    dy           dz
will produce the chain in question, which, when ϕ is algebraical and of n
dimensions, comes to a natural termination when r = n − 1. The last member
of and the number of terms in this chain are identical with the last member of
and the number of terms in Sturm’s auxiliary functions, when the variables are
reduced to two. There is some reason to anticipate that this chain of functions
may be made available in superseding Sturm’s chain of auxiliaries; and if so,
then the fatal hindrance to progress, arising from the unsymmetrical nature of
the latter, is overcome, and we shall be                                             p. 188
   able to pass from Sturm’s theorem, which relates to the theory of Kenothemes,
or Point-systems, to certain corresponding but much higher theories for lines,
surfaces, and n-themes generally.
   The restriction of space allowed to me in the present number of the Journal
will permit me only to allude in the briefest terms to the theory of Relative
Determinants, which, as it will be seen, plays an important part in the effectuation
of the reductions of the higher algebraical functions to their simplest forms. Nor

                                           192
can the effect of the processes to be indicated be correctly appreciated without a
knowledge of the circumstances under which the resultant of a given system of
equations can sink in degree below the resultant of the general type of such system.
Abstracting from the case when the equations separately, or in combination,
subdivide into factors, this lowering of degree, as may be shown by the doctrine
of characteristics, can only happen in one of two ways. Either the particular
resultant obtained is a rational root of the general resultant, or the general
resultant becomes zero for the case supposed, and the particular resultant is of a
distinct character from the general resultant, being in fact the characteristic of
the possibility not of the given system of equations being merely able to coexist
(for that is already supposed), but of their being able to coexist for a certain
system of values other than a given system or given systems. Such a resultant
may be termed a Sub-resultant; the lowest resultant in the former case may
be termed a Reduced-resultant. The theory of Sub-resultants is one altogether
remaining to be constructed, and is well worthy equally of the attention of
geometers and of analysts.
   As to the theory of Relative Determinants, the object of this theory is to
obtain the determinant resulting from eliminating as many variables as can be
eliminated, chosen at pleasure from a set of variables greater in number than the
equations containing them; and the mode of effecting this object is through the
method of the indeterminate multiplier. To avoid the discussion of the theory of
sub-resultants and other particularities, I shall content myself with giving the
rule applicable to the case (the only one of which as yet a practical application
has offered itself to me in the course of my present inquiries) when all but one of
the functions are linear.
   If U, L1 , L2 . . . Lm be the first an nc and the others linear functions of n
variables, and it be desired to find the determinant of the resultant arising from
the elimination of any m out of the n variables, the following is the rule:
   Find the determinant, that is, the resultant of the partial differential coeffi-
cients in respect to the given variables, and of λ1 , λ2 . . . λm of

                           U + L1 λ1 + L2 λ2 + · · · + Lm λm .
                                                                                               p. 189
   This resultant, in its lowest form, will be always a rational (n − 1)th root of
the resultant of the homogeneous system of equations to which the system above
given can be referred as its type; and this reduced resultant divided by a power
(determinable by the law of homogeneity) of the resultant of L1 , L2 . . . Lm , when
all but the selected variables are made zero, will be the resultant determinant
required100 . As regards what has been said concerning the reducibility of the
general typical resultant in the case before us, this is a consequence of, and may be
 100
     The same method applies not only to the Final or Constant Determinant, but likewise to
all the Functional Determinants in the chain above described, extending upwards from this to
the Hessian, or as it ought to be termed, the first Boolian Determinant.


                                            193
brought into connexion with, the following theorem, which is easily demonstrable
by the theory of characteristics. If Q1 , Q2 . . . Qm be m homogeneous functions
of m variables of the same degree, r of which enter in each equation only as
simple powers uncombined with any of the other variables, then the degree of
the reduced resultant is equal to the number of the equations multiplied by the
(m − r − 1)th power of the number of units in the degree of each, subject to
the obvious exception that when r is m, (there being in fact but one step from
r = m − 2 to r = m), instead of r, (r − 1) must be employed in the above formula.
As an example of a sub-resultant as distinguished from a reduced-resultant, I
instance the case of three quadratics U, V, W , functions of x, y, z, in each of
which no squared power of z is supposed to enter: it may easily be shown by
my dialytic method that instead of six equations, between which to eliminate
x2 , y 2 , z 2 , xy, xz, yz, we shall have only 5, the three original ones and two instead
of three auxiliaries between which to eliminate x2 , y 2 , xy, xz, yz, the apparent
resultant is accordingly of the 9th instead of the 12th degree. But this is not
the true characteristic of the possibility of the coexistence of the given systems,
which in fact is zero, as is evidenced by the fact that they always do coexist,
since they are always satisfiable by only two relations between the variables, to
wit x = 0, y = 0. The apparent resultant is then something different, and what
has been termed by the above a Sub-resultant.
    I take this opportunity of entering my simple protest against the appropriation
of my method of finding the resultant of any set of three equations of degrees
equal or differing only by a unit, one from those of the other two, by Dr Hesse,
so far as regards quadratic functions, without acknowledgment, four years after
the publication of my memoir in the Philosophical Magazine: the fundamental
idea of Dr Hesse’s partial method is identical with that of my general one.
Still more unjustifiable is the subsequent use of the dialytic principle, by the
same author, equally without acknowledgment, and in cases where there is no
peculiarity of form of procedure to give even a plausible ground for evading such
acknowledgment. It is capable of moral proof that                                          p. 190
    what I had written on the matter was sufficiently known in Berlin and at
Königsberg, at each epoch of Dr Hesse’s use of the method.
    I now proceed to the consideration of the more peculiar branch of my inquiry,
which is as to the mode of reducing Algebraical Functions to their simplest
and most symmetrical, or as my admirable friend M. Hermite well proposes to
call them, their Canonical forms. Every quadratic function of any number of
variables may always be linearly transformed into any other quadratic functions
of the same, and that too in an infinite variety of ways; but in every other
instance there will be only a limited number of ways, whereby, when possible,
one form will admit of being transmuted into any other: and with the sole
exception of a cubic function of two letters, such transmutation will never be
possible, unless a certain condition, or certain conditions, be satisfied between


                                           194
the constants of the forms proposed for transmutation. The number of such
conditions is the number of parameters entering into the canonical form, and
is of course equal to the number of terms in the general form of the function
diminished by the square of the number of letters. Thus there is one parameter
in the canonical form for the biquadratic function of two and the cubic function
of three letters, and no parameter in the cubic function of two letters. Hitherto
no canonical forms have been studied beyond the cases above cited, but I have
succeeded, as will presently be shown, in obtaining methods for reducing to their
canonical forms functions with two and four parameters respectively. Owing to
what has been remarked above, the theory of quadratic functions is a theory
apart. Simultaneous transformation gives definiteness to that theory, but has no
existence for any useful purpose for functions of the higher degrees. Where the
theory of simultaneous transformation ends, that of canonical forms properly
begins; and in what follows, the case of quadratic forms is to be understood
as entirely excluded. Such exclusion being understood, there is no difficulty
in assigning the canonical, that is, the simplest and most symmetrical general,
form to which every function of two letters admits of being reduced by linear
transformations. If the degree be odd, say 2m + 1, the canonical form will be

                            u2m+1
                             1    + u2m+1
                                     2    + · · · + u2m+1
                                                     m+1 ;

if the degree be even, say 2m, the canonical form will be

                    1 + u2 + · · · + um + K(u1 u2 · · · um ) ,
                   u2m   2m           2m                    2


all the u’s being linear functions of the two given variables. It is easy to extend
an analogous mode of representation to functions of any number of letters. From
the above we see that for cubic, biquadratic, and quintic functions of two letters,
the canonical forms will be respectively

               u3 + v 3 ,      u4 + v 4 + Ku2 v 2 ,            u5 + v 5 + w5 ,

with a linear relation in the last-named case between u, v, w.                        p. 191
   First as to the reduction of any 4c function to Cayley’s form

                                       u4 + v 4 + Ku2 v 2 .

This may be effected in a great variety of ways, of which the following is not
the simplest as regards the calculations required, but the most obvious. Let the
modulus of transformation, whereby the given biquadratic function, say F (x, y),
becomes transmuted into its canonical form, be called M ; let the determinant of
F be called D1 , and the determinant of the determinant in respect to ξ and η of
                                                  2
                                        d     d
                               
                                   ξ      +η           F (x, y),
                                       dx    dy

                                               195
which latter, for brevity’s sake, may be termed the Hessian of F , (although in
stricter justice the Boolian would be the more proper designation), be called
D2 . Then, by examining the canonical form itself (which is as it were the very
palpitating heart of the function laid bare to inspection), we shall obtain without
difficulty the two equations
                                                     1
                             (1 − 9m2 )2 = M 12 D1      ,
                                                     46
                                                              1
                    m2 (1 − 9m2 )2 (m2 − 1)2 = M 24 D2               .
                                                            122 44
Eliminating the unknown quantity M , we obtain

                  m2 (m2 − 1)2                   m3 − m
                               = c,       or             = c1/2 ,
                  (1 − 9m2 )2                    1 − 9m2

where c is a known quantity.
  This cubic equation for finding m is of a peculiar form; it being easy to
show à priori, by going back to the canonical form, that its three roots are
m, θ(m), θ2 (m), where
                                      m−1
                              θ(m) =        ,
                                     3m + 1
θ being a periodical form of function such that θ3 (m) = m.
   This it is which accounts for the simple expression for m, that may be obtained
by solving the cubic above given. A better practical mode is to take, instead of the
determinant of the given function and its Hessian, the two hyperdeterminants and
eliminate as before: a cubic equation having precisely the same properties, and
in fact virtually identical with the former, will result. When m and consequently
M are found, there is no difficulty whatever, calling the given function F and its
Hessian H(F ), in forming linear functions of the two, as
                                                        )
                             ϕ(m)F + ψ(m)H(F )
                             ϕ1 (m)F + ψ1 (m)H(F )

which shall be equal to, that is, identical with, (u2 + v 2 )2 and u2 v 2 , whence u
and v are completely determined.                                                     p. 192
   Another and interesting mode of solution is to take, besides the given function
F and its Hessian, either the second Hessian or the post-Hessian of the given
function, by the post-Hessian understanding the determinant in respect of ξ and
η of
                                    d     d 3
                                           
                                 ξ    +η       F :
                                   dx    dy
any three of the four functions will be linearly related, and it may be shown
that, calling either the second Hessian (that is, the Hessian of the Hessian) or

                                        196
the post-Hessian H ′ , we shall have

                               H ′ (F ) + aH(F ) + bF = 0,

where a and b will be rational and integer functions of the coefficients of F , and
numerical multiples of two quantities R and S, such that the determinant of F
will be equal R3 + S 2 ; and this, be it observed, without any previous knowledge
of the existence of these hyperdeterminants R and S.
   If now we go to Hesse’s form for a cubic function of three letters, we shall
find that precisely similar modes of investigation apply step for step. Calling
the function F and its Hessian H(F ), and the post-Hessian or second Hessian at
choice H ′ (F ), we shall find

                          H ′ (F ) + mSH(F ) + nR2 F = 0,

where m and n are numerical quantities and R3 + S 2 equal the determinant of
F . It is interesting to contrast this equation with the one previously mentioned
as applicable to the 4c functions of two letters, namely,

                          H ′ (F ) + mRH(F ) + nSF = 0.

   In both instances there is no difficulty in assigning the relations between the
original R and S, and the R and S of any adjunctive form. All Aronhold’s results
may be thus obtained and further extended without the slightest difficulty. As
regards the equation for finding the parameter in Hesse’s canonical form for the
cubic of three letters, this will be of the 4th degree in respect to the cube of the
parameter, and the roots will be functionally representable as

                          x;        θ(x);     ϕ(x);      ψ(x),

where
                               θ2 (x) = ϕ2 (x) = ψ 2 (x) = x;
                                 θϕ(x) = ϕθ(x) = ψ(x),
                                 ϕψ(x) = ψϕ(x) = θ(x),
                                 ψθ(x) = θψ(x) = ϕ(x);
owing to which property the equation is soluble under the peculiar form observed
by Aronhold.                                                                            p. 193
   I pass on now to a brief account of the method, or rather of a method (for I
doubt not of being able to discover others more practical), of reducing a function
of the 5th degree of two letters (say of x and y) to its canonical form u5 + v 5 + w5 ,
subject to the linear relation au + bv + cw = 0, where the ratios a : b : c, and
the linear relations between u, v, w and the two given variables are the objects of
research. Here I have found great aid from the method of Relative Determinants;

                                            197
and I may notice that the successful application of more compendious methods
to the question would be greatly facilitated were there in existence a theory of
Relative Hyperdeterminants, which is still all to form, but which I little doubt,
with the blessing of God, to be able to accomplish. It may some little facilitate
the comprehension of what follows, if c be considered as representing unity.
   Calling as before the given quintic function F , the modulus of transformation
M , the Hessian and post-Hessian of F , H and H ′ , and its ordinary or constant
determinant D, we shall find

                          a2 v 3 w3 + b2 w3 u3 + c2 u3 v 3 = M 2 H,

and
                                   P1 P2 P3 P4 = M 6 H ′ ,
where
                           P1 = a3/2 vw + b3/2 wu + c3/2 uv,
                           P2 = a3/2 vw − b3/2 wu − c3/2 uv,
                           P3 = −a3/2 vw + b3/2 wu − c3/2 uv,
                           P4 = −a3/2 vw − b3/2 wu + c3/2 uv;
also D = M 20 multiplied by the product of the sixteen values of

                              a5/4 + b5/4 (1)1/4 + c5/4 (1)1/4 .

From the above equations it may be shown that H ′ (a known function of the
8th degree of the given variables x, y) must be capable of being thrown under
the form
                L{(x − a1 y)(x − a2 y) × (x − a3 y)(x − a4 y)
                       × (x − a5 y)(x − a6 y) × (x − a7 y)(x − a8 y)},
where
                                                                         D
           (a1 − a2 )2 × (a3 − a4 )2 × (a5 − a6 )2 × (a7 − a8 )2 =          = K,
                                                                         L2
so that K is a known quantity101 . Accordingly the said equation of the 8th
degree, considered as an algebraical equation in xy , may by known methods be p. 194
   found by means of equations not exceeding the 4th or even the 3rd degree: in
fact, to do this it is only necessary to form the equation to the squares of the
differences of the roots of xy in the equation H ′ ÷ y 8 = 0, which new equation
will be of the 28th degree. If we then  √ form two other equations of the 378th
degree, one having its roots equal to K multiplied by the binary products of
 101
    Or in other words, the post-Hessian determinant of a given function in two letters of the
second degree, may be divided into four quadratic factors in such a way that the product of the
determinants of these several factors shall be equal to the determinant of the given function.


                                             198
                                                                  √
the twenty-eight roots of the equation last named, the other to K multiplied
by the reciprocal of such binary products, the left-hand members of these two
equations expressed under the usual form will have a factor in common, which
may be found by the process of common measure and will be of the 6th degree,
whose roots consisting of three pairs of reciprocals may be found by the solution
of cubics only.
   In this way, by means of cubics and quadratics,

                 (a1 − a2 )2 ,     (a3 − a4 )2 ,     (a5 − a6 )2 ,       (a7 − a8 )2 ,

can be found, which being known,

                                 a1 a2 ,   a3 a4 ,   a5 a6 ,   a7 a8 ,

can be determined in pairs by means of quadratics from the equation H ′ ÷ y 8 = 0.
This being supposed to be done, we have

              P1 = f L1 ,          P2 = gL2 ,          P3 = hL3 ,           P4 = kL4 ,

where L1 , L2 , L3 , L4 are known quadratic functions of x and y. To determine the
ratios of f, g, h, k, we have three equations102 obtained from the identity

                f L1 + gL2 + hL3 + kL4 = (P1 + P2 + P3 + P4 ) = 0;

f : g : h : k being known, f L1 : gL2 : hL3 : kL4 are known ratios.
   But

       P1 + P2 = 2a3/2 vw,             P1 + P3 = 2b3/2 wu,               P1 + P4 = 2c3/2 uv.

Hence
                 a3/2 vw = λP,             b3/2 wu = λQ,             c3/2 uv = λR,
where P, Q, R are known quadratic functions of x, y.                                                p. 195
  Hence a : b : c may be found by means of the identical equation

                          a2 w3 v 3 + b2 u3 w3 + c2 v 3 u3 = H(F ),

whereby the ratios a−5/2 : b−5/2 : c−5/2 can be obtained without any further
extraction of roots, showing that there is but one single true system of ratios
a5 : b5 : c5 applicable to the problem; a : b : c being thus found, λ is easily
determined, and thus finally u, v, w are found in terms of x and y 103 .
 102
     For we must have the coefficients of x2 , xy and y 2 in f L1 + gL2 + hL3 + kL4 , of all them
zero.
 103
     The problem thus solved may be stated as consisting in reducing the general function
ax5 + bx4 y + cx3 y 2 + dx2 y 3 + exy 4 + f y 5 to the form
                          (lx + my)5 + (l′ x + m′ y)5 + (l′′ x + m′′ y)5 .




                                                 199
   I have little doubt that a more expeditious mode of solution than the fore-
going104 will be afforded by an examination of the properties and relations of
the quadratic and cubic forms, adjunctive to the general quintic functions, and
indeed to every (4m + 1)c function of two letters hereinbefore adverted to.
   Sufficient space does not remain for detailing the steps whereby the general
cubic function of four letters may, by aid of equations not transcending the fifth
degree, be reduced to its canonical form u3 + v 3 + w3 + p3 + q 3 , wherein u, v, w, p, q
are connected by a linear equation

                              au + bv + cw + dp + eq = 0;

the four ratios of whose coefficients a : b : c : d : e give the necessary number
                                        4·5·6
                                              − 42
                                        1·2·3
parameters furnished by the general rule. Suffice it for the present to say, that
the analytical mode of solution depends upon a circumstance capable of the
following geometrical statement: Every surface of the 4th degree represented
by a function which is the Hessian to any given cubic function whatever of four
letters, has lying upon it ten straight lines meeting three and three in ten points,
and these ten points are the only points which enjoy the following property in
respect to the surface of the 3rd degree denoted by equating to zero the cubical
function in question, to wit, that the cone drawn from any one of them as vertex
to envelop the surface, will meet it not in a continuous double curve of the 6th
degree, but in two curves each of the 3rd degree, lying in planes which intersect
in the ten lines respectively above named; so that to each of the ten points
corresponds one of the ten lines: these ten points and lines are the intersections
taken respectively three with three, and two with two, of a single and unique
system of five principal planes appurtenant to every surface of the 3rd degree,
and these planes are no other than those denoted by

                 u = 0,       v = 0,       w = 0,        p = 0,       q = 0.
                                                                                                  p. 196
   I have found also by the theory of Sub-resultants, that the analogy between
lines and surfaces of the third degree, in regard to the existence of double and
conical points, is preserved in this wise: that in the same way as a double point
on a curve of the 3rd degree commands the existence of a double point on its
Hessian, so does a conical point in a surface of the 3rd degree command over and
above the 10 necessary, and so to speak natural conical points, at least one extra,
that is to say an 11th conical point on its Hessian. And here for the present I
must quit my brief and imperfect notice of this subject, composed amidst the
interruptions and distractions of an official and professional life.
 104
     The coefficients in the reducing recurrent equation of the 6th degree in the process above
detailed may rise to be of 541632 dimensions in respect to the original coefficients in F .


                                             200
   Observation. It may be somewhat interesting and instructive to my readers,
to have a table of the successive scalar105 determinants of a quintic function of
two letters presented to them at a single glance. Preserving the notation above
[page 193], we have the following expressions:
                             The given function = u5 + v 5 + w5 ,
                      its Hessian = M 2 (a2 v 3 w3 + b2 w3 u3 + c2 u3 v 3 ),
               its post-Hessian = M 6 × the product of the four forms of
                            a3/2 vw + b3/2 (1)1/2 wu + c3/2 (1)1/2 uv;
          its praeter-post-Hessian = M 12 × the product of the nine forms of
                 a4/3 v 1/3 w1/3 + b4/3 (1)1/3 w1/3 u1/3 + c4/3 (1)1/3 u1/3 v 1/3 ;
       and the final determinant = M 20 × the product of the sixteen forms of
                                a5/4 + (1)1/4 b5/4 + (1)1/4 c5/4 .
The success of the method applied depends (as above shown) upon the fact of a
certain function of the roots of the post-Hessian (which is an octavic function of
the variables) being known, which fact hinges upon the circumstance that
                                   (M 6 )2 × (M 2 )4 = M 20 .

   P.S. I have much pleasure in subjoining the cubical hyperdeterminant of the
12th degree function of two letters, worked out upon the principle of Compound
Permutation hinted at in the foregoing pages, for which I am indebted to the
kindness and skill of my friend Mr Spottiswoode.                               p. 197
   The function being called
                                              12 · 11 10 2
                     ax12 + 12bx11 y +               cx y + &c. . . . + ly 12 ,
                                                 2
the following is106 its cubical hyperdeterminant:
                       agm − 6ahl + 15aik + 10aj 2 − 6bf m,
                       − 24bhk + 30bal + 20bij − 24cf l + 114cgk,
                       − 145ci2 + 50chj + 15cem + 20cgi + 20ch2 ,
                       − 400dgj + 280dhi + 20del + 50df e + 10d2 k,
                       + 385egi − 135e2 k − 290eh2 + 705f gh,
                       − 330f 2 i − 50g 3 .
 105
       By which I mean the determinants in respect to ξ, η of
                                                       r
                                              d     d
                                         ξ      +η           F (x, y).
                                             dx    dy

 106
       See below, p. 202.


                                                  201
Mr Spottiswoode will I hope publish the work itself in the next number of the
Journal, in which I shall also show how the hyperdeterminants of the cubical
function of three letters, Aronhold’s S and T , may be similarly obtained.




                                    202
                                         33.
 On the General Theory of Associated Algebraical Forms
       [Cambridge and Dublin Mathematical Journal, VI. (1851), pp. 289–293]
                                                                                    p. 198
    The following brief exposition of the general theory of Associated Forms, as
far as it has been as yet developed by the labours or genius of mathematicians,
is intended as elucidatory and, to a certain extent, emendative of some of the
statements in my paper107 on Linear Transformations, in the preceding number
of the Journal.
    In the first place, let a linear equivalent of any given homogeneous function
be understood to mean what the function becomes when linear functions of the
variables are substituted in place of the variables themselves, subject to the
condition of the modulus of transformation (that is, the value of the determinant
formed by the coefficients of transformation) being unity.
    Secondly, let two square arrays of terms (the determinants corresponding to
each of which are unity) be said to be complementary when each term in the one
square is equal to the value of what the determinant represented by the other
square becomes when the corresponding term itself is taken unity, but all the
other terms in the same line and column with it are taken zero. This relation
between the two squares is well known to be reciprocal. Thus, for instance,

                          a b c                  α β γ
                          a′ b′ c′       and     α′ β ′ γ ′
                          a′′ b′′ c′′            α′′ β ′′ γ ′′

will be said to be reciprocally complementary to one another when the two
determinants which they represent are each unity, and when we have        p. 199


                              1 0 0               1 0 0
                       a=     0 β′ γ′ ,     α=    0 b′ c′ ,
                              0 β ′′ γ ′′         0 b′′ c′′

                              0 1 0               0 1 0
                       b=     α′ 0 γ ′ ,    β=    a′ 0 c′ ,
                              α′′ 0 γ ′′          a′′ 0 c′′

                              α 0 γ                a 0 c
                       b′ =   0 1 0 , β′ =         0 1 0 ,
                              α′′ 0 γ ′′           a′′ 0 c′′
                                   &c.         &c.
 107
       p. 184 above.



                                         203
Accordingly, two transformations, say of F (x, y, z) and G(u, v, w) respectively,
may be said to be concurrent when in F for x, y, z, we write
                                        ax + by + cz,
                                       a′ x + b′ y + c′ z,
                                      a′′ x + b′′ y + c′′ z;
and in G for u, v, w, we write
                                        au + bv + cw,
                                       a′ u + b′ v + c′ w,
                                      a′′ u + b′′ v + c′′ w;
but complementary when for u, v, w, we write
                                       αu + βv + γw,
                                      α′ u + β ′ v + γ ′ w,
                                     α′′ u + β ′′ v + γ ′′ w;
a, b, c, &c., α, β, γ, &c. being related in the manner antecedently explained.
    Two forms, each of the same number of variables, may be said to be associate
forms when the coefficients of the one are functions of those of the other; and
when it happens, that the coefficients of the first are all explicit functions of
those of the second, the latter may be termed the originant and the former the
derivant.
    If now all the linear equivalents of one or of two associated forms are similarly
related to corresponding linear equivalents of the other, so that each may be
derived from each by the same law, the forms so associated will be said to be
concomitant each to the other. This concomitance may be of two kinds, and very
probably, in the nature of things, only of the two kinds about to be described. p. 200
    The first species of concomitance is defined by the corresponding equivalents
of the two associated forms being deduced by precisely similar, or, as we have
expressed it, concurrent transformations or substitutions, each from its given
primitive. The second species of concomitance is defined by the corresponding
equivalents being deduced not by similar but by contrary, that is, reciprocal
or complementary substitutions. Concomitants of the first kind may be called
covariants; concomitants of the second kind may be called contravariants. When
of the two associated forms one is a constant, the distinction between co- and
contra-variants disappears, and the constant may be termed an invariant of the
form with which it is associated108 . It follows readily from these definitions
  108
      Accordingly an invariant to a given form may be defined to be such a function of the
coefficients of the form, as remains absolutely unaltered when instead of the given form any
linear equivalent thereto is substituted. Of course if the determinant of the coefficients of the
transformations correspondent to the respective equivalents be not taken unity as supposed
in this definition, the effect will be merely to introduce as a multiplier some power of the
determinant formed by the coefficients of transformation.


                                              204
that a covariant of a covariant and a contravariant of a contravariant are each
of them covariants; but a covariant of a contravariant and a contravariant of a
covariant are each of them contravariants; and also that an invariant, whether of
a covariant or of a contravariant, is an invariant of the original function109 .
   It will also readily be seen that as regards functions of two letters a contravari-
ant becomes a covariant by the simple interchange of x, y with −y, x, respectively.
Covariants are Mr Cayley’s hyperdeterminants; contravariants include, but are
not coincident with, M. Hermite’s formes-adjointes, if we understand by the
last-named term such forms as may be derived by the process described by
M. Hermite in the third of his letters to M. Jacobi, “Sur différents objets de
la Théorie des Nombres,” (which process is an extension of that employed for
determining the polar reciprocal of an algebraical locus110 . M. Hermite appears,
however, elsewhere to have used                                                        p. 201
   the term forme-adjointe in a sense as wide as that in which I employ con-
travariants. For instance, he has given a most remarkable theorem, which admits
of being stated as follows:
   If we have a function of any number of letters, say of x, y, z, as
                                             m(m − 1) m−2 2
              axm + mbxm−1 y + mcxm−1 z +             dx y + &c.,
                                                2
and if I be any invariant of this function, then will
                                                                 r
                         d         d        d           d
                                                                                
                   xm      + xm−1 y + xm−1 z + xm−2 y 2    + &c. I
                        da         db       dc          dd
be a “forme-adjointe” of the given function. It is perfectly true and admits
of being very easily proved, as I shall show in your next number, that this is
 109
     It may likewise be shown that linear equivalents of covariants and contravariants are them-
selves related to one another as covariants and contravariants respectively, the transformations
by which the equivalents are obtained being taken concurrent in the one case and contrary or
reciprocal in the other; and of course any algebraic function of any number of covariants is a
covariant and of contravariants a contravariant.
 110
     This has been further generalized by me in the theorem given in the last number of this
Journal, where I have shown in effect that any invariant in respect to ξ, η . . . θ of
                            f (ξ, η . . . θ) + (xξ + yη + · · · + tθ + ρ)n−1 ,
(f being supposed to be of the degree n) is a contravariant of f (x, y . . . t). When this invariant
is the determinant of f , it may be shown that we obtain M. Hermite’s theorem. It is somewhat
remarkable that contravariants should have been in use among mathematicians as well in
geometry as the theory of numbers (although their character as such was not recognized) before
covariants had ever made their appearance. Invariants of course first came up with the theory
of the equation to the squares of the differences of the roots of equations, the last term in such
equation being an invariant. I believe that I am correct in saying that covariants first made
their appearance in one of Mr Boole’s papers, in this Journal; but Hesse’s brilliant application
of one from among the infinite variety of these forms to the discovery of the points of inflexion
in a curve of the third order, in other words, to the Canonical Reduction of the Cubic Function
of Three Letters, appears to have been the first occasion of their being turned to practical
account. [p. 186 above.]


                                                  205
a contravariant of the given function111 ; but it is not (as far as I can see) a
forme-adjointe in the sense in which the use of that word is restricted in the
letter alluded to. If, however, we adopt as the definition of formes-adjointes
generally, that property in regard to their transformées which M. Hermite has
demonstrated of the particular class treated of by him in the letter alluded to,
then his formes-adjointes become coincident with my contravariants. It will thus
be seen that covariants and contravariants form two distinct and coextensive
species of associated forms, which divide between them the wide and fertile
empire of linear transformations so far as its provinces have been as yet laid open
by the researches of analysts. In your next number I propose to enter much more
largely into the subject generally. More particularly I shall describe the new
method of Permutants, including the theory of Intermutants and Commutants
(which latter are a species of the former, but embrace Determinants as a particular
case), and their application to the theory of Invariants. I shall also exhibit the
connexion between the theory of Invariants and that of Symmetrical Functions,
and some remarkable theorems on Relative Invariants112 .
   Some of your readers may like to be informed that a Supplement to my
last paper, under the title of “An Essay on Canonical Forms,” has been since
published113 ; and that I have there given a much simpler method of solution of
the problem of the reduction of quintic functions to their canonical form than in
the original memoir, and extended the method successfully to the                    p. 202
   reduction of all odd-degreed functions to their canonical form. I may take this
occasion to state that the Lemma given in Note (B) of the Supplement, upon
which this method of reduction is based, is an immediate deduction from the
well-known theorem for the multiplication of Determinants.
   There is a numerical error in “The Cubical Hyperdeterminant of the Twelfth
Degree,” worked out after the method of commutants by Mr Spottiswoode, given
at the end of my paper in the May Number. The correct result will be stated in
the next number of the Journal, where I hope also to be able to fix the number
of distinct solutions of the problem of reducing a Sextic Function to its canonical
form
                              u6 + v 6 + w6 + mu2 v 2 w2 .
For odd-degreed functions there is never more than one solution possible, as
shown in the Supplement referred to.
   P.S. Since the above was sent to press, I have discovered an uniform mode of
solution for the canonical reduction of functions, whether of odd or even degrees.
The canonical form however, except for the fourth and eighth degrees, requires
 111
     This is also true if I be taken any covariant instead of an invariant of the function.
 112
     It will be readily apprehended that the definitions and conceptions above stated, respecting
covariants and contravariants of two single functions, may be extended so as to comprehend
systems of functions covariantive or contravariantive to one another.
 113
     By Mr George Bell, University Bookseller, Fleet Street. [p. 203 below.]



                                              206
to be varied from that assumed in my previous paper. Thus, for the sixth degree
the canonical form will be

                au6 + bv 6 + cw6 + muvw(v − w)(w − u)(u − v),

where u, v, w are supposed to be connected by the identical equation u + v + w =
0. And there will be only two solutions—a remarkable and most unexpected
discovery. For functions of the eighth degree there are five distinct solutions, and
in general there is the strongest reason for believing (indeed it may be positively
affirmed) that when the canonical form has been rightly assumed for a function
of the even degree n, the number of solutions will be 12 (n + 2) when 12 n is even,
but 14 (n + 2) when 12 n is odd. It turns out therefore that the theory for functions
of the sixth degree is in some respects simpler than for those of the fourth. The
investigation into canonical forms here referred to has led me to the discovery
of a most unexpected theorem for finding all the invariants of a certain class,
belonging to functions of two letters of an even degree.




                                        207
                                            34.
An Essay on Canonical Forms, Supplement to a Sketch of a
Memoir on Elimination, Transformation and Canonical Forms
                           [George Bell, Fleet Street, 1851]
                                                                                                p. 203
   Since the above paper was in print I have succeeded in obtaining a canonical
representation of the quadratic and cubic functions adjunctive to the general
quintic (5th degreed) functions of two letters.
   Let F the quintic function of x, y,

                                   F = u5 + v 5 + w5 ,

and
                                    au + bv + cw = 0,
M being the modulus of the transformation, whereby transition is made from
x, y to u, v. Then the quadratic adjunctive is

                              M4 4
                                 {a vw + b4 wu + c4 uv};
                              c4
and the cubic adjunctive is simply
                                   1 6
                                      M (abc)2 uvw114 .
                                   c3
Hence we can, in accordance with what I ventured to predict in the preceding
sketch, find u, v, w, by means of a simple and practical co-process. To wit, call

              F = lx5 + 5mx4 y + 10nx3 y 2 + 10px2 y 3 + 5qxy 4 + ry 5 .

                                                                                                p. 204
 114
    The knowledge of the existence of these lower adjunctive forms is mainly a consequence of
Mr Cayley’s splendid discovery of hyperdeterminant constants. In fact, they are respectively
the quadratic and cubic hyperdeterminants in respect to ξ and η of
                                                              4
                                   1
                                           
                                                    d     d
                                               ξ      +η           F,
                               1·2·3·4·5           dx    dy

x and y being treated as constants.
   The fortunate proclaimer of a new outlying planet has been justly rewarded by the offer of
a baronetcy and a national pension, which the writer of this wishes him long life and health
to enjoy. In the meanwhile, what has been done in honour of the discoverer of a new and
inexhaustible region of exquisite analysis?
  114
      p. 184 above. See p. 201, note ‡.




                                            208
   Form the determinant
                            lx + my mx + ny nx + py
                            mx + ny nx + py px + qy .
                            nx + py px + qy qx + ry

Let this cubic function, by solving it as a cubic equation, be made equal to

                             L(x + f y)(x + gy)(x + hy),

then
             u = k(x + f y),       v = l(x + gy),               w = m(x + hy).
By means of the identity, F = u5 + v 5 + w5 , k 5 , l5 , m5 , are known by the solution
of linear equations, and thus u, v, w, are determined by solving a cubic equation
instead of one of the eighth degree, as in the method first given, and the process
of canonising a quintic function is rendered practically possible.
   For brevity sake let c represent unity. The constant determinant of the cubic
adjunctive will be found to be

                                    3M 30 (abc)10 .

Calling, then, the cubic adjunctive of F , C(F ), we have the remarkable equation

                                                C(F )
                               uvw = rn                    o.
                                                3 □C(F )
                                        5       1



It may also be shown that if we call the Hessian of F , H(F ), we shall have the
following equally remarkable equation:
                                        1
                             □H(F ) =     □F × □C(F ).
                                        3
Again, calling the quadratic adjunctive of F , Q(F ), we shall easily find

                                        a + b5/2 + c5/2 
                                       5/2                
                                      
                                                          
                                       a5/2 + b5/2 − c5/2 
                                                          
                      □Q(F ) = M 10                                   ,
                                      
                                       a5/2 − b5/2 + c5/2 
                                                           
                                       5/2
                                      
                                                5/2    5/2 
                                                           
                                            a     −b       −c

or, if we please,
                        n                                                 o
                = M 10 a10 + b10 + c10 − 2a5 b5 − 2a5 c5 − 2b5 c5 .

When u, v, w are known, a, b, c, which are the resultants of v, w; w, u; u, v
respectively are known. But their ratios, or, if we please to say so, the ratios of
a5 : b5 : c5 , may be found independently and very elegantly as follows:—

                                            209
  Let

        M 10 × product of the 4 forms of a5/2 + 11/2 b5/2 + 11/2 c5/2 = A,
        M 20 × product of the 16 forms of a5/4 + 11/4 b5/4 + 11/4 c5/4 = B,
        M 30 × a10 · b10 · c10 = C.
                                                                                       p. 205
  A, B, C are known quantities, being respectively what we have called □Q(F ),
□(F )∗ , 13 □C(F ).
  It may easily be shown that

                       B − A2 = 128M 20 a5 b5 c5 (a5 + b5 + c5 ).

Hence M 5 a5 , M 5 b5 , M 5 c5 are the roots of ρ in the cubic equation

                              1           (B − A2 )2
                                      (                 )
                  B − A2
               ρ + 7 2/3 ρ2 +
                3
                                                     − A ρ + C 1/3 = 0.
                  2 C         4             214 C

A, B, C, it will be observed, are independent and, as they may be termed, prime
or radical adjunctive constants. Hitherto much mystery and uncertainty have
attached to the theory of hyperdeterminants, from its having been tacitly assumed
that they were always either of lower dimensions than the ordinary determinant,
or else algebraical functions of such, and of the determinant. Whereas we now
see that, whilst the determinant of a function in two letters of the fifth degree
is of eight dimensions, one of its radical or primitive hyperdeterminants is of
four, but the other of twelve dimensions. This is a most valuable consequence,
and would seem to indicate that the number of radical hyperdeterminants to
a function, over and above the common determinant, is always equal to the
number of parameters entering into its canonical form. The importance of this
ascertainment of an unsuspected third radical constant, adjunctive to a quintic
function of two letters, in making to march the theory of hyperdeterminants, can
hardly be over-estimated.
   From the equation last given we are enabled to assign the conditions in order
that two functions of the fifth degree may be capable of being linearly transformed
either into the other. For if we call F and F ′ two such linearly equivalent quintic
functions, they must be capable each of being thrown under the same form
u5 + v 5 + (lu + mv)5 , where l and m shall be the same for each. Consequently
we must have the roots of ρ in the same ratio for F and F ′ , which conditions
may be expressed by means of the two equations

                                 B − A2   B ′ − A′2
                                        =           ,
                                  C 2/3    C ′2/3
                    (B − A2 )2 − 214 AC   (B ′ − A′2 )2 − 214 A′ C ′
                                        =                            ,
                          C 4/3                    C ′4/3

                                            210
115                                                                                                 p. 206
      A′ , B ′ , C ′ , of course representing the same functions of the coefficients of F ′
as A, B, C, respectively of F .
   The two conditions required in their simplest form are accordingly

                             A       A′               B       B′
                                  =        ,               =        ,
                            C 1/3   C ′1/3           C 2/3   C ′2/3
or
                                A3 : B 2 : C :: A′3 : B ′2 : C ′ ,
that is to say, all quintic functions of two letters of which the determinant is to
the subduplicate power of the radical hyperdeterminant of the twelfth order and
to the sesquiduplicate power of the radical hyperdeterminant of the fourth order
in given ratios, are mutually convertible.
   So for the quartic (that is, biquadratic) function of two letters, calling R and S
the radical adjunctive constants of the second and third orders, the condition of
convertibility between different forms of the same is, that R3 : S 2 shall be a given
ratio. And, in general, we may infer that the condition of convertibility between
different functions of any degree is, that the several radical adjunctive constants
of each raised respectively to such powers as will make them of like dimensions,
shall be to one another in given ratios. Of course all cubic functions of two
letters, according to this rule, are mutually convertible without any condition,
they having but one radical adjunctive constant; and in fact all such functions,
being representable as the sum of two cubes of new variables linearly related to
those given, are necessarily convertible.
   I have further succeeded in obtaining the canonical form of the quadratic
adjunctive to any odd degreed function of two letters, which presents a wonderful
analogy to the theory of relative determinants of quadratic functions of any
number of letters, and constitutes an important step towards the construction of
the theory of relative hyperdeterminants.
   Let a function of two letters of the odd degree m(= 2n − 1) be thrown under
its canonical form,
                                um1 + u2 + · · · + un ,
                                       m            m

 115
    More strictly speaking (and this correction should be supplied throughout in the “Sketch”),
B is the negative determinant of 15 F . After finding, by the method of characteristics, or
any special artifices, the algebraic part of the value of a resultant or determinant, a process
frequently of some complexity remains over in assigning its numerical multiplier; this part of
the operation being analogous to that which occurs in the Integral Calculus, of determining
the constant to be added after the general form of an integral has been determined. In the
“Sketch,” a correction for the numerical multiplier remains also to be applied to the expressions
given for the successive Hessian determinants.




                                               211
and let there exist the n − 2 equations,

                       a1 u1 + a2 u2 + · · · + an un = 0, (1)
                        b1 u1 + b2 u2 + · · · + bn un = 0, (2)
                       ................................
                         l1 u1 + l2 u2 + · · · + ln un = 0. (n − 2)

Then, if M be the modulus of the transformation which converts u1 , u2 into            p. 207
   x, y, and if, on making θ1 , θ2 . . . θn disjunctively equal to 1, 2 . . . n we use
(θn−1 , θn ) to denote in general the determinant

                                    aθ1 aθ2 · · · aθn−2
                                    bθ1 bθ2 · · · bθn−2
                                                                ,
                                    ··· ··· ··· ···
                                    lθ1 lθ2 · · · lθn−2

                                         1
the quadratic adjunctive of                          F will be
                                    m(m − 1) · · · 2

                             M m−1 X n                         o∗
                                      (θ r , θs ) m−1
                                                      (ur us )    .
                            (1, 2)m−1

N.B. By means of this formula, and of the theorem for finding relative deter-
minants of quadratic functions, we can obtain the general canonical form for
one set of the biquadratic adjunctive constants (hyperdeterminants of the fourth
order in Mr Cayley’s language) of any odd degreed function of two letters116 .
   Thus, for the fifth degree, preserving the notation of the “Sketch,” we have
the biquadratic adjunctive constant

                                     0 c4 b4 a
                                     c4 0 a4 b  M 10
                                =              × 10 .
                                      4
                                     b a 4 0 c   c
                                     a b c 0

For the seventh degree, if we suppose the function to be equal to

                                      u7 + v 7 + w7 + θ7 ,

and
                au + bv + cw + dθ = 0,             a′ u + b′ v + c′ w + d′ θ = 0;
 116
     The condition m = 2n − 1 is only necessary in order that Σn (um ) may be a canonical,
because a possible and determinate, form for any given function of the mth degree. But the
theorem in the text, so far as it serves to obtain the quadratic adjunctive of Σn (um ), is true for
all odd values of m, whether greater or less than 2n − 1.




                                               212
the biquadratic adjunctive constant will be
                                              M 14
                                         (cd′ − c′ d)14
multiplied by the determinant
                   0          (ab′ − a′ b)6   (ac′ − a′ c)6   (ad′ − a′ d)6   a   a′
              (ba − b′ a)6
                 ′
                                   0          (bc′ − b′ c)6   (bd′ − b′ d)6   b   b′
              (ca′ − c′ a)6   (cb′ − c′ b)6        0          (cd′ − c′ d)6   c   c′
                                                                                       .
              (da′ − d′ a)6   (db′ − d′ b)6   (dc′ − d′ c)6        0          d   d′
                   a                b               c              d          0   0
                   a′              b′              c′              d′         0   0
                                                                                                    p. 208
   The determinants of the Hessian, the post-Hessian, and the praeter-post-
Hessian of F will be found (in the case of the quintic function) to be always
multiples of powers of the determinant of the given function, and of its cubic
adjunctive; and I believe that in general for a function of two letters of any
degree the determinants of all the derived forms in the Hessian scale117 , will be
necessarily algebraical functions of any two of them.
   I hope very shortly to accomplish the reduction of functions, as high as the
seventh degree of two letters, to their canonical form, and also to present a
complete theory of the failing or singular cases of canonical forms.



   Since the above was in print I have discovered the following

                             General Theorem
 for reducing a function of two letters of any odd degree to its canonical form.

   Let the degree of the function be (2n − 1); then its canonical form is

                               u2n−1
                                1    + u2n−1
                                        2    + · · · + un2n−1 ,

with (n − 2) linear relations between u1 , u2 , . . . un .
   To find u1 , u2 , . . . un , proceed as follows. Let the given function of the (2n−1)th
degree be supposed to be
                                                     2n − 2
   a1 x2n−1 + (2n − 1)a2 x2n−2 y + (2n − 1)                 a3 x2n−3 y 2 + · · · + a2n y 2n−1 .
                                                       2
 116
    See Note (A) of Appendix.
 117
    I use the term Hessian (more properly speaking the Boolian) Scale, to denote the determi-
nants in respect of ξ and η of ξ dx d
                                      + η dy
                                           d
                                             + &c. F .
   Neither Hesse, however, nor any other writer up to the present time, had thought of
constructing, and still less of turning to account, the functions (the first only excepted) which
figure in this scale.


                                               213
Form the determinant
          a1 x + a2 y    a2 x + a3 y     a3 x + a4 y             · · · an x + an+1 y
          a2 x + a3 y    a3 x + a4 y     a4 x + a5 y             · · · an+1 x + an+2 y
                                                                                       .
              ···            ···             ···                 ···         ···
         an x + an+1 y an+1 x + an+2 y an+2 x + an+3 y           · · · a2n−1 x + a2n y

This determinant is a function of x and y of the nth degree, and by resolving an
equation of the nth degree, may be decomposed into n factors, say

                         (l1 x + m1 y)(l2 x + m2 y) · · · (ln x + mn y);
                                                                                           p. 209
   we shall then have
                                    u1 = p1 (l1 x + m1 y),
                                    u2 = p2 (l2 x + m2 y),
                                       ..
                                        .
                                    un = pn (ln x + mn y),
where the l’s and m’s are known, and the (2n − 1)th powers of the p’s may be
found linearly, by means of the identical equation Σu2n−1 = F (x, y). Thus for
example a function of the seventh degree of two letters may be reduced to its
canonical form

              (lx + my)7 + (l′ x + m′ y)7 + (l′′ x + m′′ y)7 + (l′′′ x + m′′′ y)7 ,

by the resolution of a biquadratic equation. My demonstration of this extraor-
dinary and unexpected consequence rests upon the following lemma118 , itself a
very beautiful and striking theorem (no doubt capable of much generalisation)
in the theory of determinants. Form the rectangular matrix consisting of n rows
and (n + 1) columns

                                T1  T2   T3           · · · Tn+1
                                T2  T3   T4           · · · Tn+2
                                T3  T4   T5           · · · Tn+3
                                ··· ···  ···          ··· ···
                                Tn Tn+1 Tn+2          · · · T2n

where
                       Ti = a1r−i b1s+i + a2r−i b2s+i + · · · + an−1
                                                                 r−i s+i
                                                                     bn−1 .
Then all the n + 1 determinants that can be formed by rejecting any one column
at pleasure out of this matrix are identically zero.
   In order the better to realise the proof, suppose

                               n = 4,    so that     2n − 1 = 7.
 118
       See Note (B) of Appendix.


                                              214
Let
             F (x, y) = a1 x7 + 7a2 x6 y + 21a3 x5 y 2 + 35a4 x4 y 3 + 35a5 x3 y 4
                             + 21a6 x2 y 5 + 7a7 xy 6 + a8 y 7 .
Suppose
                             t7 + u7 + v 7 + w7 = F (x, y) = G(u, v),
                                  at + bu = v,         a′ t + b′ u = w.
Then, if M is the modulus of transition from x, y to u, v the hyper-      p. 210
   determinant, or, to adopt my new expression, the permutant P4 (meaning
thereby)
                a1 x + a2 y a2 x + a3 y a3 x + a4 y a4 x + a5 y
                a2 x + a3 y a3 x + a4 y a4 x + a5 y a5 x + a6 y
                                                                ,
                a3 x + a4 y a4 x + a5 y a5 x + a6 y a6 x + a7 y
                a4 x + a5 y a5 x + a6 y a6 x + a7 y a7 x + a8 y
                                                                                        6
which is a constant adjunctive in respect to ξ and η of ξ dx
                                                           d
                                                             + η dy
                                                                  d
                                                                      F , will,
according to the principles laid down in the preceding “Sketch,” be the prod-
uct of a power of M multiplied by the corresponding adjunctive constant of
                 6
       d
    ξ du + η dv
              d
                       G(u, v), and is therefore a multiple of the determinant

             (1 + A1 )t + A2 u        A2 t + A3 u     A 3 t + A4 u   A 4 t + A5 u
                A2 t + A3 u           A 3 t + A4 u    A 4 t + A5 u   A 5 t + A6 u
                                                                                    ,
                A3 t + A4 u           A 4 t + A5 u    A 5 t + A6 u   A 6 t + A7 u
                A4 t + A5 u           A 5 t + A6 u    A6 t + A7 u A7 t + (1 + A8 )u

where

    A1 = a7 + a′7 ,        A2 = a6 b + a′6 b′ ,    A3 = a5 b2 + a′5 b′2 ,   ...   A8 = b7 + b′7 .

In this determinant the coefficient of u4 is
                                        A2   A3      A4   A5
                                        A3   A4      A5   A6
                                        A4   A5      A6   A7
                                        A5   A6      A7 1 + A8

which is numerically equal to

                         A3 A4 A5                  A2 A4 A5
                  A5     A4 A5 A6         − A6     A3 A5 A6
                         A5 A6 A7                  A4 A6 A7
                            A2 A3 A5                          A2 A3 A4
                       + A7 A3 A4 A6              − (1 + A8 ) A3 A4 A5            = 0,
                            A4 A5 A7                          A4 A5 A6

                                                   215
because the second factors of the products are all zero by the lemma. Hence the
permutant P4 vanishes when t = 0, and consequently it contains t as a factor,
and in like manner it may be proved to contain u, v, w.
  Hence t, u, v, w are the algebraical factors of P4 , and precisely the same proof
applies to show in the case of a function in x and y, say F2n−1 of any              p. 211
  odd degree (2n − 1) whatever, that the corresponding permutant Pn will
contain the factors u1 , u2 . . . un linear functions of x, y, such that

                             u12n−1 + u2n−1
                                       2    + · · · + u2n−1
                                                       n    = F2n−1

as was to be shown.
   Whenever Pn has equal roots, this will denote either (which is the more
general case) that the usual canonical form fails and gives place to a singular
form, (owing to some of the coefficients of transformation becoming infinite), or,
which is the more special supposition, that the canonical form becomes catalectic
by one or more of the linear roots119 disappearing. Thus in the cubic function, if
P2 has equal roots, and consequently its determinant (which is coincident with
that of the function itself) vanish, then the canonical form in general fails; so
that, for example, ax3 + bx2 y cannot in general be exhibited as the sum of two
cubes: if, however, certain further relations obtain between the coefficients of
F , the canonical form reappears catalectically, the function becoming in fact
representable as a single cube. So, again, for the quintic function (referring back
to the notation above [page 205]), if P3 have equal roots, that is if C = 0, the
canonical form fails, unless at the same time B − A2 = 0, in which case the
function becomes the sum of two fifth powers; but if furthermore A = 0, then this
catalectic form again gives place to a singular form, which, on the satisfaction of
a further condition between the coefficients, again in its turn gives way before a
(bicatalectic, that is) doubly catalectic form, namely, a single fifth power.
   It is remarkable, that the form to which Mr Jerrard’s method reduces the
function of the fifth degree, expressed homogeneously as ax5 + bxy 4 + cy 5 , is
a singular form, being incapable of being exhibited as the sum of three cubes;
such, however, is not the case with the form ax5 + bx3 y 2 + cy 5 . It may further
be remarked, that although the singly catalectic form of the quintic function
is expressible by two conditions only, namely, C = 0, B − A2 = 0, it will be
indicated by P3 (which being a cubic function of x and y contains four terms)
completely disappearing, so that apparently four conditions would appear to
be required or implied. But of course these must be capable of being shown
to be non-independent, and to be merely tantamount to the two independent
ones, C = 0, B − A2 = 0. The theory of the catalectic forms of functions of
the higher degrees of two variables presents many strong points of resemblance
and of contrast to that of the catalectic forms of quadratic functions of several
variables.
 119
       u1 , u2 . . . un may be termed the linear roots of the form F2n−1 .


                                                  216
   One important and immediate corollary from the General Theorem is, that
the constants which enter into the linear functions appurtenant to the canonical
form of any function of an odd degree form a single and unique system; or, in
other words, the canonical forms for such functions are void of                  p. 212
   multiplicity, a result contrary to what might have been anticipated, and to
what we know is the case for the canonical forms of functions of an even degree.
   It may further be shown that if we have the (n − 2) equations

                           a1 u1 + a2 u2 + · · · + an un = 0,
                           b1 u1 + b2 u2 + · · · + bn un = 0,
                           .............................
                            l1 u1 + l2 u2 + · · · + ln un = 0,

and call M the modulus of transformation in respect to u1 , u2 , and if we make

                                  Pn = Ku1 u2 · · · un ,

then
                                                     2 n(n − 1)
                                                     1
                               a3 , a4 · · · an
                               b3 , b4 · · · bn         K
                                      ···
                                l3 , l4 · · · ln
is equal to the product of the 12 n(n − 1) factors of the form

                                 aθ1 aθ2 · · · aθn−2
                                 bθ1 bθ2 · · · bθn−2
                                                               ,
                                 ··· ··· ··· ···
                                 lθ1 lθ2 · · · lθn−2

θ1 , θ2 . . . θn−2 being any (n − 2) numbers out of the n numbers 1, 2, 3 . . . n.
    It may hence be shown that
                                                      Pn
                       u1 u2 · · · un =                 1/(2n−1)
                                                                     120 ,
                                            1
                                            m □x,y Pn

m being a number which is a function of n, and which may be shown to be
equal to □x,y (xn−1 y + xy n−1 )÷ product of the squared differences of the roots
of l n−2 = 1, that is
                                                    )n−2
                                 (n − 1)2 − 1
                             (
                       m=                                  = (−n)n−2 ,
                                   −(n − 2)

and thus
                                                        Pn
                     u1 u2 · · · un =       q                               .
                                        2n−1
                                                   (−n)−(n−2) □x,y Pn

                                               217
                                                                                      p. 213
   As an example of the mode of finding u1 , u2 . . . un , let

                                  F = 3x5 + 20x3 y 2 + 10xy 4 ,

then
                                     3x 2y 2x
                             P3 =    2y 2x 2y         = 4x3 − 4y 2 x.
                                     2x 2y 2x
Hence
                       u = f x,       v = g(x + y),          w = h(x − y).
To find f, g, h, we have u5 + v 5 + w5 = F , hence

                  f 5 + g 5 + h5 = 3;        g 5 + h5 = 2;        g 5 − h5 = 1;

whence we have
                               F = x5 + (x + y)5 + (x − y)5 .
Again, we find
                                  □(4x3 − 4y 2 x) = −44 × 12,
                                           1
                                                   1/5
                                              □P3          = 4,
                                         (−3)
and accordingly
                                                       P3
                            x(x + y)(x − y) =        −1
                                                                 ,
                                                {(−3) · □P3 }1/5
according to the general formula above given.
   As a second example let

                        F = 3x7 + 42x5 y 2 + 70x3 y 4 + 14xy 6 + y 7 ;

then
                  3x   2y    2x   2y
                  2y   2x    2y   2x
          P4 =                               = 4(x3 y − xy 3 ) = 4xy(x − y)(x + y),
                  2x   2y    2x   2y
                  2y   2x    2y 2x + y

and accordingly we shall find

                             x7 + y 7 + (x − y)7 + (x + y)7 = F.

Moreover
                                     □(4x3 y − 4xy 3 ) = 49 ,
 120
       □x,y means the determinant in respect to x and y.


                                               218
and
                                     44−2 = 42 .
Thus                                         s
                                    P4    7 □P4
                                        =        ,
                                   tuvw     44−2
agreeable to the general formula.                                                    p. 214
   As a corollary to our general proposition, it may be remarked, that if F2n−1
be a symmetrical function of x, y of the (2n − 1)th degree, Pn (F2n−1 ) will be also
a symmetrical function of x and y, and may therefore be resolved into its factors
by solving a recurring equation of the nth degree, which may, by well-known
methods, be made to depend on the solution of an equation of the 12 nth or
2 (n − 1)th degree, according as n is even or odd.
1

   Hence the reduction of a function of two letters of the degree 4m ± 1 to its
canonical form as the sum of powers may be made to depend on the solution of
an equation of the mth degree; so that, for example, a symmetrical function of
x, y, as high as the fifteenth or seventeenth degree, may be reduced by means of
a biquadratic equation only.
   In a short time I hope to present to the public a complete solution of the
canonical forms of functions of two letters of even degrees, and possibly to exhibit
some important applications of the principles of the method to the theory of
numbers.



                                     Appendix.
                                     Note (A).

   The permutants (meaning, in Mr Cayley’s language, the hyperdeterminants)
of F2n+1 (x, y) of the fourth dimension in respect to the coefficients of F , may
be all obtained by taking the quadratic permutant in respect to x and y of the
quadratic permutant in respect of ξ and η of
                                            2l
                                  d     d
                           
                               ξ    +η            F2n+1 (x, y),
                                 dx    dy
l having any integer value from 1 to n.
   In extension of a theorem in the foregoing Supplement, which applies only to
the case of l = n, I am able to state the following more general theorem, in which
the same notation is preserved as above [page 207]. The quadratic permutant in
respect to ξ and η of

                          1
                                                             2l
                                              d     d
                                             
                                           ξ    +η                 F2n+1 (x, y),
            (2n + 1)2n · · · (2n − 2l + 2)   dx    dy

                                        219
is equal to
                        M 2l X
                                  {(θr , θs )2l (ur .us )2n+1−2l }.
                       (1, 2)2l
If now we proceed to form the quadratic permutant of the above sum in respect
to x and y, we know à priori, by reason of Mr Cayley’s invaluable researches,
that we shall not get radically distinct results for all values, but only for certain
periodically changing values of l.                                                    p. 215
   I have not yet had leisure to seek for an explicit demonstration of this
remarkable law, founded upon the above given canonical representation.

                                         Note (B).

   The lemma, upon which the general method for reducing odd degreed functions
to their canonical form is founded, may be stated rather more simply and more
generally as follows:—
   The determinant
                             Tr1         Tr2           ···     Trn
                            Tr1 +t1     Tr2 +t1        · · · Trn +t1
                            Tr1 +t2     Tr2 +t2        · · · Trn +t2     ,
                              ···         ···          ···      ···
                           Tr1 +tn−1   Tr2 +tn−1       · · · Trn +tn−1

where Tθ denotes A1 aθ1 + A2 aθ2 + · · · + Am aθm provided that m is less than n, is
identically zero. In the theorem, as thus stated, there is no substantial loss of
generality arising from the omission of the b’s.
   Thus stated the theorem and its extensions evidently repose upon the same
or the like basis as the theory of partial fractions.

              Note (C), referring to the original “Sketch.”

   The Boolo-Hessian scale of determinants furnishes a very pretty general
theorem of geometrical reciprocity in connexion with the doctrine of successive
polars. Let F (x, y, z), a cubic homogeneous function of x, y, z equated to zero,
express in general a curve of the third degree; then
                                        d    d  d
                                                           
                                   a      +b +c    F
                                       dx   dy  dz
will express its first polar in respect to the point a, b, c, that is, the conic which
passes through the six points in which the tangents drawn from a, b, c to touch
the given curve meet the same.
   Again, if we take l, m, n the coordinates of any new point,
                           d    d  d                    d    d  d
                                                                      
                      l      +m +n                 a      +b +c    F
                          dx   dy  dz                  dx   dy  dz

                                             220
will express the polar, that is the chord of contact of the above conic, in respect
to the last named point. If now we eliminate l, m, n between the three equations
                   d    d  d        d       d       d
                                                       
                l    +m +n       a      +b +c            F = 0,
                  dx   dy  dz      dx      dy      dz
                   d    d  d         d        d       d
                                                      
                l    +m +n       a′     + b′     + c′      F = 0,
                  dx   dy  dz       dx       dy       dz
                   d    d  d          d        d        d
                                                        
                l    +m +n       a′′    + b′′     + c′′      F = 0,
                  dx   dy  dz        dx       dy        dz
                                                                                        p. 216
  it is easily seen that the resultant of the elimination is the square of the
determinant
                                  a b c
                                  a′ b′ c′ ,
                                  a′′ b′′ c′′
multiplied by the Hessian of the given function. And, moreover, that if we
eliminate x, y, z we shall obtain precisely the same result with the letters l, m, n
substituted for x, y, z. Hence it follows, that if we take the doubly infinite system
of first polars to a given curve of the third degree, in respect to all the points
lying in its plane, and then from any point in the Hessian to the given curve,
draw pairs of tangents to each conic of the system so generated, then all the
chords of contact will meet in one and the same point, which will itself be also a
point situated upon the Hessian and conjugate to the former.
   So, in general, for a function of any degree of any number of letters, viewed
with relation to the doctrine of successive polars, the determinants of the Boolo-
Hessian scale take one another up in pairs; namely the first takes up the last but
one, the second the last but two, and so on; and consequently, if the degree of
the function be odd, that function which (making abstraction of the constant
determinant at the end) lies in the middle of the scale pairs with itself, and, in a
sense analogous to that above exhibited for a function of the third degree, may
be said to be always its own reciprocal.
   P.S. I have just discovered the method of reducing functions of two letters of
even degrees to their canonical form, which will shortly be published in a second
Supplement.
   At present I offer the annexed theorem (which strikingly contrasts with the
law of uniqueness demonstrated of functions of an odd degree) as a foretaste of
the enchanting developments with which I hope shortly to present my readers:—
   If a given homogeneous function of x and y of the degree 2n be supposed to be
thrown under its canonical form,

                      1 + u2 + · · · + un + K(u1 u2 · · · un ) ,
                     u2n   2n           2n                    2


then will K n have n2 −1 in general distinct values, to each of which will correspond

                                         221
a single distinct system of the linear functions of x and y,

                    11/n u1 ,    11/n u2 ,    ...    11/n un .




                                        222
                                         35.
 Explanation of the Coincidence of a Theorem Given by Mr
Sylvester in the December Number of this Journal, with One
Stated by Professor Donkin in the June Number of the Same
             [Philosophical Magazine, Fourth Series, I. (1851), pp. 44–46]
                                                                                     p. 217
   I wish to state, without loss of time, that in the theorem given by me121 for
the composition of two successive rotations about different axes, I have been
anticipated by Prof. Donkin in the June Number of your Journal.
   To my shame I must confess, that, although an occasional contributor to,
I am not invariably a constant reader of your valuable miscellany, otherwise I
should not have introduced the theorem in question without due acknowledgment
of Professor Donkin’s claims to whatever merit may attach to the priority of
publication. The fact is, that I made out the theorem for myself nine years
ago, and had some communication on the subject with Professor De Morgan,
who was then writing the seventeenth chapter of his Differential Calculus. A
recent conversation with this gentleman has brought back to my mind a vivid
recollection of the course of that communication. I brought under Professor De
Morgan’s notice the analytical memoir of Sr Gabrio Pola on the subject in the
Memoirs of the Italian Society of Modena, and satisfied myself of the existence of
the single axis of displacement by compounding the two rotations in the manner
given in my paper, which, for the case of two axes fixed in space, is the same
as Professor Donkin’s, and for two axes fixed in the rotating body is materially,
although not formally the same.
   It then occurred to me that a more simple demonstration ought to be deducible
from the possibility of always finding the point on a sphere, by revolution about
which, as a pole, one equal arc could actually be shown to be transportable
into the place of another. But in proceeding to work out this idea I fell into a
remarkable blunder, in which I have since been followed by more than one able
friend to whom I have proposed the question. The                                     p. 218
   blunder was of this kind:—Two arcs have to be drawn, bisecting at right
angles the arcs joining the extremities of two equal arcs; the point of intersection
of the two bisecting arcs must in all cases fall outside the quadrilateral formed
by the equal and joining arcs. I supposed it to fall inside. There appears to be a
fatal tendency to do so in all who take the subject in hand. In consequence of
this error, the cause of which I did not at the moment perceive, I was driven to
deny and admit in one breath the same proposition. Mr De Morgan sent me the
correct proof after this method (the same as that given by him at page 489 of
his Calculus), I am inclined to think after I had myself detected my error; but of
this I cannot feel certain.
 121
       p. 158 above.


                                         223
   This is the method alluded to by me in the words “it is right to bear in mind,
&c.,” at the time of writing which all recollection of the same thing having been
published by Mr De Morgan had vanished from my memory.
   The proof of the triangle of rotations is so simple, that, as Professor Donkin
states (in a letter which he has done me the favour of addressing me on the
subject) was the case with himself, I thought it incredible that it should not
have appeared in some elementary work, and I was therefore at no pains to
publish it as my own; nor should I have written at all on the subject, had it not
been for the surprise occasioned to my mind by falling in with Professor Stokes’s
article in the Cambridge and Dublin Mathematical Journal, to demonstrate the
existence of an instantaneous axis, which proceeds in apparent unconsciousness
of the so simply demonstrable law, that any number of rotations of any kind
(and therefore those that take place in an instant of time) are representable by a
single rotation about a single axis. I shall feel obliged by the early insertion of
this explanation, more in justice to myself than to Professor Donkin, whose high
and worthily earned reputation, not to speak of the disinterested love of truth
for its own sake, apart from personal considerations, which animates the labours
of the genuine votary of science, must make him indifferent to whatever credit
might be supposed to result from the first authorship or publication of the very
simple (however important) theorem in question.




                                       224
                                              36.
 An Enumeration of the Contacts of Lines and Surfaces of
                   the Second Order
         [Philosophical Magazine, Fourth Series, I. (1851), pp. 119–140]
                                                                                                      p. 219
   It is well known that in general any two homogeneous quadratic functions of
the same system of variables may be simultaneously transformed, so as to be
expressed each of them as pure quadratic functions of a new system of variables
equal in number and linearly connected with the original ones; a pure quadratic
function meaning one in which only the squares of the variables are retained.
   Every homogeneous quadratic function may be treated as the characteristic122
of a locus of the second degree: if the function be of two letters, the locus is a
binary system of points in a line wherein the distances of two fixed points from
either point of the given system or given multiples of such distances correspond
to the variables; if of three letters, the locus is a conic, the distances or given
multiples of the distances of every point in which from three given lines in the
plane of the conic are represented by the variables; if of four letters, the locus is
a surface of the second order, the coordinates being the distances or multiples of
the distances of any point therein from four planes drawn in the space in which
the surface is contained, and so on for loci of four and higher dimensions.
   I propose, however, in the present paper to restrict myself to the theory of
the contacts of loci not transcending the limits of vulgar space, by which I mean
the space cognizable through the senses123 , and shall accordingly be                 p. 220
   almost exclusively concerned in determining the singular cases of conjugate
systems of quadratic forms of two, three, and four letters respectively.
   In order that the reduction of any such system, say U and V , to a pure
quadratic form may be possible (as it generally is), it is necessary that none of
the roots of the complete determinant of U + λV shall be equal; if any relation
of equality exist between these roots, the general reduction is generally no longer
possible; under peculiar conditions, however, as will hereafter appear, in spite
 122
      According to the definition stated by me in a previous paper, the characteristic of a locus
is the function which, equated to zero, constitutes the equation thereto.
  123
      If the impressions of outward objects came only through the sight, and there were no sense of
touch or resistance, would not space of three dimensions have been physically inconceivable? The
geometry of three dimensions in ordinary parlance would then have been called transcendental.
But in very truth the distinction is vain and futile. Geometry, to be properly understood, must
be studied under a universal point of view; every (even the most elementary) proposition must
be regarded as a fact, and but as a single specimen of an infinite series of homologous facts.
   In this way only (discarding as but the transient outward form of a limited portion of an
infinite system of ideas, all notion of extension as essential to the conception of geometry,
however useful as a suggestive element) we may hope to see accomplished an organic and vital
development of the science.



                                               225
of the equality of certain of the roots, the irreducibility in its turn will cease,
and the ordinary reduction be capable of being effected. It is easily seen, that
to every relation of equality between the roots of the determinant of U + λV
must correspond a particular species of contact between the loci which U and
V characterize. But we should make a great mistake were we to suppose that
every such relation of equality corresponded with but one species of contact;
for instance, the characteristics of U and V of two conics are functions of three
letters, and □(U + λV ) will be a cubic function of λ. Such a function may
have two roots, or all its roots equal: this would seem to give but two species
of contact, whereas we well know that there are no less than four species of
contact possible between two conics. Accordingly we shall find, that, in order
to determine the distinctive characters of each species of contact, we must look
beyond the complete determinant, and examine into the relations (in themselves
and to one another) of the several systems of minor determinants that can be
formed from U + λV .
   By pursuing this method, we may assign à priori all the possible species of
contact between any two loci of the second degree. How important this method
is will be apparent from the fact, that not only have the distinctive characters of
the various contacts possible between surfaces of the second order never been
determined, but their number and the nature of certain of them have remained
until this hour unknown and unsuspected.
   The method which we shall pursue is an exhaustive one, and will conduct us
by a natural order to a systematic arrangement of all the different modes and
gradations of such contacts.
   In a paper124 in this Magazine for November 1850, I explained the decline
of minor determinants, and stated a law, called the homaloidal law, concerning
them.
   If U and V be characteristics of the two loci whose contacts are to be considered,
U + λV will be the function, the properties of whose complete determinant, and
of the minor systems of determinants belonging to it, will serve to specify the
nature of the contact.
   It will be remembered, that, whatever be the number of variable letters in
any quadratic function U , three of its first minor determinants being zero,          p. 221
   makes all the first minors zero; six of its second minors being zero, makes
all the second minors zero; and so on for the third, fourth, &c. minor systems
according to the progression of the triangular numbers.
   It is well known that whatever linear transformations be applied to a quadratic
function W , the complete determinant thereof will remain unaltered, except by
a multiplier depending upon the coefficients introduced into the equations of
transformation; consequently the roots of λ in the equation obtained by making
 124
       p. 150 above.



                                         226
the determinant of U + λV zero remain unaffected by such transformation; and
any relation or relations of equality among the roots of the equation □(U +
λV ) = 0 is an immutable property of the system U, V , which is unaffected by
linear transformations. Another and more general kind of immutable property
(comprehending the above as a particular case), to which I shall have occasion
to refer, is the following.
   Suppose all the minors of any order of U + λV have a factor λ + e in common;
this factor will continue common to the same system of minors when U and V
are simultaneously transformed. This is a very important proposition, and easily
demonstrated; for if λ + e be a common factor to all the rth minors of U + λV ,
(U − eV ) will have its rth minors zero, and therefore, as explained by me in the
paper above referred to, U − eV will be degraded r orders below U or V . This
is clearly a property independent of linear transformation, consequently λ + e
will remain a factor of the transformed rth minors.
   In like manner it is demonstrable that any number of distinct factors λ + e1 ,
λ + e2 , . . . common to the rth minors of one form of U + λV , will remain
common factors of any other linearly derived form of the same. It is consequently
necessary that each rth minor of one form of any quadratic function W shall be
a syzygetic125 function of all the rth minors of any other form of the same; and
consequently a function of λ of any degree, whether all its factors be or be not
distinct, which is common to the rth minors of one form of U + λV , will remain
so to the rth minors of any other form of the same.
   The law exhibiting the connexion of each rth minor of one form of W (any
homogeneous quadratic function) with all the rth minors of any other form of
W , will form the subject of a distinct communication.
   Finally, to fully comprehend the annexed discussion, the following principle
must be apprehended.                                                                 p. 222
   If any factor K enter into all the rth minors of W , and if K be the highest
                     e                                               i

power of K common to all the (r + 1)th minors, then K 2e−i will be a common
factor to all the (r − 1)th minors.
   Let r be taken unity; it is easily proved126 that the complete determinant of any
square matrix may be expressed by the difference between two products127 , each
 125
     If A = pL + qM + rN + &c., where p, q, r . . . do not any of them become infinite when
L, M, N . . . or any of them become zero, A may be termed a syzygetic function of L, M, N . . ..
  In the theorem above alluded to, it will be shown (as might be expected) that the syzygy in
the case concerned is of the simplest kind, that is, that each rth minor of a quadratic function
of any number of letters is a homogeneous linear function of all the rth minors of the same
quadratic function linearly transformed.
 126
     This will appear in my promised paper on Determinants and Quadratic Functions.
 127
     When the matrix is symmetrical about one of its diagonals (as it is in the case which we
are concerned with), one of these products becomes a square. I may take this occasion of
hinting, that the theory of quadratic functions merges in a larger theory of binary functions,
consisting of the sum of the multiples of binary products formed by combining each of one set of
quantities, x, y, z . . . with each of the same number of quantities of another set, as x′ , y ′ , z ′ . . ..


                                                   227
of two first minor determinants divided by a certain second minor determinant.
The proposition is therefore demonstrated for this case, and thereby in fact
implicitly for every case, inasmuch as the first minors of every rth minor are
(r + 1)th minors of the original matrix. Hence it follows, that if any system
of rth minor determinants have a common factor εi , the complete determinant
must contain at lowest the factor ε(r+1)i , and any system of (r − s)th minor
determinants thereunto will contain at lowest the factor ε(s+1)i .
   I now proceed to apply these principles to the determination of the relative
forms of conjugate quadratic functions representing geometrical loci of the second
order. I shall begin with two binary systems of points in a right line.
   The general characteristics U and V of two such systems may be thrown under
the form
                                U = x2 + y 2 ,
                                               )
                                                 .
                                V = ax2 + by 2
When □(V + λU ) = 0 has its two roots equal, these systems have a point in
common. The above forms cease to be applicable, and convert into
                                            U = xy,
                                                                     )

                                            V = ax2 + bxy
where x = 0 represents the common point.                                      p. 223
   Let U and V now represent two conics. When there is no contact, we have as
the types of their characteristics
                        U = x2 + y 2 + z 2 ,                V = ax2 + by 2 + cz 2 .
The three roots of □(V + λU ) = 0 are
                                λ = −a,            λ = −b,               λ = −c,
showing that there are three distinct pairs of lines in which the intersections of
U and V are contained, the equations to three pairs being respectively
                                     (b − a)y 2 + (c − a)z 2 = 0,
                                      (c − b)z 2 + (a − b)x2 = 0,
                                     (a − c)x2 + (b − c)y 2 = 0;
For instance,
                axx′ + bxy ′ + cxz ′ + a′ yx′ + b′ yy ′ + c′ yz ′ + a′′ zx′ + b′′ zy ′ + c′′ zz ′
would be a binary function, and its determinant (no longer, as in a quadratic function,
symmetrical about either diagonal) would correspond to the square matrix
                                                a      b       c
                                                a′    b′      c′ .
                                                a′′   b′′     c′′
Almost all the properties of quadratic apply, with slight modifications, to binary functions.


                                                      228
the four points of the intersection being defined by the equations corresponding
to the proportions
                                      q            q                q
                       x : y : z ::    (b − c) :       (c − a) :        (a − b).

   Now let □(U + λV ) have two equal roots; the characteristics assume the form

                                  U = x2 + y 2 + xz,
                                                                )
                                                                    .
                                  V = ax2 + by 2 + cxz

Two of the pairs of lines become identical, that is, two of the four points of
intersection coincide.128                                                      p. 224
   This may be termed “Simple Contact.” The tangent at the point of contact is
x = 0; this equation making U and V each become of only one order.
   The intersections are
                                           x           =   0,       y = 0,    (1)
                                           x           =   0,       y = 0,    (2)
                       (a       +   (b                 =   0,       z = 0,    (3)
                     p            p
                          − c)x        − c)y
                       (a − c)x − (b − c)y             =   0,       z = 0.    (4)
                     p            p


These are obtained by making V − aU = 0, which gives x = 0 or z = 0.
 128
     We may if we please make a = b; for it may be shown that the equations, in their present
forms, contain an arbitrariness of 10 degrees; namely, 9 on account of x, y, z being arbitrary
linears of ξ, η, θ; 2 on account of the ratios a : b : c; together 11 reduced by one degree on
account of x, y, z, changed into lx, ly, lz, leaving U = 0, V = 0 unaffected. Now the degrees
of arbitrariness in two conics, subject to satisfy only one condition, is 2 × 5 − 1 or 9. Hence
there is one degree of arbitrariness to spare. In fact, on making a = b, the axis z becomes the
line joining the two points of intersection distinct from the point of contact; x remaining the
tangent at the point of contact, and y, strange to say, still arbitrary, subject only to passing
through the point of contact; if, however, y be made to pass through the point of contact, and
either one of the distinct intersections, this form,

                                      U = x2 + y 2 + xz,
                                      V = ax2 + ay 2 + cxz,

becomes no longer tenable, but gives place to

                                      U = y 2 + yx + xz,
                                      V = ay 2 + ayx + cxz,

where x is the tangent at the point of contact, z the line joining the two intersections with one
another, and x, x + y respectively the lines joining either of them with the point of contact; if
the multiplier of yx in V in the above be made b instead of a, x remains the tangent as before,
y becomes any line through the point of contact, and z any line through one of the distinct
intersections. A systematic view of similar modulations of form and the study of the laws of
arbitrariness connected with them, as applicable to the general subject-matter of this paper,
must be deferred to a subsequent occasion.



                                              229
   x = 0 gives y 2 = 0, that is, y = 0 twice over, and z = 0 gives

                             (a − c)x2 + (b − c)y 2 = 0.

The number of conditions to be satisfied in this case is one only.
   Next let □(U + λV ) have all its roots equal. This condition will be satisfied
(still leaving U and V as general as they can remain consistent with these
conditions) by making

                  U = x2 + yz + yx,         V = ax2 + ayz + byx.

Here only one distinct pair of lines can be drawn to contain the intersections,
showing that three out of the four points come together.
   This may be termed “Proximal Contact.” The number of affirmative conditions
to be satisfied is two, and the contact is therefore entitled of the second degree.
   The tangent at the point of contact is y = 0, and the four intersections become

                                   x = 0,   y = 0,
                                   x = 0,   y = 0,
                                   x = 0,   y = 0,
                                   x = 0,   z = 0.

These may be obtained from the equation V − aU = 0, which gives y = 0 or
z = 0; the former implying concurrently with itself x2 = 0, and the latter yz = 0.
   Thus we obtain three systems,

                                   x = 0,   y = 0,

and one
                                   x = 0,   z = 0,
corresponding to three consecutive points and the single distinct one.                p. 225
   The determinant of U + λV being only of the third degree in λ, we have
exhausted the singularities of the system U, V dependent on the form of the
complete determinant of U + λV .
   Let now the first minors of U + λV have a factor in common; this will indicate
that U + λV may be made to lose two orders by rightly assigning λ, in other
words, that the intersections of U and V are contained upon a pair of coincident
lines. Here it is remarkable that the original forms of U and V reappear, but
with a special relation of equality between the coefficients: we shall have, in fact,

                   U = x2 + y 2 + z 2 ,     V = ax2 + ay 2 + bz 2 .




                                          230
This gives the law for double, or, as I prefer to call it, diploidal contact129 . By
virtue of the Homaloidal law, we know that if three first minors of U + λV be
zero, all are zero; we have therefore to express that three quadratic functions of λ
have a root in common. This implies the existence of two affirmative conditions;
the contact of the two conics taken collectively may therefore be still entitled of
the second degree, although the contact at each of the two points where it takes
place is simple, or of the first degree.
   These points are evidently defined by the equation
                                  √
                              x + −1 y = 0, z = 0,
                            (
                                  √
                              x − −1 y = 0, z = 0,

and the ordinary algebraical solution of the equations U = 0, V = 0 would
naturally lead to the four systems
                                         √
                                   x + √−1 y = 0, z = 0,
                                   x + √−1 y = 0, z = 0,
                                   x − √−1 y = 0, z = 0,
                                   x − −1 y = 0, z = 0;
                                                          √                √
the two tangents at the point of contact are x + −1 y = 0, x − −1 y = 0, and
the coincident pair of lines containing the intersections is z 2 = 0.                     p. 226
   It may at first view appear strange, that whilst no condition is required in
order that U and V may be simultaneously metamorphosed into the forms of
x2 + y 2 + z 2 , ax2 + by 2 + cz 2 , a, b and c being all unequal, for this metamorphosis
to be possible when any two become equal, not one but two conditions must be
satisfied. The reason of this is, that the coefficients of transformation, which, as
well as a, b, c, are functions of the coefficients of the given quadratic functions,
become infinite on constituting between the said coefficients such relations as are
necessary for satisfying the equation a = b, or a = c, or b = c, except upon the
assumption of some further particular relations between them over and above
that implied in such equality.
 129
     See my remarks on the conditions which express double contact in the Cambridge Journal,
Nov. 1850; see p. 129 above. If n functions, being all zero, be the condition of a fact, but r
independent syzygetic equations admit of being formed between these functions, the number
of affirmative conditions required is not n, but (n − r); because the fact may be expressed by
affirming (n − r) equations and denying certain others. Thus if P = 0, Q = 0, R = 0, S = 0
express a fact, and
                                 P P ′ + QQ′ + RR′ + SS ′ = 0,
                               P P ′′ + QQ′′ + RR′′ + SS ′′ = 0,
the fact is expressible by affirming P = 0, Q = 0, and denying R′ S ′′ − R′′ S ′ = 0, for then
P = 0, Q = 0 will imply R = 0, S = 0; or, in like manner, by affirming any other two out of
the four necessary equations, and denying the other equations. Observe, however, that all the
required equations may coexist in the absence of such right of denial.
 129
     p. 129 above.


                                             231
    In the ordinary case of diploidal contact, the first minors having a factor in
common, this factor will enter twice into the complete determinant of U +λV , but
it may enter three times: this will indicate, that not only do the four intersections
lie on a coincident pair of lines, but furthermore, that there is but one pair of
lines of any kind on which they lie.
    In the ordinary case of diploidal contact, it will be observed that this latter
condition does not obtain; the four intersections lie on a coincident pair of lines;
but they lie also on a crossing pair, namely, in the two tangents at the points
of contact. In this higher species of diploidal contact, it is clear that the two
points of contact, which are ordinarily distinct, come together, and that all four
intersections coincide.
    This I call confluent contact; the forms of U and V corresponding thereto will
be
                       U = x2 + y 2 + xz,      V = ay 2 + axz;
the common tangent at the point of contact being x = 0, and the four coincident
points x2 = 0, y 2 = 0.
   The number of affirmative conditions to be satisfied being three, the contact
is to be entitled of the third degree.
   Observe, that it is of no use to descend below the first minors in this case;
because the second minors, being linear functions of λ, could not have a factor
in common, unless V : U becomes a numerical ratio, which would imply that the
conics coincided130 .
   Fortified by the successful application of our general principles to the preceding
more familiar cases of contact, we are now in a condition to apply with greater
confidence the same à priori method to the exhaustion and characterization of
all the varied species of contact possible between surfaces                           p. 227
   of the second order; a portion of the subject comparatively unexplored, and
never before thought susceptible of reduction to a systematic arrangement.
   When there is no contact, we may write
              U = x2 + y 2 + z 2 + t 2 ,    V = ax2 + by 2 + cz 2 + dt2 ,
and the intersection of the surfaces will lie in each of the four cones,
                       (a − d)x2 + (b − d)y 2 + (c − d)z 2 = 0,
                        (a − b)x2 + (c − b)z 2 + (d − b)t2 = 0,
                        (a − c)x2 + (b − c)y 2 + (d − c)t2 = 0,
                        (b − a)y 2 + (c − a)z 2 + (d − a)t2 = 0.
Whenever the surfaces are in contact, certain of these cones will coincide with
certain others, so that their number will be always less than four. Also, as
 130
     No-contact and complete coincidence may be conceived as the two extreme cases in the
scale of relative conjugate forms.


                                           232
we shall find in such event, they may degenerate into pairs of intersecting or
coincident planes.
   Let us begin with considering the cases of contact for which the first minors
(and consequently à fortiori the minors inferior to the first) have no factor in
common.
   Here □(V + λU ) is a biquadratic function.
   If λ have all its roots unequal, we have U and V as above given.
   If two roots are equal, the characteristics assume the form
                          U = x2 + y 2 + z 2 + xt,
                                                      )
                                                          .
                          V = ax2 + by 2 + cz 2 + dxt
The touching plane is x = 0; the point of contact is x = 0, y = 0, z = 0; the
curve of intersection is one of the fourth degree, with a double point at the point
of contact.
   There is but one condition to be satisfied, and the contact may be entitled
“simple” and of the first degree.
   Next let λ have three equal values, the equations become

              U = x2 + yz + t2 + xy,       V = x2 + yz + at2 + bxy.

The tangent plane at the point of contact y = 0, and the point itself x = 0, y = 0,
t = 0. The curve of intersection is a curve of the fourth order, with a cusp at the
point of contact. The number of affirmative conditions to be satisfied is two; the
contact is of the second degree, and may be termed “proximal” or cuspidal.          p. 228
   Next let □(U + λV ) have two pairs of equal roots, we shall find

                  U = x2 + xy + zt,       V = ayz + bxy + czt.

The line x = 0, z = 0 will be common to both surfaces. The curve of intersection
will therefore break up into a right line and a line of the third order.
   The former will meet the latter in two points, which will be each of them
points of contact. The contact is therefore diploidal; but as there is another
species of diploidal contact to which we shall presently come, it will be expedient
to characterize each of them by the nature of the intersections of the two surfaces;
accordingly this may be termed unilinear-intersection contact, or more briefly,
unilinear contact.
   The number of affirmative conditions to be satisfied being two, it may be said
to be collectively of the second degree, but (obviously?) the contact at each of
the two points is of the nature of simple contact.
   Lastly, let us suppose that all four roots of U + λV are equal; we shall find,
as the most simple expressions of the most general forms of the two surfaces,

               U = x2 + xy + yz + zt,         V = axy + bz 2 + azt.

                                        233
In this case the two points of intersection of the curve of the third degree, and
the right line on which the surfaces intersect, come together, so that the right
line becomes a tangent to the curve. The number of conditions to be satisfied is
three: there is but one point of contact which may be considered as the union of
two which have coalesced, and the species may be defined as confluent-unilinear
contact.
   If we throw the equations to the conoids having an unilinear contact into the
form
                    x(x + y) + zt = 0,     xy + z(y + ct) = 0,
we obtain
                            (x + y)(y + ct) − yt = 0,
which last equation is no longer satisfied by x = 0, z = 0, these systems of roots
having been made to disappear by the process of elimination.
   The curve of the third degree, in which the two given conoids intersect, may
thus be defined as their common intersection with the new conical surface defined
by the third of the above equations.                                               p. 229
   More generally, it is apparent that the three conoids,
                                               
                                 xu − yt = 0,
                                             
                                             
                                 yv − zu = 0, ,
                                               
                                  zt − xv = 0 
                                               

in which x, y, z, t, u, v may any of them be considered as a homogeneous linear
function of four others, intersect in the same line of the third degree. Besides
which, the first and second intersect in the right line y, u; the second and
third in z, v; the third and first in x, t; each of which lines it is evident is a
chord of the common curve of intersection. For instance, y = 0, u = 0 may
be satisfied concurrently with all the above three equations by satisfying the
equation zt − xv = 0, which, as two linear relations exist originally between the
six letters, and two more have been thrown in, becomes a quadratic equation
between any two of the letters.
   The only case of exception to this reasoning is, when y = 0, u = 0 can be
satisfied concurrently with z = 0, v = 0, and with x = 0, t = 0; but in this case
the surfaces all become cones; and as there is no longer a curve of the third
degree, “Cadit quaestio.” Even here, however, the intersection of any two of the
surfaces becomes a conic, and two coincident generating lines on the two cones;
so that if we take one of these and the conic to represent a degenerate form of a
line of the third degree, the remaining straight line passes through a double point
of such degenerate form, and the case passes into that of confluent-unilinear
contact.
   The two double points in the intersection of the two conoids
              U = x(x + y) + zt = 0,         V = xy + z(y + ct) = 0,

                                       234
by which I mean the points of intersection of the conic with the right line common
to them, are found by making x = 0, z = 0, and substituting in the derived
equation
                            (x + y)(y + ct) − ty = 0,
which gives y = 0, or y + (c − 1)t = 0; so that the two points required are

                              x = 0,      y = 0,   z = 0,

                          x = 0,    y = (1 − c)t,     z = 0.
It appears also that the entire intersection is contained in each of the two cones,

                     U − V,    that is,     x2 + z{(1 − c)t − y}

and
                   cU − V,     that is,    cx2 + y{(c − 1)x − z},
the respective vertices of which are at the points above determined.                    p. 230
   The equations for confluent-unilinear contact,

                              x(x + y) + z(y + t) = 0,

                                xy + z(cz + t) = 0,
give
                          (x + y)(cz + t) − (y + t)y = 0;
which, on making x = 0, z = 0, is satisfied by y 2 = 0; showing that the confluence
takes place at the point

                              x = 0,      y = 0,   z = 0.

   The number of terms in the two equations for ordinary unilinear contact being
six, and in those given for confluent unilinears seven, and the empirical rule
in all other cases being that the terms tend to diminish and never increase in
number as the degree of the contact (expressed by the number of conditions to
be satisfied) rises, I am led to suspect that the conjugate system for the latter
species of contact may admit of being reduced to some more simple form.
   I must state here once for all, that all the distinct systems of (at least
consecutive) conjugate forms that have been, and will be given, are mutually
untransformable. This it is which distinguishes singular from particular forms.
   A particular form is included in its primitive; but a singular form is one, which,
while it responds to the same conditions as some other more general form, is
incapable of being expressed as a particular case of the latter, on account of the
additional condition or conditions which attach to it.


                                           235
   I pass now to the singularities which arise from the first minor determinants
of U + λV having a factor in common, the second minors being supposed to be
still without a common factor.
   When this common factor is linear in respect to λ, let it be supposed to enter
not more than twice (twice, we know, by the general principle enunciated at the
commencement of this paper, it must enter) into the complete determinant.
   Two of the cones containing the intersection of U and V then become coincident,
and degenerate each into the same pair of crossing planes. This may be termed
biplanar-contact. The characteristics of such contact are

             U = x2 + y 2 + z 2 + t 2 ,      V = ax2 + ay 2 + bz 2 + ct2 ;

the points of contact are two in number, being at the intersection of the two
plane conics into which the curve of intersection breaks up. The two planes     p. 231
   in which these lie are given by the equation (b − a)z + (c − a)t = 0; these
                                                         2          2

intersect in the right line z = 0, t = 0, which meets both surfaces in the same
two points,                                   √
                         z = 0, t = 0, x + −1 y = 0,
                                              √
                         z = 0, t = 0, x − −1 y = 0,
the two common tangent planes at these points being
                       √                    √
                   x + −1 y = 0,        x − −1 y = 0

respectively.
   This, then, is another species of double contact between two conoids, and,
as far as I know, the only kind hitherto recognized as such. The number of
conditions to be satisfied remains two, as in the former species.
   Next suppose that the common factor of the first minor enters three times
into the complete determinant instead of twice only, as in the last case.
   The corresponding characteristics will be found to be

             U = x2 + zt + y 2 + z 2 ,       V = ax2 + azt + by 2 + cz 2 .

The intersection of U, V still lies in two planes,

                             (b − a)y 2 + (c − a)z 2 = 0;

but the intersection of these two planes,

                                   y = 0,     z = 0,

meets the surfaces in the two coincident points,

                              y = 0,      z = 0,   x2 = 0.

                                           236
This, therefore, I call confluent-biplanar contact; the two conics constituting the
complete intersection, instead of cutting, touch and at their point of contact the
two conoids have a contact of a superior order. The conditions to be satisfied for
this case are three in number.
   Next suppose that the common factor of the first minors enters only twice
into the complete determinant, but that the remaining two factors become equal.
   Here the analytical characters of unilinear and biplanar contact are blended;
in fact, the intersection consists of a conic and a pair of right lines meeting one
another and the conic. The characteristics are

            U = x2 + y 2 + z 2 + zt,        V = ax2 + ay 2 + bz 2 + czt.
                                                                                      p. 232
  The intersection is contained in the two planes

                       z = 0,       (b − a)z + (c − a)t = 0,

and consists of the two lines z = 0, x2 + y 2 = 0, lying in the common tangent
plane z = 0, and the conic
                                    (b − a)z + (c − a)t = 0,
                                                               )
                                                                   .
                    (a − c)x2 + (a − c)y 2 + (b − c)z 2 = 0
There are three points of contact, namely, the point x = 0, y = 0, z = 0, where
the two right lines cut, and x2 + y 2 = 0, t = 0, z = 0, where these lines meet
the conic. This, then, is a case of triple contact. I distinguish it by the name of
bilinear-contact. The number of conditions is still three.
   Now all else remaining as before, let the two pairs of equal roots in the
complete determinant become identical, or, in other words, let the common
factor of the first minors be contained four times in the complete determinant.
The characteristics become

            U = xz + xt + y 2 + z 2 ,       V = axz + bxt + by 2 + bz 2 .

The intersection becomes the two right lines

                                x = 0,    y 2 + z 2 = 0,

and the conic
                                z = 0,    x2 + y 2 = 0.
All these meet in the same point,

                             x = 0,      y = 0,   z = 0;

so that instead of contact in three points, the contact takes place about one only,
in which the three may be conceived as merging. This I call confluent-bilinear
contact. It requires the satisfaction of four conditions.

                                          237
    Next let us suppose that the two distinct factors are common to each of the
first minors. This will imply the existence of four affirmative conditions.
    The complete determinant will of necessity contain each of these factors
twice, so that no additional singularity can enter through this determinant. The
characteristics assume the form

                  U = x2 + y 2 + z 2 + t2 ,          V = ax2 + ay 2 + bz 2 + bt2 .

The two surfaces will meet in four straight lines, forming a wry quadrilateral,
whose equations are
                         √                    √
                     x ± −1 y = 0,        z ± −1 t = 0.
                                                                                                       p. 233
   These intersect each other in the four points

                                 x = 0,     y = 0,     z 2 + t2 = 0,

                                 z = 0,     t = 0,     x2 + y 2 = 0,
each of which will be a distinct point. This I term quadrilinear contact.
   Now let the two factors common to each of the first minors become identical;
so that a squared function, instead of an ordinary quadratic function of λ, is
now their common measure.
   The factor which enters twice into each of the first minors will enter four times
into the complete determinant; the number of conditions to be satisfied is one
more than in the preceding case, namely five, and the characteristics become

                 U = x2 + y 2 + xz + yt,             V = ax2 + by 2 + cxz + cyt.

    Here arises a singularity of form in the intersections utterly unlike anything
which has been remarked in the preceding cases. For it will not fail to have been
observed, that the intersection in the nine preceding cases was always a line or
system of lines of the fourth degree, so as to be cut by any plane in four points.
    But in this case, the fact of the first minors having a factor in common, shows
that the intersection is contained in two planes (which is of course to be viewed
as a degenerate species of cone); and the fact of the complete determinant having
all its roots equal, shows that there is but one system of a pair of planes in which
the intersection is contained, and no more.
    So that the two pairs of planes, into which the wry quadrilateral was divisible
in the case immediately preceding, now become a single pair. This can only be
explained by two of the opposite sides of the quadrilateral becoming indefinitely
near to one another, but still not coinciding in the same planes; so that the
actual visible or quasi-visible131 intersection will be in three right lines, of which
the middle one meets each of the two others.
 131
       I use the term quasi-visible, because the intersection may become in part or whole imaginary.


                                                 238
  This will further appear by proceeding regularly to solve the equations

                                U = 0,         V = 0.

                                        a−c
                                          r
V − cU = 0 gives y = ±kx, where k =           , and therefore xz + kxt = 0, or
                                        c−b
xz − kxt = 0; whence we see that the complete intersection is represented by the
lines
                 (x = 0, y = 0); (z + kt = 0, y − kx = 0),
                 (x = 0, y = 0); (z − kt = 0, y + kx = 0),
                                                                                         p. 234
   showing that there are but three physically distinct lines, as already premised.
   This, then, may be considered as derived from the preceding case of a wry
quadrilateral intersection, by conceiving two opposite sides of the quadrilateral
to come indefinitely near, but without coinciding.
   Let these two lines be called P and P ′ ; take any point in P and any two
points in P ′ indefinitely near to one another and the point first taken, then this
indefinitely small plane will be common to both surfaces, and consequently they
ought to touch along every point in the line P . This is again confirmed by the
forms given to U and V . For at any point where the coordinates are 0, 0, ξ, θ the
equations to the tangent planes to the two surfaces respectively are

                         ξx + θy = 0,       cξx + cθy = 0,

that is to say, are identical.
   Whilst, therefore, certain grounds of geometrical, and still stronger grounds
of analytical analogy, might seem to justify this species of contact taking the
name of confluent quadrilinear, yet as, in fact, the intersection is trilinear, and
as, moreover, the two indefinitely proximate lines must be considered, not as
coincident, but as turned away from one another through an indefinitely small
angle and out of the same plane, I prefer to take advantage of this striking
property of contact at every point along a line (a property entirely distinct from
any that we have yet considered), and confer upon the species of contact we have
been considering the designation of unilinear-indefinite contact.
   Where the line of indefinite contact meets the two other lines of the intersection,
the contact is of course of a higher order; thus offering a parallel to what takes
place in ordinary unilinear contact, in which there is no contact, except only at
two points of the right line forming part of the complete intersection.
   I believe that this kind of contact, which forms a natural family with two
others about to be described, and which will close the list, has never before been
imagined, and would at first sight have been rejected as impossible.
   Having now exhausted the cases of the first class, in which the minors have
no factor in common, and the two sections of the second class, in which the
second minors have no common factor, but the first minors of U + λV a linear

                                         239
or quadratic function of λ in common, I descend to the third class, in which
the second minors, which are quadratic functions of λ, are supposed to have a
common factor.
   This common factor must enter twice into each of the first minors by virtue
of the law previously indicated, and cannot enter more than twice, as           p. 235
   otherwise the first minors of U + λV could only differ from one another by a
numerical multiplier, which is obviously impossible, except when U + λV is of
the form (k + λ)U , that is, when the two surfaces coincide.
   Again, the common factor of the first minor must enter three times into the
complete determinant; but there is no reason why it may not enter four times,
and thus two cases arise. In the first, the characteristics take the form

             U = x2 + y 2 + z 2 + t2 ,      V = ax2 + ay 2 + az 2 + bt2 .

The second determinant having a factor in common, shows that the intersection
U, V is contained in a pair of coincident planes; but the complete determinant,
having two distinct factors, evidences that these plane intersections, viewed as
indefinitely near but still distinct, lie in the same cone, which will be a cone
enveloping both the surfaces U and V all along their mutual intersections. This
is also seen easily from the forms of U and V ; for we have V − aU = (b − a)t2 ,
which proves that the intersection lies in the coincident, or, to speak more strictly,
consecutive planes t2 = 0; and at any point x = ξ, y = η, z = ζ, the tangent
plane to each surface becomes

                                 ξx + ηy + ζz = 0.

As there are six independent, that is, non-necessarily co-evanescent second
minors, that the second minor systems shall all have a common factor, implies
the satisfaction of five conditions. This species of contact I call curvilineo-
indefinite; it is, I believe, the only kind of indefinite contact between two surfaces
of the second order hitherto taken account of.
   There is still, however, a higher species of contact, videlicet, when all the four
roots of the complete determinant of U + λV are identical with the root common
to each of its second minors. In this case the common enveloping cone becomes
identical with the plane (considered as a coincident pair of planes) in which the
surfaces intersect.
   The characteristics take the form

                       U = x2 + xy + zt,         V = xy + zt.

The intersection is contained completely in the common tangent plane x = 0,
and consists of the two right lines,

                                  (x = 0,    z = 0),

                                         240
                                      (x = 0,     t = 0).
This, the highest and crowning species of contact, I call bilineo-indefinite. It is
defined by six conditions.
   At each point of the two lines of intersection of U and V there is contact, and
a very peculiar species of contact at the intersection of these two lines themselves. p. 236
   To form a distinct idea of this, let the physical visible or quasi-visible intersec-
tion of U, V take place along the two lines L, M ; the rational intersection must
be conceived as made up of the wry quadrilateral, L, M ; L′ , M ′ , in which L is
indefinitely near to L′ , and M to M ′ . It follows, therefore, that there is contact
at the four angles of the quadrilateral; but as there is nothing to fix the relative
directions of the diagonal joining the intersection of L and M to that of L′ and
M ′ , because there is nothing to restrict the position of the latter point, except
that it shall lie upon either surface132 , it appears that not only is there contact
at the junction of the two lines constituting the complete intersection of the two
surfaces, but that these surfaces continue to touch at consecutive points taken
all round this first, and indefinitely near to it in any direction133 .
   Bilineo-indefinite (the highest) contact for two conoids is strictly analogous to
confluence, the highest species of contact between conics. For this latter may be
conceived as an intersection made up of two coincident pairs of coincident points;
and the former, as an intersection made up of two coincident pairs of crossing
right lines; and a pair of crossing lines is to a plane locus of the second degree
what a coincident pair of points is to a rectilinear locus of the same degree.
   In the subjoined table I have brought under one point of view the characters
and algebraic forms which I call the condensed forms corresponding to each
species of contact above detailed.
 132
      This will be better seen by reference to the analogy presented by the case when the two
conoids touch all along a curve. The rational intersection is made up of this curve and another
indefinitely near it. The two curves, whatever be the position of their node, will lie in the same
enveloping cone, so that the position of the node is indeterminate.
  133
      As the two surfaces jut one close into the other at this point, it would perhaps be not
improper to designate the contact at such point as umbilical.




                                              241
                               A. Quadratic loci in a right line.
                                                                   (
                                                            xy
             Simple contact.             One condition.
                                                            x2 + xy
                                 B. Quadratic loci in a plane.
             1st Class.                                            (
                                                                       x2 + y 2 + xz
             Simple contact.             One condition.
                                                                       ax2 + by 2 + cxz
                                                                   (
                                                                       x2 + yx + yz
             Proximal contact.           Two conditions.
                                                                       ax2 + byx + ayz
             2nd Class.                                            (
                                                                       x2 + y 2 + z 2
             Diploidal contact.          Two conditions.
                                                                       ax2 + ay 2 + bz 2
                                                                   (
                                                                       x2 + y 2 + xz
             Confluent contact.          Three conditions.
                                                                       y 2 + xz
                                                                                                               p. 237

                                              C.   Quadratic loci in space.
1st Class.
                                                                 x2 + y 2 + z 2 + xt
                                                             n                              o
Simple contact.                       One condition.
                                                                 ax2 + by 2 + cz 2 + dxt
                                                                 x2 + y 2 + xt + zt
                                                             n                              o
Proximal contact.                     Two conditions.
                                                                 ax2 + by 2 + cxt + azt
                                                                 x2 + xy + zt
                                                             n                      o
Unilinear contact.                    Two conditions.
                                                                 ayz + bxy + czt
1st species of diploidal.
                                                                 x2 + yz + xy + zt
                                                             n                          o
Confluent-unilinear, or triple con-   Three conditions.
                                                                 az 2 + bxy + bzt
tact.
2nd Class, 1st Section.
                                                                 x2 + y 2 + z 2 + t2
                                                             n                              o
Biplanar contact.                     Two conditions.
                                                                 ax2 + ay 2 + bz 2 + ct2
2nd species of diploidal.
                                                                 x2 + zt + y 2 + z 2
                                                             n                              o
Confluent-biplanar contact.           Three conditions.
                                                                 ax2 + azt + by 2 + cz 2
                                                                 x2 + y 2 + z 2 + zt
                                                             n                              o        n
                                                                                                         xz + yt
Bilinear contact.                     Three conditions.                                         or
                                                                 ax2 + ay 2 + bz 2 + czt                 axt + byz
                                                                 xz + xt + y 2 + z 2
                                                             n                              o
Confluent-bilinear contact.           Four conditions.
                                                                 axz + bxt + by 2 + bz 2
2nd Class, 2nd Section.
                                                                 x2 + y 2 + z 2 + t2
                                                             n                              o        n
                                                                                                         xy + zt
Quadrilinear, or quadruple con-       Four conditions.                                          or
                                                                 ax2 + ay 2 + bz 2 + bt2                 axy + bzt
tact.
                                                                 x2 + y 2 + xz + yt
                                                             n                              o
Unilineo-indefinite contact.          Five conditions.
                                                                 ax2 + by 2 + cxz + cyt
3rd Class.
                                                                 x2 + y 2 + z 2 + t2
                                                             n                              o
Curvilineo-indefinite contact.        Five conditions.
                                                                 ax2 + ay 2 + az 2 + bt2
                                                                 x2 + xy + zt
                                                             n                  o
Bilineo-indefinite contact.           Six conditions.
                                                                 xy + zt
                                                                                                               p. 238



                                                   242
   Another (and, in a physical sense, more) natural mode of grouping the
twelve species of conoidal contact, which, without observing the same lines of
demarcation, leaves intact the sequence of the species, is into the three families.
The first, or definite-continuous, for which the surfaces touch in a single point,
and intersect in an unbroken curve, comprises simple and cuspidal contact.
   The second definite-discontinuous, for which the surfaces touch in one, two,
three or four points, but intersect in a curve more or less broken up into distinct
parts, comprises all the species from the third to the ninth inclusive. The third
natural family is that of indefinite contact, and comprises the three last species.
It will of course be observed that there are five species of single contact, that is,
contact at one point, namely, simple, cuspidal, and the three confluent species,
two of double, one of treble, one of quadruple, and three of indefinite contact;
the last being distinguishable inter se lineo-indefinite as being special at two
points, curvilineo-indefinite as having no speciality, and bilineo-indefinite as
being special at one point only.
   I might now proceed to discuss more particularly the nature of the contact
taken, not collectively, but with reference to each single point where it exists.
This, however, must be reserved for a future communication; as also, among
other important and curious matter, the ascertainment of the singular forms of
quadratic conjugate functions of five or more letters. At present I shall content
myself with stating the following general proposition, which naturally suggests
itself from a consideration of the cases already considered.
   In a conjugate quadratic system of any number of letters, the lowest and also
the highest degree of singularity will be always unique; the conditions to be
satisfied in the former case being only one in number, and, in the latter 12 r(r − 1),
where r denotes the number of the letters. The first part of this proposition is
self-apparent, the latter part may be inferred from the homaloidal law; for the
(r − 2)nd minors will be quadratic functions, and the highest degree of contact
will correspond to those having a factor in common, which would involve the
satisfaction of 12 r(r − 1) − 1 conditions only; but over and above this, that the
complete determinant, instead of containing this common factor, as it needs
must, (r − 1) times, shall contain it r times: this gives one condition more,
making up the entire number to 12 r(r − 1).
   The total number of different species of singularity for conjugate functions of
a given number of letters can only be expressed by aid of formulae containing
expressions for the number of various ways in which numbers admit of being
broken up into a given number of parts.
   The computation of this number in particular cases, upon the principle of the
foregoing method, is attended with no difficulty.                                       p. 239
   We have seen that this number for two, three and four letters, is respectively
one, four, twelve.
   I have found that for five letters the number is twenty-four, for six letters fifty,


                                          243
for seven letters a hundred, and (subject to further examination) for eight letters
one hundred and ninety-three. The series, therefore, as far as I have yet traced it,
is 1, 4, 12, 24, 50, 100, 193. The last number must not be relied upon at present.
   It will be observed, that the foregoing table for the contacts of surfaces of
the second order contains no form corresponding to a complete intersection in
two non-intersecting lines and an undegenerated conic. In fact, if two such lines
form part of the intersection, at least one other right line intersecting them both,
must go to make up the remaining part. This is easily verified; for it is readily
seen that the most general representation of two conoids intersecting in two
non-meeting lines will be

                       U = xy + zt,       V = axy + bzt + cxt + eyz,

where the two lines in question are

                          (x = 0,     z = 0),         (y = 0,   t = 0).

Now it will be found that the first minors of V + λU formed from the above
equation will all contain the common factor (a + λ)(b + λ) − ce, showing that
the contact is quadrilinear or linear-indefinite, that is bilinear, according as the
roots of
                           λ2 + (a + b)λ + ab − ce = 0
are distinct or equal; which explains how it is that only one species of bilinear
contact (that is to say, the case corresponding to the two conoids agreeing in the
two right lines in which each is cut by a common tangent plane) comes to find a
place in the preceding enumeration.
   It may not be uninteresting, under an euristic point of view, to state that
the above theory, which, as well in what it accomplishes as in what it suggests
(the author cannot but feel conscious), constitutes a substantial accession to
analytical science, arose out of a theorem which occurred to him as likely to be
true, in the act of reviewing for the press his paper “On Certain Additions” in
the last November Number134 of this Magazine, and which he had only then
time to throw into a foot-note as a probable conjecture.
   Wishing to subject it to an analytical test, he found it necessary to obtain
the condensed forms which serve to characterize the confluent contact of           p. 240
   conies. In this way he became aware of the great utility of these condensed
forms, and of the desideratum to be supplied in obtaining a complete list of them
applicable to all varieties of contact. The happy thought then occurred to him
of inverting the process which he had applied in the treatment of the contacts of
conies, in the November Number135 of the Cambridge and Dublin Mathematical
Journal; for whereas the nature of the contacts was there assumed and translated
 134
       p. 148 above.
 135
       p. 119 above.


                                                244
into the language of determinants, he soon discovered that it was the more easy
and secure course to assume the relations of every possible immutable kind that
could exist between the complete and minor determinants corresponding to the
characteristics, by aid of these relations to construct the characteristics, and
from the characteristics so obtained, determine the geometrical character of each
resulting species of contact. Thus he has been able to effect the very results
stated by himself as desiderata at the close of the paper in this Magazine above
referred to.

                                     Note.

   It is proper to remark, that all the condensed forms given in this paper have
actually been obtained by the author in the way above pointed out. The limits
imposed by the objects to which the Magazine is devoted have restricted him
from exhibiting the method at full; but any of his readers will be able without
difficulty to make it out for himself.
   The process consists in finding U + λV by means of solving for each case a
problem of position (a kind of chess-board problem) on a square table, containing
three places in length and breadth for conies, four places by four for surfaces,
and so on (if need be) according to the number of variable letters involved.
U + λV being thus determined in form, U and V become readily cognizable.
It is right also to add, that some of the condensed forms here set forth have
been incidentally noticed and employed by previous authors, as Plücker and Mr
Cayley.
   The conditions in each case to which the position-problem is subject are
immediately deducible from the laws which the complete determinant, and the
successive minor systems of determinants of U + λV , are required to satisfy.




                                      245
                                               37.
        On the Relation between the Minor Determinants of
             Linearly Equivalent Quadratic Functions
                       [Philosophical Magazine, I. (1851), pp. 295–305]
                                                                                             p. 241
   I showed in the preliminary part of my paper on Contacts in the February
Number of this Magazine 136 , by a priori reasoning, that if a quadratic function
(U ) be linearly converted into another (V ), any minor determinant of any order
of V must be a syzygetic function of all the minor determinants of U of the same
order.
   The object of my present communication is to exhibit the syzygy in question,
which, as I indicated, is linear; by which I mean that a determinant of the
one function is equal to the sum of the pari-ordinal determinants of the other
affected respectively with multipliers formed exclusively out of the coefficients
of the equations of transformation. In order that a clear enunciation of the
theorem in view may be possible, it is necessary to premise a new but simple,
and, as experience has proved to me, a most powerful, because natural, method
of notation applicable to all questions concerning determinants.
   Every determinant is obtained by operating upon a square array of quantities,
which, according to the ordinary method, might be denoted as follows:

                                    a1,1 ,   a1,2   · · · a1,n ,
                                    a2,1 ,   a2,2   · · · a2,n ,
                                    a3,1 ,   a3,2   · · · a3,n ,
                                     ···     ···           ···
                                    an,1 ,   an,2   · · · an,n .

My method consists in expressing the same quantities biliterally as below:

                                  a1 α1 , a1 α2 · · · a1 αn ,
                                  a2 α1 , a2 α2 · · · a2 αn ,
                                   ···     ···         ···
                                  an α1 , an α2 · · · an αn .
                                                                                             p. 242
   where of course, whenever desirable, instead of a1 , a2 . . . an and α1 , α2 . . . αn ,
we may write simply a, b . . . l, and α, β . . . λ respectively. Each quantity is
now represented by two letters; the letters themselves, taken separately, being
symbols neither of quantity nor of operation, but mere umbrae or ideal elements
of quantitative symbols. We have now a means of representing the determinant
above given in a compact form; for this purpose we need but to write one set of
 136
       p. 221 above.



                                               246
                                                   (                    )
                                           a1 , a2 . . . an
umbrae over the other as follows:                           . If we now wish to obtain the
                                           α1 , α2 . . . αn
algebraic value of this determinant, it is only necessary to take a1 , a2 . . . an in all
its 1, 2, 3 . . . n different positions, and we shall have
                (                      )
                    a1 , a2 . . . an
                                           =
                                               X
                                                   ±{a1 αθ1 × a2 αθ2 × · · · × an αθn },
                    α1 , α2 . . . αn

in which expression θ1 , θ2 . . . θn represents some order of the numbers 1, 2 . . . n,
and the positive or negative sign is to be taken according to the well-known
dichotomous law. Thus, for example,
                                                                     
                                                                     
                                                                      aa × bβ × cγ
                                                                     
                                                                       + aβ × bγ × ca
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                       + aγ × ba × cβ
                         (         )                                 
                                                                     
                             abc                                     
                                       will represent
                             αβγ                                     
                                                                      − aβ × ba × cγ
                                                                     
                                                                     
                                                                       − aa × bγ × cβ
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                       − aγ × bβ × ca.

   Although not necessary for our immediate object, it may not be inopportune
to observe how readily this notation lends itself to a further natural extension of
its application.
    (             )                                              (     )     (    )      (   )   (       )
      ab cd                                                      ab          cd   ab   cd
                          will naturally denote                            ×    −    ×    ;
      αβ γδ                                                      αβ          γδ   γδ   αβ

that is
        (                  )       (                   )     (                   )       (           )
             aa × bβ      cγ × dδ      aγ × bδ      ca × dβ
                       ×            −            ×            .
            −(aβ × ba)   −(cδ × dγ)   −(aδ × bγ)   −(cβ × da)

And in general the compound determinant
                     (                                                                       )
                          a1 , b1 . . . l1 , a2 , b2 . . . l2 · · · ar , br . . . lr
                         α1 , β1 . . . λ1 , α2 , β2 . . . λ2 · · · αr , βr . . . λr

will denote
              (                        )       (                       )             (                   )
     X          a1 , b1 . . . l1               a2 , b2 . . . l2                      ar , br . . . lr
            ±                              ×                               × ··· ×                           ,
              aθ1 , βθ1 . . . λθ1            aθ2 , βθ2 . . . λθ2                   aθr , βθr . . . λθr
                                                                                                                 p. 243
   where, as before, we have the disjunctive equation

                                           θ1 , θ2 . . . θr = 1, 2 . . . r.


                                                           247
As an example of the power of this notation, I will content myself with stating
the following remarkable theorem in compound determinants, one of the most
prolific in results of any with which I am acquainted, but which is derived from
a more particular case of another vastly more general. The theorem is contained
in the annexed equation
        (                                                                                     )
            a1 , a2 . . . ar , ar+1 ; a1 , a2 . . . ar , ar+2 . . . a1 , a2 . . . ar , ar+s
           α1 , α2 . . . αr , αr+1 ; α1 , α2 . . . αr , αr+2 . . . α1 , α2 . . . αr , αr+s
                   (                      )s−1     (                                          )   (1)
                       a1 , a2 . . . ar             a1 , a2 . . . ar , ar+1 , ar+2 . . . ar+s
               =                                 ×                                            .
                       α1 , α2 . . . αr            α1 , α2 . . . αr , αr+1 , αr+2 . . . αr+s

It is obvious, that, without the aid of my system of umbral or biliteral notation,
this important theorem could not be made the subject of statement without an
enormous periphrasis, and could never have been made the object of distinct
contemplation or proof.
   To return to the more immediate object of this communication, suppose that
we have any binary function of two sets of quantities, x1 , x2 . . . xn ; ξ1 , ξ2 . . . ξn , of
which the general term will be of the form cr,s × xr ξs ; according to the principles
of notation above laid down, nothing can be more natural than to represent cr,s
by the biliteral group ar αs ; the function in question will then take the form

                                                   ar αs · xr ξs ;
                                               X


the x’s and ξ’s denoting quantities, but the a’s and α’s mere umbrae. The
function may then be thrown under the convenient symbolical form

               (a1 x1 + a2 x2 + · · · + an xn )(α1 ξ1 + α2 ξ2 + · · · + αn ξn ).

So if we confine ourselves to quadratic functions, for which x1 , x2 . . . xn ; ξ1 , ξ2 . . . ξn
become respectively identical, the general symbolical representation of any such
will be
                            (a1 x1 + a2 x2 + · · · + an xn )2 .
The complete determinant will be denoted by
                                            (                  )
                                                a1 , a2 . . . an
                                                                 ,
                                                α1 , α2 . . . αn

and any minor determinant of the rth order by
                                           (                       )
                                              a1 , a2 . . . ar
                                                                       ,
                                             aθ1 , aθ2 . . . aθr
                                                                                                        p. 244




                                                       248
    where θ1 , θ2 . . . θr are some certain r distinct numbers taken out of the series
1, 2, 3 . . . r. Suppose now that we have

                                    U = (a1 x1 + a2 x2 + · · · + an xn )2

linearly transformable into

                                     V = (b1 y1 + b2 y2 + · · · + bn yn )2 ,

by means of the n equations

                           x1 = a1 b1 · y1 + a1 b2 · y2 + · · · + a1 bn · yn ,
                           x2 = a2 b1 · y1 + a2 b2 · y2 + · · · + a2 bn · yn ,
                                ············
                          xn = an b1 · y1 + an b2 · y2 + · · · + an bn · yn ,

in which equations, be it observed, each coefficient ar bs is a single quantity,
perfectly independent of the quantities denoted generally by ar as , br bs , which
enter into U and V . Our object is to be able to express the minor determinant
                                                  (                          )
                                                      bk1 , bk2 . . . bkr
                                                                                 ,
                                                       bl1 , bl2 . . . blr

in which the one group of distinct numbers k1 , k2 . . . kr may either differ wholly
from, or agree wholly or in part with the other group of distinct numbers
l1 , l2 . . . lr , under the form of
                                            ((                             )            )
                                        X            aθ1 , aθ2 . . . aθr
                                                                                 ×Q .
                                                     bϕ1 , bϕ2 . . . bϕr
                                                                                                                        
                                                                                                         θ1 ,θ2 ...θr
The particular value of Q corresponding to each double group,                                            ϕ1 ,ϕ2 ...ϕr        , may
                                           
                             θ1 ,θ2 ...θr
be denoted by Q              ϕ1 ,ϕ2 ...ϕr       ; so that our problem consists in determining the
                               
                 θ1 ,θ2 ...θr
value of Q       ϕ1 ,ϕ2 ...ϕr       in the equation
        (                           )            (                               !        (                   ))
         bk1 , bk2 . . . bkr                           θ1 , θ2 . . . θr                aθ1 , aθ2 . . . aθr
                                        =
                                            X
                                                     Q                               ×                               .
          bl1 , bl2 . . . blr                          ϕ1 , ϕ2 . . . ϕr                aϕ1 , aϕ2 . . . aϕr

Accordingly I enunciate that
                                        !        (                           )        (                   )
              θ1 , θ2 . . . θr                     ak1 , ak2 . . . akr             al1 , al2 . . . alr
            Q                               =                                    ×
              ϕ1 , ϕ2 . . . ϕr                     bθ1 , bθ2 . . . bθr             bϕ1 , bϕ2 . . . bϕr
                                                     (                       )         (                     )                 (2)
                                                   al1 , al2 . . . alr                 a , a . . . akr
                                                 +                                   × k1 k2                     ,
                                                   bθ1 , bθ2 . . . bθr                 bϕ1 , bϕ2 . . . bϕr

                                                              249
subject to one sole exception in the case of θ1 , θ2 . . . θr being identical with
ϕ1 , ϕ2 . . . ϕr ; namely, that for the terms (for such case) of the form          p. 245

                                                                  !
                                               θ , θ . . . θr
                                             Q 1 2              ,
                                               θ1 , θ2 . . . θr
the value to be taken is not that which the general formula would give, namely,
                                            (                         )2
                                             a , a . . . akr
                                           2 k1 k2                            ,
                                             bθ1 , bθ2 . . . bθr
but the half of this, that is simply the square of
                                            (                         )
                                                ak1 , ak2 . . . akr
                                                                          .
                                                aθ1 , aθ2 . . . aθr
                                      
The value of Q ϕθ11,ϕ,θ2 ...θr
                       2 ...ϕr
                               , it is obvious, contains only quantities of the form
ar · bs , which are coefficients in the equations of transformation, but none of
the form ar · as or br · bs ; showing that the syzygetic connexion between the
minor determinants of U and V of the same order is linear, as has been already
anticipatively announced.
   The problem which I have treated above is only a particular case of a more
general one, which may be stated as follows: given
                                   U = (a1 x1 + a2 x2 + · · · + an xn )2 ,
and supposing m linear equations to be instituted between x1 , x2 . . . xn , so that U
may be made a function of (n − m) letters only, to express any minor determinant
of the reduced form of U without performing the process of elimination between
the given equations. Let the given equations be written under the form
                    a1 an+1 x1 + a2 an+1 x2 + · · · + an an+1 xn = 0,
                    a1 an+2 x1 + a2 an+2 x2 + · · · + an an+2 xn = 0,
                                                                                      ············
                 a1 an+m x1 + a2 an+m x2 + · · · + an an+m xn = 0,
and let it be convened (which takes nothing away from the generality of these
equations) that an+r an+s shall signify zero for all values of r and s concurrently
greater than zero. Suppose that x1 , x2 . . . xm , being eliminated, U becomes of
the form
                    (bm+1 xm+1 + bm+2 xm+2 + · · · + bn xn )2 ;
and suppose that we wish to determine the value of the complete determinant of
this last function; it will be found to be
(                          )       (                                              )   (              )2
    bm+1 , bm+2 . . . bn            a1 , a2 . . . an , an+1 . . . an+m      a1 , a2 . . . am
                               =                                       ÷                                  ,
    bm+1 , bm+2 . . . bn            a1 , a2 . . . an , an+1 . . . an+m   an+1 , an+2 . . . an+m

                                                       250
the squared divisor being, as is obvious, a function only of the coefficients of the
transforming equations, and depending for its value upon the particular               p. 246
   m quantities selected for elimination. The dividend, on the contrary, is
independent of this selection, but involves the coefficients of the function combined
with the coefficients of transformation. This is the symbolical representation of
the theorem given by me in the postscript to my paper in the Cambridge and
Dublin Mathematical Journal for November 1850137 .
   Suppose, now, more generally that we wish to find any minor determinant.
The solution is given138 by the equation
(                               )   (                                       ) (
    bθm+1 , bθm+2 . . . bθm+s   aθ1 , aθ2 . . . aθn , aθm+1 , aθm+2 . . . aθm+s      a1 , a2 . . . am
                           =                                                    ÷
    bϕm+1 , bϕm+2 . . . bϕm+s  aϕ1 , aϕ2 . . . aϕm , aϕm+1 , aϕm+2 . . . aϕm+s    an+1 , an+2 . . . an+m
                                                                                      (3)
If we make n = 2γ and m = γ, and aγ+r aγ+s = 0 for all positive values of either
r or s, and aγ−i an+e = 0 for all values of i and e differing from one another, and
for equal values aγ−e aγ+e = −1, it will readily be seen that this last theorem
reduces to the one first considered; and on careful inspection it will be found,
that the solution given of the general question includes within it that presented
for the particular case in question. Such inclusion, however, I ought in fairness
to state is far from being obvious; and to demonstrate it exactly, and in general
terms, requires the aid of methods which my readers would probably find to
exceed their existing degree of knowledge or familiarity with the subject.
   The theorem above enunciated was in part suggested in the course of a
conversation with Mr Cayley (to whom I am indebted for my restoration to
the enjoyment of mathematical life) on the subject of one of the preliminary
theorems in my paper on Contacts in this Magazine.
   It is wonderful that a theory so purely analytical should originate in a geomet-
rical speculation. My friend M. Hermite has pointed out to me, that some faint
indications of the same theory may be found in the Recherches Arithmétiques of
Gauss. The notation which I have employed for determinants is very similar to
that of Vandermonde, with which I have become acquainted since writing the
above, in Mr Spottiswoode’s valuable treatise On the Elementary Theorems of
Determinants. Vandermonde was evidently on the right road. I do not hesitate
to affirm, that the superiority of his and my notation over that in use in the
ordinary methods is as great and almost as important to the progress of analysis,
as the superiority of the notation of the differential calculus over that of the
fluxional system. For what is the theory of determinants? It is an algebra upon
algebra; a                                                                                 p. 247
   calculus which enables us to combine and foretell the results of algebraical
operations, in the same way as algebra itself enables us to dispense with the
 137
       p. 136 above.
 138
       see p. 251 below.


                                           251
performance of the special operations of arithmetic. All analysis must ultimately
clothe itself under this form139 .
   I have in previous papers defined a “Matrix” as a rectangular array of terms,
out of which different systems of determinants may be engendered, as from
the womb of a common parent; these cognate determinants being by no means
isolated in their relations to one another, but subject to certain simple laws of
mutual dependence and simultaneous deperition. The condensed representation
of any such Matrix, according to my improved Vandermondian notation, will be
                                          (                    )
                                               a1 , a2 . . . an
                                                                .
                                               a1 , a2 . . . am

To return to the theorems of the text. Theorem (2) admits of being presented in
a more convenient form for the purposes of analytical operation, so as to become
relieved from all cases of exception appertaining to particular terms.
   The limitation to the generality of the expression for Q arises from our treating
                                          (                         )
                                              aθ1 , aθ2 . . . aθr
                                              aϕ1 , aϕ2 . . . aϕr

as identical with its equal,
                                          (                         )
                                          aϕ1 , aϕ2 . . . aϕr
                                                                        .
                                          aθ1 , aθ2 . . . aθr

If, however, we now convene to treat these two forms as distinct, so that in
theorem (2)        (                      ! (                     ))
                 X       θ1 , θ2 . . . θr     aθ1 , aθ2 . . . aθr
                     Q                     ×
                        ϕ1 , ϕ2 . . . ϕr     aϕ1 , aϕ2 . . . aϕr
will contain                                                                2
                           n(n − 1) · · · (n − r + 1)
                                

                                  1 · 2···r
terms, then we may write simply
                                  !       (                         )       (                   )
               θ1 , θ2 . . . θr               ak1 , ak2 . . . akr         al1 , al2 . . . alr
             Q                        =                                 ×                           ,
               ϕ1 , ϕ2 . . . ϕr               bθ1 , bθ2 . . . bθr         bϕ1 , bϕ2 . . . bϕr
                                                                                                        p. 248
 139
    Perhaps the most remarkable indirect question to which the method of determinants has
been hitherto applied is Hesse’s problem of reducing a cubic function of 3 letters to another
consisting only of 4 terms by linear substitutions–a problem which appears to set at defiance all
the processes and artifices of common algebra. I have succeeded in applying a method founded
upon this calculus to the linear reduction of a biquadratic function of two letters to Cayley’s
form x4 + mx2 y 2 + y 4 , and of a 5c function of two letters to the new form x5 + y 5 + (ax + by)5 .
This last reduction is effected by means of the properties of a certain other function of the 8th
degree connected with the given function of the 5th degree. See a paper on this subject in the
forthcoming May Number of the Cambridge and Dublin Mathematical Journal. [p. 191 above.]


                                                     252
   which equation is subject to no exception for the case of the θ’s and ϕ’s
becoming identical. As regards this theorem, it will not fail to strike the reader
that it ought to admit of verification; for that U may be derived from V in the
same manner as V from U if we express y1 , y2 . . . yn in terms of x1 , x2 . . . xn , by
solving the system of equations (2), which there is no difficulty in doing. In fact,
if we write
                    y1 = α1 β1 x1 + α1 β2 x2 + · · · + α1 βn xn ,
                                    y2 = α2 β1 x1 + α2 β2 x2 + · · · + α2 βn xn ,
                                       ············
                                    yn = αn β1 x1 + αn β2 x2 + · · · + αn βn xn ,
we shall obtain
                               (                                                    )       (               )
                                   a1 , a2 . . . ar−1 , ar+1 , ar+2 . . . an              a , a . . . an
                αr βs =                                                                 ÷ 1 2              .
                                    b1 , b2 . . . bs−1 , bs+1 , bs+2 . . . bn             b1 , b2 . . . bn

Accordingly we shall find
          (                             )              (                            !       (                   ))
             am1 , am2 . . . amr                             ψ1 , ψ2 . . . ψr             b , b . . . bψr
                                            =                                           × ψ1 ψ2
                                                 X
                                                           Q                                                         ,
              ap1 , ap2 . . . apr                            ω1 , ω2 . . . ωr             bω1 , bω2 . . . bωr

and                                        !       (                            )       (                    )
                 ψ1 , ψ2 . . . ψr                      am1 , am2 . . . amr            ap1 , ap2 . . . apr
               Q                               =                                    ×                            ;
                 ω1 , ω2 . . . ωr                      βψ1 , βψ2 . . . βψr            βω1 , βω2 . . . βωr
substituting for the α’s and β’s their symbolical equivalents given above, and
applying the theorem given below, we shall easily obtain
                                       !       (                                )       (                            )
             ψ1 , ψ2 . . . ψr                     am+1 , am+2 . . . amr               apr+1 , apr+2 . . . an
           Q                               =                                        ×
             ω1 , ω2 . . . ωr                     bψr+1 , bψr+2 . . . bψn             βpr+1 , βpr+2 . . . βpn
                                                   (                   )2
                                                 a , a . . . an
                                               ÷ 1 2                        .
                                                 b1 , b2 . . . bn

If, now, in the expression
(                          )            ((                           )(                         )(                         ))
    bk1 , bk2 . . . bkr                        ak1 , ak2 . . . akr        al1 , al2 . . . alr        aθ1 , aθ2 . . . aθr
                               =
                                   X
                                                                                                                                ,
     bl1 , bl2 . . . blr                       bθ1 , bθ2 . . . bθr        bϕ1 , bϕ2 . . . bϕr        aϕ1 , aϕ2 . . . aϕr
                                   (                         )
                                     aθ1 , aθ2 . . . aθr
we resubstitute for                                              its value in the form of
                                     aϕ1 , aϕ2 . . . aϕr
                                                   ((                           )    )
                                            X            bω1 , bω2 . . . bωr
                                                                                    Q ,
                                                         bψ1 , bψ2 . . . bψr



                                                                 253
                       (                        )
                         bk1 , bk2 . . . bkr
we shall obtain                                     under the form of
                          bl1 , bl2 . . . blr
                               (                           !     (                      ))
                                     ω1 , ω2 . . . ωr            b , b . . . bωr
                                                               × ω1 ω2                       ;
                         X
                                   R
                                     ψ1 , ψ2 . . . ψr            bψ1 , bψ2 . . . bψr
                                                                                                                p. 249
                ω1 ,ω2 ...ωr
   and R        ψ1 ,ψ2 ...ψr       must = 0, except for the case of ω1 , ω2 . . . ωr ; ψ1 , ψ2 . . . ψr
                                                                                                                 
being respectively identical with k1 , k2 . . . kr ; l1 , l2 . . . lr , for which case R kl11,k 2 ...kr
                                                                                             ,l2 ...lr
must be unity. I have gone through this calculation and verified the result; in
order to effect which, however, the following important generalization of theorem
(1) must be apprehended.
   Suppose two sets of umbrae,

                                    a1 , a2 . . . am+n ,         b1 , b2 . . . bm+n ,

and let r be any number less than m, and let any r-ary combination of the m
numbers 1, 2, 3 . . . m be expressed by q θ1 , q θ2 . . . q θm , where q goes through all
the values intermediate between 1 and µ, µ being
                                         m(m − 1) · · · (m − r + 1)
                                                                    ;
                                               1 · 2···r
then I say that the compound determinant,

a1 θ1 , a1 θ2 . . . a1 θm , am+1 , am+2 . . . am+n               a2 θ1 , a2 θ2 . . . a2 θm , am+1 , am+2 . . . am+n
b1 θ1 , b1 θ2 . . . b1 θm , bm+1 , bm+2 . . . bm+n               b2 θ1 , b2 θ2 . . . b2 θm , bm+1 , bm+2 . . . bm+n
                                                       ·········
                               aµ θ1 , aµ θ2 . . . aµ θm , am+1 , am+2 . . . am+n
                               bµ θ1 , bµ θ2 . . . bµ θm , bm+1 , bm+2 . . . bm+n

is equal to the following product,
                        (                                  )µ′ (                        )µ′′
                            am+1 , am+2 . . . am+n                 a1 , a2 . . . am+n
                                                                                                 ,          (4)
                            bm+1 , bm+2 . . . bm+n                 b1 , b2 . . . bm+n

where
           (m − 1)(m − 2) · · · (m − r + 1)                                 (m − 1)(m − 2) · · · (m − r)
   µ′′ =                                    ,                        µ′ =                                ;
                  1 · 2 · · · (r − 1)                                               1 · 2···r
when r = 1, we have the case already given in theorem (2), and of course µ′′ is
to be taken unity.
   This very general theorem is itself several degrees removed from my still
unpublished Fundamental Theorem which is a theorem for the expansion of the
products of determinants.                                                       p. 250


                                                           254
    Obs. The analogy upon which the extension of the Vandermondian notation
from simple to compound determinants is grounded, would be better apprehended
if the biliteral symbols of simple quantities were written with the umbral elements
disposed vertically, as ab , instead of horizontally, as ab; which latter is the method
for the purposes of typographical uniformity adopted in the text above. The
other mode is, however, much to be preferred, and is what I propose hereafter to
adhere to. For my two general umbrae, a, b, Vandermonde uses two numbers, one
set a-cock upon the other, as 54 . The objection to the use of numbers is apparent
as soon as it becomes necessary to treat of the mutual relations of diverse
systems of determinants, and his mode of writing the umbrae militates against
the perception of the most valuable algebraical analogies. The one important
point in which Vandermonde has anticipated me, consists in expressing a simple
determinant by two horizontal rows of umbrae one over the other. But the idea
upon which this depends is so simple and natural, that it was sure to reappear
in any well-constructed system of notation.




                                         255
                                                    38.
      Note on Quadratic Functions and Hyper-determinants
                       [Philosophical Magazine, I. (1851), p. 415]
                                                                                                        p. 251
  Permit me to correct an error of transcription in the MS. of my paper “On
Linearly Equivalent Quadratic Functions” in the last number of the Magazine.
The theorem [p. 246 above] marked (3), should read as follows:—
(                               )       (                                                                         )
    bθm+1 , bθm+2 . . . bθm+s             a1 , a2 . . . am , aθm+1 , aθm+2 . . . aθm+s , an+1 , an+2 . . . an+m
                                    =
    bϕm+1 , bϕm+2 . . . bϕm+s             a1 , a2 . . . am , aϕm+1 , aϕm+2 . . . aϕm+s , an+1 , an+2 . . . an+m
                                            (                       )2
                                             a1 , a2 . . . am
                                        ÷                                .
                                          an+1 , an+2 . . . an+m

I may take this opportunity of mentioning, that by extending to algebraical
functions generally a multiliteral system of umbral notation, analogous to the
biliteral system explained in the paper above referred to as applicable to quadratic
functions, I have succeeded in reducing to a mechanical method of compound
permutation the process for the discovery of those memorable forms invented by
Mr Cayley, and named by him hyper-determinants, which have attracted the
notice and just admiration of analysts all over Europe, and which will remain a
perpetual memorial, as long as the name of algebra survives, of the penetration
and sagacity of their author.




                                                    256
                                        39.
     On a Certain Fundamental Theorem of Determinants
               [Philosophical Magazine, II. (1851), pp. 142–145]
                                                                                    p. 252
   The subjoined theorem, which is one susceptible of great extension and
generalization, appears to me, and indeed from use and acquaintance (it having
been long in my possession) I know to be so important and fundamental, as
to induce me to extract it from a mass of memoranda on the same subject;
and as an act of duty to my fellow-labourers in the theory of determinants,
more or less forestall time (the sure discoverer of truth) by placing it without
further delay on record in the pages of this Magazine. Its developments and
applications must be reserved for a more convenient occasion, when the interest
in the New Algebra (for such, truly, it is the office of the theory of determinants
to establish), and the number of its disciples in this country, shall have received
their destined augmentation. In a recent letter to me, M. Hermite well alludes
to the theory of determinants as “That vast theory, transcendental in point of
difficulty, elementary in regard to its being the basis of researches in the higher
arithmetic and in analytical geometry.”
   The theorem is as follows:—Suppose that there are two determinants of the
ordinary kind, each expressed by a square array of terms made up of n lines and
n columns, so that in each square there are n2 terms. Now let n be broken up
in any given manner into two parts p and q, so that p + q = n. Let, firstly, one
of the two given squares be divided in a given definite manner into two parts,
one containing p of the n given lines, and the other part q of the same; and
secondly, let the other of the two given squares be divided in every possible way
into two parts, consisting of q and p lines respectively, so that on tacking on
the part containing q lines of the second square to the part containing p lines of
the first square, and the part containing p lines of the second square to the part
containing q of the first, we                                                       p. 253
   get back a new couple of squares, each denoting a determinant different from
the two given determinants; the number of such new couples will evidently be
                             n(n − 1) · · · (n − p + 1)
                                                        ;
                                    1 · 2···p
and my theorem is, that the product of the given couple of determinants is equal
to the sum of the products (affected with the proper algebraical sign) of each of
the new couples formed as above described. Analytically the theorem may be
stated as follows.
   Let               (                  )   (                  )
                       a1 , a2 . . . an       a1 , a2 . . . an
                                          ,                      ,
                       b1 , b2 . . . bn       β1 , β2 . . . βn

                                        257
according to the notation heretofore140 employed by me in the preceding numbers
of this Magazine, denote any two common determinants, each of the nth order,
and let the numbers θ1 , θ2 . . . θn be disjunctively equal to the numbers 1, 2 . . . n
and p + q = n; then will
                               (                      )     (                  )
                                   a1 , a2 . . . an         a1 , a2 . . . an
                                                          ×
                                   b1 , b2 . . . bn         β1 , β2 . . . βn
            (                                                ) (                                )
                a1 , a2 . . . ap , ap+1 . . . an             a1 , a2 . . . aq , aq+1 . . . an
=
    X
         ±                                            ×                                            .
           b1 , b2 . . . bp , βθp+1 , βθp+2 . . . βθn   βθ1 , βθ2 . . . βθq , bp+1 , bp+2 . . . bn
The general term under the sign of summation may be represented by aid of the
disjunctive equations

                  ϕ1 , ϕ2 . . . ϕn = 1, 2 . . . n,           ψ1 , ψ2 . . . ψn = 1, 2 . . . n,

under the form of
(aϕ1 b1 × aϕ2 b2 × · · · × aϕp bp )(aψp+1 bp+1 × aψp+2 bp+2 × · · · × aψn bn )
    × (aϕp+1 βθp+1 × aϕp+2 βθp+2 × · · · × aϕn βθn )(aψ1 βθ1 × aψ2 βθ2 × · · · × aψp βθp ).

1st. When ϕ1 , ϕ2 . . . ϕp = ψ1 , ψ2 . . . ψp , it will readily be seen, that for given
values of ϕ1 , ϕ2 . . . ϕp , the product of the third and fourth factors becomes
substantially identical with the general term of the determinant
                                            (                    )
                                                a1 , a2 . . . an
                                                                 ,
                                                β1 , β2 . . . βn

and consequently, making the system ϕ1 , ϕ2 . . . ϕp (or, which is the same thing,
its equivalent ψ1 , ψ2 . . . ψp ) go through all its values, we get back for the sum of
the terms corresponding to the equation

                                    ϕ1 , ϕ2 . . . ϕp = ψ1 , ψ2 . . . ψp ,
                                                                                                  p. 254
    the product of the determinants
                          (                     )               (                  )
                             a1 , a2 . . . an                    a1 , a2 . . . an
                                                      and                         .
                             b1 , b2 . . . bn                    β1 , β2 . . . βn

2nd. When we have not the equality above supposed between the ϕ’s and the
ψ’s, let
                   ϕp−η = ψp+k and ϕp+η = ψp−ζ ;
the corresponding term included under the Σ will contain the factor

                                      aϕp+η βθp+η × aψp−ζ βθp−ζ .
 140
       p. 242 above.


                                                      258
Now leaving ϕ1 , ϕ2 . . . ϕp , and ψ1 , ψ2 . . . ψp unaltered, we may take a system of
values θ1′ , θ2′ . . . θn′ , such that
                               ′                            ′
                              θp+η = θp−ζ ,                θp−ζ = θp+η ,

and for all other values of q except p + η, or p − ζ, θq′ = θq . The corresponding
new value of the general term so formed by the substitution of the θ′ for the θ
series, will be identical with that of the term first spoken of, but will have the
contrary algebraical sign, because the θ′ arrangement of the figures 1, 2, 3 . . . p is
deducible by a single interchange from the θ arrangement of the same, the rule
for the imposition of the algebraical sign plus or minus being understood to be,
that the term in which

                         βθp+1 , βθp+2 . . . βθn ;          βθ1 , βθ2 . . . βθp

enter into the symbolical forms of the respective derived couples of determinants,
has the same sign as, or the contrary sign to, that in which

                         βθp+1
                           ′   , βθp+2
                                   ′   . . . βθn′ ;         βθ1′ , βθ2′ . . . βθp′

so enter, according as an odd or an even number of interchanges is required to
transform the arrangement

                              θp+1 , θp+2 . . . θn ;        θ1 , θ2 . . . θp

into the arrangement
                             ′      ′
                            θp+1 , θp+2 . . . θn′ ;         θ1′ , θ2′ . . . θp′ .

I have therefore shown that all the terms arising from the expansion of the
products included under the sign of summation, for which the disjunctive identity
ϕ1 , ϕ2 . . . ϕp = ψ1 , ψ2 . . . ψp does not exist, enter into the final sum in pairs, equal
in quantity and differing in sign, which consequently mutually destroy, and that
the terms for which the said identity does exist together make up the sum
                          (                      )     (                       )
                              a1 , a2 . . . an         a1 , a2 . . . an
                                                     ×                  ;
                              b1 , b2 . . . bn         β1 , β2 . . . βn
                                                                                               p. 255
    which proves, upon first principles drawn direct from that notion of polar
dichotomy of permutation systems which rests at the bottom of the whole theory
of the subject, the fundamental, and, as I believe, perfectly new theorem, which
it is the object of this communication to establish.
    In applying the theorem thus analytically formulized, it is of course to be
understood that, under the sign Σ, permutations within the separate parts of a
given arrangement,
                           θp+1 , θp+2 . . . θn ; θ1 , θ2 . . . θp ,

                                                     259
are inadmissible, the total number of terms so included being restricted to

                           n(n − 1) · · · (n − p + 1)
                                                      .
                                  1 · 2···p
The theorem may be extended so as to become a theorem for the expansion of
the product of any number of determinants, and adapted so as to take in that
far more general class of functions known to Mr Cayley and myself under the
new name of commutants, of which determinants present only a particular, and
that the most limited instance.




                                      260
                                           40.
    On Extensions of the Dialytic Method of Elimination
                [Philosophical Magazine, II. (1851), pp. 221–230]
                                                                                         p. 256
   The theory about to be described is a natural extension of the method of
elimination presented by me ten years ago (in June, 1841) in the pages of
this Magazine, which I have been induced to review in consequence of the
flattering interest recently expressed in the subject by my friend M. Terquem,
and some other continental mathematicians, and because of the importance of
the geometrical and other applications of which it admits, and of the inquiries to
which it indirectly gives rise. We shall be concerned in the following discussion
with systems of homogeneous rational integral functions of a peculiar form, to
which for present purposes I propose to give the name of aggregative functions,
consisting of ordinary homogeneous functions of the same variables but of different
degrees, brought together into one sum made homogeneous by means of powers
of new variables entering factorially.
   Thus if F, G, H . . . L be any number of functions of any number of letters
x, y . . . t of the degrees m, m − ι, m − ι′ . . . m − (ι) respectively,
                                               ′
                           F + Gλι + Hµι + · · · + Lθ(ι)

will be an aggregative function of the variables entering into F, G, &c. and of
λ, µ . . . θ. I shall further call such a function binary, ternary, quaternary, and so
forth, according to the number of variables contained in the functions F, G, H,
&c. thus brought into coalition.
   It will be convenient to recall the attention of the reader to the meaning of
some of the terms employed by me in the paper above referred to.
   If F be any homogeneous function of x, y, z . . . t, the term augmentative of F
denotes any function obtained from F of the form

                                  xα y β z γ · · · tδ × F.

Again, if we have any number of such functions F, G, H . . . K of as many            p. 257
  variables x, y, z . . . t, and we decompose F, G, H . . . K in any manner so as to
obtain the equations

                   F = xa P1 + y b P2 + z c P3 + &c. · · · + td (P ),
                   G = xa Q1 + y b Q2 + z c Q3 + &c. · · · + td (Q),
                   H = xa R1 + y b R2 + z c R3 + &c. · · · + td (R),
                     ············
                   K = xa S1 + y b S2 + z c S3 + &c. · · · + td (S),

                                           261
and then form the determinant
                             P1 P2 P3 · · ·        (P )
                             Q1 Q2 Q3 · · ·        (Q)
                             R1 R2 R3 · · ·        (R) ,
                             ··· ··· ···            ···
                             S1 S2 S3 · · ·        (S)

this determinant, expressed as a function of x, y, z . . . t, is what, in the paper
referred to, I called a secondary derivee, but which for the future I shall cite by
the more concise and expressive name of a connective of the system of functions
F, G, H . . . K from which it is obtained. One prevailing principle regulates all
the cases treated of in this and the antecedent memoir, namely that of forming
linearly independent systems of augmentatives or connectives, or both, of the
given system whose resultant is to be found, of the same degree one with the
other, and equal in number (when this admits of being done) to the number of
distinct terms in the functions thus formed. The resultant of these functions,
treated as linear functions of the several combinations of powers of the variables
in each term, will then be the resultant of the given system clear of all irrelevant
factors. If the number of terms to be eliminated exceed the number of the
functions, the elimination of course cannot be executed. If the contrary be the
case, but the equality is restored by the rejection of a certain number of the
equations, the resultant so obtained will vary according to the choice of the
equations retained for the purpose of the elimination. The true resultant will not
then coincide with any of the resultants so obtained, but will enter as a common
factor into them all.
   The following simple arithmetical principles will be found applicable and
useful for quotation in the sequel:–
   (a) The number of terms in a homogeneous function of p letters of the mth
degree is
                             m(m + 1) · · · (m + p − 1)
                                                        .
                                     1 · 2···p
                                                                                       p. 258
   (b) The number of augmentatives of the (m + n)th degree belonging to a
function of p letters of the mth degree is

                          (n + 1)(n + 2) · · · (n + p − 1)
                                                           .
                                    1 · 2···p

(c) The number of solutions in integers (excluding zeros) of the equation a1 +
a2 + · · · + ap = k is
                        (k − 1)(k − 2) · · · (k − p + 1)
                                                         .
                               1 · 2 · · · (p − 1)




                                        262
To begin with the case of binary aggregatives. Let
                                                          ′
       Fm (x, y) + Fm−ι (x, y)λι + Fm−ι′ (x, y)µι + &c. · · · + Fm−(ι) (x, y)θ(ι) 
                                                                                         
                                                                                        
                                                     ι′
       Gn (x, y) + Gn−ι (x, y)λ + G
                                ι
                                             (x, y)µ + &c. · · · + Gn−(ι) (x, y)θ      (ι) 
                                                                                         
                                      n−ι′
                                                                                           
                                                                                               (A)
       ··················                                                                
                                                                                         
                                                                                         
                                                                                         
                                                     ′
       Kp (x, y) + Kp−ι (x, y)λι + Kp−ι′ (x, y)µι + &c. · · · + Kp−(ι) (x, y)θ(ι)
                                                                                         
                                                                                         

be a system of functions (whose Resultant it is proposed to determine) equal in
number to the variables x, y, λ, µ . . . θ, and similarly aggregative, that is having
only the same powers of λ, µ, &c. entering into them, but of any degrees equal
or unequal m, n . . . p. Let the number of the functions be r. Raise each of the
given functions by augmentation to the degree s, where

                     s = (m + n + · · · + p) − (ι + ι′ + · · · + (ι)) − 1,

the number of augmentatives of the several functions will be

                (s + 1) − m,        (s + 1) − n,              ··· ,     (s + 1) − p,

and the total number will therefore be

                               r(s + 1) − (m + n + · · · + p),

which
                   = (r − 1)(m + n + · · · + p) − r(ι + ι′ + · · · + (ι)).
Again, the number of terms to be eliminated will be the sum of the numbers
of terms in functions respectively of the sth, (s − ι)th, (s − ι′ )th, . . . (s − (ι))th
degrees, which are respectively

                  s + 1,    s + 1 − ι,   s + 1 − ι′ ,          ··· ,   s + 1 − (ι),
                                                                                                     p. 259
   and the number of these partial functions is r − 1. Hence the number of terms
to be eliminated is
         (r − 1){m + n + &c. + p − (ι + ι′ + &c. + (ι))} − (ι + ι′ + &c. + (ι))
               = (r − 1)(m + n + &c. + p) − r(ι + ι′ + · · · + (ι)),
which is exactly equal to the number of the augmentative functions. Hence the
Resultant141 of the given functions can be found dialytically by linear elimination,
and the exponent of its dimensions in respect to the coefficients of the given
functions will be the number

                                     (r − 1)Σm − rΣι,
 141
    The Resultant of a system of functions means in general the same thing as the left-hand
side of the final equation (clear of extraneous factors) resulting from the elimination of the
variables between the equations formed by equating the said functions severally to zero.


                                               263
as above found.
   The method above given may be replaced by another more compendious,
and analogous to that known by the name of Bezout’s abridged method for
ordinary functions of two letters. As the method is precisely the same whatever
the number of the functions employed may be, I shall for the sake of greater
simplicity restrict the demonstration to the case of three functions, U, V, W ,
whose degrees (if unequal, written in ascending order of magnitude) are m, n, p
respectively. Let
                         U = Fm (x, y) + Fm−ι (x, y)z ι ,
                          V = Gn (x, y) + Gn−ι (x, y)z ι ,
                          W = Hp (x, y) + Hp−ι (x, y)z ι .
Let θ, ω be taken any two numbers which satisfy in integers greater than zero
the equation θ + ω = m + 1, and let

                        Fm (x, y) = ϕm−θ · xθ + ϕm−ω · y ω ,
                        Gn (x, y) = γn−θ · xθ + γn−ω · y ω ,
                        Hp (x, y) = ηp−θ · xθ + ηp−ω · y ω ,
where the ϕ’s, γ’s, η’s may be always considered rational integer functions of x
and y; for every term in each of the functions Fm , Gn , Hp must either contain xθ
or y ω , since, if not, its dimensions in x and y would not exceed

                                 (θ − 1) + (ω − 1),

that is m − 1, whereas each term is of m conjoined dimensions, at least, in x
and y. Hence from the equations

                         U = 0,       V = 0,         W = 0,
                                                                                       p. 260
  by eliminating xω , y θ and z ι we obtain the connective determinant

                               ϕm−θ ϕm−ω Fm−ι
                               γn−θ γn−ω Gn−ι ,
                               ηp−θ ηp−ω Hp−ι

which will be of the degree

                              m + n + p − (θ + ω + ι),

that is of the degree (n + p − ι − 1) in x and y; and the number of such connectives
by principle (c) is p.
   Again, by augmentation we can raise each of the functions U, V, W to the
same degree as the connectives, and by principle (b) the number of such will be

                       n + p − m − ι,       p − ι,      n − ι,

                                        264
from U, V, W respectively, together making up the number

                                 2n + 2p − m − 3ι.

Hence in all we have 2n + 2p − 3ι equations; and the number of terms to be
eliminated will be, n + p − ι arising from Fm , Gn , Hp , and n + p − 2ι from
Fm−ι , Gn−ι , Hp−ι ; together making up the proper number 2n + 2p − 3ι.
   Each connective contains ternary combinations of the coefficients, namely one
of the coefficients belonging to that part of U, V, W which contains z ι , and two
coefficients from the other part: the dimensions of the resultant in respect of the
coefficients of the former will hence be readily seen to be equal to the number of
connectives + the number of terms in the augmentatives into which z ι enters,
that is, will equal m + n + p − 2ι; the total dimensions of the resultant in respect
to all the coefficients of U, V, W will be

                            3m + (2n + 2p − m − 3ι),

that is,
                                2m + 2n + 2p − 3ι;
and consequently, in respect to the coefficients of Fm ; Gn ; Hp , will be of

                    (2m + 2n + 2p − 3ι) − (m + n + p − 2ι),

that is, of m + n + p − ι dimensions. This result, which is of considerable
importance, may be generalized as follows.
  Returning to the general system (A), for which we have proved that the total
dimensions of the resultant are

                 (r − 1)(m + n + · · · + p) − r(ι + ι′ + · · · + (ι)),
                                                                                       p. 261
   let the coefficients of the column of partial functions
                                               ..
                               Fm ,    Gn ,     .,     Kp ,

be called the first set; the coefficients of the column
                                                 ..
                            Fm−ι ,    Gn−ι ,      .,    Kp−ι ,

the second set, and so forth; then the dimensions in respect of the 1st, 2nd
. . . (r − 1)th sets respectively are s, s − ι, s − ι′ . . . s − (ι), where

                   s = m + n + &c. + p − (ι + ι′ + &c. + (ι)).

The important observation remains to be made, that all the above results remain
good although any one or more of the indices of dimension of the partial functions

                                         265
in the system (A), as m − ι, m − ι′ , n − ι , &c., should become negative, provided
that the terms in which such negative indices occur be taken zero, as will be
apparent on reviewing the processes already indicated upon this supposition. If
we take
               m = n = · · · = p,       ι = ι′ = &c. = (ι) = m − ϵ,
the exponent of the total dimensions of the resultant becomes

                   (r − 1)rm − r(r − 2)(m − ϵ) = rm + r(r − 2)ϵ,

when ϵ = 0, this becomes mr, which is made up of 2m units of dimension
belonging to the coefficients of the first column, and of m belonging to each of
the (r − 2) remaining columns. Consequently, if we have

                                Fm (x, y) + ξλ + ξ ′ λ′ = 0,
                                Gm (x, y) + ηλ + η ′ λ′ = 0,
                                Hm (x, y) + ζλ + ζ ′ λ′ = 0,
                                Km (x, y) + θλ + θ′ λ′ = 0,

or any other number of equations similarly formed, the result of the elimination
is always of m dimensions only in respect of ξ, η, ζ, θ, or of ξ ′ , η ′ , ζ ′ , θ′ , and of 2m
in respect of the coefficients in F, G, H, K.
   I now proceed to state and to explain some seeming paradoxes connected with
the degree of the resultant of such systems of defective functions as have been
previously treated of in this memoir, as compared with the degree                               p. 262
   of the general resultant of a corresponding system of complete functions of
the same number of variables.
   In order to fix our ideas, let us take a system of only three equations of the
form                                                      
                           Fm (x, y) + Fm−ι (x, y)z ι = 0,
                                                          
                                                          
                             Gn (x, y) + Gn−ι (x, y)z ι = 0,                               (B)
                                                           
                              Hp (x, y) + Hp−ι (x, y)z = 0 
                                                         ι 

The resultant of this system found by the preceding method is in all of 2m +
2n + 2p − 3ι dimensions. But in general, the resultant of three equations of the
degrees m, n, p is of mn + mp + np dimensions.
   Now in order to reason firmly and validly upon the doctrine of elimination,
nothing is so necessary as to have a clear and precise notion, never to be let go
from the mind’s grasp, of the proposition that every system of n homogeneous
functions of n variables has a single and invariable Resultant. The meaning
of this proposition is, that a function of the coefficients of the given functions
can be found, such that, whenever it becomes zero, and never except when it
becomes zero, the functions may be simultaneously made zero for some certain
system of ratios between the variables. The function so found, which is sufficient

                                             266
and necessary to condition the possibility of the coexistence of the equality to
zero of each of the given functions, is their resultant, and by analogy they may
be termed its components. It follows that if R be a resultant of a given system
of functions, any numerical multiple of any power of R or of any root of R when
(upon certain relations being supposed to be instituted between the coefficients
of its components) R breaks up into equal factors, will also be a resultant. This
is just what happens in system (B) when m = n = p = ι; the resultant found by
the method in the text is of the degree 3m; the general resultant of the system
of three equations to which it belongs is of the degree 3m2 ; the fact being, that
the latter resultant becomes a perfect mth power for the particular values of
the coefficients which cause its components to take the form of the functions in
system (B).
    Suppose, however, that we have still m = n = p, but ι less than m, 6m − 3ι
will express the degree of the resultant of system (B); but this is no longer in
general an aliquot part of 3m2 , and consequently the resultant of system (B)
that we have found is no longer capable in general of being a root of the general
resultant. The truth is, that on this supposition the general resultant is zero; as
it evidently should be, because the values
                                  x              y
                                    = 0,           =0
                                  z              z
satisfy the equations in system (B), except for the case of m = ι; consequently the
resultant furnished in the text, although found by the same process, is something
of a different nature from an ordinary resultant; it                                     p. 263
   expresses, not that the system of equations (B) may be capable of coexisting,
but that they may be capable of coexisting for values of xz , yz other than 0 and
0. This is what I have elsewhere termed a sub-resultant. But there is yet a
further case, to which neither of the above considerations will apply. This is
when m, n, p are not equal, but p − ι = 0.
   On this supposition the degree of the resultant of (B) becomes 2m + 2n − p,
which in general will not be a factor of mn + mp + np; and in this case it will no
longer be true that the values xz = 0, yz = 0 will satisfy the system (B), inasmuch
as the last equation therein cannot so be satisfied. Now, calling the general
resultant R and the particular resultant R′ , if R′ should break up into factors so
as to become equal to (r′ )a × (s′ )b · · · (t′ )ν , it might be the case that R should
equal (r′ )α · (s′ )β · · · (t′ )γ , and there would be nothing in this fact which would
be inconsistent with the theory of the resultant as above set forth; but suppose
that R′ is indecomposable into factors, then it is evident that we must have
R = R′ · R′′ , and consequently that the existence of such a particular resultant
as R′ will argue the necessity of the existence of another resultant R′′ ; in other
words, the resultant so found cannot be in a strict sense the true and complete
resultant for the particular case assumed, and yet the process employed appears
to give the complete resultant, or at least it is difficult to see how the wanting

                                           267
factor escapes detection. To make this matter more clear, take a particular and
a very simple case, where m = 2, n = 2, p = ι = 1, so as to form the system of
equations
                      Ax2 + Bxy + Cy 2 + (Dx + Ey)z = 0
                                                           
                                                           
                                                           
                        A′ x2 + B ′ xy + C ′ y 2 + (D′ x + E ′ y)z = 0           (C)
                                                                   
                                                   lx + my + nz = 0
                                                                   

By virtue of my theorem, the degree of the resultant R′ is

                                   2(2 + 2 + 1) − 3 · 1 = 7,

but the resultant R of the system

                        Ax2 + Bxy + Cy 2 + (Dx + Ey)z + F z 2 = 0
                                                                           
                                                                 
                                                                 
                    A′ x2 + B ′ xy + C ′ y 2 + (D′ x + E ′ y)z + F ′ z 2 = 0     (D)
                                                                       
                                                       lx + my + nz = 0
                                                                       


which becomes identical with the former when F = 0, F ′ = 0 is of

                                    2 × 2 + 2 × 1 + 2 × 1,

that is, of 8 dimensions. Hence it is evident that when F = 0, F ′ = 0, R must
become R′ × R′′ .                                                                    p. 264
    It will be found in fact , that on the supposition of F = 0, F = 0, R
                               142                                         ′

becomes equal to N × R′ ; and accordingly, besides the portion R′ of the resultant
of system (C), found by the method in the text, there is another portion N
which has dropped through; but it may be asked, is N truly a relevant factor?
were it not so, the theory of the resultant would be completely invalidated; but
in truth it is; for N = 0 will make the equations in system (C), considered as a
particular case of system (D), capable of co-existing; the peculiarity, which at
first sight prevents this from being obvious, consisting in the fact that the values
of xz , yz which satisfy the three equations when N = 0 become infinite.
    Thus, finally, we have arrived at a clear and complete view of the relation of
the particular to the general resultant.
    The general resultant may be zero, in which case the particular resultant
is something altogether different from an ordinary resultant; or the particular
resultant may be a root of the general resultant, or it may be more generally
the product of powers of the simple factors, which enter into the composition of
the general resultant; or lastly, it may be an incomplete resultant, the factors
wanting to make it complete being such as when equated to zero, will enable the
components of the resultant to coexist, but not for other than infinite values of
certain of the ratios existing between the variables.
 142
       See the Author’s remarks below, p. 283.


                                                 268
   Without for the present further enlarging on the hitherto unexplored and
highly interesting theory of Particular Resultants, I will content myself with
stating one beautiful and general theorem relating to them; to wit, “if F = 0,
G = 0, &c. be a given system of equations with the coefficients left general, and
R be the resultant of F, G, &c., and if now the coefficients in F, G be so taken
that R comes to contain as a factor or be coincident with Rm , then will R′ = 0
indicate that (when the coefficients are so taken as above supposed) F = 0,
G = 0, &c. will be capable of being satisfied, not, as in general, by one only, but
by m distinct systems of values of the variables in F, G, &c., subject of course
to the possibility, in special cases, of certain of the systems becoming multiple
coincident systems.”
   I pass on now143 to the more recondite and interesting theory of the resultant
of Ternary Aggregative Functions, that is to say, functions of the form

               Fm (x, y, z) + Fm−ι (x, y, z)tι + &c. · · · + Fm−(ι) (x, y, z)t(ι) ,

which will be seen to admit of some remarkable applications to the theory of
reciprocal polars.




 143
       See the Author’s remarks below, p. 283.


                                                 269
                                                  41.
   On a Remarkable Discovery in the Theory of Canonical
             Forms and of Hyperdeterminants
                       [Philosophical Magazine, II. (1851), pp. 391–410]
                                                                                           p. 265
   In a recently printed continuation144 of a paper which appeared in the Cam-
bridge and Dublin Mathematical Journal, I published a complete solution of the
following problem. A homogeneous function of x, y of the degree 2n + 1 being
given, required to represent it as the sum of n + 1 powers of linear functions of
x, y. I shall prepare the way for the more remarkable investigations which form
the proper object of this paper, by giving a new and more simple solution of this
linear transformation.
   Let the given function be

   a0 x2n+1 + (2n + 1)a1 x2n y + 12 (2n + 1)(2n)a2 x2n−1 y 2 + · · · + a2n+1 y 2n+1 ,

and suppose that this is identical with

           (p1 x + q1 y)2n+1 + (p2 x + q2 y)2n+1 + &c. + (pn+1 x + qn+1 y)2n+1 .

The problem is evidently possible and definite, there being 2n + 2 equations to
be satisfied, and (2n + 2) quantities p1 , q1 , &c. for satisfying the same.
  In order to effect the solution, let

            q1 = p 1 λ 1 ,        q2 = p2 λ2 ,    &c. = &c.,        qn+1 = pn+1 λn+1 ,
                                                                                           p. 266
   we have then
                                        p2n+1
                                         1    + p2n+1
                                                 2    + · · · + p2n+1
                                                                 n+1 = a0 ,
                             p2n+1
                              1    λ1 + p2n+1
                                         2    λ2 + · · · + p2n+1
                                                            n+1 λn+1 = a1 ,
                             p2n+1
                              1    λ21 + p2n+1
                                          2    λ22 + · · · + p2n+1
                                                              n+1 λn+1 = a2 ,
                                                                   2

                                                                      ············
                             p2n+1
                              1    λn1 + p2n+1
                                          2    λn2 + · · · + p2n+1
                                                              n+1 λn+1 = an ,
                                                                   n

                    p2n+1
                     1    λn+1
                           1   + p2n+1
                                  2    λn+1
                                        2              n+1 λn+1 = an+1 ,
                                            + · · · + p2n+1 n+1

                                                                      ············
                p2n+1
                 1    λ2n+1
                       1    + p2n+1
                               2    λ2n+1
                                     2               n+1 λn+1 = a2n+1 .
                                          + · · · + p2n+1 2n+1


Eliminate p1 , p2 . . . pn+1 between the 1st, 2nd, 3rd . . . (n + 2)th equations, and it
is easily seen that we obtain

                  an+1 − an Σλ1 + an−1 Σλ1 λ2 · · · ± a0 λ1 λ2 · · · λn+1 = 0.
 144
       p. 203 above.


                                                  270
Again, eliminating in like manner p2n+11   λ1 , p2n+1
                                                 2              n+1 λn+1 between the
                                                      λ2 . . . p2n+1
2nd, 3rd . . . (n + 3)th equations, we obtain
                   an+2 − an+1 Σλ1 + · · · ∓ a1 λ1 λ2 · · · λn+1 = 0;
and proceeding in the same way until we come to the combination of (n + 1)th
. . . (2n + 2)th equations, and writing
          Σλ1 = s1 ,      Σλ1 λ2 = s2 ,            ··· ,       λ1 λ2 · · · λn+1 = sn+1 ,
we find
                   an+1 − an s1 + an−1 s2 · · · + a0 sn+1 = 0,
                   an+2 − an+1 s1 + an s2 · · · ∓ a1 sn+1 = 0,
                 an+3 − an+2 s1 + an+1 s2 · · · + a2 sn+1 = 0,
                                                                   ············
                a2n+1 − a2n s1 + a2n−1 s2 · · · + an sn+1 = 0.
Hence it is obvious that
                         (x + λ1 y)(x + λ2 y) · · · (x + λn+1 y)
is a constant multiple of the determinant
                       xn+1 , −xn y, xn−1 y 2 ,            · · · , ±y n+1
                       an+1 ,  an ,   an−1 ,               ··· ,    a0
                       an+2 , an+1 ,   an ,                ··· ,    a1 .
                         ···    ···     ···                         ···
                       a2n+1 , a2n ,  a2n−1 ,              ··· ,    an
                                                                                           p. 267
  Hence λ1 , λ2 . . . λn+1 are known, and consequently
                            p1 , p2 . . . pn+1 ,    q1 , q2 . . . qn+1
are known, by the solution of an equation of the (n + 1)th degree.
   Thus suppose the given function to be
             F = ax5 + 5bx4 y + 10cx3 y 2 + 10dx2 y 3 + 5exy 4 + 10f y 5
               = (p1 x + q1 y)5 + (p2 x + q2 y)5 + (p3 x + q3 y)5 ,
we shall have, by an easy inference from what has preceded,
                          (p1 x + q1 y)(p2 x + q2 y)(p3 x + q3 y)
equal to a numerical multiple of the determinant
                               x3 , −x2 y, xy 2 , −y 3
                               d,    c,     b,     a
                                                       .
                               e,    d,     c,     b
                               f,    e,     d,     c

                                              271
The solution of the problem given by me in the paper before alluded to presents
itself under an apparently different and rather less simple form. Thus, in the
case in question, we shall find according to that solution,

                        (p1 x + q1 y)(p2 x + q2 y)(p3 x + q3 y)

equal to a numerical multiple of the determinant

                            ax + by bx + cy cx + dy
                            bx + cy cx + dy dx + ey .
                            cx + dy dx + ey ex + f y

The two determinants, however, are in fact identical, as is easily verified, for the
coefficients of x3 and y 3 are manifestly alike; and the coefficient of x2 y in the
second form will be made up of the three determinants,

                        a b d         a c c          b b c
                        b c e ,       b d d ,        c c d ,
                        c d f         c e e          d d e

of which the latter two vanish, and the first is identical with the coefficient of x2 y
in the first solution. The same thing is obviously true in regard of the coefficients
of xy 2 in the two forms, and a like method may be applied to show that in all
cases the determinant above given is identical with the determinant of my former
paper, namely

                a0 x + a1 y    a1 x + a2 y   · · · an x + an+1 y
                a1 x + a2 y    a2 x + a3 y   · · · an+1 x + an+2 y
                                                                   .
                    ···            ···                   ···
               an x + an+1 y an+1 x + an+2 y · · · a2n x + a2n+1 y
                                                                                          p. 268
   Thus, then, we see that for odd-degreed functions, the reduction to their
canonical form of the sum of (n + 1) powers depends upon the solution of one
single equation of the (n + 1)th degree, and can never be effected in more than
one way.
   This new form of the resolving determinant affords a beautiful criterion for
a function of x, y of the degree 2n + 1 being composed of n instead of, as in
general, (n + 1) powers. In order that this may be the case, it is obvious that
two conditions must be satisfied; but I pointed out in my supplemental paper
on canonical forms, that all the coefficients of the resolving determinant must
vanish, which appears to give far too many conditions. Thus, suppose we have

    ax7 + 7bx6 y + 21cx5 y 2 + 35dx4 y 3 + 35ex3 y 4 + 21f x2 y 5 + 7gxy 6 + hy 7 .

The conditions of catalecticism, that is, of its being expressible under form of
the sum of three (instead of, as in general, four) seventh powers, requires that all

                                         272
the coefficients of the different powers of x and y must vanish in the determinant
                                y 4 −y 3 x y 2 x2 −yx3 x4
                                a    b        c    d   e
                                 b   c        d    e   f ;
                                 c   d        e    f   g
                                d    e       f     g   h
in other words, we must have five determinants,
                       a   b   c d       a       c d e         a   b   c e
                       b   c   d e       b       d e f         b   c   d f
                                   ,                   ,                   ,
                       c   d   e f       c       e f g         c   d   e g
                       d   e   f g       d       f g h         d   e   f h

                                a   b   d e            b   c d e
                                b   c   e f            c   d e f
                                            ,                    ,
                                c   d   f g            d   e f g
                                d   e   g h            e   f g h
all separately zero. But by my homaloidal law145 , all these five equations amount
only to (5−4)(5−3), that is, to 2. I may notice here, that a theorem substantially
identical with this law, and another absolutely identical with the theorem of
compound determinants given by me in this Magazine, and afterwards generalized
in a paper also published146 in this Magazine, entitled                             p. 269
    “On the Relations between the Minor Determinants of Linearly Equivalent
Quadratic Forms,” have been subsequently published as original in a recent
number of M. Liouville’s journal.
    The general condition of mere singularity, as distinguished from catalecticism,
that is, of the function of the degree 2n + 1, being incapable of being expressed
as the sum of n + 1 powers, is that the resolving resultant shall have two equal
roots; in other words, that its determinant shall be zero.
    Mr Cayley has pointed out to me a very elegant mode of identifying the two
forms of the resolving resultant, which I have much pleasure in subjoining. Take
as the example a function of the fifth degree, we have by the multiplication of
determinants,
 y 3 −y 2 x yx2 −x3   1 0                    0    0         y3       a       b        c
  a       b   c   d   x y                    0    0          0 ax + by bx + cy cx + dy
                    ×                                  =                                ,
   b      c   d   e   0 x                    y    0          0 bx + cy cx + dy dx + ey
   c      d   e   f   0 0                    x    y          0 cx + dy dx + ey ex + f y
which dividing out each side of the equation by y 3 , immediately gives the identity
required, and the method is obviously general.
 145
       p. 150 above.
 146
       p. 241 above.


                                                 273
   Turn we now to consider the mode of reducing a biquadratic function of two
letters to its canonical form, videlicet

                 (f x + gy)4 + (hx + ky)4 + 6m(f x + gy)2 (hx + ky)2 .

Let the given function be written

                             ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 .

Let
           g = f λ1 ,   k = hλ2 ,      mf 2 h2 = µ,    λ1 + λ2 = s1 ,      λ1 λ2 = s2 ,
then we have
                                                f 4 + h4 + 6µ = a,
                                   4f 4 λ1 + 4h4 λ2 + 6µ(2s1 ) = 4b,
                             6f 4 λ21 + 6h4 λ22 + 6µ(s21 + 2s2 ) = 6c,
                                4f 4 λ31 + 4h4 λ32 + 6µ(2s1 s2 ) = 4d,
                                        f 4 λ41 + h4 λ42 + 6µs22 = e.
                                                                                          p. 270
  Eliminating f and h between the first, second and third; the second, third
and fourth; and the third, fourth and fifth equations successively, we obtain

                                as2 − bs1 + c − µ(8s2 − 2s21 ) = 0,
                               bs2 − cs1 + d − µ(4s1 s2 − s31 ) = 0,
                             cs2 − ds1 + e − µ(8s22 − 2s21 s2 ) = 0.

Let now
                                       (2s21 − 8s2 )µ = v,
and we shall have
                                     as2 − bs1 + (c + v) = 0,
                                               v
                                                
                                   bs2 − c −       s1 + d = 0,
                                               2
                                     (c + v)s2 − ds1 + e = 0.
Hence v will be found from the cubic equation

                                    a     b   c+v
                                    2b 2c − v  2d            = 0,
                                   c+v    d     e

that is,
                                                a b c
                        v − v(ae − 4bd + 3c ) + b c d
                         3                       2
                                                                    = 0,
                                                c d e


                                               274
in which equation it will not fail to be noticed that the coefficient of v 2 is zero,
and the remaining coefficients are the two well-known hyperdeterminants, or, as
I propose henceforth to call them, the two Invariants of the form

                          ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 ;

be it also further remarked that
                                          1 2
                                                      
                                      v=8  s − s2 µ,
                                          4 1
in which equation the coefficient of 8µ is the Determinant or Invariant of

                                      x2 + s1 xy + s2 y 2 .

When v is thus found, s1 , s2 , and µ, being given by the equations in terms of v,
are known, and by the solution of a quadratic λ1 , λ2 become known in terms of
s1 , s2 , and f, h in terms of λ1 , λ2 , µ, and the problem is completely determined.
The most symmetrical mode of stating this method of solution is to suppose the
given function thrown under the form

                (f x + gy)4 + (f1 x + g1 y)4 + 6ε(f x + gy)2 (f1 x + g1 y)2 .

Then writing
                       (f x + gy)(f1 x + g1 y) = Lx2 + M xy + N y 2 ,
                                                                                                p. 271
   −v, the quantity to be found by the solution of the cubic last given, becomes
                                                       !
                                              M2
                                      8ε LN −              .
                                              4

   I shall now proceed to apply the same method to the reduction of the function

a0 x8 +8a1 x7 y+28a2 x6 y 2 +56a3 x5 y 3 +70a4 x4 y 4 +56a5 x3 y 5 +28a6 x2 y 6 +8a7 xy 7 +a8 y 8 ,

under the form of
             (p1 x + q1 y)8 + (p2 x + q2 y)8 + (p3 x + q3 y)8 + (p4 x + q4 y)8
                 + 70ε(p1 x + q1 y)2 (p2 x + q2 y)2 (p3 x + q3 y)2 (p4 x + q4 y)2 .

It will be convenient to begin, as in the last case, by taking

      q1 = p 1 λ 1 ,   q2 = p2 λ2 ,   q3 = p3 λ3 ,   q4 = p4 λ4 ,      εp21 p22 p23 p24 = m,

and

(x + λ1 y)(x + λ2 y)(x + λ3 y)(x + λ4 y) = x4 + s1 x3 y + s2 x2 y 2 + s3 xy 3 + s4 y 4 = U,

                                               275
we shall then have nine equations for determining the nine unknown quantities
of the general form

                     p81 λι1 + p82 λι2 + p83 λι3 + p84 λι4 + Mι m = aι ,

where ι has all values from 0 to 8 inclusive, and where
                                      1 · 2 · · · ι · 1 · 2 · · · (8 − ι)
                         Mι = 70
                                                  1 · 2···8
multiplied into the coefficient of y ι x8−ι in U 2 .
   Taking these nine equations in consecutive fives, beginning with the first,
second, third, fourth, fifth, and ending with the fifth, sixth, seventh, eighth,
ninth, we obtain the five equations following:

                   a0 s4 − a1 s3 + a2 s2 − a3 s1 + a4 s0 − mN1 = 0,
                   a1 s4 − a2 s3 + a3 s2 − a4 s1 + a5 s0 − mN2 = 0,
                   a2 s4 − a3 s3 + a4 s2 − a5 s1 + a6 s0 − mN3 = 0,
                   a3 s4 − a4 s3 + a5 s2 − a6 s1 + a7 s0 − mN4 = 0,
                   a4 s4 − a5 s3 + a6 s2 − a7 s1 + a8 s0 − mN5 = 0,

where
                    N1 = M0 s4 − M1 s3 + M2 s2 − M3 s1 + M4 ,
                    N2 = M1 s4 − M2 s3 + M3 s2 − M4 s1 + M5 ,
                    N3 = M2 s4 − M3 s3 + M4 s2 − M5 s1 + M6 ,
                    N4 = M3 s4 − M4 s3 + M5 s2 − M6 s1 + M7 ,
                    N5 = M4 s4 − M5 s3 + M6 s2 − M7 s1 + M8 .
                                                                                   p. 272
  Developing now U 2 , we have
                            35            5           5      5
        M0 = 70,     M1 =       M2 = 5s2 + s21 , M3 = s3 + s1 s2 ,
                               s1 ,
                             2            2           2      2
                                   5      5                     5
     M4 = 2s4 + 2s1 s3 + s2 , M5 = s1 s4 + s2 s3 , M6 = 5s2 s4 + s23 ,
                          2
                                   2      2                     2
                              35
                         M7 = s3 s4 , M8 = 70s24 .
                               2
Hence
                           N1 = 72s4 − 18s1 s3 + 6s22 ,
                                          9          3
                           N2 = 18s1 s4 − s21 s3 + s1 s22 ,
                                          2          2
                           N3 = 12s2 s4 − 3s1 s2 s3 + s32 ,
                                          9          3
                           N4 = 18s3 s4 − s1 s23 + s22 s3 ,
                                          2          2
                           N5 = 72s4 − 18s1 s3 s4 + 6s22 s4 .
                                   2



                                              276
Hence we have
                                s1                 s2                 s3
       N1 = 72I,     N2 = 72I      ,   N3 = 72I       ,   N4 = 72I       ,   N5 = 72Is4 ,
                                4                  6                  4
where it will be observed that I is the quadratic invariant of U .
  Making now
                                   72mI = v,
we shall have the five following equations:

                         a0 s4 − a1 s3 + a2 s2 − a3 s1 + (a4 − v) = 0,
                                                       v
                                                        
                      a1 s4 − a2 s3 + a3 s2 − a4 +         s1 + a5 = 0,
                                                       4
                                               v
                                                
                      a2 s4 − a3 s3 + a4 −         s2 − a5 s1 + a6 = 0,
                                               6
                                       v
                                        
                      a3 s4 − a4 +         s3 + a5 s2 − a6 s1 + a7 = 0,
                                       4
                         (a4 − v)s4 + a5 s3 − a6 s2 − a7 s1 + a8 = 0;

so that the problem reduces itself to finding v, which is found from the equation
of the fifth degree:

                       a0     a1           a2        a3      a4 − v
                       a1     a2           a3      a4 + v4     a5
                       a2     a3         a4 − v6     a5        a6        = 0,
                       a3   a4 + v4        a5        a6        a7
                     a4 − v   a5           a6        a7        a8
                                                                                              p. 273
   v, it will be observed, being 72 times the quadratic invariant of

                     (p1 x + q1 y)(p2 x + q2 y)(p3 x + q3 y)(p4 x + q4 y),

the function being supposed to be thrown under the form of

       Σ(p1 x + q1 y)8 + 70ε(p1 x + q1 y)2 (p2 x + q2 y)2 (p3 x + q3 y)2 (p4 x + q4 y)2 .

It is obvious that in the equation for finding v, all the coefficients being functions
of the invariable quantities p1 , q1 , &c., and ε, must be themselves invariants of
the given function; so that the determinant last given will present under one
point of view four out of the six invariants belonging to a function of the eighth
degree, and these four will be of the degrees 2, 3, 4, 5 respectively.147
    I shall now proceed to generalize this remarkable law, and to demonstrate the
existence and mode of finding 2n consecutively-degreed independent invariants
 147
    The reasoning in this paragraph seems of doubtful conclusiveness. It may be accepted,
however, as a fact of observation confirmed and generalized by the subsequent theorem, that
the coefficients are invariants.


                                             277
of any homogeneous function of the degree 4n, and of n + 1 consecutively-
even-degreed independent invariants of any homogeneous function of the degree
4n + 2; a result, whether we look to the fact of such invariants existing, or
to the simplicity of the formula for obtaining them, equally unexpected and
important, and tending to clear up some of the most obscure, and at the same
time interesting points in this great theory of algebraical transformations.
   In the first place, let me recall to my readers in the simplest form what is
meant by an invariant148 of a homogeneous function, say of two variables x and
y. If the coefficients of the function f (x, y) be called a, b, c . . . l, and if when for
x we put lx + my, and for y, nx + py, where lp − mn = 1, the coefficients of the
corresponding terms become a′ , b′ . . . l′ , and if

                                        I(a, b . . . l) = I(a′ , b′ . . . l′ ),

then I is defined to be an invariant of f .
   Let now f (x, y) be a homogeneous function in x, y of the 2ιth degree, and
write
                         d     d ι
                                
                      ξ    +η      f (x, y) + λ(ηx − ξy)ι = P,
                        dx    dy
                                   ι
                         d     d
                
                    ξ      +η           f (lx + my, nx + py) + λ(ηx − ξy)ι = P ′ ,
                        dx    dy
where ξ and η are independent of x, y, and lp − mn = 1.
  Let
                       x′ = lx + my,      y ′ = nx + py,
then
                     d     d    dx′ d      dy ′ d      dx′ d      dy ′ d
                ξ      +η    =ξ        + ξ         + η        + η         ,
                    dx    dy    dx dx′     dx dy ′     dy dx′     dy dy ′
                                                                                                   p. 274
   and if we now write

                                  lξ + mη = ξ ′ ,              nξ + pη = η ′ ,

we find
                                         d     d      d      d
                                    ξ      +η    = ξ′ ′ + η′ ′ .
                                        dx    dy     dx     dy
Again, from the equations between x′ , y ′ , x, y, we find

                        px′ − my ′                                     ly ′ − nx′
              x=                   = px′ − my ′ ,               y=                = ly ′ − nx′ ;
                         pl − mn                                       pl − mn
therefore
                        ηx − ξy = (pη + nξ)x′ − (mη + lξ)y ′ = η ′ x′ − ξ ′ y ′ .
 148
       Olim, Hyperdeterminant, Constant derivative.



                                                        278
Hence                                                             ι
                                           d        d
                                   
                         P ′ = ξ′            ′
                                               + η′ ′                  f (x′ , y ′ ) + λ(η ′ x′ − ξ ′ y ′ )ι .
                                          dx       dy
Again,
                               d      d     d                                 d     d     d
                                  = l ′ + n ′,                                  = m ′ + p ′.
                               dξ    dξ    dη                                dη    dξ    dη
Hence
                  ι                           ι                                   ι−1
             d                          d    d ′                   d ι ′      d
                                                                                                                        
                       P ′ = lι                  P + P ′ + ιlι−1 n
                                                     &c. +  n ι
                                                                           P,
             dξ                        dξ ′ dη ′                  dη ′       dξ ′
  ι−1
  d      d ′           d ι ′                                d ι−1 d ′
                                                             
           P =l m
               ι−1
                             P + {l p + (ι − 1)l mn}
                                      ι−1        ι−2
                                                                            P
  dξ    dη            dξ ′                                 dξ ′        dη ′
                                d ι ′
                                   
              + &c. + nι−1 p           P,
                               dη ′
                                               ···························
            ι                            ι                                    ι−1                                    ι
        d                           d                                 d                  d ′               d
                                                                                                               
                  ′                              ′
                 P =m      ι
                                               P + ιm        ι−1
                                                                   p                         P + &c. + pι                    P ′.
       dη                          dξ ′                              dξ ′               dη ′              dη ′
But P ′ being of ι dimensions in ξ ′ and η ′ , and also in x and y, each of the
equations above written will be of ι dimensions in x and y, and of no dimensions
in ξ ′ , η ′ ; in fact, the successive terms of the right-hand members of the above
ι + 1 equations will be multiples of the (ι + 1) quantities

                                   (x′ )ι ,          (x′ )ι−1 y ′ ,        (x′ )ι−2 y ′2 . . . (y ′ )ι .

Consequently a linear resultant may be taken of
                                          ι                      ι−1                           ι
                                   d                         d               d ′      d
                                                                                          
                                               P ′,                            P ...                  P ′,
                                   dξ                        dξ             dη       dη

treating xι , xι−1 y . . . y ι as independent, and as quantities to be eliminated; and
this, according to a well-known principle of elimination, will prove                   p. 275
   the linear resultant of the foregoing equations to be equal to the linear resultant
of
                           d ι ′          d ι−1 d ′             d ι ′
                                                               
                                 P  ,                P . . .          P,
                         dξ ′            dξ ′   dη ′           dη ′
multiplied by the determinant

                                                                            2 ι(ι − 1)n l
                                                                            1
                  lι         ιnlι−1                                                    2 ι−2               ···     nι
             lι−1 m lι−1 p + (ι − 1)mnlι−2                                                                 · · · nι−1 p
                                                                                                                        .
               ···             ···                                                    ···                  ···    ···
                                                                           2 ι(ι − 1)m
                                                                           1
               mι           ιmι−1 p                                                   ι−2 p2               ···     pι



                                                                      279
This last written determinant may be shown from the method of its formation to
                       ι(ι+1)
be equal to (lp − mn) 2 , that is, to unity, because lp − mn = 1. Again, since

                         x′ι = lι xι + ιlι−1 mxι−1 y + &c. + mι y ι ,

       x′ι−1 y ′ = lι−1 nxι + {lι−1 n + (ι − 1)lι−2 mn}xι−1 y + &c. + mι−1 py ι ,
                                     ···························
                           y ′ι = nι xι + ιnι−1 pxι−1 y + · · · + pι y ι ,
the resultant of                                 ι                     ι
                                            d                d
                                                                
                                                          ′
                                                      P ...                  P ′,
                                            dξ              dη
obtained by treating xι , xι−1 y . . . y ι as the eliminables, will be equal to the resul-
tant of the same functions when x′ι , x′ι−1 y ′ . . . y ′ι are taken as the eliminables149
multiplied by a power of the determinant

                                              lι          ···    mι
                                            lι−1 n        · · · mι−1 p
                                                                       ,
                                             ···          ···    ···
                                             nι           ···    pι

which determinant, like the last, is unity. Thus, then, we have succeeded in
showing that the resultant obtained by eliminating xι , xι−1 y . . . y ι between
                                ι                        ι−1                      ι
                           d                         d               d ′      d
                                                                             
                                        ′
                                      P,                               P ...             P′
                           dξ                        dξ             dη       dη

is equal to the resultant obtained by eliminating (x′ )ι , (x′ )ι−1 y ′ . . . y ′ι between
                                ι                        ι−1                        ι
                          d                       d                  d ′        d
                                                                              
                                     P ′,                                P ...             P ′;
                         dξ ′                    dξ ′               dη ′       dη ′
                                                                                                   p. 276
   or, which is evidently the same thing, the resultant obtained by eliminating
x , xι−1 y . . . y ι between
 ι

                                  ι                      ι−1                      ι
                             d                       d               d        d
                                                                            
                                       P,                              P ...             P;
                             dξ                      dξ             dη       dη
that is to say, this last resultant remains absolutely unaltered in value when for
x, y we write respectively

                                        lx + my,                    nx + py,
   For the statement of the general principle of the change of the variables of elimination, see
 149

my paper in the March Number, 1851, of the Camb. and Dub. Math. Jour. [p. 186 above].



                                                              280
provided that lp − mn = 1.
   Hence by definition this resultant is an invariant f (x, y), and λ being arbitrary,
all the separate coefficients of the powers of λ in this resultant must also be
invariants. I proceed to express this resultant in terms of λ and the coefficients
of (x, y). Let ϖ = 1 · 2 · 3 · · · ι and

                1 d ι          d ι
                                           
                        P =         f + λ(−y)ι = E1 ,
               ϖ dξ           dx
           1 d ι−1 d
                              ι−1
                               d        d
               
                        P =               f + λ(−y)ι−1 x = E2 ,
           ϖ dξ      dη       dx       dy
         1 d ι−2 d 2
                              ι−2  2
                               d          d
                
                        P =                   f + λ(−y)ι−2 x2 = E3 ,
         ϖ dξ     dη          dx         dy
                          ············
               1
                  ι         ι
                   d           d
                        P =         f + λxι = Eι+1 ;
               ϖ dη           dy
and
                                        1
      f (x, y) = a0 x2ι + 2ιa1 x2ι−1 y + (2ι)(2ι − 1)a2 x2ι−2 y 2 + &c. + a2ι y 2ι .
                                        2
We find, writing σλ for λ, where σ = 2ι(2ι − 1) · · · (ι + 1),
          1                              1
            E1 = a0 xι + ιa1 xι−1 y + ι(ι − 1)a2 xι−2 y 2 . . .
          σ                              2
                    1
                 + ι(ι − 1)aι−2 x2 y ι−2 + ιaι−1 xy ι−1 + aι y ι + λ(−y)ι ,
                    2
          1                              1
            E2 = a1 xι + ιa2 xι−1 y + ι(ι − 1)a3 xι−2 y 2 . . .
          σ                              2
                    1
                 + ι(ι − 1)aι−1 x2 y ι−2 + ιaι xy ι−1 + aι+1 y ι + λ(−y)ι−1 x,
                    2
          1
            E3 = a2 xι + ιa3 xι−1 y . . .
          σ
                    1
                 + ι(ι − 1)aι x2 y ι−2 + ιaι+1 xy ι−1 + aι+2 y ι + λ(−y)ι−2 x2 ,
                    2
               ············
       1
         Eι+1 = aι xι + &c. + λxι ;
       σ
                                                                                         p. 277
   accordingly, by eliminating
                                              1
                        xι ,       ιxι−1 y,     ι(ι − 1)xι−2 y 2 . . . y ι ,
                                              2




                                               281
we obtain as the required resultant150 ,

                       aι ± λ aι−1                  aι−2         ···      a0
                        aι+1 aι + λι                aι−1         ···      a1
                                                         λ
                        aι+2  aι+1              aι ± 1 ι(ι−1)    ···      a2       .
                                                      2
                           ···     ···               ···         ···     ···
                           a2ι    a2ι−1              ···         · · · aι + λ

 Inasmuch as all the coefficients of λ in this expression are invariants of f (x, y),
and there are no invariants of the first order, it is clear that the coefficient of λι
must be always zero, which is easily verified.
   Again, if ι is odd, the determinant remains unaltered if we write −λ for λ;
hence when f (x, y) is of the degree 4e + 2, all the coefficients of the odd powers
of λ disappear. Thus, then, our theorem at once demonstrates that a function of
x, y of the degree 4e has 2e invariants of all degrees from 2 up to 2e + 1 inclusive,
and that a function of x, y of the degree 4e + 2 has e + 1 invariants whose degrees
correspond to all the even numbers in the series from 2 to 2e + 2.
   But in order that the proposition, as above stated, may be understood in its
full import and value, it is necessary to show that these invariants are independent
of one another, which is usually a most troublesome and difficult task in inquiries
of this description, but which the peculiar form of our grand determinant enables
us to accomplish with extraordinary facility. In order to make the spirit of the
demonstration more apparent, take the case of a function of the twelfth degree,
whose coefficients, divided by the successive binomial numbers 1, 12, 12·11   2 , &c.
may be called
                             a, b, c, d, e, f, g, h, i, j, k, l, m.
Our grand determinant then takes the form                                                             p. 278

               g+λ   f               e            d           c          b       a
                h  g − λ6           f              e         d           c       b
                i    h            g + 15
                                       λ
                                                  f           e          d       c
                                                     λ
                j    i              h           g − 20       f           e       d  .
                k    j               i            h        g + 15
                                                                λ
                                                                         f       e
                l    k               j             i         h         g − λ6    f
                m    l              k              j          i          h      g+λ
 150
     Mr Cayley has made the valuable observation, that λ (given by equating to zero the above
determinant) may be defined by means of the equation
                                          ι
                            d d     d d
                                 −              {f (x, y) × ϕ(ξ, η)} = λϕ(x, y),
                           dx dη   dy dξ

ϕ being itself a certain rational integral form of a function of the ιth degree, the ratio of whose
coefficients would be given by virtue of the above equations as functions of λ and the coefficients
of f (x, y).


                                                   282
Here it will be observed that
                         a and m appear only             1 time,
                           b and l   ···                 2 times,
                          c and k    ···                 3 ...
                          d and j    ···                 4 ...
                          e and i    ···                 5 ...
                         f and h     ···                 6 ...
                                g    ···                 7 ...

Let now the coefficients be called

                              H2 , H3 , H4 , H5 , H6 , H7 .

H2 and H3 manifestly are independent.
  Again, if possible, let H4 = pH22 , then a and m would appear twice in H4 ,
contrary to the rule.
  Hence H4 is independent of H2 , H3 .
  For a similar reason H5 cannot depend on H2 , H3 .
  Again, if possible, let

                              H6 = pH23 + qH2 H4 + rH32 ;

H23 will contain b6 l6 , which by the rule cannot appear in H2 H4 or in H32 .
   Hence p = 0.
   Also H4 will contain b2 l2 × the coefficient of λ3 in
                              λ               λ        λ
                                                           
                           g+              g−       g+    ,
                              15              20       15
                                                                                       p. 279
   which is not zero. And H2 also contains bl; hence H2 H4 will contain b3 l3 .
But H3 will evidently not contain b3 or l3 , or b2 l or bl2 , nor can H6 contain b3 l3 ;
hence q = 0. Finally, H32 will contain c6 and k 6 , but H6 can only contain as to
these letters the combination c3 k 3 ; hence r = 0.
   Consequently H6 does not depend on H2 , H4 , H3 . As regards H2 , H3 , H4 , H5 , H6
not vanishing, this may be made at once apparent by making all the letters but
g vanish; the H’s then become identical with the coefficients of
                                         2            2 
                                     λ             λ              λ
                                                                   
                     (g + λ)  2
                                  g−            g+             g−    ,
                                     6             15             20

none of which are zero except that of λ6 . The same or a similar demonstration
may be extended to H7 and easily generalized; hence, then, this most unexpected



                                            283
and surprising law is fully made out.151
   To return to the subject of canonical forms, I have not found the method so
signally successful in its application to the 4th and 8th degrees, conduct to the
solution of other degrees, such as the 6th, 12th, or 16th, of all of which I have
made trial; possibly another canonical form must be substituted to meet the
exigency of these cases;152 and it may be remarked in general, that if we have a
function of the (2n)th degree, the canonical form assumed may be taken,

                                     Σ(p1 x + q1 y)2n + V ;

where V , in lieu of being the squared product of

                        (p1 x + q1 y), (p2 x + q2 y), . . . , (pn x + qn y),
                                                                                                        p. 280
   may be any hyperdeterminant, or (as I shall in future call such functions)
covariant of this product, understanding P (x, y) to be a covariant of f (x, y) when
P (lx + my, nx + py) stands in precisely the same relation to f (lx + my, nx + py)
as P (x, y) to f (x, y), provided only that lp − mn = 1. For the relation and
distinction between covariants and contravariants, see a short article of mine153 in
the Cambridge and Dublin Mathematical Journal for this month. In endeavouring
to apply the method of the text to the Sextic Function

            ax6 + 6bx5 y + 15cx4 y 2 + 20dx3 y 3 + 15ex2 y 4 + 6f xy 5 + gy 6 ,

thrown under the form
                                    Σ(px + qy)6 + 20εU 2 ,
 151
    This demonstration, however, does not extend to show that the coefficients of the powers
of λ may not possibly be dependents, that is, explicit functions of one another combined with
other invariants not included among their number, or of these latter alone. For example, in
the case of the 12th degree, we know by Mr Cayley’s law that there must be two invariants of
the 4th order. Our determinant gives only one of these. Call the other one K4 ; by the above
reasoning it is not disproved but that we may have

                              H6 = pH23 + qH2 H4 + rH32 + sH2 K4 .

I believe, however, that the H’s may be demonstrated without much difficulty to be primitive
or fundamental invariants. The law of Mr Cayley here adverted to admits of being stated in the
following terms:—The number of independent invariants of the 4th order belonging to a function
of x, y of the nth degree is equal to the number of solutions in integers (not less than zero) of the
equation 2x + 3y = n − 3. Vide his memorable paper (in which several numerical errors occur
against which the reader should be cautioned) “On Linear Transformations,” vol. I. Cambridge
and Dublin Mathematical Journal, new series. There is no great difficulty in showing, by aid of
the doctrine of symmetrical functions, that there can never be more than one quadratic or one
cubic invariant, and in what cases there is one or the other, or each, to any given function of
two variables. The general law, however, for the number of invariants of any order other than 2,
3, 4 remains to be made out, and is a great desideratum in the theory of linear transformations.
  152
      See the Postscript [p. 283] for a verification of this conjecture.
  153
      p. 200 above.


                                                284
where

       U = (p1 x + q1 y)(p2 x + q2 y)(p3 x + q3 y) = s0 x3 + s1 x2 y + s2 xy 2 + s3 y 3 ,

I obtain the following equations:

                as3 − bs2 + cs1 − ds0 = ε(162s20 s3 − 54s0 s1 s2 + 12s31 ),
                bs3 − cs2 + ds1 − es0 = ε(54s0 s1 s3 + 6s21 s2 − 36s0 s22 ),
               cs3 − ds2 + es1 − f s0 = ε(−54s0 s2 s3 − 6s1 s22 + 36s21 s3 ),
               ds3 − es2 + f s1 − gs0 = ε(−162s0 s23 + 54s1 s2 s3 + 12s32 ).

In these equations, if we call the quantities multiplied by ε respectively L, M, N, P ,
we shall find
                                 1         1
                         s3 L − s2 M − s1 N + s0 P = 0,
                                 3         3
and
                           s3 L − s2 M − s1 N + s0 P = I;
where I denotes the determinant, or, as I shall in future call such function (in
order to avoid the obscurity and confusion arising from employing the same word
in two different senses), the Discriminant,154 which is the biquadratic (and of
course sole) invariant of the cubic function

                               s0 x3 + s1 x2 y + s2 xy 2 + s3 y 3 .

The reduction of the function of the fourth degree to its canonical form may be
effected very easily by means of the properties of the invariants of                     p. 281
   the canonical form, as I have shown in the Cambridge and Dublin Mathematical
Journal. Accordingly I have endeavoured to ascertain whether the reduction of
the sixth degree might not be effected by a similar method.
   If we start with the form ax6 + by 6 + cz 6 + 90mx2 y 2 z 2 , where x + y + z = 0,
which is only another mode of representing the canonical form previously given,
we shall find that there are four independent invariants, of the second, fourth,
sixth and tenth degrees. Calling these H2 , H4 , H6 , H10 , and writing s1 , s2 , s3 for
 154
    “Discriminant,” because it affords the discrimen or test for ascertaining whether or not equal
factors enter into a function of two variables, or more generally of the existence or otherwise
of multiple points in the locus represented or characterized by any algebraical function, the
most obvious and first observed species of singularity in such function or locus. Progress in
these researches is impossible without the aid of clear expression; and the first condition of a
good nomenclature is that different things shall be called by different names. The innovations
in mathematical language here and elsewhere (not without high sanction) introduced by the
author, have been never adopted except under actual experience of the embarrassment arising
from the want of them, and will require no vindication to those who have reached that point
where the necessity of some such additions becomes felt.




                                              285
a + b + c, ab + ac + bc, abc it will be found, after performing some extremely
elaborate computations, that

              H2 = s2 − 270m2 ,
              H4 = 6ms3 + 45m2 s2 + 216m3 s1 + 891m4 ,
              H6 = 4s23 + 120s2 s3 m − (684s22 + 432s1 s3 )m2
                     + (13 · 27 · 64s3 − 64 · 81s1 s2 )m3 + 8 · 81 · 169s2 m4
                     + 7 · 128 · 729s1 m5 + 16 · 729 · 239m6 .

H10 is too enormously long to attempt to compute; but we can easily prove its
independent existence by making m = 0, in which case the (determinant, or, to
use the new term proposed, the) discriminant of ax6 + by 6 + cz 6 becomes the
product of the twenty-five forms of the expression
                                 1         1    1         1    2
                            (ab) 5 + (ac) 5 · 1 5 + (bc) 5 · 1 5 .155
                                                           1       1    1   1   2
 Now in general the value of such a product for α 5 + β 5 · 1 5 + γ 5 · 1 5 is obviously
of the form

              (α + β + γ)5 + αβγ{f (α + β + γ)2 + g(αβ + αγ + βγ)};

for when α = 0 or β = 0 or γ = 0, the product must become respectively
(β + γ)5 , (γ + α)5 and (α + β)5 . Moreover, without caring to calculate f, g,156
it is enough for our present purpose to satisfy ourselves that g cannot be zero,
as then the product would have a factor (α + β + γ)2 . Hence, then, on putting
α = bc, β = ac, γ = ab, we see that the discriminant, when m is 0, will be of p. 282
the form
                               s52 + f s22 s23 + gs33 s1 .
 155
     Such a product in the language of the most modern continental analysis is, I believe, termed
a Norm. If we suppose the general function of x, y of the 4th degree thrown under the form
Au4 + Bv 4 + Cw4 , where u + v + w = 0, and the general function of x, y, z of the 3rd degree
thrown under the form Au3 + Bv 3 + Cw3 + Dθ3 , where u + v + w + θ = 0, the theory of norms
will afford an instantaneous and, so to speak, intuitive demonstration of the respective related
theorems, and the discriminant (aliter determinant) of each such function is decomposable into
the sum of a square and a cube. Each of these forms is indeterminate, in either case there being
but two relations fixed between the coefficients A, B, C; A, B, C, D; and we may easily establish
the following singular species of algebraical porism. In the first case

                                 (ABC)2 : (AB + AC + BC)3 ,

and in the second case

                           (ABCD)3 : (ΣA2 B 2 C 2 − 2ABCDΣAB)2

are invariable ratios.
 156
     f = −625, g = 3125.



                                               286
But when m is 0, H4 vanishes, and there is no term s1 or s3 in H2 . Hence
evidently the discriminant H10 just found cannot be dependent on H2 , H4 , or
H6 ; nor is it possible to make
                                       H10 + pH25 + qH22 H6 ,
that is,
                                  (p + 1)s52 + f s22 s23 + gs33 s1 ,
a perfect square on account of g not vanishing; so there is no H5 upon which
H10 can depend. Hence, admitting, as there seems every reason to do, that the
number of invariants of a function of x, y of the degree m is m − 2, we find that
the four invariants in the case of the first degree are respectively of the second,
fourth, sixth, and tenth dimensions, a determination in itself, as a step to the
completion of the theory of invariants, of no minor importance.
   But it seems hopeless by means of these forms to arrive at the desired canonical
reduction. The forms, however, of H2 , H4 , H6 are very remarkable as not rising
above the first, first and second degrees respectively in s1 , s2 , s3 . Also H4 vanishes
when m = 0 and H4 has been obtained by putting
                                 ax6 + by 6 + cz 6 + 90mx2 y 2 z 2
under the form of
           Ax6 + 6Bx5 y + 15Cx4 y 2 + 20Dx3 y 3 + 15Ex2 y 4 + 6F xy 5 + Gy 6 ,
and taking the determinant
                                           A    B     C     D
                                           B    C     D     E
                                                              .
                                           C    D     E     F
                                           D    E     F     G
Consequently in general the vanishing of the above-written determinant will
express the condition that a function of the sixth degree may be decomposable
into three sixth powers. This also is true more generally. If F (x, y) be a function
of 2i dimensions, the vanishing of the resultant in respect to
                                           xi , xi−1 y . . . y i
(taken dialytically) of
                                 i                 i−1                 i
                             d                  d           d        d
                                                                 
                                      F,                      F ...           F
                            dx                 dx          dy       dy
will indicate that F admits of being decomposed into i powers of linear functions
of x, y.157
 157
     Such a function so decomposable may be termed meio-catalectic. Meio-catalecticism for
even-degreed functions is the analogue of singularity for odd-degreed functions.


                                                    287
   In consequence of the greater interest, at least to the author, of the preceding
investigations, I have delayed the insertion of the promised continuation of my
paper on extensions of the dialytic method, which will                               p. 283
   appear in a subsequent Number. I take this opportunity of correcting a
trifling slip of the pen which occurs towards the end158 of the paper alluded
to. The values of xz and yz become zero, and not infinite, when N = 0; and the
antepenultimate paragraph should end with the words “an incomplete resultant.”
The theorem also, in the last paragraph but one, should be stated more distinctly
as subject to an important exception as follows.
   Whenever the resultant of a system of equations F = 0, G = 0, &c. contains a
factor Rm , this will indicate that, on making R = 0, the given system of equations
will admit of being satisfied by m algebraically distinct systems of values of the
variables, except in those cases where there is a singularity in the forms of F, G,
&c., taken either separately, or in partial combination with one another. An
example will serve to make the meaning of the exception apparent. Let F, G, H
denote three quadratic equations in x and y, so that F = 0, G = 0, H = 0
may be conceived as representing three conic sections. Let R be the resultant
of F, G, H, and suppose the relations of the coefficients in F, G, H to be such
that R = R′2 ; then R′ = 0 will imply the existence of one or the other of the
three following conditions: namely, either that the three conics have a chord in
common, which is the most general inference; or, which is less general, that two
of the conics touch one another; or, which is the most special case of all, that
one of the conics is a pair of right lines.
   So, again, if we have two equations in x, and their resultant contains F 2 , this
may arise either from one of the functions containing a square factor, or from
their being susceptible, on instituting one further condition, namely of F = 0, of
having a quadratic factor in common between them.
   P.S. The conjecture made in the preceding pages has been since confirmed by
the discovery of a modification in the canonical form applicable to functions of
the sixth degree, which simplifies the theory in a remarkable manner. Assume
f (x, y), a function of the sixth degree, as equal to

                       au6 + bv 6 + cw6 ± muvw(u − v)(v − w)(w − u),

where u, v, w, linear functions of x and y, satisfy the equation

                                      u + v + w = 0;

then will the product of uvw be capable of being determined by means of
the solution of a quadratic equation, of the square root of whose roots the
 158
       p. 264 above.


                                           288
coefficients of uvw will be known linear functions. Thus by an affected quadratic,
a pure quadratic, and a cubic equation, the values of u, v, w may be completely
ascertained. The discussion of this theory, and of a general inverse method for
assigning the true (in the sense of the most manageable) Canonical Form for
functions of any even degree, will form the subject of a subsequent communication.




                                       289
                                              42.
             On the Principles of the Calculus of Forms
       [Cambridge and Dublin Mathematical Journal, VII. (1852), pp. 52–97]
                                                                                                     p. 284

                          Part I. Generation of Forms159
                       Section I. On Simple Concomitance.

   The primary object of the Calculus of Forms is the determination of the
properties of Rational Integral Homogeneous Functions or systems of functions:
this is effected by means of transformation; but to effect such transformation
experience has shown that forms or form-systems must be contemplated not
merely as they are in themselves, but with reference to the ensemble of forms
capable of being derived from them, and which constitute as it were an unseen
atmosphere around them. The first part of this essay will therefore be devoted
to the theory of the external relations of forms or form-systems; the second part
to the analysis of forms: that is to say, the first part will treat of the Generation
and affinities, and the second part of the Reduction and equivalences of forms.
   In its most crude and absolute, or, so to speak, archetypal condition a Rational
Integral Homogeneous Function may be regarded as a linear function of several
distinct and perfectly independent classes of variables.                              p. 285
   The first step towards the limitation of this very general but necessary con-
ception consists in imagining the total number of classes to become segregated
into groups, and certain correspondences to obtain between the variables of a
class in any group with some the variables in each other class of the same group.
The investigations in this and the subsequent section will be confined exclusively
to the theory of functions where the several classes of variables, if more than one,
all belong to a single group, so that the variables in one class have each their
respective correspondents in the remaining classes. Such a group may again be
conceived to become subdivided into sets each of the same number of variables,
and the corresponding variables in the different sets to become absolutely identi-
cal. This leads to the conception of a homogeneous function of related classes of
 159
     It may be well at the outset to give notice to my readers of the exact meaning to be attached
to the following terms: 1. The linear-transformations are supposed to be always taken such that
the modulus, that is, the determinant of the coefficients of transformation, is unity; or, as it
may be phrased, the transformations are uni-modular. 2. The word Determinant is restricted in
all cases to signify the alternate function formed in the usual manner from a group of quantities
arranged in square order. 3. The word Discriminant (typified by the prefix-symbol D) is used
to denote the determinant (usually but most perplexingly so called) of a homogeneous function
of variables. 4. The resultant of two or more homogeneous functions of as many variables is
the left-hand side of the final equation (in its complete form and free from extraneous factors)
which results from eliminating the variables between the equations obtained by making each of
the functions zero.


                                              290
variables of various degrees of exponency in respect to the several classes. The
relation of the different classes, if containing the same number of variables (in
which case the relation may be termed Simple) will be understood to be defined
by their being simultaneously subject to similar or contrary operations of linear
substitution; so that, for example, if x, y, z; ξ, η, ζ are two such classes, when
x, y, z are replaced by ax + by + cz, a′ x + b′ y + c′ z, a′′ x + b′′ y + c′′ z, respectively,
ξ, η, ζ will be, according to the species of the relation, subject to be at the same
time replaced either by

                  aξ + bη + cζ,      a′ ξ + b′ η + c′ ζ,     a′′ ξ + b′′ η + c′′ ζ,

or otherwise by

                αξ + βη + γζ,        α′ ξ + β ′ η + γ ′ ζ,   α′′ ξ + β ′′ η + γ ′′ ζ,

where

               1 0 0                     0 1 0                        0 0 1
        α=     0 b′ c′ ,         β=      a′ 0 c′ ,           γ=       a′ b′ 0 ,         &c.
               0 b′′ c′′                 a′′ 0 c′′                    a′′ b′′ 0

On the former supposition the related classes x, y, z, ξ, η, ζ will be said to be
cogredient, and on the latter supposition contragredient.160161162 If now we have
one or more functions of classes of variables so related, such function or system
of functions may have associated with it a concomitant, also made up of distinct
but related classes of variables, such classes being capable of being either greater
or fewer in number than the classes of the given function or system of functions.
   In the primitive function or system, as also in the concomitant, the related
classes may be all of the same species, or some of one and the others of the
contrary species. Even if we limit ourselves to the conception of a                  p. 286
   primitive function or system of functions with only one class of variables, its
concomitant may be composed of various classes of variables, in respect to some
of which it will be covariant with, and in respect to the others contravariant to,
the primitive function or system.163 This is an immense and most important
extension of the conception of a concomitant given in my preceding paper in this
Journal, and will be shown to have the effect of reducing the whole existing theory
under subjection to certain simple abstract and universal laws of operation.
 160
     See my paper in the previous number of this Journal [p. 199 above].
 161
     The germ of the notion of contragredience will be found in the immortal Arithmetic of the
great and venerable Gauss.
 162
     The relation here spoken of will be observed to be of a dynamical character, not referring to
the systems as they are in themselves, but to the movements to which they are simultaneously
subject.
 163
     And of course the concomitant may be an invariant to its originant in respect of one or
more systems of variables entering into the former.


                                               291
   The relation of concomitance is purely of form. A being a given form, B is
its concomitant, when A′ being derived from A by simultaneous substitutions
impressed upon the class of variables or upon each of the classes (if there be
more than one) in A, and B ′ from B by corresponding (coincident or contrary)
substitutions impressed upon the class or classes of variables in B, B ′ is capable
of being derived from A′ after the same law as B from A; or, as it may be
otherwise expressed, “functions are concomitant when their correlated linear
derivatives are homogeneous in point of form”.164
   This definition implies that one at least of the forms must be the most general
possible of its kind: in a secondary but very important sense, however, functions
obtained by impressing particular values or relations upon the quantities entering
into the primitive and its associate form, will still be called concomitant. Thus
x3 − y 3 will be termed a concomitant to x3 + y 3 , not that we can affirm that
(ax + by)3 − (cx + dy)3 , that is

          (a3 − c3 )x3 + 3(a2 b − c2 d)x2 y + 3(ab2 − cd2 )xy 2 + (b3 − d3 )y 3 ,

treated as a function of x and y, can be derived from (ax + by)3 + (cx + dy)3 ,
that is

          (a3 + c3 )x3 + 3(a2 b + c2 d)x2 y + 3(ab2 + cd2 )xy 2 + (b3 + d3 )y 3 ,

when ad − bc = 1 by the same law as (x3 − y 3 ) from (x3 + y 3 ), for the elements
for forming such comparison are wanting, but because x3 + y 3 and x3 − y 3 are
the correspondent particular values respectively assumed by

                               ax3 + 3bx2 y + 3cxy 2 + dy 3 ,

and its concomitant
                (ad2 + 2c3 − 3bcd)x3 − (6b2 d − 3c2 b − 3acd)x2 y
                   + (6ac2 − 3cb2 − 3cba)xy 2 − (a2 d + 2b3 − 3bca)y 3 ,

when
                        a = 1,        b = 0,         c = 0,     d = 1.
With the aid of this extended signification of the term concomitant (whether it
be a covariant or contravariant) we can in all cases speak (as otherwise we in
general could not) of the concomitant of a concomitant.                         p. 287
   The relation between systems of variables has been stated to be Simple
(whether they be cogredient or contragredient) when each variable in one system
corresponds with some one in each other. Compound relation arises as follows:–
Suppose x, y; ξ, η two independent systems of two variables each, and that the
  164
      Or, more generally, it may be said that concomitance consists in the persistence of morpho-
logical affinity.


                                               292
system of four variables u, v, w, t is subject to linear variations imitating, in the
way of cogredience or contragredience, those to which xξ, xη, yξ, yη are subject;
then u, v, w, t may be said to be cogredient or contragredient to the continued
systems x, y; ξ, η. If x, y; ξ, η be themselves cogredient, then a system of three
variables u, v, w may be cogredient or contragredient in respect to xξ, xη + yξ, yη,
and if x, y; ξ, η be coincident, u, v, w may be similarly related to x2 , xy, y 2 . The
illustration may easily be generalized, and it will be seen in the sequel that
its conception of compound-relation between systems of a differing number of
variables will greatly extend the power and application of the methods about
to be developed. Without having recourse to a formal definition, it is obvious
that the notion of a concomitant conveyed in my former paper in this Journal
lends itself without difficulty to the most general supposition which can be
made of functions between which any number of systems of related variables
are distributed, whatever such relation be, whether simple or compound, and
whether of cogredience or of contragredience. The proposition stated in my
last paper relative to a concomitant of the concomitant of a function being a
concomitant of the original still applies to concomitants in the wider sense in
which we now understand that term, and the species of each system of variables
in the second concomitant with respect to the species or either species (if there
be systems of both kinds in the primitive) will be determined upon the general
principle which determines the effect of concurrence and contrariety being made
to operate each upon itself or one in either order upon the other.
    The highest law and the most powerful in its applications which I have yet
discovered in the theory of concomitants may be expressed by affirming that
when several related classes of variables are present in any concomitant, a new
concomitant, derived from the former by treating one or any number of these
classes as independent of the remaining classes, will still be a concomitant of the
primitive. I shall quote this hereafter as the Law of Succession. This law, to
which I have been led up inductively, requires an extended examination and a
rigorous proof. It is the keystone of the subject, and any one who should suppose
that it is a self-evident proposition (as from the simplicity of the enunciation it
might be supposed to be) will commit no slight error.
    If ϕ(x, y . . . z) be any homogeneous form of function of x, y, . . . , z, every
homogeneous sum in the expansion by Taylor’s theorem of
                            ϕ(u + u′ , v + v ′ . . . w + w′ ),
                                                                                          p. 288
  which in fact, on making u′ = x, v ′ = y . . . w′ = z, becomes identical (to a
numerical factor pres) with
                                                      ι
                                    d     d    d
                             
                                 u    +v    +w             ϕ,
                                   dx    dy    dz
is what I have elsewhere termed an Emanant, and by a partial method I had
demonstrated that every invariant of such an emanant in respect to u, v . . . w,

                                          293
in which x, y . . . z are treated as constants, or vice versa, would give a covariant
of ϕ. The reason of this is now apparent. For it may easily be shown165 that
every emanant is in fact itself a covariant of the function to which it belongs
with respect to each of the related classes of variables which enter into it, or is
as it may be termed a double covariant. The law of Succession shows therefore
that a concomitant to an emanant from which one of the classes has disappeared
will be a covariant of the primitive in respect to the remaining class.
   In applying the law of Succession, great use can be made of a function of two
classes of letters which may be termed a Universal Mixed Concomitant; this is
xξ + yη + · · · + zζ, which has the property of remaining unaltered when any
linear substitution (for which the modulus is unity) is impressed upon x, y . . . z,
and the contrary one upon ξ, η . . . ζ.166
   If f (x, y) be any function of x, y, of the degree m, f + λ(xξ + yη)m will         p. 289
   be a mixed concomitant of f , it being evident that every function of concomi-
tants of a function is itself a concomitant of the same.
 165
     To demonstrate this it is only necessary to observe that if u, v, . . . , w, u′ , v ′ , . . . , w′ be
cogredient with themselves and with x, y, . . . , z, then

                                   ϕ(u + λu′ , v + λv ′ , . . . , w + λw′ )

will evidently be a concomitant of ϕ(x, y, . . . , z); and, λ being arbitrary, the coefficients of the
different powers of λ must be separately concomitants of ϕ(x, y, . . . , z); but these coefficients
are the emanants of ϕ.
 166
     Thus, if

                x = ax′ + by ′ + cz ′ ,       y = f x′ + gy ′ + hz ′ ,        z = lx′ + my ′ + nz ′ ,
                           ξ = (gn − hm)ξ ′ + (hl − f n)η ′ + (f m − gl)ζ ′ ,
                         η = (−nb + mc)ξ ′ + (−lc + na)η ′ + (−ma + lb)ζ ′ ,
                            ζ = (bh − cg)ξ ′ + (cf − ah)η ′ + (ag − bf )ζ ′ ,

then
                                  a       b     c
             xξ + yη + zζ =       f       g     h    (x′ ξ ′ + y ′ η ′ + z ′ ζ ′ ) = x′ ξ ′ + y ′ η ′ + z ′ ζ ′ .
                                  l       m     n
When the coefficients of transformation correspond to the direction-cosines between one system
of rectangular axes and another, the reciprocal system is identical with the direct system; so
that x, y, z; ξ, η, ζ on this particular supposition may be regarded indifferently as contragredient
or as cogredient; accordingly they may be made identical, and then x2 + y 2 + z 2 remains
invariable, which is the well-known characteristic of orthogonal transformation. It may be
observed here that there exists a special theory of concomitance limited to such species of
linear transformations, which may be termed Conditional Concomitance, and I have found in
several cases that the invariants of conditional concomitants turn out to be absolute invariants
of the primitive. Much more important is the remark that there exists a theory of universal
concomitants for an indefinite number instead of merely two systems of variables, as used in
the text. In the sequel it will be seen that the application of this universal concomitant (like
the touch of an enchanter’s wand) serves to transmute covariants into contravariants, and back
again, and causes single invariants to germinate and fructify into complete connected systems
of forms.



                                                        294
   Suppose now
                                    1
                f = axm + mbxm−1 y + m(m − 1)cxm−2 y 2 + &c.,
                                    2
the concomitant becomes
                                       1
(a + λξ m )xm + m(b + λξ m−1 η)xm−1 y + m(m − 1)(c + λξ m−2 η 2 )xm−2 y 2 + &c.
                                       2
Consequently if P be any concomitant of f , P ′ obtained from P by writing
a + λξ m , b + λξ m−1 η, &c. for a, b, &c., will still be a concomitant of f ; and by
Taylor’s theorem P ′ evidently equals

          m d               d           1      d          d      2
                                                                         
 P+ ξ           +ξ   m−1
                           η + &c. P +     ξm    + ξ m−1 η + &c. P + &c.
           da               db         1·2    da          db

If we take P an invariant of f , we have M. Hermite’s theorem167 for f (x, y), and
precisely the same demonstration applies to the general case of f (x, y . . . z). P ′
is, by virtue of the general rule, a contravariant of f in respect to ξ, η . . . ζ: if P
be taken a function containing one single system, and is also a contravariant to
f in respect to that system, P ′ will be a double contravariant; and if we make
the two systems in P ′ identical, we have the extension of M. Hermite’s theorem
alluded to by me in one of the notes168 to my last paper, wherein I have stated
that “I may be taken any covariant of the function”: as regards the purpose of
that statement, the word covariant was used in error for contravariant.
    The preceding method may be viewed as a particular application of the general
principle, that if U1 , U2 . . . Um be any m functions (whether concomitants any of
them of the others or not), then any concomitant of λ1 U1 + λ2 U2 + · · · + λm Um
being expressed as a function of λ1 , λ2 . . . λm , every coefficient in such expression
will be a concomitant of the system U1 , U2 . . . Um . Thus, for example, if U and V
be two quadratic functions of n variables x, y . . . z, the discriminant □(λU + µV )
will contain n + 1 terms, of which the coefficients of the first and last will be
□U and □V ; and every one of the (n + 1) coefficients will be a concomitant (of
course an invariant) of U and V . These (n + 1) invariants will in fact constitute
the fundamental scale of invariants to the system U and V , and every other
invariant of U                                                                           p. 290
    and V will be an explicit rational function of the (n + 1) terms of the scale.
In connexion with this principle may be stated another relative to any system
of homogeneous functions of a greater number of variables of the same class,
  167
      This theorem was first stated to me by Mr Cayley, who, I understand, derived it from
M. Eisenstein, under the form of a theorem of covariants, which of course it becomes on
interchanging x, y with −y, x. But as a theorem of covariants it could not be extended to
functions of more than two variables. M. Hermite appears to have discovered this theorem,
under its more eligible form, subsequently to, but independently of, M. Eisenstein.
  168
      p. 201 above, note *.


                                          295
namely, that if any set of the variables one less in number than the number of the
functions be selected at will, and any invariant of a given kind be taken of the
resultant of the functions in respect to the variables selected, all such invariants
so formed will have an integral factor in common, and this common factor will
be an invariant of the given system of functions.
   It will be convenient to speak hereafter of systems for which the march of the
linear substitutions is coincident as cogredient, and those for which the march is
contrary as contragredient systems.
   Suppose m cogredient classes of m variables, the determinant formed by
writing the m × m quantities in square order will evidently be a universal
covariant. Thus, take the two systems x, y; ξ, η. xη − yξ is a universal covariant,
and evidently therefore F , which I use to denote

                              ϕ(x, y) × ϕ(ξ, η) + λ(xη − yξ)m ,

will be a covariant to ϕ(x, y). Let ϕ(x, y) be of m dimensions; any invariant of
F will be an invariant of ϕ; thus, let the two systems x, y; ξ, η be treated as
perfectly independent, and take the discriminant of F (viewed as a function of
x, y, ξ, η), that is the resultant of the four functions
                                   dF      dF       dF       dF
                                      ,       ,        ,        ;
                                   dx      dy       dξ       dη
this resultant will be an invariant of ϕ; and λ being arbitrary, all the coefficients
of its different powers will be invariants of ϕ. We thus fall upon another theorem
of M. Hermite, namely that if
                                          ϕ(x, y) × ϕ(ξ, η)
                                    λ=                      ,
                                            (xξ − yη)m
the coefficients of the equation which will give the minimum values of λ are
invariants of ϕ. So more generally, any invariant of f (x, y, ξ, η) − λ(xξ − yη)m , f
being of the degree m in x, y and in ξ, η, will be an invariant of f ; and among
other invariants may be taken the discriminant obtained by treating x, ξ, y, η as
absolutely unrelated.
   If f be a function of various classes each containing n covariables, and if not
less than n of these classes be covariable classes, and after selecting at will any
n of such systems, as x1 , y1 . . . z1 ; x2 , y2 . . . z2 ; . . . ; xn , yn . . . zn , the symbolical
determinant
                                  d       d                 d
                                                 ···
                                dx1 dy1                   dz1
                                  d       d                 d
                                                 ···
                                dx2 dy2                   dz2
                                 ···     ··· ··· ···
                                  d       d                 d
                                                 ···
                                dxn dyn                   dzn

                                                296
                                                                                                    p. 291
    be expanded and written equal to D, then Dt f will be a concomitant of
f ; and, more generally, by selecting different combinations of the covariable
systems n and n together in every way possible, and forming corresponding
                                                                     ′
symbols of operation E, F . . . H, we shall have Dt E t . . . H (u) f , for all values of
t, t′ . . . (u), a covariant of f in respect to the classes so combined. This explains
and contains the whole pith and marrow of Mr Cayley’s simple but admirable
method of obtaining covariants and invariants (or, as termed by their author,
hyperdeterminants) to a function ϕ1 of a single system x1 , y1 . . . z1 ; he forms
similar functions ϕ2 . . . ϕµ of x2 , y2 . . . z2 ; . . . xµ , yµ . . . zµ , and uses the product
ϕ1 × ϕ2 × · · · × ϕµ as a function f of µ systems: the multiple covariant obtained
by operating thereupon becomes a simple covariant on identifying the different
classes of covariables introduced in the procedure.

                    Section II. On Complex Concomitance.

   We have hitherto been engaged in considering only a particular case of
concomitance, the true idea of which relates not to an individual associated form
(as such), but to a complex of forms capable of degenerating into an individual
form. Such a complex may be called a Plexus. A plexus of forms is concomitant
to a given form or combination of forms under the following circumstances.
   If (O) be the originant, meaning thereby the primitive form or system of forms,
and P the concomitant plexus made up of the µ forms P1 , P2 . . . Pµ , and if, when
by duly related linear substitutions, O becomes O′ , the plexus P becomes P ′ ,
made up of the forms P1′ , P2′ . . . Pµ′ , and if the plexus ′ P formed from O′ after the
same law as P from O be made up of the forms ′ P1 ,′ P2 . . .′ Pµ , then will each
form in either of the plexuses ′ P, P ′ be a linear function of all the forms in the
other plexus, and the connecting constants in every such linear function will be
functions of the coefficients of the substitution whereby O and P have become
transformed into O′ and P ′ .
   A function forming part of a concomitant plexus may be termed a concomi-
tantive. Concomitantives therefore usually have a joint relation to a common
plexus and a concomitant is only another name for an unique concomitantive.
Every plexus contains a definite number of concomitantives; in place of any one
of these may be substituted an arbitrary linear function of all the rest, but the
total number of independent forms sufficient and necessary to make the complete
plexus respond to the requirements of the definition will remain constant.
   If now we combine together the whole number of functions contained in one or
more plexuses concomitant to any given originant, all of the same degree relative
to any given selected system or systems of variables, and if the number of the
concomitantives so combined be exactly equal to the                                       p. 292
   number of terms in each, arranged as a function of the selected class or classes
of variables, then the dialytic resultant (obtained by treating each combination

                                              297
of the selected variables as an independent variable, and forming a determinant
in the usual manner), will be a concomitant to the given originant. This, which
is only the partial expansion of some much higher law, may be termed the “Law
of Synthesis.”
   Let f be any function of a single class of variables x1 , x2 . . . xn . Let χ represent
any product of these variables or of their several powers of any given degree r;
the number of different values of χ will be µ, where
                                         n(n + 1) · · · (n + r − 1)
                                µ=                                  ,
                                                1 · 2···r
and χ1 f, χ2 f . . . χµ f will form a covariantive plexus to f .
  Again, let D represent any product of the degree r of the symbols
                                          d          d       d
                                             ,          ...     ;
                                         dx1        dx2     dxn
D1 f, D2 f . . . Dµ f will also form a covariant plexus to f .
   The coefficients of connexion between the forms of either plexus depend in
an analogous manner upon the coefficients of the substitution supposed to be
impressed upon the variables, with the sole difference that every coefficient taken
from the line r and column s of the determinant of substitution which appears
in any coefficient of connexion of the one plexus is replaced by the coefficient
taken from the line s and column r in the corresponding coefficient of connexion
for the other plexus.
   Let f (x, y) be any function of x, y of the degree 2m; then
                                m                  m−1                 m
                            d                   d           d        d
                                                                  
                                     ,                        ,...,
                           dx                  dx          dy       dy
will form a covariantive plexus; thus, suppose

                f (x, y) = a1 x2m + 2ma2 x2m−1 y + · · · + a2m+1 y 2m ;

omitting numerical factors, the plexus will be composed of the (m + 1) lines
following:
                a1 xm      +ma2 xm−1 y      + · · · +am+1 y m ,
                a2 x m     +ma3 x  m−1 y    + · · · +am+2 y m ,
                  ···           ···          ···       ···
               am+1 xm +mam+2 xm−1 y + · · · +a2m+1 y m ,
and consequently, by the law of synthesis, the determinant

                                 a1              a2       · · · am+1
                                 a2              a3       · · · am+2
                                 ···             ···      ···     ···
                                am+1 am+2                 · · · a2m+1

                                                    298
is an invariant of f .                                                                p. 293
   When this determinant is zero, I have proved in my paper        169 on Canonical
Forms, in the Philosophical Magazine for November last, that f is resoluble into
the sum of m powers of linear functions of x and y. I shall hereafter refer to a
determinant formed in this manner from the coefficients of f as its catalecticant.
Mr Cayley was, I believe, the first to observe that all catalecticants170 are
invariants.
   Again, more generally, let f (x, y, ξ, η) be a function of the mth degree of x, y,
and of a like degree in respect of ξ, η, which are supposed to be cogredient with
x and y; then
                            f (x, y, ξ, η) + λ(xη − yξ)m
(say F ) will be a concomitant of f ; and therefore if we take the system
                                m                 m−1                 m
                            d                  d           d        d
                                                                
                                     F,                      F ...           F,
                           dx                 dx          dy       dy
which will be functions of ξ and η alone, and take their resultant, this resultant
will be an invariant of f . As a particular case of this theorem, let
                                                            m
                                            d     d
                                          
                                      f= ξ    +η                 ϕ,
                                           dx    dy
where ϕ is supposed to be a function of x and y only and of 2m dimensions, f
is a concomitant of ϕ, and therefore the invariant of f , obtained in the manner
just explained, will be an invariant of ϕ. Thus then we have an instantaneous
demonstration of the theorem given171 by me in the paper of the Philosophical
Magazine before named, namely, if

                  ϕ(x, y) = a1 x2m + 2ma2 x2m−1 y + · · · + a2m+1 y 2m ,

say, in order to fix the ideas,

                           = ax6 + 6bx5 y + 15cx4 y 2 + · · · + gy 6 ;

then the determinant
                                 a     b      c     d+λ
                                 b     c   d − 31 λ  e
                                 c  d + 3λ
                                         1
                                              e      f
                                d−λ    e      f      g
 169
      See p. 282 above.
 170
      But the catalecticant of the biquadratic function of x, y was first brought into notice as an
invariant by Mr Boole; and the discriminant of the quadratic function of x, y is identical with
its catalecticant, as also with its Hessian. Meicatalecticizant would more completely express
the meaning of that which, for the sake of brevity, I denominate the catalecticant.
  171
      p. 277 above.


                                                    299
(and the analogously formed determinant for the general case) will be an invariant
of ϕ. The general determinant so formed is peculiarly interesting, because it
furnishes when equated to zero the one sole equation necessary to be solved
in order to be able to effect the reduction of ϕ(x, y) to its canonical form, and
gives the means, irrespective of any other view of the theory of invariants, of
determining completely and absolutely the condition                                  p. 294
   of the possibility of two given functions of the same degree of x, y being
linearly transformable one into the other. This theorem will be obtained in a
more general manner in the following section. I only pause now to make the very
important observation, that not only is the determinant an invariant, but every
minor system172 of determinants that can be formed from it (there are of course
m such systems) is an invariantive plexus to the given function ϕ.
   The form under which this theorem presents itself suggests a theorem vastly
more general and of peculiar interest, as showing a connexion between the theory
of functions of a certain degree and of a certain number of variables with other
functions of a lower degree but of a greater number of variables. Here again, under
a different aspect, is reproduced the great principle of dialysis, which, originally
discovered in the theory of elimination, in one shape or another pervades the
whole theory of concomitance and invariants.
   Let ϕ represent any function of the degree pq (of any number, or, to fix the
ideas, say of three variables x, y, z); let the general term of ϕ be represented by
                                     pq(pq − 1) · · · 1
                                                                      (α, β, γ)xα y β z γ ,
                        (1 · 2 · · · α)(1 · 2 · · · β)(1 · 2 · · · γ)
where α + β + γ = pq, and (α, β, γ) represents a portion of the coefficient of
xα y β z γ .
 172
     These minor systems mean as follows:–the system of rth minors comprises all the distinct
determinants that can be got by striking out from the square array (which I call the Matrix)
from which the complete determinant is formed, any r lines and any r columns selected at will.
The last, or mth minor, is of course a system consisting of the coefficients of ϕ(x, y), and it is
evident that if ϕ(x, y . . . z) be any function of any number of variables x, y . . . z, the coefficients
will form an invariantive plexus to ϕ. The following remark as to the changes undergone by
the coefficients of ϕ when the variables undergo any substitution, is not without interest and
importance for the theory. Let
                                        x become f x + f ′ y + · · · + (f )z,
                                        y become gx + g ′ y + · · · + (g)z,
                                                   ············
                                        z become hx + h′ y + · · · + (h)z.
Then the coefficient of the highest power of x becomes ϕ(f, g . . . h), and the coefficient of the
term containing xr . . . z s becomes
                                         r                                           s
            ′ d   ′ d            d  ′                       d     d                 d
           f    +g    + ··· + h                &c. ×   (f ) + (g)    + · · · + (h)            ϕ(f, g . . . h).
             df    dg           dh                         df     dg               dh



                                                           300
   Let
                                         1 · 2···p
                                                                       xr y s z t = θr,s,t ,
                         (1 · 2 · · · r)(1 · 2 · · · s)(1 · 2 · · · t)
where r + s + t = p, so that there are as many θ’s as there are modes of                            p. 295
    subdividing p into three integral parts (zeros being admissible); that is 6 (p +              1

1)(p + 2)(p + 3). Then any product such as xα y β z γ may be divided in a variety
of ways into the product of q of these θ’s, and it may be shown that the entire
quantity
                                         pq(pq − 1) · · · 1
                                                                           (xα y β z γ )
                            (1 · 2 · · · α)(1 · 2 · · · β)(1 · 2 · · · γ)
                                          1 · 2···q
                                                                                               
           =Σ                                                                  (θ θ · · · θµr ) ,
                                                                                  m1 m2      mr
                    (1 · 2 · · · m1 )(1 · 2 · · · m2 ) · · · (1 · 2 · · · mr ) µ1 µ2
where m1 + m2 + · · · + mr = q. Consequently ϕ may be represented under the
form of a function of the degree q of 16 (p + 1)(p + 2)(p + 3) (say ι) variables
θ1 , θ2 . . . θι , and its general term will be of the form
                               1 · 2···q
                                                                    (α, β, γ){θµm11 θµm22 · · · θµmrr },
         (1 · 2 · · · m1 )(1 · 2 · · · m2 ) · · · (1 · 2 · · · mr )
where α, β, γ are the indices respectively of x, y, z, when the last factor is ex-
pressed as a function of these variables.173 Now if D be used to denote this
new representation of ϕ when θ1 , θ2 . . . θι are treated as absolutely independent
variables, and if we attach to it any universal concomitant, as (xξ + yη + zζ)p
admitting of being written under the form ω(θ1 , θ2 . . . θι ), wherein the coefficients
will be functions of ξ, η, ζ, then any invariant to D and ω, treated as two systems
of ι variables, will be a concomitant to ϕ, the original function in x, y, z.174 D
and ω may be termed respectively, for facility of reference, the Particular and
Absolute functions. Thus, for example, we take ϕ a function of x, y of the degree
4n, say
                      a0 x4n + 4na1 x4n−1 y + &c. + a4n+1 y 4n ,
and make p = 2n, q = 2, so that D becomes a quadratic function of (2n + 1)
variables obtained by making x2n = θ1 , x2n−1 y = θ2 . . . y 2n = θ2n+1 ,175 and the
concomitant ω, formed from (ξx + ηy)2n , becomes
                             θ1 ξ 2n + 2nθ2 ξ 2n−1 η + · · · + θ2n+1 η 2n ;
 173
      See Note (1) in Appendix. [p. 322 below.]
 174
      In fact D is a concomitant to ϕ, and ω to a power of the universal concomitant; the θ’s
forming a system of variables cogredient with the compound system xr y s z t , &c.; and it must
be well observed that the same substitutions which render D and ω respectively identical with
ϕ and a power of the universal concomitant, would render an infinite number of other functions
also coincident with the same; but none of these other functions would be concomitants. Herein
we see the importance of the definition and conception of compound relation; the θ system
being compound by relation with the x, y, z system, after the manner of cogredience.
  175
      A slight variation upon the method as above explained for the general case has been here
introduced inadvertently by writing x2n−1 y = θ2 , &c., in lieu of 2nx2n−1 y = θ2 , &c., which, as
it does not in any degree affect the reasoning, I have not deemed it worth while to alter.


                                                     301
then if we take R the quadratic invariant of ω, that is

                                                      1 · 2 · 3 · · · (2n) 1
                 R = θ1 θ2n+1 − 2nθ2 θ2n &c. ±                               (θn+1 )2 ,
                                                        (1 · 2 · · · n)2 2
                                                                                            p. 296
   it will readily be seen that the determinant of D + λR, treated as a quadratic
function of (2n + 1) variables, will give an invariant of ϕ, and this will be the
same as that obtained by the particular method above given. Thus, suppose

                        ϕ(x, y) = ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 .

Let
                                   x2 = θ1 , 2xy = θ2 , y 2 = θ3 ,
so that
                       D = aθ12 + 2bθ1 θ2 + cθ22 + 2cθ1 θ3 + 2dθ2 θ3 + eθ32 ,
                                                                                    θ22
               ω = (xξ + yη)2 = x2 θ1 + xyθ2 + y 2 θ3 ,              R = θ1 θ3 −        .
                                                                                    4
Then ∆ the discriminant of D + 2λR in respect to θ1 , θ2 , θ3

                                        a     b     c+λ
                                  =     b  c − 21 λ  d  ,
                                       c+λ    d      e

and I may remark that the relations between the several transformees of the
invariantive plexuses formed by the minor determinant systems of ∆ (in this
and in general for the case of an evenly-even index) may be found by treating
D + 2λR as a quadratic function of the variables (in this case θ1 , θ2 , θ3 ) and
applying the rule given by me in the Philosophical Magazine in my paper “On
the relation between the Minor Determinants of linearly-equivalent Quadratic
Forms.”176 This second method, however, is not immediately applicable to the
case of indices oddly even, that is of the form 4n + 2, to which the first method
applies, equally as to the case 4n; for if we make 2n + 1 = p and q = 2, ω being of
an odd degree, has no quadratic invariant; it has however a quadratic covariant,
which will be of the second degree in respect to θ1 , θ2 . . . θp+1 as well as in respect
to ξ, η; and if we call this R and take the discriminant of D + λR in respect to
the variables θ1 , θ2 . . . θp+1 , we shall obtain, as I am indebted to a remark of my
valued friend M. Hermite for bringing under my notice, a very beautiful and
interesting function of λ, of which all the coefficients will be contravariants of ϕ.
Thus, let

           ϕ = ax6 + 6bx5 y + 15cx4 y 2 + 20dx3 y 3 + 15ex2 y 4 + 6f xy 5 + gy 6 ,
                                                                                            p. 297
 176
       p. 241 above.


                                                302
   make
                   x 3 = θ1 ,   3x2 y = θ2 ,    3xy 2 = θ3 ,   y 3 = θ4 ,
so that
            D = aθ12 + 2bθ1 θ2 + cθ22 + 2cθ1 θ3 + 2dθ2 θ3 + 2dθ1 θ4 + gθ42
                  + 2f θ3 θ4 + eθ32 + 2eθ2 θ4 ,

                  ω = (xξ + yη)3 = θ1 ξ 3 + θ2 ξ 2 η + θ3 ξη 2 + θ4 η 3 ,
                                  3θ1 ξ + θ2 η θ2 ξ + θ3 η
                           R=                               ,
                                   θ2 ξ + θ3 η θ3 ξ + 3θ4 η

           −R = ξ 2 θ22 + η 2 θ32 + ξηθ2 θ3 − 9ξηθ1 θ4 − 3ξ 2 θ1 θ3 − 3η 2 θ2 θ4 .
Consequently the discriminant in respect to θ1 , θ2 , θ3 , θ4 of D − 2λR becomes

                         a         b     c − 3λξ 2 d − 9λξη
                         b     c + 2λξ 2 d + λξη e − 3λη 2
                                                            .
                     c − 3λξ 2 d + λξη e + 2λη 2      f
                     d − 9λξη e − 3λη 2     f          g

If this determinant be expanded as a function of λ, all the coefficients of the
various powers of λ will be contravariants to the given function ϕ. The term
involving λ4 is zero. Let ξ become −y and η become x, then the remaining terms
(abstraction made of the powers of λ) become covariants of ϕ. The first term
(the coefficient of λ3 ) becomes ϕ itself; the last term is the catalecticant, and
thus we see, in general, that for functions of x and y of an oddly-even degree,
a whole series of covariants may be interpolated between the function and its
catalecticant, the dimensions in respect of the coefficients of ϕ in arriving at each
step increasing by 1 unit and the degree in respect of the variables diminishing
by 2 units. This is consequently a much simpler and more available scale than
one with which I have been long previously acquainted, and which applies alike
to functions of any even degree.
   Thus, let ϕ(x, y) be of 2k dimensions; form all the even emanants of ϕ, which
will be all of the form
                                     d      d 2ι
                                             
                                  ξ    +η        ϕ,
                                    dx     dy
and take their respective catalecticants in respect to ξ and η. We shall in this
way obtain a regular scale of covariants interpolated between the Hessian of ϕ
(corresponding to ι = 1) and the catalecticant of ϕ (corresponding to ι = k). If ϕ
be of the degree 2k + 1, we shall have an analogous scale interpolated between the
Hessian of ϕ and its canonizant; the latter term denoting the function which is
the product of the k + 1 linear functions of x and y, the sum of whose (2k + 1)th
powers is identically equal to ϕ.177


                                            303
   By means of the Theory of the Plexus we may obtain various representa-          p. 298
   tions of the same invariant; thus, for example, if we take F a function of x, y
of the fifth degree and form its Hessian H, that is

                                         d2 F      d2 F
                                          dx2      dxdy    ,
                                         d2 F      d2 F
                                         dydx       dy 2
this will be a function of the sixth degree in x, y, and of the two orders in the
coefficients. If we combine the two plexuses

                             dF dF            d2 H d2 H d2 H
                               ,   ;              ,    ,     ,
                             dx dy            dx2 dxdy dy 2

we shall have five equations between which x4 , x3 y, x2 y 2 , xy 3 , y 4 may be elimi-
nated dialytically; the resultant will be of the 2 + 3 · 2, that is the eighth order in
the coefficients, and of the form □F − I2 , where □F and I4 are respectively the
determinant and quintic invariant of F , each affected with a proper numerical
multiplier (the “B − A2 ” of my supplemental178 essay on canonical forms) which,
as Mr Cayley has remarked, may also be represented by the resultant of P ; dQ       dx ,
dQ
 dy where  P  and  Q are respectively the quadratic  and    cubic invariants in respect
to ξ and η of
                                     d       d 4
                                              
                                  ξ     +η        F.
                                    dx      dy
   It will be well at this point to recapitulate in brief a method of elimination
applicable to certain systems of functions published by me many years since
in the Philosophical Magazine, and to compare this method with that afforded
by the theory of the plexus for finding an invariant for each of the very same
systems, possessing all the external characters, formed in a precisely similar
manner and not impossibly identical with, the resultant of every such system. I
shall devote my first moments of leisure to the ascertainment of this last most
important point, as to the identity or otherwise of the plexus-invariant with the
resultant. Take the case of three functions of x, y, z (say ϕ, ψ, ω) each of the
same degree n; to fix the ideas, suppose n = 3: there are two purely algebraical
processes (modifications of the same method and leading to identical results) by
which the resultant of ϕ, ψ, ω may be found. I shall call these processes the first
and second respectively.
   First process: Write

 ϕ = x2 P + yQ + zR,             ψ = x2 P ′ + yQ′ + zR′ ,      ω = x2 P ′′ + yQ′′ + zR′′ ,
 177
       See Note (2) in Appendix. [p. 322 below.]
 178
       p. 205 above.



                                                304
decompositions which may be effected in an infinite variety of manners, so that
P, Q, R shall be integer functions of x, y, z; take the linear resultant of ϕ, ψ, ω,
in respect to x2 , y, z, which call H2,1,1 ; this will evidently be                    p. 299
   of 9−4, that is, of 5 dimensions. Form analogously the functions H1,2,1 , H1,1,2 ; H2,1,1 , H1,2,1
constitute an auxiliary system of functions which vanish when ϕ, ψ, ω vanish
together; combine this auxiliary system with the augmentative system

                          x2 ϕ, y 2 ϕ, z 2 ϕ, xyϕ, yzϕ, zxϕ,
                          x2 ω, y 2 ω, z 2 ω, xyω, yzω, zxω,
                          x2 ψ, y 2 ψ, z 2 ψ, xyψ, yzψ, zxψ.

We shall thus have in all 3 + 3 × 6, that is, 21 functions into which the 21 terms
x5 , x4 y, x4 z, &c. enter linearly: the linear resultant of these 21 functions is the
resultant of ϕ, ψ, ω, clear of all extraneousness.
    Second process: Write

 ϕ = x3 P + yQ + zR,            ψ = x3 P ′ + yQ′ + zR′ ,             ω = x3 P ′′ + yQ′′ + zR′′ ,

and, as before, take the linear resultant H3,1,1 , which will however be of 9 − 5,
that is, of only 4 dimensions.
  Again, take

ϕ = x2 L + y 2 M + zN,          ψ = x2 L′ + y 2 M ′ + zN ′ ,         ω = x2 L′′ + y 2 M ′′ + zN ′′ ,

and form the determinant H2,2,1 ; we shall thus have the auxiliary system

               H3,1,1 ,     H1,3,1 ,    H1,1,3 ,   H2,2,1 ,     H2,1,2 ,    H1,2,2 .

Let this be combined with the augmentative system

                          xω, yω, zω;     xϕ, yϕ, zϕ;         xψ, yψ, zψ.

Between these 6 + 9, that is, 15 functions, the 15 terms x5 , x4 y, x4 z, &c. may be
linearly eliminated, and the resultant thus obtained will be precisely the same as
that got by the preceding process.
   Here we have 6 auxiliaries and 6 augmentatives; the auxiliaries are of three
dimensions in respect to the coefficients of ϕ, ψ, ω; the augmentatives of one di-
mension only; in the former process there were 3 auxiliaries and 18 augmentatives,
6 × 3 + 9 = 27 = 3 × 3 + 18.
   Now let this method be compared with the following:
   First process: Take the 18 augmentatives x2 ϕ, x2 ω, x2 ψ, &c. as in the first
process of the algebraical method above explained; but in place of the 3 auxiliaries
therein given, take another system of 9 as follows:                                  p. 300




                                               305
   Write the determinant
                                        dϕ      dϕ       dϕ
                                        dx      dy       dz
                                        dψ      dψ       dψ
                                                                  = R;
                                        dx      dy       dz
                                        dω      dω       dω
                                        dx      dy       dz
                                         dR           dR      dR
                                            ,            ,
                                         dx           dy      dz
form a concomitantive plexus; the 18 augmentatives form another; the linear
resultant of these two plexuses will be an invariant of ϕ, ψ, ω, and of precisely
the same dimensions as the resultant last found; if they are not identical it will
be indeed a matter of exceeding wonder, and even more interesting than if they
should be proved so to be.
   Second process: Combine the augmentative plexus

                        xω, yω, zω;           xϕ, yϕ, zϕ;         xψ, yψ, zψ,

with the differential plexus

                  d2 R        d2 R           d2 R        d2 R          d2 R         d2 R
                       ,           ,              ,           ,             ,            ,
                  dx2         dxdy           dy 2        dydz          dz 2         dzdx
we thus obtain a linear resultant in a manner precisely similar to that afforded
by the second process of our algebraical method.
   In general, if ϕ, ψ, ω be of the degrees n, n, n, as there are two algebraical
varieties of the linear method for finding the resultant, so are there two varieties of
the concomitantive method for finding the resembling invariant. In both methods
the augmentatives are identical; the only difference being in the auxiliary system.
   In the first process the augmentative system will be got by operating upon
each of the functions ϕ, ψ, ω, with the multipliers xn−1 , y n−1 , z n−1 , and the other
homogeneous products of x, y, z; the auxiliary system by operating upon R with
the symbolical multipliers
                                  n−2                 n−2                 n−2
                              d                    d                   d
                                                                 
                                         ,                    ,                     ,
                             dx                   dy                   dz

and the other homogeneous products of dx         , dy , dz of the degree n − 2.
                                               d d d

  In the second process the augmentative system is formed by the aid of the
multipliers xn−2 , y n−2 , z n−2 , &c., and the auxiliary system by aid of
                             n−1                 n−1                 n−1
                         d                    d                   d
                                                            
                                    ,                    ,                      ,   &c.
                        dx                   dy                   dz

                                                      306
   For the particular case of n = 2 the first process of the concomitantive method
is merely an application under its most symmetrical form of the first                 p. 301
   process of the general algebraical method. The second process of the con-
comitantive method for this same case (at least when ϕ, ψ, ω are the partial
differential coefficients of the same function of the third degree) has been shown
by Dr Hesse to give the resultant, so that for this case, at all events, we know that
each concomitantive auxiliary must be a linear function of the augmentatives
and the algebraical auxiliaries.
   Again, if we go to the system where ϕ, ψ, ω are of the respective degrees
n, n, n + 1. In the algebraical method (for applying which there are no longer two,
but one only process), the augmentative system is obtained by multiplying ϕ by
the homogeneous products of xn−1 , xn−1 y, xn−1 z, &c., ψ by the like products,
and ω by the homogeneous products xn−2 , xn−2 y, &c. The auxiliary system is
made up of functions of the general form
                          Hp,q,r        where p + q + r = n + 2,
Hp,q,r being the determinant obtained by writing
ϕ = Lxp +M y q +N z r ,            ψ = L′ xp +M ′ y q +N ′ z r ,       ω = L′′ xp +M ′′ y q +N ′′ z r .
And in like manner for the case of ϕ, ψ, ω being of the respective degrees n, n, n−1,
the augmentative system is obtained by affecting ϕ, ψ each with multipliers
xn−2 , xn−2 y, &c., and ω with the multipliers xn−1 , xn−1 y, &c.
   The number of functions (for either case) in the augmentative and auxiliary
plexuses thus obtained will be found to be exactly equal to the number of terms
in each such function, as shown by me in the paper alluded to. Let this be
compared with the transcendental method (I use this word at this point in
preference to concomitantive, because in fact the algebraical and differential
auxiliary systems are both alike concomitantive plexuses to ϕ). For the case of
n, n, n + 1, the Jacobian determinant R of ϕ, ψ, ω will be of the degree 3n − 2,
and the system
                        n−1           n−2
                          d               d        d
                                 R,                  R, &c.
                         dx              dx       dy
combined with the augmentative systems
                                     xn−2 ω, xn−3 yω, &c.
                                     xn−1 ϕ, xn−2 yϕ, &c.
                                     xn−1 ψ, xn−2 yψ, &c.
will give an invariant resembling (at least in generation and form) if not identical
with the resultant of ϕ, ψ, ω. For the case of ϕ, ψ, ω being of the degrees n, n, n−1,
the Jacobian R is of the degree 3n − 4 and
                                   n−2                 n−3
                               d                    d           d
                                              
                                          R,                      R,   &c.
                              dx                   dx          dy

                                                 307
                                                                                                     p. 302
   is the system which, combined with the augmentative systems

                                  xn−2 ϕ, xn−3 yϕ, &c.
                                  xn−2 ψ, xn−3 yψ, &c.
                                  xn−1 ω, xn−2 yω, &c.

will produce the resembling invariant.
    Finally, for the last and more special case which the algebraical method applies
to, namely of ϕ, ψ, ω, θ, four quadratic functions of x, y, z, t, there can be here
little doubt (upon the first impression) that in place of the algebraically obtained
plexus
                        H2,1,1,1 , H1,2,1,1 , H1,1,2,1 , H1,1,1,2 ,
may be substituted the differential plexus
                                  dR       dR      dR      dR
                                     ,        ,       ,       ,
                                  dx       dy      dz      dt
which, combined with the augmentatives

         xϕ, xψ, xω, xθ;       yϕ, yψ, yω, yθ;      zϕ, zψ, zω, zθ;       tϕ, tψ, tω, tθ,

will render possible the dialytic elimination of the 20 homogeneous products

                           x3 , x2 y, x2 z, x2 t, xyz, y 3 , &c. &c.179


   Upon precisely the same principles may be verified instantaneously the method
given by Hesse (without demonstration) for finding the polar reciprocal of lines
of the third and fourth orders, at least to the extent of seeing that the functions
obtained by his methods are contravariants (of the right degree and order) of
the function from which they are derived. The polar reciprocal to a surface of
the third degree may be obtained in the same manner.
   Let ϕ(x, y, z, t) be the characteristic of such a surface. If we form a differential
plexus of the first emanant of ϕ taken together with the concomitant w =
xξ + yη + zζ + tθ, by operating with
                                     d       d      d      d
                                       ,       ,       ,
                                    dx      dy      dz     dt
 179
    Subsequent reflection induces me to reject as very improbable the (at first view likely)
conjecture of the identity of the resultant with the invariant which simulates its form, except in
the proved cases of three quadratic functions and the strongly resembling case of four quadratic
functions last adverted to in the text above. Did this identity obtain, analogy would indicate
that the catalecticant of the Hessian of two homogeneous functions of the same degree in x, y
should be identical with their resultant, which is easily demonstrated to be false, except when
the functions are of the third degree.


                                              308
upon
                                ′ d   ′ d        ′ d            ′ d
                                                                     
                            ξ        +η      +ζ        +θ       (ϕ + λw),
                                 dx      dy       dz        dt
and combining this plexus with xξ ′ + yη ′ + zζ ′ + tθ′ , the resultant taken in respect
to ξ ′ , η ′ , ζ ′ , θ′ (say R) will (according to the law of synthesis) be a                   p. 303
    contravariant to the system ϕ + λw; and w, and therefore to ϕ, because w
is itself a concomitant to ϕ. R is of the third degree in x, y, z, t, as also in the
coefficients of ϕ. If we form a differential plexus of R + µw analogous to that
formed above with ϕ+λw, and combine these two plexuses with the augmentative
system xw, yw, zw, tw, there will be 4 + 4 + 4, that is, 12 functions containing the
12 terms x2 , y 2 , z 2 , t2 , xy, xz, xt, yz, yt, zt, λ, µ, and the dialytic resultant, which
will be found to be a contravariant of the twelfth degree in ξ, η, ζ, θ, and of the
twelfth order in respect of the coefficients of ϕ, will be (there can be little doubt)
the polar reciprocal to the characteristic ϕ.
    A few remarks upon the analytical character of a polar reciprocal may be
not out of place here. If ϕ be any homogeneous function of the degree m of any
number (n) of variables (x, y . . . z), the object of the theory of polar reciprocals is
to discover what is the relation between ξ, η . . . ζ expressed in the simplest terms
such that, when this equation is satisfied, ξx + ηy + · · · + ζz = 0 will be tangential
to ϕ = 0. In order for this to take effect it is necessary that when any one of the
variables z is expressed in terms of the others . . . y, x, and this value established
in ϕ, the discriminant of ϕ, so transformed, should be zero. Consequently the
characteristic of the polar reciprocal to ϕ is that rational integral function which
is common to all the discriminants obtained by expressing ϕ (by aid of the
equation ξx + ηy + · · · + ζz) as a function of any (n − 1) of the variables. Let
Ix be any invariant whatever of the order r of ϕx (meaning by this last symbol
what ϕ becomes when x is eliminated), and Iy . . . Iz the corresponding invariants
when y . . . z respectively are eliminated; Ix will evidently be of the form (ξ)Emr        x
                                                                                              ,
the numerator being an integer of r dimensions in the coefficients of ϕ and of mr
dimensions in respect of ξ, η . . . ζ; and by the fundamental definition of invariants
it may easily be shown that
                                                 1          1                1
                      Ix : Iy : · · · : Iz ::          :          : ··· :          , 180
                                                ξ mr       η mr             ζ mr
and therefore
                  Ex   Ey        Ez                                         m(n − 2)r
                      = p = ··· = p ,                      where p =                  .
                  ξ p  η         ζ                                           n−1
 Consequently all these quotients must be essentially integer, and any one of
them will be of the order r in respect of the coefficients of ϕ and of the    p. 304
 180
     We see indirectly from this, that for a function of (n − 1), say γ, variables of the degree
m, an invariant of the order r must be subject to the condition that mr γ
                                                                           = an integer. This is
easily shown upon independent grounds; when γ = 2, mr  γ
                                                          must be not merely an integer but an
even integer, and doubtless some analogous law applies to the general case.


                                                  309
    degree n−1
           mr
                in respect of ξ, η . . . ζ. Consequently the polar characteristic of ϕ,
which is the common factor of the discriminants of Ix , Iy . . . Iz (for which species
of invariant r evidently is equal to (n − 1)(m − 1)n−2 , the function being in fact
the discriminant of a function of the mth degree of (n − 1) variables), will be of
the order (n − 1)(m − 1)n−2 in respect of the coefficients of ϕ and of the degree
m(m − 1)n−2 in respect of the contragredients ξ, η . . . ζ.
    As to what relates to the reciprocity which exists between ϕ and its polar
reciprocal ψ, this is included in a much higher theory of elimination, one propo-
sition of which may be enunciated somewhat to the effect following, namely that
if ϕ be a homogeneous function of x, y . . . z, and ω of x, y . . . z, u, v . . . w, and if,
by aid of the equations
                dϕ    dω                 dϕ    dω                             dϕ    dω
  ϕ = 0,           +λ    = 0,               +λ    = 0,            ··· ,          +λ    = 0,
                dx    dx                 dy    dy                             dz    dz
x, y . . . z be eliminated and the resultant be called ψ, then the effect of performing
a similar operation upon ψ, ω, with respect to u, v . . . w, as that just above
indicated for the system ϕ, ω, with respect to x, y . . . z, will be to give a resultant,
one factor of which will be the primitive function ϕ over again. There is some
reason for supposing that polar reciprocals, which are scarcely ever (if ever,
except indeed for quadratic functions) the simplest contravariants to a given
function, may be expressed algebraically by means of the simpler contravariants,
in the same way as discriminants admit (in many, if not in all cases, with the same
exception as above) of being represented as algebraical functions of invariants of
a lower order or simpler form.
   I close this section with the remark that every complete and unambiguous
system of functions of the constants in a given form or set of forms characteristic181
of any singularity absolute or relative in such form or forms must                        p. 305
   constitute an invariantive plexus or set of invariantive plexuses. The system
unambiguously characteristic of a singularity of an order n will (except when
n = 1) almost universally consist of far more than n functions, subject of course
to the existence of syzygetic182 relations between any (n + 1) of such functions.
The existence of multiple roots of a function of two variables is a specific, but
by no means a peculiar case of singularity, and requires, for its complete and
 181
     I repeat here that a function or system of functions which severally equated to zero
express unequivocally and completely the existence of any position or negation, is termed
the characteristic of such position or negation. Thus for example the resultant of a group
of equations is the characteristic of the possibility of their coexistence. The discriminant of
a function of two variables is the characteristic of its possession of two equal factors; the
catalecticant is the characteristic of its decomposability into the sum of a defined number of
powers of linear functions of the variables, &c.
 182
     Rational integer functions which admit of being multiplied severally by other rational integer
functions such that the sum of the products is identically zero, are said to be “syzygetically
related.”



                                               310
systematic elucidation, to be treated in connexion with the general theory of the
subject.

                          Section III. On Commutants.

  The simplest species of commutant is the well-known common determinant.
  If we combine each of the n letters a, b . . . l with each of the other n, α, β . . . λ,
we obtain n2 combinations which may be used to denote the terms of a determi-
nant of n lines and columns, as thus:

                                  aα, aβ · · · aλ,
                                  bα, bβ · · · bλ,
                                  ··· ···      ···
                                  lα, lβ · · · lλ.

It must be well understood that the single letters of either set are mere umbrae,
or shadows of quantities, and only acquire a real signification when one letter of
one set is combined with one of the other set. Instead of the inconvenient form
above written, we may denote the determinant more simply by the matrix

                                   a, b, c · · · l,
                                   α, β, γ · · · λ;

and to find the expanded value of such a matrix the rule is evidently to take one
of the lines in all its 1, 2, 3 . . . n different forms, arising from the permutations
of the letters (or umbrae) which it contains; and then form the product of the
n quantities formed by the combination of the respective pairs of letters in the
same vertical column, affecting such product with the sign of + or − according to
the rule, that all products corresponding to arrangements of the terms subject to
the permutation derivable from one another by an even number of interchanges
are of the same, and by an odd number of interchanges of a contrary sign. If both
lines are permuted and a similar rule applied, with the additional circumstance
that the sign of the products                                                             p. 306
    is made to depend on the product of the algebraical signs due to the respective
arrangements in the two lines of umbrae, it is evident that the result will be
the same as when only one line is put into motion, save and except that a
numerical factor 1 · 2 · 3 · · · n will affect each term. If the two sets of umbrae
a, b, c . . . l; α, β, γ . . . λ be taken identical, and if it be convened that the order
of the combination of any two letters shall not affect the value of the quantity
thereby denoted,
                                          a, b, c · · · l
                                          a, b, c · · · l
will denote a symmetrical determinant.


                                          311
   If instead of two lines of umbrae, three or more be taken, the same principle
of solution will continue to be applicable. Thus, if there be a matrix of any even
number r of lines each of n umbrae,

                                     a1 , b1 · · · l1 ,
                                     a2 , b2 · · · l2 ,
                                     ··· ···       ···
                                     ar , br · · · lr ,

the first may be supposed to remain stationary, and the remaining (r − 1)
lines each be taken in 1, 2 . . . n different orders; every order in each line will be
accompanied by its appropriate sign + or −; and each different grouping in each
line will give rise to a particular grouping of the letters read off in columns. The
value of the commutant expressed by the above matrix will therefore consist of
the sum of (1 · 2 · · · n)r−1 terms, each term being the product of n quantities
respectively symbolized by a group of r letters and affected with the sign + or
− according as the number of negative signs in the total of the arrangements of
the lines (from the columnar reading off of which each such term is derived) is
even or odd.
   For example, the value of
                                          a, b,
                                          c, d,
                                          e, f,
                                          g, h,
will be found by taking the (1 · 2)3 arrangements, as below,

                  a, b,   a, b,   a, b,   a, b,   a, b,   a, b,   a, b,   a, b,
                  c, d,   d, c,   c, d,   d, c,   c, d,   d, c,   c, d,   d, c,
                  e, f,   e, f,   f, e,   f, e,   e, f,   e, f,   f, e,   f, e,
                  g, h,   g, h,   g, h,   g, h,   h, g,   h, g,   h, g,   h, g.

The signs of c, d; e, f ; g, h being supposed +, those of d, c; f, e and h, g will be
each −. Consequently the sum of the terms will be expressed by

          aceg × bdf h − adeg × bcf h − acf g × bdeh + adf g × bceh
             − aceh × bdf g + adeh × bcf g + acf h × bdeg − adf h × bceg.
                                                                                                 p. 307
   Commutants thus formed may be termed total commutants, because the entire
of each line is made to pass through all its possible forms of arrangement. In
total commutants it is necessary that the number of lines r be even; for if taken
odd, on making all the r lines to change, instead of obtaining 1 · 2 · · · n lines, the
result obtained when all but one are made to change, it will be found that the
latter will be repeated 12 (1 · 2 · · · n) times with the sign +, and 12 (1 · 2 · · · n) times
with the sign −, so that the algebraical sum of the terms will be zero. Moreover

                                              312
the commutants of the species above described, besides being total, are simple,
inasmuch as all the umbræ to be termed consist of single letters.
   My first proposition in the application of the theory of commutants to that of
forms is as follows:
   Let ϕ be a function homogeneous and linear in respect to an even number r
of any systems whatever of variables, as
                    x1 , y1 . . . t1 ;        x2 , y2 . . . t2 ;     xr , yr . . . tr .
Form the commutant
                               d     d          d
                                   ,     , ···     ,
                              dx1 dy1          dt1
                               d     d          d
                                   ,     , ···     ,
                              dx2 dy2          dt2
                               ···   ···     ···
                               d     d          d
                                   ,     , ···     .
                              dxr dyr          dtr
Let the general term of this commutant, expanded, be called
                                         Fθ1 × Fθ2 × · · · × Fθr ,
then is
                                ΣFθ1 ·ϕ × Fθ2 ·ϕ × · · · × Fθr ·ϕ
a covariant or invariant183 as the case may be, of ϕ.
    Be it observed that the march of the substitution for the different sets of
variables in the above proposition is supposed to be perfectly independent. All
the systems but one may undergo linear transformation, or they may all undergo
distinct and disconnected transformations at the same time, and the proposition
still continue applicable. It will however evidently be no less applicable should the
march of substitution for any of the systems become cogredient or contragredient
to that of any other systems.
    If we suppose ϕ to be a function of an even degree r of a single system of
n variables x, y . . . t, so that the r systems x1 , y1 , &c., x2 , y2 , &c. . . . xr , yr , &c.
become identical, we can at once infer from the above scheme the existence and
mode of forming an invariant to ϕ of the order n. This last appears                              p. 308
    for the case n = 2, and ought, for all other values of n, to have been known             184

to the author of the immortal discovery of invariants, termed by him hyperdeter-
minants, in the sense which, according to the nomenclature here adopted, would
be conveyed by the term hyperdiscriminants.
 183
    See below, p. 324.
 184
    That this was not known explicitly to and should have escaped the penetration of the
sagacious author of the theory, and those who had studied his papers, must be attributed to the
imperfection of the notation heretofore employed for denoting the coefficients of a homogeneous
polynomial function. The umbral method of denoting such a function ϕ of the degree r under the
form of (ax + by + · · · + cz)r , which is equivalent to, but a more compendious and independent


                                                    313
   Before proceeding to discuss the theory of compound total commutants, or
enlarging upon that of partial commutants, I shall make an interesting application
of the preceding general proposition to the discovery of Aronhold’s S and T ,
the two invariants respectively of the fourth and sixth orders appertaining to
a homogeneous cubic function (say F ) of three variables x, y, z. These may be
termed respectively H4 and H6 . As to H6 a theoretically possible but eminently
prolix and ungraceful method immediately presents itself, namely to take F 2 = G,
and after forming the commutant with six lines,
                                        d      d     d
                                          ,      ,      ,
                                       dx     dy     dz
                                        d      d     d
                                          ,      ,      ,
                                       dx     dy     dz
                                        d      d     d
                                          ,      ,      ,
                                       dx     dy     dz
                                        d      d     d
                                          ,      ,      ,
                                       dx     dy     dz
                                        d      d     d
                                          ,      ,      ,
                                       dx     dy     dz
                                        d      d     d
                                          ,      ,      ,
                                       dx     dy     dz

to operate with the 65 ternary products of which this is made up upon G: the
result being an invariant of G, will be so to F , and being of the third degree in
respect to the coefficients of G, will be of the sixth in respect to those of F . It
will evidently therefore be H6 , or at least a numerical multiple of H6 , the form of
which, inasmuch as the only other invariant is H4 , we know inform to be unique.
But the general theorem affords another and probably the                              p. 309
   most practically compendious   185 solution as regards H6 , of which the question
admits.
   Let G186 represent the mixed concomitant to F formed by the bordered
mode of mentally conceiving and handling the representation
                                                           r
                                     d     d           d
                                  x    +y    + ··· + z           ϕ,
                                    dx    dy           dz

exhibits the true internal constitution of such functions, and necessarily leads to the discovery
of their essential properties and attributes.
 185
     Having since this was printed been favoured with a view of some of the proof-sheets of Mr
Salmon’s most valuable Second Part of his System of Analytical Geometry (about to appear,
and which is calculated, in my opinion, to awaken a higher idea of and excite a new taste
for geometrical researches in this country), I find that I am mistaken in this point; the less
symmetrical method operated with by Mr Salmon being decidedly of the shortest for practically
obtaining S and T in the general case. Symmetry, like the grace of an eastern robe, has not
unfrequently to be purchased at the expense of some sacrifice of freedom and rapidity of action.
 186
     G is the mixed concomitant to the given cubic function, which is halfway (so to speak)
between it and its polar reciprocal. In fact, when the operation is repeated upon G, which was


                                              314
determinant
                                 d2 F      d2 F        d2 F
                                                              ξ
                                 dx2      dx dy       dx dz
                                 d2 F      d2 F        d2 F
                                                              η
                                dy dx      dy 2       dy dz       .
                                 d2 F      d2 F        d2 F
                                                              ζ
                                dz dx     dz dy        dz 2
                                  ξ         η           ζ     0
G is a function of the second order as to x, y, z, and of the like order in respect
to ξ, η, ζ, which two systems will be respectively cogredient and contragredient
in respect to the x, y, z system in F . In other words, which is all we need to look
to, G is a concomitant of F , and so also will be

                                  G + λ(xξ + yη + zζ)2 ,

which may be termed H. Form now the commutant
                                         d      d      d
                                           ,      ,       ,
                                        dx     dy      dz
                                         d      d      d
                                           ,      ,       ,
                                        dx     dy      dz
                                         d      d      d
                                           ,      ,       ,
                                        dξ     dη      dζ
                                         d      d      d
                                           ,      ,       ,
                                        dξ     dη      dζ
this being applied to H will give an invariant (the fact that the march of
the substitutions for the systems x, y, z; ξ, η, ζ is contrary, being completely
immaterial to the applicability of the general theorem above given);                p. 310
   the commutant so formed will be a cubic function of λ, in which the coefficient
of λ3 is a numerical quantity, that of λ2 is zero, that of λ is H4 and the constant
term is H6 .
   Thus for example let F = x3 + y 3 + z 3 + 6mxyz, then

                                         x mz my ξ
                                         mz y mx η
                                G=
                                         my mx z ζ
                                          ξ  η ζ 0

and therefore

            H = Σ{(λ − m2 )x2 ξ 2 + (λ + m2 )2yzηζ + yzξ 2 − 2mx2 ηζ},
executed upon the given function to obtain G (that is, when we border the Hessian of G in
respect to x, y, z, vertically and horizontally with the column and line ξ, η, ζ) the determinant
thereby represented becomes the polar reciprocal to the given function.


                                               315
the Σ implying the sum of similar terms with reference to the interchanges
between x, ξ; y, η; z, ζ.
   In developing the commutant above, the first line may be kept in a fixed
position; for the sake of brevity, (x), (y), (z); (ξ), (η), (ζ) may be written in the
place of
                        d     d     d          d       d      d
                          ,     ,      ;         ,       ,       ,
                       dx    dy     dz        dξ      dη      dζ
and it will readily be seen that the only effective arrangements will be as
underwritten:
                (x)(y)(z)   (x)(y)(z)   (x)(y)(z)
                (x)(y)(z)   (x)(y)(z)   (x)(y)(z)
                (ξ)(η)(ζ)   (η)(ζ)(ξ)   (ζ)(ξ)(η)
                (ξ)(η)(ζ)   (ζ)(ξ)(η)   (η)(ζ)(ξ)
                (x)(y)(z)   (x)(y)(z)   (x)(y)(z)   (x)(y)(z)   (x)(y)(z)   (x)(y)(z)
                (x)(z)(y)   (x)(z)(y)   (ξ)(η)(ζ)   (z)(y)(x)   (y)(x)(z)   (y)(x)(z)
                (ξ)(η)(ζ)   (ξ)(ζ)(η)   (ξ)(η)(ζ)   (ζ)(η)(ξ)   (ξ)(η)(ζ)   (η)(ξ)(ζ)
                (ξ)(ζ)(η)   (ξ)(η)(ζ)   (ξ)(η)(ζ)   (ξ)(η)(ζ)   (η)(ξ)(ζ)   (ξ)(η)(ζ)
                (x)(y)(z)   (x)(y)(z)   (x)(y)(z)   (x)(y)(z)   (x)(y)(z)   (x)(y)(z)
                (y)(z)(x)   (z)(x)(y)   (y)(z)(x)   (y)(z)(x)   (z)(x)(y)   (z)(x)(y)
                (ξ)(η)(ζ)   (η)(ζ)(ξ)   (ζ)(η)(ξ)   (η)(ζ)(ξ)   (ξ)(η)(ζ)   (ζ)(ξ)(η)
                (ξ)(ζ)(η)   (η)(ξ)(ζ)   (η)(ζ)(ξ)   (ξ)(η)(ζ)   (ζ)(η)(ξ)   (ξ)(η)(ζ).
                                                                                                    p. 311
   The signs of the four lines in each of these arrangements are two alike, and
two contrary to the signs of the correspondent lines in the first arrangement;
hence the effective sign is the same for all, and the result, after rejecting from
each term the common factor −16, is seen, from inspection, to be

 4(λ − m2 )3 − 8m3 + 6(λ − m2 )(λ + m2 )2 − 12m(λ + m2 )2 + 2(λ + m2 )3 + 1,

which is equal to

                  12λ3 + 0 · λ2 − 12(m − m4 )λ + 1 − 20m3 − 8m6 ;

here the coefficients m−m4 and 1−20m3 −8m6 are the two invariants (Aronhold’s
S and T ) for the canonical form operated upon; and it will be observed that

                  (1 − 20m3 − 8m6 )2 + 64(m − m4 )3 = (1 + 8m3 )3 ,

which is easily proved to be the discriminant of

                                   x3 + y 3 + z 3 + 6mxyz.

   It may however be observed, that this is not the discriminant of the function
in λ just found, as reasons of analogy187 might have suggested it probably would
 187
     The biquadratic function of x, y having only one parameter, and therefore two invariants,
its theory possesses striking analogies to the theory of the cubic function of three letters. The
function in λ which gives these invariants for the first-named function, according to the method
given in the first section, has the same discriminant as the function itself.


                                                316
be: in order that this might be the case, the coefficient of λ3 should be 4 instead
of 12, and of λ, m − m4 instead of m4 − m. There is ground for supposing that
another function of λ may be found by a different method, in which this relation
will take effect.
   The theorem above given for simple total commutants admits of an interesting
application to the general case of a function F of the nth degree, in respect to
each of two independent systems of two variables x, y; ξ, η. Let F be symbolically
represented by (ax + by)n (αξ + βη)n , so that an αn represents the coefficient of
xn ξ n , nan−1 bαn of xn−1 yξ n , &c. &c.; then the commutant

                                                a, b, (1)
                                                a, b, (2)
                                                ··· ···
                                                a, b, (n)
                                                α, β, (1)
                                                α, β, (2)
                                                ··· ···
                                                α, β, (n)

will represent a quadratic invariant of F , which will contain (n + 1)2 coefficients.
By expanding this commutant we obtain a general expression for the invariant
under a very interesting form.                                                        p. 312
   I now proceed to give the general theorem for compound total commutants as
applicable to the discovery of invariants.
   Let there be a function of m disconnected classes of systems of variables; let
the systems in the same class be supposed all distinct but cogredient with one
another. The function is supposed to be linear in respect to each system in each
class, and the number of systems is the same for all the classes, and the number
of variables the same in each system. This function may then be represented
symbolically under the form

       (1 a1 1 x1 + 1 b1 1 y1 + · · · + 1 l1 1 t1 )(1 a2 1 x2 + 1 b2 1 y2 + · · · + 1 l2 1 t2 )
          · · · (1 an 1 xn + 1 bn 1 yn + · · · + 1 ln 1 tn )
       × (2 a1 2 x1 + 2 b1 2 y1 + · · · + 2 l1 2 t1 )(2 a2 2 x2 + 2 b2 2 y2 + · · · + 2 l2 2 t2 )
          · · · (2 an 2 xn + 2 bn 2 yn + · · · + 2 ln 2 tn ) × &c.
       × (p a1 p x1 + p b1 p y1 + · · · + p l1 p t1 )(p a2 p x2 + p b2 p y2 + · · · + p l2 p t2 )
          · · · (p an p xn + p bn p yn + · · · + p ln p tn ).

In this expression the x, y . . . t’s are all real, but the a, b . . . l’s all umbral; in fact,
f a , f b , &c. may be understood to denote
   g     g

                                        d              d
                                                ,               ,   &c.
                                     d fx   g        d fy   g


                                                    317
The n systems of variables in each of the sets above written are supposed to be
cogredient inter se.
  Take the symbolical product of the first set, first making for the moment
         1
             x1 = 1 x2 = · · · 1 xn = x,       &c. &c.,             1
                                                                         t1 = 1 t2 = · · · 1 tn = t;

and let the coefficients of the several terms

                                    xn ,       xn−1 y . . . &c.,

be called
                                    1            1
                                        A1 ,         A2 , . . . 1 Aµ ,
where µ is the number of terms contained in a homogeneous function of the nth
degree of the m variables x, y . . . t. In like manner proceed with each of the lines,
and then write down the commutant
                                   1A ,        1A ,        ···   1A
                                     1           2                    µ,
                                   2A ,        2A ,        ···   2A
                                     1           2                  µ,
                                    ···         ···               ···
                                   pA ,        pA , · · ·        pA .
                                      1           2                 µ

This commutant is an invariant of F : it will of course be remembered that, unless
p is even, the commutant vanishes.                                                 p. 313
   Thus, for example, take two sets of two systems of two variables: in all four
systems,
                          x, y; ξ, η : p, q; ϕ, ψ,
each couple of systems on either side of the colon (:) being cogredient inter se:
and let F be symbolically represented by

                         (ax + by)(αξ + βη)(lp + mq)(λϕ + µψ);

then the invariant given by the theorem will be the commutant

                                    aα; aβ + αb; bβ,
                                    lλ; lµ + λm; mµ.

The six positions of this are as below written (the first three being positive and
the second three negative)

               aα; aβ + αb; bβ, aα; aβ + αb; bβ, aα; aβ + αb; bβ,
              lλ; lµ + λm; mµ, lµ + λm; mµ; lλ, mµ; lλ; lµ + λm,
               aα; aβ + αb; bβ, aα; aβ + αb; bβ, aα; aβ + αb; bβ,
              lµ + λm; lλ; mµ, lλ; mµ; lµ + λm, mµ; lµ + λm; lλ.



                                                     318
If we write F under its explicit form,

                      Axξpϕ + Bxξpψ + Cxξqϕ + Dxξqψ
                       + A′ xηpϕ + B ′ xηpψ + C ′ xηqϕ + D′ xηqψ
                       + A′′ yξpϕ + B ′′ yξpψ + C ′′ yξqϕ + D′′ yξqψ
                       + A′′′ yηpϕ + B ′′′ yηpψ + C ′′′ yηqϕ + D′′′ yηqψ,

we have identically the relations following,

                 aαlλ = A,        aαlµ = B,         aαmλ = C,           aαmµ = D,
                 aβlλ = A′ ,      aβlµ = B ′ ,      aβmλ = C ′ ,        aβmµ = D′ ,
                 bαlλ = A′′ ,     bαlµ = B ′′ ,     bαmλ = C ′′ ,       bαmµ = D′′ ,
                 bβlλ = A′′′ ,    bβlµ = B ′′′ ,    bβmλ = C ′′′ ,      bβmµ = D′′′ .

and the commutant expanded becomes

               A(B ′ + C ′′ + C ′ + B ′′ )D′′′ + (B + C)(D′ + D′′ )A′′′
                  + D(A′ + A′′ )(B ′′′ + C ′′′ ) − (B + C)(A′ + A′′ )D′′′
                  − A(D′ + D′′ )(B ′′′ + C ′′′ ) − D(B ′ + C ′′ + C ′ + B ′′ )A′′′ .

In the foregoing the x’s in the several lines were for the moment taken identical,
in order the more easily to explain the law of formation of the                    p. 314
   quantities A. But suppose that they become actually identical for the same
line. F then becomes a function of the nth degree in respect to each of p systems
of variables, and may be represented symbolically under the form

(1 a 1 x+1 b 1 y+· · ·+1 l 1 t)n ×(2 a 2 x+2 b 2 y+· · ·+2 l 2 t)n · · ·×(p a p x+p b p y+· · ·+p l p t)n .

We may still further limit the generality of the theorem by supposing
1
    x = 2 x = · · · p x = x,      1
                                      y = 2 y = · · · p y = y,       ··· ,      1
                                                                                    t = 2 t = · · · p t = t;

F then becomes
                                        (ax + by + · · · + lt)np .
Accordingly, as many different factors as can be found contained an even number
of times in the exponent of the function, so many invariants can be formed
immediately from a function of any number of variables m by the method of
total commutation.
   If one of these factors be called n, the commutant corresponding thereto will
be of the order
                           (n + 1)(n + 2) · · · (n + m − 1)
                                   1 · 2 · · · (m − 1)
in respect to the coefficients. Thus take m = 2, so that

                                           F = (ax + by)np .

                                                   319
The general form of such a commutant will be found by taking A1 , A2 . . . An+1
the coefficients of the several combinations of x, y in (ax + by)n , from which the
numerical coefficients n, 21 n(n − 1), &c. may be rejected, as only introducing a
numerical factor into the result; the commutant will therefore be expressed by
means of the form
                        an ; an−1 b; an−2 b2 . . . ; bn ,      (1)
                        an ; an−1 b; an−2 b2 . . . ; bn ,      (2)
                                   ···············
                        an ; an−1 b; an−2 b2 . . . ; bn .      (p)

If p = 2, the compound commutant

                               an ; an−1 b; · · · ; bn ,
                               an ; an−1 b; · · · ; bn ,
                                                                                        p. 315
  will easily be seen to be only another form for the catalecticant of (ax + by)2n .
Thus, let n = 2,

              (ax + by)4 = Ax4 + 4Bx3 y + 6Cx2 y 2 + 4Dxy 3 + Ey 4 ;

so that

          a4 = A,      a3 b = B,       a2 b2 = C,       ab3 = D,           b4 = E.

The commutant (which is of the form of the matrix to an ordinary determinant,
with the exception that the umbræ enter compoundly instead of simply into the
several terms separated by the marks of punctuation), will be

                                     a2 ; ab; b2 ,
                                     a2 ; ab; b2 :

this, written in the six forms
                               )                    )                      )
                a2 ; ab; b2          a2 ; ab; b2            a2 ; ab; b2
                a2 ; ab; b2          a2 ; b2 ; ab           ab; a2 ; b2
                               )                    )                      )
                a2 ; ab; b2          a2 ; ab; b2            a2 ; ab; b2
                b2 ; ab; a2          ab; b2 ; a2            b2 ; a2 ; ab

gives the expression

    a4 × a2 b2 × b4 − a4 × (ab3 )2 − b4 × (a3 b)2 − (a2 b2 )3 + 2a3 b × ab3 × a2 b2 ;

that is
                        ACE − AD2 − EB 2 − C 3 + 2BCD.


                                           320
    One important observation may here be made of a fact which otherwise might
easily escape attention, which is, that commutants, where the same terms simple
or compound are found in all or several of the lines, in general give rise to
products, some of them equal and with the same sign, and others equal but with
the contrary sign.
    This last phenomenon does not manifest itself in commutants appertaining
to functions of two variables of the two particular and different species which
first and most naturally present themselves, namely where there are only two
lines or only two columns188 —I believe that it displays itself in every other
case of commutantives to functions of two variables. Thus it is that algebraical
expressions derived from given functions disguise their symmetry; to make
which come to light it becomes necessary to add terms of contrary sign to such
expressions. As an example, the reader is invited to develope the cubic invariant
of a function of x and y, symbolically expressed by (ax + by)8 , where

                      a8 = A,         a7 b = B . . . ab7 = H,         b8 = I,
                                                                                                        p. 316
   by means of the commutant

                                        a2 ,   ab,  b2 ,
                                        a2 ,   ab,  b2 ,
                                        a2 ,   ab,  b2 ,
                                        a2 ,   ab, b .189
                                                    2



 188
     These commutants give respectively the quadrinvariant and the catalecticant, the former of
which alone was formerly recognised by Mr Cayley as a commutant.
 189
     See p. 346 below. The number of terms resulting from the independent permutation of
each of the 3 linear lines is 63 , that is 216; but the actual result is (using small letters instead
of large) P − Q, where

                    P = aei + 3ag 2 + 12beh + 3c2 i + 24cf 2 + 24d2 g + 15e3 ,
                    Q = 4af h + 4bid + 8bgf + 22ceg + 8chd + 36def,
so that the effective number of permutations is only 164. The difference between this and 216
divided by 216 may be termed the Index of Demolition, which we see in this case is 216    52
                                                                                             or 13
                                                                                                 54
                                                                                                    ;
that is, somewhat less than 14 . For the cubic invariant of the function of the fourth degree this
index is zero, all the permutations being effective. If we take the cubic invariant of the function
ax12 + 12bx11 y + 66cx10 y 2 + &c. + my 12 under the form P − Q, we shall find
        P = 6ahl + 10ajj + 60f m + 54bhk + 54cf l + 156cii + 10ddm + 430djj
             + 155eek + 520ehh + 520f f l + 280ggg,
        Q = agm + 15aik + 30bgl + 50bij + 15cem + 4cgk + 150chj + 30del + 210df k
             + 250dhi + 230ef j + 555egl + 660f gh.
The number of terms in P and Q is of course the same, and will be found to be 2200 for each;
so that out of the 65 , that is 7776 permutations of the 5 lower rows, only 4400 are effective,
and the index of demolition becomes 3376
                                       7776
                                            , that is 211
                                                      486
                                                          , or rather greater than 12
                                                                                   5
                                                                                      . The Index of
Demolition thus goes on constantly increasing as the degree of the function rises; probably (?)


                                                321
  Suppose F to be the general even-degreed function of two variables of the
degree 2np.
  Let
                          d      d np
                                  
                 H= ξ       −η        F + λ(xξ + yη)np ,
                         dy     dx
and express H umbrally under the form

                                  (ax + by)np (αξ + βη)np .
                                                                                                      p. 317
   The commutant
                                an ,    an−1 b · · ·    bn ,   (1)
                                an ,    an−1 b · · ·    bn ,   (2)
                                       ············
                                an ,    an−1 b · · ·   bn , (p)
                                αn ,   αn−1 β · · ·    β n , (1)
                                αn ,   αn−1 β · · ·    β n , (2)
                                       ············
                                αn ,   αn−1 β · · ·    β n , (p)
will be a function of λ, and all the several coefficients will be invariants of F 190 .
   When p = 1 we obtain the Λ given in the preceding section, and originally
published by me in the Philosophical Magazine for the month of November, 1851.
The Λ obtained on this supposition has for its coefficients a series of independent
invariants, commencing with the catalecticant and closing with the quadratic
it converges either towards 12 or else towards unity. In arranging the terms it will be found
most convenient to adopt, as I have done above, the dictionary method of sequence. The
computations are greatly facilitated by the circumstance of the effect of any arrangement of
each of the 5 lower lines not being altered when these lines are permuted with one another;
this gives rise to the subdivision of the 7776 permutations into groups as follows: 6 of 120
identical terms, 60 of 60, 36 of 20, 60 of 30, 24 of 20, 30 of 10, 30 of 5, and 6 of 1. So that the
total number of permutational arrangements to be constructed is only 252. Other methods
of abridging the labour will readily suggest themselves to the practical computer. The total
number of the groups of terms is of course always known à priori, and, for instance, in the
case before us, must be equal to the number of ways in which 12 (12 × 3), that is the number
18, can be divided into 3 parts, none of which is to exceed the number 12, that is 25; for the
cubic invariant of the function of the eighth degree of two variables it is the number of ways in
which 12 can be divided into 3 parts, of which none shall exceed 8, and so forth, zeros being
always understood to be admissible; and of course in general for an invariant of the order r to
a function of the degree n of i variables, the number of distinct terms is in general the number
of ways in which nri
                      can be divided into r parts, of which none shall exceed n, subject however
always to the possibility in particular cases of a diminution in consequence of some of the
groups assuming zero for their coefficient.
  190
      By substituting the symbols dx d    d
                                       , dy , &c. in place of the umbræ a, b, &c., the theorem is
easily stated for covariants generally. But in applying the commutantive method to obtain
covariants, or rather in the statement of the results flowing from each application, it is never
necessary to go beyond the case of invariants, because the commutantive covariants of any
given homogeneous function are always identical with commutantive invariants of emanants of
the same function.


                                               322
invariant. When p has any other value, we observe a similar series commencing
with a commutantive invariant of a lower order than the catalecticant, but always
closing with the quadratic invariant. Thus, for example, when 2np = 8, we may
obtain by the preceding theorem three different quadratic functions; one giving
the invariants of the orders 5, 4, 3, 2, the second those of the orders 3, 2, the
third the invariant of the order 2.
   In this case the invariants of the same order given by the different Λ’s are the
same to numerical factors près. Whether this is always necessarily the case is a
point reserved for further examination.
   The commutants applied in the preceding theorems have been called by me
total commutants, because the total of each line of umbræ is permuted in every
possible manner. If the lines be divided into segments, and the permutation be
local for each segment instead of extending itself over the whole line, we then
arrive at the notion of partial commutants, to which I have also (in concert with
Mr Cayley) given the distinctive name of Intermutants. In order to find the
invariants of functions of odd degrees, the theory of total commutants requires
the process of commutation to be applied, not immediately to the coefficients of
the proposed function, but to some derived concomitant form. I became early
sensible of this imperfection, and stated to the friend above named, to whom I
had previously                                                                      p. 318
   imparted my general method of total commutation, my conviction of the
existence of a qualified or restricted method of permutation, whereby the invari-
ants of the cubic function, for instance, of two and of three letters would admit,
without the aid of a derived form, of being represented. Many months ago, when
I was engaged in this important research, and had made some considerable steps
towards the representation of the invariant, that is, the discriminant of the cubic
function of x and y, under the form of a single permutant, I was surprised by
a note from the friend above alluded to, announcing that he had succeeded in
fixing the form of the permutant of which I was at that moment in search. It is
with no intention of complaining of this interference on the part of one to whose
example and conversation I feel so deeply indebted, (and the undisputed author
of the theory of Invariants,) that I may be permitted to say that, independent
of the intervention of this communication, I must inevitably have succeeded in
shaping my method so as to furnish the form in question; and that with greater
certainty, after my theory of commutants had furnished me with the precedent
of permutable forms giving rise to terms identical in value but affected with
contrary signs. As I have understood that Mr Cayley is likely to develop this
part of the subject in the present number of the Journal, it will be the less
necessary for me to enter at any length into the theory of partial commutants
on the present occasion.
   The method of partial commutation is a simple but most important corollary
from that of total commutation hereinbefore explained. To fix the ideas, conceive


                                        323
a class of p cogredient systems, and that there are qr such classes perfectly
independent. Proceed to divide these qr classes in any manner whatever into
r sets, each containing q classes; and form the symbol of the total commutant
corresponding to each such set. Now let these commutants be placed side by side
against one another, and transpose the terms in each compound line thus formed
once for all, but in any arbitrary manner. Then permute in every possible way
all those symbols in each line, inter se, which belong to the same class, and
operate with the symbols thus produced by reading off the vertical columns and
attending to the rule of the + and − signs, as in the case of a total commutant;
the result will be a commutant of the form operated upon. For instance, let
p = 1, q = 3, r = 2, and let the number of variables in each system be 2. Form
the commutant operators
                                d       d        d       d
                                  ,       ,        ,       ,
                               dx      dy       dξ      dη
                                d       d        d       d
                                  ,       ,        ,       ,
                               dp      dt       dϕ      dθ
                                d       d        d       d
                                  ,       ,        ,       .
                               dr      ds       dρ      dσ
                                                                                      p. 319
  Interchange in any manner but once for all the symbols in each line, as thus:
                                 d       d      d        d
                                   ,       ,       ,       ,
                                dx      dy      dξ      dη
                                 d       d      d        d
                                   ,       ,       ,       ,
                                dϕ      dp      dt      dθ
                                 d       d      d        d
                                   ,       ,       ,       .
                                ds      dρ      dr      dσ
Now permute, inter se, the variables of each system, as
                               d d             d d
                                , ;              , ,      &c.;
                              dx dy            dp dt

the total number of the operative forms resulting will be (1 · 2)6 , and the sum of
the (1 · 2)6 quantities, half positive and half negative, formed after the type of
               d d d       d d d      d d d        d d d
                                                                        
            Σ          U×          U×          U×          U ,
              dx dϕ ds    dy dp dρ    dξ dθ dr    dη dt dσ
U being supposed to be a function homogeneous in

                      x, y;   ξ, η;    p, t;    ϕ, θ;    r, s;   ρ, σ,

will be a covariant of U .
   The proof of the truth of this proposition is contained in what is shown in the
Notes of the Appendix for total commutants, it being only necessary to make the

                                           324
systems which are independent vary consecutively, and then apply the inference
to the supposition of their varying simultaneously.
   It may be extended to the more general supposition of classes of an unequal
number of cogredient systems of unequal numbers of variables in each, the only
condition apparently required being that the number of distinct terms shall be
the same in each line of the final commutantive operator. The important remark
to be made is, that in applying this theorem there is nothing to prevent any of
the systems being made identical; or, in other words, a given function of one
system of variables may be regarded as a function of as many different, although
coincident, sets as we may choose to suppose. Thus, suppose
                            U = Ax2 + 2Bxy + Cy 2 ;
we may take the partial commutant formed of the two total commutant operators
                                       d       d
                                         ,       ,
                                      dx      dy
                                       d       d
                                         ,       ,
                                      dx      dy
                                                                                     p. 320
   combined with itself. If we write them in the same order,
                               d˙    d˙   d′   d′
                                  ,     ,    ,    ,
                              dx dy dx dy
                               d˙    d˙   d′   d′
                                  ,     ,    ,    ,
                              dx dy dx dy
(where I use the dots and dashes to distinguish those in the same line which are
considered as belonging to the same class, and therefore as permutable, inter se),
we shall evidently obtain 4{AC − B 2 }2 ; if we commence with a permutation, so
as to have the form of operation
                               d˙   d˙      d′   d′
                                  ,    ,       ,    ,
                              dx dy dx dy
                               d˙   d′      d′   d˙
                                  ,    ,       ,    ,
                              dx dy dx dy
it will be found that we obtain 2{AC − B 2 }2 .
   Again, suppose that we have
                      U = Ax3 + 3Bx2 y + 3Cxy 2 + Dy 3 .
If we write
                               d˙      d˙     d′     d′
                                  ,       ,      ,      ,
                              dx      dy      dx     dy
                               d˙      d˙     d′     d′
                                  ,       ,      ,      ,
                              dx      dy      dx     dy
                               d˙      d˙     d′     d′
                                  ,       ,      ,      ,
                              dx      dy      dx     dy

                                         325
the value of the commutant would come out zero; but if we make a permutation,
and write
                             d˙   d˙   d′    d′
                                ,    ,    ,     ,
                            dx dy dx dy
                             d˙   d′   d′    d˙
                                ,    ,    ,     ,
                            dx dy dx dy
                             d˙   d˙   d˙    d′
                                ,    ,    ,     ,
                            dx dy dx dy
the operation indicated by the above performed upon U , will give a multiple of
the discriminant of U .                                                         p. 321
   In like manner we may represent Aronhold’s Sextic Invariant of the form
(x, y, z)3 by means of the partial commutant

                            d˙        d˙           d˙      d′       d′       d′
                               ,         ,            ,       ,        ,        ,
                           dx        dy            dz      dx       dy       dz
                            d˙        d˙           d˙      d′       d′       d′
                               ,         ,            ,       ,        ,        ,
                           dx        dy            dz      dx       dy       dz
                            d˙        d˙           d˙      d′       d′       d′
                               ,         ,            ,       ,        ,        .
                           dx        dy            dz      dx       dy       dz
If we make
                                                                                       2
                   ′ d        ′ d            ′ d             d     d    d
                                                  
          V = ξ          +η         +ζ                    ξ    +η    +ζ                     (x, y, z)3 ,
                   dξ         dη             dζ             dx    dy    dz
and use H to signify the determinant

                                               x y z
                                               ξ η ζ ,
                                               ξ′ η′ ζ ′

which is evidently an universal triple covariant, and make

                                             W = V + λH,

and apply to W the partial commutantive symbol

                          d˙         d˙         d˙         d′        d′       d′
                              ,          ,          ,           ,        ,         ,
                         dx         dy         dz          dx       dy        dz
                          d          d          d          d′        d′       d′
                              ,          ,          ,           ,        ,         ,
                         dξ         dη         dζ          dξ       dη        dζ
                          d          d          d          d′        d′       d′
                              ,          ,          ,           ,        ,         ,
                         dξ ′       dη ′       dζ ′        dξ ′     dη ′      dζ ′
we shall obtain a function of λ of which all the odd powers and the second power
will disappear, and such that the coefficients of λ2 and the constant term will be

                                                      326
Aronhold’s S and T , and the discriminant of the entire function in respect to λ2
(if not for the distribution assigned to the dots and dashes in the foregoing, at
least for some other distribution) may not improbably be the discriminant of
the given function (x, y, z)3 .                                                   p. 322


                                Notes in Appendix.

   (1) [p. 295 above.] More generally, in as many ways as the number n can be
divided into parts, in so many ways can a given function of one set of variables
be as it were unravelled so as to furnish concomitant forms.
   For instance, the form ax3 + 3bx2 y + 3cxy 2 + dy 3 has for a concomitant

                        aux + buy + bvx + cvy + cwx + dwy,

where u, v, w are cogredient with x2 , 2xy, y 2 ; and also

         auu′ x + buuy + buv ′ x + bvu′ x + cvv ′ x + cvu′ y + cuv ′ y + dvv ′ y,

where u, v; u′ , v ′ are cogredient with each other and with x and y; and the
proposition in the text may be best derived from this more general theorem by
dividing the index into equal parts, forming as many systems as there are such
parts, and then identifying the systems so formed.
   (2) [p. 297 above.] The following additional example will illustrate the power
of this method.
   Let ϕ = (x, y, z)4 be the general function of the fourth degree. Form by
unravelment the concomitant form (u, v, w, p, q, r)2 (say P ) where u, v, w, p, q, r
are cogredient with x2 , y 2 , z 2 , 2zy, 2xz, 2yx.
   Again, the universal concomitant (xξ + yη + zζ)2 will have for its concomitant

                        uξ 2 + vη 2 + wζ 2 + pηζ + qζξ + rξη,

where ξ, η, ζ are contragredient to x, y, z. Now take the reciprocal polar of this
last form with respect to ξ, η, ζ; that is,
                   1            1     1
                                                   
             Σ vw − p2 x21 + 2Σ   qr − pu y1 z1 ,                  (say G),
                   4            4     2
where x1 , y1 , z1 , being contragredient to ξ, η, ζ, will be cogredient with x, y, z. P +
λG is a quadratic function of the six variables u, v, w, p, q, r, and its discriminant
will give a function of λ of the sixth degree, all of whose even coefficients will
be covariants of ϕ. If we replace x1 , y1 , z1 by x, y, z, these even coefficients will
be respectively (understanding that order refers to the dimensions quoad the
coefficients of ϕ and degree to the dimensions quoad x, y, z) as follows:                  p. 323




                                            327
                                Of order 6 degree 0,
                                   ”5        ”     2,
                                   ”4        ”     4,
                                   ”3        ”     6,
                                   ”2        ”     8,
                                   ”1        ”    10,
                                   ”0        ”    12.
The two last coefficients must evidently be identically zero. It is possible that
some of the others may be so too: as regards the one of the third order and
sixth degree, this is of the same form as, and may be identical with, the Hessian
of ϕ; as regards the one of the fourth order and fourth degree, this may be ϕ
itself multiplied by the cubic invariant (which the theory of Section III. proves
to exist) of ϕ. But the covariants of the fifth order and second degree, and of the
second order and eighth degree, if they are not identically zero, and if the latter
is not ϕ2 (which a trial or two of some very simple cases will easily establish one
way or the other) are probably irreducible forms. The existence of a correlated
conic section to a curve of the fourth order, if established, would be particularly
interesting, and its geometrical meaning would well deserve being elicited.
   (3) [p. 303 above.] If any form (f ) of the degree n be written symbolically,

                              (a1 x1 + a2 x2 + · · · + aι xι )n ,

where x1 , x2 . . . xι are real but a1 , a2 . . . aι umbral, and if Ir be any invariant of
the order r in respect of the real coefficients of (f ), it is easily seen by reason of
Ir remaining unaltered when x1 , x2 . . . xι become respectively f1 x1 , f2 x2 . . . fι xι ,
provided that f1 , f2 . . . fι = 1, that each term in Ir expressed by means of the
umbræ, must contain an equal number of times a1 , a2 . . . aι , so that each such
term will contain nr  ι of each of them, of course differently subdivided and grouped;
hence we have the universal condition that nr          ι must be an integer; but this is
less stringent than the actual condition, which is that nr      ι must be an integer of a
certain form; for instance, as before observed, when ι = 2, nr           ι must be an even
integer.
    (4) [p. 307 above.] To prove the theorem given in the text for total simple
commutants it is only necessary to bear in mind that whenever two columns
in any total commutant become identical, the commutant vanishes. To fix the
ideas, take the commutant formed of lines similar to dx           , dy , dz , written
                                                                d d d
                                                                                             p. 324
    under one another; let there be r such lines, the total number of terms will
be (1 · 2 · 3)r : the 1 · 2 · 3 positions of the line written above will correspond to
(1 · 2 · 3)r−1 several groupings of the remaining lines. Now when x, y, z undergo
a unimodular linear substitution, dx        , dy , dz will undergo a related substitution
                                          d d d

not coincident with that of x, y, z, but still unimodular; let x, y, z change, all the


                                             328
other systems remaining fixed, and suppose
                                         d      d       d
                                           ,      ,
                                        dx     dy       dz
to become respectively
         d     d   d                   d       d     d                       d         d      d
    f      +g    +h ,            f′      + g′    + h′ ,              f ′′      + g ′′    + h′′ ,
        dx    dy   dz                 dx      dy     dz                     dx        dy      dz
then each of the (1 · 2 · 3)r−1 groups of the terms arising from the permutation of
dx , dy , dz will subdivide into 27 groups, of which we may reject those in which
 d d d
                                
any of the terms dx   , dy , dz occurs twice or three times; accordingly there will
                    d d d

be left only the six effective orders of permutations,
                    d       d     d                    d     d        d
                                                                           
               f      , g ′ , h′′    ;           f       , h′ , g ′′    ;          &c.
                   dx      dy     dz                  dx     dz      dy
consequently each of the (1 · 2 · 3)r−1 groups gives rise to 6 times 6 products
whose sum will be
             f ′′ g ′′ h′′
             f ′ g ′ h′       × the sum of the 6 products corresponding
             f g h

to the permutations of dx    , dy , dz ; and therefore, the transformations being
                           d d d

unimodular, the sum of the products corresponding to the entire (1 · 2 · 3)r
permutations remains constant when x, y, z change. In like manner, all the
systems may change one after the other, and consequently all of them at the
same time without affecting the value of the commutant: and in like manner for
the general case.                                                          Q.E.D.
   (5) [p. 312 above.] The truth of the proposition relative to compound commu-
tants and the mode of the demonstration will be apparent from the subjoined
example.
   Let the function be supposed to be
                     (ax + by)(a′ x′ + b′ y ′ )(αξ + βη)(α′ ξ ′ + β ′ η ′ ),
                                                                                                   p. 325
   where x, y; x′ , y ′ are cogredient and ξ, η; ξ ′ , η ′ cogredient; the a, b, α, β, &c.
are of course mere umbræ. Now take the compound commutant
                                  aa′ , ab′ + a′ b, bb′ ,
                                  αα′ , αβ ′ + α′ β, ββ ′ .
Let x, y; x′ , y ′ undergo a linear substitution, and, accordingly,
                                 a      become        f a + gb,
                                 a′        ”          f a′ + gb′ ,
                                  b        ”          ha + kb,
                                 b′        ”          ha′ + kb′ ,

                                               329
f, g, h, k being of course actual and not umbral; then the above commutant will
be easily seen to decompose into 6 others, which will be equal to the original
commutant multiplied by the determinant
                                       f2     2f g  g2
                                       f h f k + gh gk
                                       h2     2hk   k2
which is equal to (f k − gh)3 , that is = 1.
   And so in general, which shows, as in the preceding note, that all the classes
of cogredient systems may be transformed successively one after the other, and
therefore simultaneously, without altering the value of the commutant.
   (6) In the last May Number191 of the Journal, Mr Boole, to whose modest
labours the subject is perhaps at least as much indebted as to any one other
writer, has given a theorem192 , (14) p. 94, the excellent idea contained in which
there is no difficulty in shaping so as to render it generalizable by aid of the
theory of contravariants. It may be regarded in some sort a pendant or reciprocal
to the Eisenstein-Hermite theorem, presented by me under a wider aspect in the
First Section of this paper.                                                       p. 326
   Let ϕ(x, y . . . z) have any contravariant θ(x, y . . . z); then will
                                 d d      d
                                            
                            ϕ     ,   ...      · θ(x, y . . . z)
                                dx dy     dz
be a contravariant of ϕ. For orthogonal transformations the terms contravariant
and covariant coincide, and the theorem for this case appears to have been
known to Mr Boole, see (15), same page. More generally, if ψ and θ be any
two concomitants of ϕ, the algebraical product ψθ will also be a concomitant
of ϕ, provided that the systems of variables in ψ and θ have all distinct names,
or that those which bear the same names are cogredient with one another. If
this proviso does not hold good, the product in question will evidently be no
longer a concomitant of ϕ. Let however Ψ denote what ψ becomes, and ϑ what
θ becomes, when in place of the variables x, y . . . z of every two contragredient
synonymous systems in ψ and θ we write
                                     d d         d
                                       ,    ... ,
                                    dx dy       dz
 191
    Camb. and Dub. Math. Journ. Vol. vi. (1851), pp. 87–106.
 192
    Mr Boole applied his theorem to obtain          the cubic invariant of (x, y) , say ϕ(x,2n
                                                                                   4
                                                                                              y), by
operating upon its Hessian with ϕ dy , − dx . More generally, when ϕ(x, y) = (x, y) , the
                                            d    d

catalecticant of the antepenultimate        emanant of ϕ is also of the degree 2n; and this, when
operated upon by ϕ dy               , will give an invariant of the order n + 1, which is probably
                         d      d
                                  
                           , − dx
identical with the catalecticant of ϕ itself. There exists a most interesting transformation of the
catalecticant of any emanant of a function of any degree in x, y, whether even or odd, under the
form of a determinant some of the lines of which contain combinations only of x and y, without
any of the coefficients, and all the rest the coefficients only of the given function without x or y.
The Hessian being the catalecticant of the second emanant is of course included within this
statement.


                                                330
then will ϑψ and Ψθ be each of them concomitants of ϕ, the synonymous systems
becoming cogredient with ψ in the one case and with θ in the other.
   (7) There is one principle of paramount importance which has not been
touched upon in the preceding pages, which I am very far from supposing to
exhaust the fundamental conceptions of the subject, (indeed, not to name other
points of enquiry, I have reason to suppose that the idea of contragredience itself
admits of indefinite extension through the medium of the reciprocal properties
of commutants; the particular kind of contragredience hereinbefore considered
having reference to the reciprocal properties of ordinary determinants only).
   The principle now in question consists in introducing the idea of continuous
or infinitesimal variation into the theory. To fix the ideas, suppose C to be a
function of the coefficients of ϕ(x, y, z), such that it remains unaltered when
x, y, z become respectively f x, gy, hz, provided that f gh = 1. Next, suppose
that C does not alter when x becomes x + εy + ϵz, when ε and ϵ are indefinitely
small: it is easily and obviously demonstrable that if this be true for ε and ϵ
indefinitely small, it must be true for all values of ε and ϵ. Again, suppose that C
alters neither when x receives such an infinitesimal increment, y and z remaining
constant, nor when y nor z separately receive corresponding increments, z, x
and x, y in the respective cases remaining constant; it then follows from what
has been stated above that this remains true for finite increments to x or y or z
separately; and hence it may easily be shown that C will remain constant for
any concurrent linear transformations of x, y, z, when the modulus is unity. This
all-important principle enables
                                  us at once to fix the form of the symmetrical
functions of the roots of ϕ xy , 1 which represent invariants of ϕ(x, y) when the
coefficient of the                                                                   p. 327
   highest power of x is made unity. It also instantaneously gives the neces-
sary and sufficient conditions to which an invariant of any given order of any
homogeneous function whatever is subject, and thereby reduces the problem of
discovering invariants to a definite form. But as these conditions coincide with
those which have been stated to me as derived from other considerations by the
gentleman whose labours in this department are concomitant with my own, I
feel myself bound to abstain from pressing my conclusions until he has given his
results to the press.
   (8) By aid of the general principle enunciated in Note (6) above, we can easily
obtain Aronhold’s S and T . Let U be the given cubic function of x, y, z, and let
G(x, y, z; ξ, η, ζ) be the polar reciprocal in respect to ξ, η, ζ of

                                    d     d    d
                                                 
                                 ξ    +η    +ζ    U,
                                   dx    dy    dz

then G(ξ, η, ζ; x, y, z) as well as the former G will be a concomitant to U , but
the homonymous systems of variables in the two G’s will be contragredient; and,



                                        331
accordingly,
                    d d d d d d
                                               
                 G   , , ; , ,                      · G(ξ, η, ζ; x, y, z)
                   dx dy dz dξ dη dζ
will be a concomitant to U ; this concomitant is readily seen to be an invariant
of the fourth order; that is, Aronhold’s S. Again, from S, by means of the
Eisenstein-Hermite theorem, we may derive a form K(x, y, z) of the third degree
in x, y, z, and whose coefficients will be of three dimensions; and, accordingly, if
the Hessian of U be called H(U ),

                                      d d d
                                               
                             K         , ,          · H(U )
                                     dx dy dz
will be a Sextic Invariant of U , that is, Aronhold’s T .




                                         332
                                            43.
           On the Principles of the Calculus of Forms
   [Cambridge and Dublin Mathematical Journal, VII. (1852), pp. 179–217]



  Part I.       Section IV. Reciprocity, also Properties and
                 Analogies of certain Invariants, &c.
                                                                                         p. 328
   It will hereafter be found extremely convenient to represent all systems of
variables cogredient with the original system in the primitive form by letters of
the Roman, and all contragredient systems by letters of the Greek alphabet; the
rules for concomitance may then be applied without paying any regard to the
distinction between the direction of the march of the substitutions, the variables
at the close of each operation as it were telling their own tale in respect of being
cogredients or contragredients. This distinction has not (as it should have) been
uniformly observed in the preceding sections; as, for instance, in the notation for
emanants which have been derived by the application of the symbol
                                                   2
                                     d     d
                                                            
                                  ξ    +η    + &c. ,
                                    dx    dy
instead of the more appropriate one
                                                              2
                                   ′ d          ′ d
                              
                               x         +y            + &c.       .
                                   dx           dy

   The observations in this section will refer exclusively to points of doctrine which
have been started in the preceding sections in such order as they more readily
happen to present themselves. And, first, as to some important applications of
the reciprocity method referred to in Notes (6) and (8) of the Appendix [pp. 325,
327 above].
   The practical application of this method will be found greatly facilitated by
the rule that x, y, z, &c. may always in any combination of concomitants be
replaced respectively by
                               d          d           d
                                  ,         ,            ,   &c.,
                               dξ        dη           dζ
and vice versâ. I shall apply this prolific principle of reciprocity to elucidate some
of the properties and relations of Aronhold’s S and T , and certain other kindred
forms. This S and T are the quartinvariant and sextinvariant respectively of a
cubic of three variables. I give the names of s and t to the quadrinvariant and
cubinvariant of the quartic function of two variables. Furthermore, whoever will

                                            333
consider attentively the remarks made in Section II. of the foregoing relative to
reciprocal polars, will apprehend without any difficulty that to every invariant
of a function of any degree of any number of variables will                                p. 329
   correspond a contravariant of a function of the same degree of variables one
more in number, and that between such invariants, whatever relations exist
expressed independently of all other quantities, precisely the same relations must
exist between the corresponding contravariants. Thus, then, to s and t the
two invariants of (x, y)4 will correspond two contravariants σ and τ of (x, y, z)4 ,
and to S and T the two invariants of (x, y, z)3 will correspond Σ and S two
contravariants of (x, y, z, t)3 . Calling r the resultant of (x, y)4 , R the resultant of
(x, y, z)3 , ρ the polar reciprocal, or, more briefly, the reciprocant of (x, y, z)4 , and
(R) the reciprocant of (x, y, z, t)3 , we have the following equations (presuming that
all the quantities are previously affected with the proper numerical multipliers),
namely
                              r = s3 + t2 ,     ρ = σ3 + τ 2,
                          R = S3 + T 2,        (R) = Σ3 + S2 .

   I propose in this First Annotation to point out the remarkable analogies which
exist between the modes of generating the four pairs of quantities s, t, &c., the
functions severally corresponding to which I shall call u, ω, U, Ω. The Hessian
corresponding to any of these functions will be denoted by an H prefixed, and
when we have to consider, not the pure Hessian, but the matrix formed from
it by adding a vertical and horizontal border of variables, the same in number
but contragredient to the variable of the function (as, for instance, the Hessian
of u bordered with ξ, η horizontally and vertically, or of U with ξ, η, ζ), then
I shall denote the result by the ruled symbol H, and if there be occasion to
add two borders, as ξ, η, ζ; ξ ′ , η ′ , ζ ′ , both repeated in the horizontal and vertical
directions, the result will be typified by the doubly ruled H, H.
   Now, in the first place, as observed by me in Note (8) of the Appendix in the
last number; if we call the coefficients of U (10 in number) a, b, c, d, &c., we have

                             d d d d d d
                                                   
                 S=H           , , ; , ,       H(x, y, z; ξ, η, ζ),
                             dξ dη dζ dx dy dz
also
                          dS d3 H   dS d3 H     dS d3 H
                    T =           +           +           + &c.
                          da dx3    db dx2 dy   dc dx2 dz
I will now add the further important relation

                         dT d3 H   dT d3 H     dT d3 H
                  S2 =           +           +           + &c.193
                         da dx3    db dx2 dy   dc dx2 dz




                                           334
                                                                                                    p. 330
   so that it will be observed if all the derivatives of S are zero, T is zero, and
vice versâ.
   Precisely in the same way, using h and h to denote respectively the Hessian
of u and the same bordered with ξ, η, we have
                               d d d d
                                                         
                           s=h   , ; ,     h(x, y; ξ, η),
                               dξ dη dx dy

                           ds d4 h ds d4 h     ds d4 h
                      t=          +          +            + &c.,
                           da dx4   db dx3 dy dc dx2 dy 2
                               dt d4 h dt d4 h   dt d4 h
                      s2 =            +        +            + &c.
                               da dx4 db dx3 dy dc dx2 dy 2

  Again, taking (H) the second bordered Hessian of Ω; that is, Ω bordered as well
horizontally as vertically with the double lines and columns ξ, η, ζ, θ; ξ ′ , η ′ , ζ ′ , θ′ ,
                        d d d d d d d d ′ ′ ′ ′
                                                                                   
                Σ = (H)   , , , ; , , , ;ξ ,η ,ζ ,θ
                        dξ dη dζ dθ dx dy dz dt
                        × (H)(x, y, z, t; ξ, η, ζ, θ; ξ ′ , η ′ , ζ ′ , θ′ ),

                   dΣ d3 H   dΣ d3 H    dΣ d3 H    dΣ d3 H
             S=          3
                           +      2
                                      +      2
                                                 +           + &c.,
                   da dx     db dx dy   dc dx dz   dd dx2 dt
                                      dS d3 H   dS d3 H
                             Σ2 =             +           + &c.
                                      da dx3    db dx2 dy
In like manner again
                                   d d d d d d ′ ′ ′
                                                                               
                       σ = (h)      , , ; , , ;ξ ,η ,ζ
                                  dξ dη dζ dx dy dz
                                × h{x, y, z; ξ, η, ζ; ξ ′ , η ′ , ζ ′ },

                           dσ d4 (h)                             dτ d4 h
                     τ=              + &c.,              σ2 =            + &c.
                           da dx4                                da dx4
   σ and τ are the same quantities as are calculated by Mr Salmon, in his
inestimable work On Higher Plane Curves, but are there expressed under the
names of S and T , with the sole difference that in place of x, y, z, used by Mr
 193
     It will be found hereafter convenient to designate contravariants formed in this manner
from invariants as Evects of such invariants or contravariants, and according to the number of
times that such process of derivation is applied, 1st, 2nd, 3rd, &c. evects. Such evects form a
peculiar class, and when considered generally, without reference to the base to which they refer,
they may be termed evectants. Evectants will be again distinguishable according as their base
is an invariant simply or a contravariant. Perhaps the terms pure and affected evectants may
serve to mark this distinction.



                                                  335
Salmon, the contragredient variables ξ, η, ζ are used in the expressions above.
Mr Salmon has also pointed out to me that σ may be obtained by operating with

                             4 d     d         d
                                                            
                           ξ    + ξ η + ξ 2 ζ 2 + &c.
                                     3
                             da      db        dc

directly upon I a cubic invariant of the function u, or (x, y, z)4 . This I is no other
than the simple commutant obtained by operating upon u with the commutantive
symbol formed by taking four times over the line
                                    d      d     d
                                      ,      ,      ,
                                   dx     dy     dz
agreeable to the remark made in the third section that                                p. 331
   every function of an even degree of n variables possesses an invariant of the
nth order in extension of Mr Cayley’s observation that every such function of
two variables possesses a quadrinvariant, that is an invariant of the second order.
   I need hardly remark that σ is of 2 dimensions in the coefficients and of 4 in the
contragredient variables, τ of 3 in the coefficients and of 5 in the contragredients,
Σ of 4 in the constants and 4 in the contragredients, S of 6 in the constants and
6 in the contragredients, or that the single-bordered Hessians of u and U and
the double-bordered Hessians of ω and Ω are each of them quadratic in respect
of the x &c. as well as of the ξ &c. systems.
   If the right numerical factors be attributed to S, T , Aronhold has shown that

                        H{H(U )} + T · H(U ) + S 2 U = 0,

and in my paper in the last May Number194 , I gave the equation

                             h{h(u)} + s · h(u) + tu = 0.

I think it highly probable that it will be found that the analogous equations
obtain, namely
                      H{H(Ω)} + S · H(Ω) + Σ2 Ω = 0,
                             h{h(ω)} + σ · h(ω) + τ ω = 0.
These remarkable equations, if verified (of which I can scarcely doubt), will
be most powerful aids to the dissection of the forms ω, Ω, and thereby to the
detection of the fundamental properties of curves of the fourth and surfaces of
the third degree, of which at present so little is known. It will have been observed
that in the preceding developments the contravariants of ω and Ω were derived
in precisely the same way from ω and Ω as the corresponding invariants of u
and U from u and U , with the sole difference that the Hessian used in the two
latter cases is replaced by a single-bordered Hessian in the two former cases,
 194
       p. 192 above.


                                          336
and a single-bordered Hessian in the two latter by a double-bordered Hessian in
the two former. The analogies are not even yet stated exhaustively; for it will
be remembered (as shown in the third section), that T and S can be derived
directly and concurrently by means of operating with the commutantive symbol
                 d      d      d
                                  
                   ,      ,       
                dx     dy      dz 
                                  
                                  
                 d      d      d
                                  
                                  
                   ,      ,       
                                  
                dx     dy      dz
                                  
                 d      d      d      upon H(U ) + λ(xξ + yη + zζ)2 ,
                   ,      ,       
                                  
                dξ     dη      dζ 
                                  
                                  
                 d      d      d 
                   ,      ,
                                  
                                  
                                  
                dξ     dη      dζ
                                                                                            p. 332
  which gives a result of the form m(λ3 + Sλ + T ), m being a number; and I
conjecture that if
                              d    d   d    d
                                               
                                ,    ,   ,     
                             dx dy dz       dt 
                                               
                                               
                              d    d   d    d
                                               
                                               
                                ,    ,   ,     
                                               
                             dx dy dz       dt
                                               
                              d    d   d    d
                                ,    ,   ,     
                                               
                             dξ   dη dζ dθ    
                                               
                              d    d   d    d 
                                ,    ,   ,
                                               
                                               
                                               
                             dξ   dη dζ dθ
be made to operate upon

                               HΩ + λ(xξ + yη + zζ + tθ)2 ,

and the result be put under the form

                              m(λ4 + Aλ3 + Bλ2 + Cλ + D),

that A will be zero, B and C will be respectively Σ and S, and perhaps D
(a contravariant, if it effectively exist, of 8 dimensions in the coefficients of Ω,
and of a like number in the contragredients ξ ′ , η ′ , ζ ′ , θ′ ), also zero. But of the
evanescence of D I do not speak with any degree of assurance.
   Mr Salmon has made an excellent observation to the effect that if we call (σ)
what σ becomes when ξ ′ , η ′ , ζ ′ are replaced by
                                       d      d     d
                                         ,      ,      ,
                                      dx     dy     dz

(σ)h(ω) will represent a covariant to ω of 3 + 2, that is, 5 dimensions in the
coefficients, and of 6 − 4, that is, of 2 dimensions in x, y, z, h(ω) being of 3 and
6 dimensions in these respectively, and σ of 2 and 4 dimensions respectively in
the same. Now these resulting dimensions 5 and 2 precisely agree with the form

                                             337
especially noticed by me in Note195 (2) of the Appendix, where it was derived as
one of a group by the method of unravelment. There can be little doubt that
these two conics each of them indissolubly connected with every curve of the
fourth degree are identical. The form (σ)h(ω) enables us to prove readily (thanks
to Mr Salmon’s calculation of σ, given in his Higher Plane Curves, under the
name of S) that this is a bonâ fide existent conic.
   For if we take a particular case of ω, say

                                ω = a1 x4 + b2 y 4 + c3 z 4 + 6dy 2 z 2 ,

we find
                                       a1 x2        0             0
                           h(ω) =        0   b2 y 2 + dz 2      dyz
                                         0        dyz      c3 z 2 + dy 2
                       = a1 (b2 c3 + d2 )x2 y 2 z 2 + a1 b2 dx2 y 4 + a1 c3 dx2 z 4 ,
                                                                                            p. 333
   and σ becomes
                                                  a1 dη 2 ζ 2 ,
and consequently (σ) is
                                                      2         2
                                               d             d
                                              
                                         a1 d                          ;
                                              dy             dz
and therefore
                                   (σ)h(ω) = 4a21 d(b2 c3 + d2 )x2 ,
the conic here reducing to a pair of coincident straight lines. This example
demonstrates that the conic is in general actually existent.
   As I have said so much upon S and T it may not be irrelevant to state in
this place how I obtained the conditions for U , the characteristic of the curve
of the third degree becoming the characteristic of a conic and a straight line,
that is breaking up into a linear and a quadratic factor, which Mr Salmon has
inserted in the notes to his work above referred to. When U is of this form it
may obviously by linear transformations be expressed by ax3 + 6dxyz, but when
starting with the general form,

                               a1 x3 + b2 y 3 + c3 z 3 + &c. + 6Dxyz,

we form two contravariants from S and T , to wit
                        d        d        d               d
                                                                          
                     ξ 3
                           + η3     + ζ3     + &c. + ξηζ    S,                  say S ′ ,
                       da1      db2      dc3             dD
                        d        d        d               d
                                                                          
                     ξ 3
                           + η3     + ζ3     + &c. + ξηζ    T,                  say T ′ ,
                       da1      db2      dc3             dD
 195
       p. 323 above.


                                                     338
and then make a1 = a, D = d, and all the other coefficients zero, it will easily
be seen on examining the forms of S and T , given by Mr Salmon, that (S ′ ) and
(T ′ ) (the evectants of S and T ) become respectively

                                 4d2 ξηζ,          31d5 ξηζ;

we have therefore (T ′ ) + λ(S ′ ) = 0: and (T ′ ) and (S ′ ), although contravariantive
to their primitive U , are covariantive with one another, so that (T ′ ) + λ(S ′ ) = 0
is a persistent relation unaffected by linear transformations; it follows therefore
that when U is of, or reducible to, the form supposed,
             dS dS dS            dS   dT dT dT            dT
                :   :    : &c. :    =    :   :    : &c. :    ,
             da1 db2 dc3         dD   da1 db2 dc3         dD

which is the criterion given in the note referred to.196
   I am also able to obtain these equations more directly by another method
founded upon a New View of the Theory of Elimination, an account of which, p. 334
   however, I must reserve for another occasion, but which, I may mention, serves
to fix not merely the conditions, as in the ordinary restricted theory, that a
given set of equations may be simultaneously satisfiable by some one system of
values of the variables, but the conditions that such set of equations may be
simultaneously satisfiable by any given number of distinct systems of variables.
   Mr Salmon has remarked to me to the effect that if in τ we write
                                     d       d       d
                                       ,       ,        ,
                                    dx      dy       dz

in place of the contragredients, and call τ so altered (τ ), then (τ )h(ω) will be an
invariant of 6 dimensions in the coefficients of ω. This sextinvariant I have little
doubt is identical with that obtained by operating upon ω with the commutantive
symbol
                                                                          
                d 2       d d     d 2           d d    d 2          d d
                                                           
                     ,         ,        ,            ,        ,           
                                                                          
              dx 2     dx dy  dy           dy dz  dz          dz dx .
                                                                          
                                      2                     2
                d         d d     d             d d    d            d d
                     ,         ,        ,            ,        ,           
                                                                          
               dx        dx dy   dy            dy dz   dz           dz dx
                                                                          

This, like every other commutant of 2 lines only, is of course capable of being
expressed under the form of an ordinary determinant, and the remark is not
without interest, as showing how the proposition, known with respect to quadratic
functions of any number of variables, namely of every such having an invariantive
determinant, lends itself to the general case of functions of any even degree of
any number of variables which also have always an invariantive determinant
 196
     Mr Salmon has remarked to me that the two evectants (S) and (T ) intersect in the nine
cuspidal points of the polar reciprocal to the curve.


                                            339
attached to them, of which the terms are simple coefficients of such functions.
The only peculiarity (if it be one) of quadratic functions in this respect being
that they have each but one invariant of such form and no other. In the case
before us, if we write

          ω = a1 x4 + b2 y 4 + c3 z 4 + 4a2 x3 y + 4a3 x3 z + 4b1 y 3 x + 4b3 y 3 z
                 + 4c1 z 3 x + 4c2 z 3 y + 6dy 2 z 2 + 6ez 2 x2 + 6f x2 y 2
                 + 12lx2 yz + 12mxy 2 z + 12nxyz 2 ,

the sextinvariant in question becomes representable under the form of the
determinant
                         a1 a2 f l e a 3
                         a2 f b1 m n l
                         f b1 b2 b3 d m 197
                                                .
                          l m b3 d c2 n
                          e n d c2 c3 c1
                         a3 l m n c 1 e
                                                                                                  p. 335
    Before quitting the subject of S and T the two invariants of the cubic function
of 3 variables, or, as it may be termed, of the cubic curve, it may not be amiss
to give the complete table which I have formed corresponding to all the singular
cases which can befall such curve, which will be seen below to be eight in number;
it is of the highest importance to push forward the advanced posts of geometry,
and for this purpose to obtain the same kind of absolute power and authority
over, and clear and absolute knowledge of, the properties and affections of cubic
forms as have been already attained for forms of the second degree.
    Let
                          U = ax3 + 4bx2 y + 4cxz 2 + &c.

   (1) When U has one double point

                                        S 3 + T 2 = 0.

   (2) When U has two double points, that is becomes a conic and right line

                             dS dT   dS dT
                                   −       = 0,           &c. &c.
                             da db   db da

   (3) When U has a cusp

                                     S = 0,         T = 0.
 197
    This determinant is identical with the determinant formed by taking the second differential
coefficients of the function and arranging in the usual manner the coefficients of the several
powers and combinations of powers of the variables treated as if they were independent
quantities.


                                              340
   (4) When U has two coincident double points, that is, is a conic and a tangent
line thereto, which comprises the two preceding cases in one,
                           dT            dT
                              = 0,          = 0,     &c.,
                           da            db
and also therefore
                                      S = 0.
  (5) When U becomes three right lines forming a triangle

                        d2 S d2 T   d2 T d2 S
                                  −            = 0,        &c.,
                       da db dc de da db dc de
where a, b, c, e each represent any of the coefficients arbitrarily chosen, whether
distinct or identical.
   Another, and lower in degree system of equations, may be substituted for the
above, obtained by affirming the equality of the ratios between the coefficients
of U and the corresponding coefficients of its Hessian.
   (6) When U represents a pencil of three rays meeting in a point
                           dS            dS
                              = 0,          = 0,     &c.
                           da            db
and also therefore
                                      T = 0.
Also in place of this system may be substituted the system obtained by taking
all the coefficients of the Hessian zero.                                     p. 336
   (7) When U becomes a line, and two other coincident lines,
                           dS            dS
                              = 0,          = 0,     &c.
                           da            db
and also
                          d2 T           d2 T
                               = 0,           = 0, &c.
                           da2          da db
I have not ascertained whether this second system necessarily implies the first; I
rather think that it does not. In the preceding case also it would be interesting
to show the direct algebraical connexion between the system formed by the
coefficients of the Hessian and the system consisting of the first derivatives of S.
   (8) When U becomes a perfect cube representing three coincident right lines

                          d2 S           d2 S
                               = 0,           = 0,    &c.
                          da2           da db
and
                         d2 T           d2 T
                              = 0,            = 0,    &c.
                         da2            da db

                                       341
The first of these systems of equations necessarily implies the equations
                           dT              dT
                              = 0,            = 0,         &c.,
                           da              db
as is obvious from the equation

                               dS d3 H   dS d3 H
                         T =           +           + &c.
                               da dx3    db dx2 dy
but not necessarily the second and lower system

                                  d2 T
                                       = 0,        &c.
                                  da2
above written.
  So if we take

                   u = ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4

when 2 roots are equal
                                    s3 + t2 = 0,
when 2 pairs of roots are equal
                           ds dt ds dt
                                −      = 0,                &c.,
                           da db db da
when 3 roots are equal
                                  s = 0,          t = 0,
and when all 4 roots are equal
                           dt               dt
                              = 0,             = 0,        &c.
                           da               db
   Before closing this Section I may make a remark, in reference to the sextic
invariant of ω, which admits of being extended to all commutants formed by
operating upon the function with a commutantive symbol obtained by writing
over one another lines consisting of powers of
                                   d        d
                                     ,        ,     &c.
                                  dx       dy
and                                                                            p. 337
  their combinations (to which, in the Third Section, I gave the name of
compound commutants, a qualification which, for reasons that will hereafter be




                                           342
adduced, I think it advisable to withdraw). The remark I have to make is this,
namely that the invariant obtained by operating upon ω with
                                                                       
               d 2       d d     d 2           d d    d 2          d d
                                                        
                    ,         ,        ,            ,        ,           
                                                                         
             dx 2     dx dy  dy           dy dz  dz          dz dx ,
                                                                         
                                     2                     2
               d         d d     d             d d    d            d d
                    ,         ,        ,            ,        ,           
                                                                         
              dx        dx dy   dy            dy dz   dz           dz dx
                                                                         

is precisely the same as may be obtained by operating with
                          d     d       d     d      d      d
                                                              
                            ,      ,      ,      ,      ,      
                         du     dv     dw     dp     dq     dr
                                                               
                          d     d       d     d      d      d
                            ,      ,      ,      ,      ,      
                                                               
                         du     dv     dw     dp     dq     dr
upon the concomitant quadratic function to ω obtained by the method of
unravelment, as in Note (2) of the Appendix [p. 322 above]; and so, in general,
every commutant obtained by operating upon a function of any number of
variables of the degree 2mp with a symbol consisting of 2p lines in which the
mth powers of dx   , dy , &c. and their mth combinations occur, will be identical
                 d d

with the commutant obtained by operating with a symbol also of 2p lines, in
which only the simple powers occur of du  , dv , &c. (where u, v, &c. are cogredient
                                         d d

with x , x y, &c.), upon a function of u, v, &c., formed by the method of
       p   p−1

unravelment from the given function.
   Finally, before quitting the subject of reciprocity, I may state, it follows from
the general statement made at the commencement of this Section, that inasmuch
as
                               (xξ + yη + zζ + &c.)2
is a universal concomitant form, so also must
                                                    2
                            d d    d d   d d
                                                            
                                 +     +      + &c.
                            dξ dx dη dy dζ dz
be a universal concomitant symbol of operation; accordingly it is certain that
any concomitant in which x, y, z, &c., ξ, η, ζ, &c. enter, operated upon with
this symbol, will remain a concomitant; in several cases which I have examined,
the effect of this operation will be to produce an evanescent form, but I see no
ground for supposing that this is other than an accidental, or at all events for
supposing that it is a necessary and universal consequence of the operation. It
may also be observed that in the case of as many cogredient sets of variables as
variables in each set, as for instance 3 sets                                    p. 338
   of 3 variables each, the determinant which may be formed by arranging them
in regular order, as
                                    x y z
                                    x′ y ′ z ′ ,
                                    x′′ y ′′ z ′′

                                           343
is evidently a universal concomitant, and moreover an equivocal concomitant,
possessing the property of remaining a concomitant when the variables are respec-
tively but simultaneously exchanged for their contragredients ξ, η, ζ; ξ ′ , η ′ , ζ ′ ; ξ ′′ , η ′′ , ζ ′′ ;
which shows also that in place of the variables may be written the differential
operators
                        d d d          d d d           d      d    d
                          , , ;          ′
                                           , ′, ′;       ′′
                                                            , ′′ , ′′ ;
                       dx dy dz       dx dy dz       dx dy dz
a remark which leads us to see the exact place in the general theory occupied by
Mr Cayley’s method of generating covariants given in the concluding paragraph
of the First Section [p. 290 above]. I may likewise add, that inasmuch as
(x′ ξ + y ′ η + z ′ ζ + &c.)2 is a universal concomitant,
                                                                      r
                                               ′ d        ′ d
                                      
                                          x          +y         + &c.
                                               dx         dy
will be so too, by virtue of the general law of interchange, which conducts
immediately to the theory of emanation, showing that this last symbol, operating
upon any function, furnishes covariants thereunto for any integer value of r.
   One additional interesting remark presents itself to be made concerning U ,
the cubic function of x, y, z, which is, that calling as before T its sextic invariant,
and a, 3b, 3c, d, &c. the coefficients, the formula
                                                                  2
                                 d        d       d      d
                                                                          
                           ξ3      + ξ 2 η + ξ 2 ζ + ξηζ    + &c. T
                                da        db      dc     dd
will give the polar reciprocal, or, as it has been agreed to term it, the reciprocant
of U . I believe the remark of the probability of this being the case originated
with myself, but Mr Cayley first verified it by actual calculation, using for that
purpose the value of T , given by Mr Salmon in his work On the Higher Plane
Curves, already frequently alluded to, which is an indispensable manual equally
for the objects of the higher special geometry as for the new or universal algebra,
being in fact a common ground where the two sciences meet and render mutual
aid.
   Mr Salmon also observed, that the first evect of T , namely
                                            d        d
                                                                     
                                      ξ3      + ξ 2 η + &c. T,
                                           da        db
                                                                                                        p. 339
   was identical in form with what may be termed the first devect of the polar
reciprocal, that is, the result of operating upon the polar reciprocal with what
U becomes when
                                   d      d    d
                                      ,     ,    ,
                                   dξ    dη   dζ
are substituted in the stead of x, y, z. And inasmuch as, by Euler’s law,
           (        3                   2                )
                d               d               d              d        d
                                                                                     
              a            + 3b                   + &c. × ξ 3    + ξ 2 η + &c. T
                dξ              dξ             dη             da        db

                                                      344
                                      d    d
                                                                 
                          =6 a          + b + &c. T = 36T,
                                     da    db
it follows that T is the second devect of the polar reciprocal, or at least identical
with it in point of form. But, since the preceding matter was printed, I have
discovered in the course of a most instructive and suggestive correspondence
with Mr Salmon, the principle upon which these and similar identifications
depend, thereby dispensing with the necessity for the excessively tedious labour
of verification which, even in the simple example before us, would be found to
extend over several pages of work.
    The theory in which this principle is involved will be given, along with other
very important matter, in the next number of the Journal.

Supplementary Observations on the Method of Reciprocity.

   It has been observed, that ξ, η, &c. may always be inserted in place of
                                           d           d
                                             ,           ,       &c.,
                                          dx          dy
and vice versâ, in a concomitant form, without destroying its concomitance.
Accordingly, instead of the evector symbol
                                           d        d
                                     ξ3      + ξ 2 η + &c.,
                                          da        db
we may employ
                                    3                     2
                                d         d            d          d d
                                                 
                                            +                          + &c.;
                               dx        da           dx         dy db
and operating with this upon any concomitant, the result will be a concomitant.
Hence we see, for example, that if we take the concomitant SH formed by the
product of the invariant S and the covariant H,
                     (        3                      2                      )
                           d          d            d          d d
                                              
                                        +                          + &c. SH
                          dx         da           dx         dy db

will be a covariant; in fact this will be found to be T , the difference between this
and the expression before given for T , namely
                               3                          2
                           d             dS            d          d dS
                                                 
                                    H       +                      H   + &c.,
                          dx             da           dx         dy db
being                 (                                                             )
                                         3                           2
                           d         d           d                d         d
                                                            
                 S×                           H+                              H + &c. ,
                          da        dx           db              dx        dy
                                                                                          p. 340



                                                      345
    which is zero, there being no invariant to (x, y, z)3 of the 3rd degree in a, b, c,
&c., as the factor multiplied by S would be were it not evanescent. The same
observation may be extended to analogous equations given previously.
    I have chiefly, however, made the above observation with a view to making
more clear the enunciation of the theorem which I am now about to state, the
most important perhaps in its application of any yet brought to light on the
subject, but the consequences of which, as I have but quite recently discovered
it, must be reserved for a future number of the Journal.
    Let any function of any number of variables be supposed to have for its
coefficients the letters a, b, &c. affected with the ordinary binomial or multinomial
coefficients; and let another function be taken identical with the former in all
respects, except in the circumstance that all their numerical multipliers are
suppressed. Let this function or form be termed the respondent to the primitive:
furthermore, by the inverse of any form understand what that form becomes
when, in place of x, y, z, &c. ξ, η, ζ, &c.,
                 d      d     d                    d        d     d
                   ,      ,      ,    &c.,            ,       ,      ,   &c.,
                dx     dy     dz                   dξ      dη     dζ
are respectively substituted (and so for all the systems of the variables), and
likewise at the same time similar substitutions are made of
                                d      d       d
                                  ,       ,       ,       &c.,
                               da      db      dc
in place of a, b, c, &c.; then we have this grand and simple law—The inverse of
any concomitant to a respondent is a concomitant to its primitive. When the
inverse of any concomitant to the respondent is made to operate upon the same
concomitant of the primitive, it will be found that the result is a power of the
universal concomitant. If the concomitant to the respondent be an invariant
thereof, the rule indicates that on merely replacing in the respondent a, b, c, &c.
by da , db , dc , &c., the result operating on any invariant or other concomitant of
    d d d

the primitive, leaves it still an invariant or other concomitant. For instance, if
we take the function
                ax5 + 5bx4 y + 10cx3 y 2 + 10dx2 y 3 + 5exy 4 + f y 5 ,
which has three invariants L, M, N , of the degrees 4, 8, 12, respectively: and if
we call λ, µ, ν what L, M, N become when, in place of a, b, c, d, e, f respectively,
we write
                    d    1 d     1 d      1 d      1 d      d
                      ,       ,       ,         ,       ,     ,
                   da    5 db   10 dc     10 dd    5 de    df
we shall find that
                             λM = L,       µN = L,
and
                       λN = a linear function of M and L2 .

                                             346
                                                                                        p. 341
   Again, if in the case of any function of x, y, z, &c., we take, instead of any
other concomitant to the respondent, the respondent itself, its inverse gives the
symbol of operation
                                        3                   2 
                          d         d          d         d           d
                                                                     
                                             +                               + &c.,
                         da        dx          db       dx          dy
just previously treated of. If again, in the case of a function of x, y, say
                     axn + nbxn−1 y + · · · + nb′ xy n−1 + a′ y n ,
we take the inverse of the polar reciprocal of the respondent, we get the operator
                                       n                   n−1
                          d        d          d         d          d
                                                  
                                            −                         + &c.;
                         da       dη          db       dη          dξ
and replacing dη , dξ by y, x, we find that
               d d


                                  d           d
                                    − y n−1 x + &c.,
                                   yn
                                 da          db
operating on any concomitant, leaves it still a concomitant, which is M. Eisen-
stein’s theorem before adverted to, only generalized by the introduction of any
concomitant in lieu of the discriminant.
   This extraordinary theorem of respondence will be found on reflection to
favour the notion of treating the coefficients of a general function as themselves
a system of variables, in a manner contragredient to the terms to which they are
affixed.
   Finally, there is yet another mode of applying the principle of reciprocity,
which must be carefully distinguished from any previously stated in these pages.
   I have said that in place of the quantitative symbols of one alphabet, as x, y, z,
&c., we may always substitute the operation symbols dξ        , dη , dζ , &c. of the
                                                             d d d

opposite alphabet. But now I say, in place of the quantitative symbols x, y, z, &c.
occurring in the concomitant to any form f , may be substituted the quantities
(observe, no longer operative symbols but quantities)
                                  dF         dF         dF
                                     ,          ,          ,       &c.,
                                  dξ         dη         dζ
F being itself any concomitant to f . Thus, for instance, taking F identical with
f , we see that
                                    df df df
                                                  
                               f      , , , &c.
                                    dξ dη dζ
is concomitant to f : or again, if f be a function of x, y only, say f (x, y), taking
F the polar reciprocal of f , that is f (−η, ξ), we see that
                                                   df df
                                                             
                                            f −      ,
                                                   dy dx

                                                   347
will be a                                                                       p. 342
   concomitant to f : this concomitant, by the way it may be observed, will
                                                    df     df
always contain f as a factor, because when f = 0, x dx + y dy = 0. Possibly it
may be true that, when f is a function of any number of variables x, y, z, &c.,
and F (ξ, η, ζ, &c.) its polar reciprocal,
                            dF (x, y, z, &c.) dF (x, y, z, &c.)
                                                                                     
                   f                         ,                  , &c. ,
                                   dx                dy
which is a concomitant to f , contains f as a factor; but I have not had time to
see how this is. It is rather singular that Mr Cayley and Professor Borchardt of
Berlin have both independently made to me the observation that, when f (x, y)
is taken a cubic function of x and y,
                                                    df    df
                                                               
                                            f          ,−
                                                    dy dx
is equal to the product of f by the first evectant of the discriminant of f . The
general consideration of the consequences of this new and important application
of the idea of reciprocity must be reserved for a future section.

 Section V.         Applications and Extension of the Theory of
                              the Plexus.

  If
                    ϕ = ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 ,
we can obtain, by operating catalectically with x′ , y ′ upon
                                            2                                       4
                           ′ d        ′ d                           ′ d        ′ d
                                                           
                       x         +y              ϕ,             x         +y              ϕ,
                           dx         dy                            dx         dy
the two concomitants
                        ax2 + 2bxy + cy 2 bx2 + 2cxy + dy 2
                                                            ,                                  (1)
                        bx2 + 2cxy + dy 2 cx2 + 2dxy + ey 2

                                                 a b c
                                                 b c d ,                                       (2)
                                                 c d e
the one in fact being the Hessian, the other the catalecticant of ϕ itself. Again, if
                       ϕ = ax5 + 5bx4 y + 10cx3 y 2 + · · · + f y 5 ,
by operating catalectically with x′ , y ′ upon the second and fourth emanants, as
in the last case, we obtain the two covariants

            ax3 + 3bx2 y + 3cxy 2 + dy 3 bx3 + 3cx2 y + 3dxy 2 + ey 3
                                                                       ,                       (1)
            bx3 + 3cx2 y + 3dxy 2 + ey 3 cx3 + 3dx2 y + 3exy 2 + f y 3

                                                      348
                                                                                       p. 343
                               ax + by bx + cy cx + dy
                               bx + cy cx + dy dx + ey ,                        (2)
                               cx + dy dx + ey ex + f y
which are in fact the Hessian and canonizant respectively of ϕ. So in general, for
a function of x, y of the degree 2ι or 2ι + 1, we can obtain ι covariantive forms,
the first being the Hessian, and the last the catalecticant on the first supposition
and the canonizant on the second: calling the index of the function for either
case n, the forms appearing in this scale will be of the degree (r + 1) in the
constants, and of the degree (r + 1)(n − 2r) in x and y.
   It has previously198 been intimated that all these determinants admit of a
remarkable transformation.
   This transformation may be expressed more elegantly by dealing not directly
with the covariant forms as above given, but with their polar reciprocants
obtained immediately by writing ξ for −y and η for x.
   (1) Suppose
                         ϕ = ax3 + 2bx2 y + 3cxy 2 + dy 3 ;
                                        a 2b c
                                         b 2c d
                                        ξ 2 2ξη η 2
will be found to be the reciprocant of its Hessian.
   (2) Let
                          ϕ = ax4 + 4bx3 y + · · · + ey 4 ;
the reciprocant of its Hessian will be found to be

                                     a 3b 3c d
                                      b 3c 3d e
                                                   .
                                     ξ 2 2ξη η 2 0
                                     0 ξ 2 2ξη η 2

(3) Let
                               ϕ = ax5 + 5bx4 y + · · · + f y 5 ;
the reciprocant of its Hessian will be

                                 a 4b 6c 4d e
                                  b 4c 6d 4e f
                                 ξ 2 2ξη η 2  0  0 ;
                                 0 ξ 2 2ξη η 2 0
                                 0    0  ξ 2 2ξη η 2
                                                                                       p. 344
 198
       p. 325 above, note †.


                                             349
   and the reciprocant of its canonizant is

                                     a    3b     3c   d
                                      b   3c     3d    e
                                                          .
                                      c   3d     3e   f
                                     ξ 3 3ξ 2 η 3ξη 2 η 3

The numerical coefficients in this and in the first case are inserted for the sake of
uniformity, but it will of course be readily observed that when there is but one
line of ξ and η, that the numerical coefficients being the same for each column
may be rejected without affecting the form of the result.
   So again, if
                           ϕ = ax6 + 6bx5 y + · · · + gy 6 ,
the reciprocant of the Hessian is

                            a 5b 10c 10d 5e f
                             b 5c 10d 10e 5f g
                            ξ 2 2ξη η 2  0    0  0
                                                     ,
                            0 ξ 2 2ξη η 2     0  0
                            0    0  ξ 2 2ξη η 2 0
                            0    0  0    ξ 2 2ξη η 2

and the reciprocant of the second form in the scale, which comes between the
Hessian and the catalecticant, is

                                 a     b    c   d   e
                                  b    c    d    e  f
                                  c    d    e   f   g ,
                                 ξ 3 ξ 2 η ξη 2 η 3 0
                                 0 ξ 3 ξ 2 η ξη 2 η 3

and so in general. The rule of formation is sufficiently plain not to need formu-
lating in general terms. It is easy to see that all these forms are concomitants to
the function from which they are formed; for example, take

                            ϕ = ax6 + 6bx5 y + · · · + gy 6 ;

then                            2                                   2
                            d                d d                 d
                                                           
                                     ϕ,           ϕ,                      ϕ
                           dx               dx dy               dy
form a plexus.                                                                          p. 345
   So likewise if we take ψ = (xξ + yη)4 ,

                                          dψ           dψ
                                             ,
                                          dξ           dη

                                                 350
form a plexus. But ψ and ϕ are concomitantive, ψ being a universal concomitant.
Hence we may combine together these two plexuses, that is
                                                                       
                     ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 
                     bx4 + 4cx3 y + 6dx2 y 2 + 4exy 3 + f y 4 ,
                     cx4 + 4dx3 y + 6ex2 y 2 + 4f xy 3 + gy 4 
                                                              

                                                                       )
                      ξ 3 x4 + 3ξ 2 ηx3 y + 3ξη 2 x2 y 2 + η 3 xy 3
                                                                           ,
                      ξ 3 x3 y + 3ξ 2 ηx2 y 2 + 3ξη 2 xy 3 + η 3 y 4
and, by the principle of the plexus, x4 , x3 y, x2 y 2 , xy 3 , y 4 may be eliminated
dialytically, and the resultant will be the determinant last given, which is
therefore a contravariant to ϕ.
   The manner in which I was led to notice this singular transformation is
somewhat remarkable.
   In the supplemental part of my essay On Canonical Forms [p. 203 above],
my method of solution of the problem of throwing the quintic function of two
variables under the form u5 + v 5 + w5 , led me to see that u, v, w are the three
factors of
                          ax + by bx + cy cx + dy
                          bx + cy cx + dy dx + ey ;
                          cx + dy dx + ey ex + f y
the more simple mode of the solution of the same problem, given by me in the
Philosophical Magazine for the month of November last [p. 266 above], led to

                                a    b     c    d
                                 b   c     d    e
                                 c   d     e    f
                                y 3 −xy 2 x2 y −x3

as the product of the same three factors; whence the identity of the two forms
becomes manifest. In the paper last named I gave two proofs, one my own, the
other Mr Cayley’s, of a like kind of identity for the canonizant of any odd-degreed
function of x, y in general. The proof of the identity of the corresponding forms
in the much more general proposition above indicated [p. 325 above, footnote
†] must be reserved until more pressing and important matters are disposed of.
In the footnote referred to I ought to have added, in order to make the sense
more clear, that the degree of the catalecticant there referred to in respect of
the coefficients would be n.                                                        p. 346
   I regret to think that there are many other typographical errors in the earlier
sections; the most unfortunate of these is in the note at page [316], in the values
of P and Q belonging to the cubic commutant dodecadic function of x and y, the
corrected values of which will be given in my next communication. I ought also
to observe, in correction of the remark made in the footnote to page [302], that

                                            351
it follows as a consequence of a recent paper by Dr Hesse in Crelle’s Journal,
that the method given by me in the text applied (according to what I have there
termed the 1st process for obtaining an invariant resembling the resultant) to
a system of three cubic equations (in which application only the 1st powers of
dx , dy , dz enter) produces for that case also, as well as for the cases specified in
 d d d

the note, not a counterfeit resemblance of, but the actual resultant itself.
    Returning to the theory of the plexus of which I am about to enunciate a most
important extension, I beg to refer my readers to the last paragraph, p. [291],
in the last number of the Journal, where I have shown how to form, under
certain conditions, a determinant by combining together various concomitants
and eliminating dialytically one set of the variables, which determinant will be
concomitantive to the concomitants out of which it is formed, and of course also
therefore to their common original.
    Now the extension of this theorem, to which I wish to call attention, is this,
that not only such determinant as a whole is a concomitant to such original,
but every minor system of determinants that can be formed out of it will
form a concomitantive plexus complete within itself to the same original. But,
much more generally, it should be observed that there is no occasion to begin
with a square determinant; it is sufficient to have a rectangular array of terms
formed by taking the several terms of one plexus or of several plexuses combined,
provided that they are of the same degree in respect to the variables (or to the
selected system of variables, if there be several systems), and forming out of such
rectangular array any minor system of determinants at will. Every such system
will be a concomitantive plexus. The simple illustrations which follow will make
my meaning clear.
    Suppose

        ϕ = ax6 + 6bx5 y + 15cx4 y 2 + 21dx3 y 3 + 15ex2 y 4 + 6f xy 5 + gy 6 .

I have previously remarked, in the foregoing sections, that a, b, c, d, e, f, g, the
coefficients form an invariantive plexus to ϕ; so also we know that the catalecticant

                                         a        b   c d
                                         b        c   d e
                                         c        d   e f
                                         d        e   f g
                                                                                         p. 347
   is an invariant to ϕ. But we are now able to couple together these facts and
see the law which is contained between them; for if we take
                               ι                 ι−1                ι
                           d                  d           d       d
                                                             
                                    ϕ,                      ϕ...           ϕ,
                          dx                 dx          dy      dy



                                                  352
ι being any number, as for instance, if we take ι = 3, we shall have as a plexus
                               ax3 + 3bx2 y + 3cxy 2 + dy 3 ,
                               bx3 + 3cx2 y + 3dxy 2 + ey 3 ,
                               cx3 + 3dx2 y + 3exy 2 + f y 3 ,
                               dx3 + 3ex2 y + 3f xy 2 + gy 3 ;
accordingly not only is the determinant
                                              a    b   c d
                                              b    c   d e
                                              c    d   e f
                                              d    e   f g
an invariant, but also the system obtained by striking out any one line and one
column, being what I term the first minors, will be an invariantive plexus, so
too will the system of second minors
       ac − b2 ,   bd − c2 ,    ce − d2 ,          ad − bc,       ae − bd,       be − cd,   &c.
form an invariantive plexus, as well as the last minors, that is, the simple terms
a, b, c, d, e, f, g. Again, we might have taken the plexus
                                    2                                 2
                                d                  d d             d
                                                             
                                         ϕ,             ϕ,                  ϕ,
                               dx                 dx dy           dy
which would give the array
                                         a, b, c, d, e
                                         b, c, d, e, f
                                         c, d, e, f, g;
but the minor systems of determinants herein comprised will be found to be
identical with those last considered, with the exception that the highest system,
containing a single determinant only, will now be wanting. So in general it will
easily be seen that a similar method in general, when ϕ is of 2ι dimensions,
will lead to ι + 1 invariantive plexuses comprising the given coefficients grouped
together at one extremity of the scale, and the catalecticant alone at the other;
and if ϕ is of 2ι + 1 dimensions, there will still be ι + 1 such plexuses, commencing
with the coefficients as one group and ending with a system of combinations
of the (ι + 1)th degree in regard to the coefficients, which system accordingly
takes the place of the catalecticant of the former case, which for this case is
non-existent.                                                                         p. 348
   As a profitable example of the application of this law of synthesis, in its
present extended form, let it be required to determine the conditions that a
function of x, y of the fifth degree may have three equal roots. In general, let
              ϕ = ax5 + 5bx4 y + 10cx3 y 2 + 10dx2 y 3 + 5exy 4 + f y 5 ,

                                                    353
then ϕ has a quadratic and cubic covariant of which I have written at large
in my supplemental essay above referred to, being in fact the s and t (that is
the quadrinvariant and cubinvariant) in respect to x′ , y ′ (x, y being treated as
constants) of
                                           4
                                ′ d    ′ d
                            
                              x     +y        ϕ.
                                 dx     dy
Let these covariants respectively be called

          Ax2 + 2Bxy + Cy 2 = u,              αx3 + 3βx2 y + 3γxy 2 + δy 3 = v;

then                                    (
                                          Ax + By
                                          Bx + Cy
forms a plexus, and             (
                                    αx2 + 2βxy + γy 2
                                    βx2 + 2γxy + δy 2
will form another.
   Now when a = 0, b = 0, c = 0, ϕ will have three equal roots, and
                                                         4
                                        ′ d        ′ d
                                 
                                    x         +y              ϕ
                                        dx          dy
becomes
                   6dy x′2 y ′2 + 4(dx + ey)x′ y ′3 + (ex + f y)y ′4 ,
of which the quadrinvariant in respect to x′ , y ′ is easily seen to be d2 y 2 and the
cubinvariant d3 y 3 . Accordingly the grouping

                              A B                             0 0
                                          becomes                  ,
                              B C                             0 d2

and the grouping
                         α β γ                            0 0 0
                                         becomes                 .
                         β γ δ                            0 0 d3
Accordingly, we see that the determinant

                                              A B
                                              B C

and all the first minors of
                                         α β γ
                                               ,
                                         β γ δ




                                              354
that is αγ − β 2 , βδ − γ 2 , αδ − βγ, become zero; but the former single quantity

                                            A B
                                            B C

being an invariant, and this last system being an invariantive plexus, all the
quantities so affirmed to be zero will remain zero, notwithstanding any linear
transformations to which ϕ may be subjected; thus then we obtain an immediate
proof of the theorem that                                                          p. 349
   when a function of x and y of the fifth degree contains three equal roots the
determinant of its quadratic covariant, which in fact is its sole quart-invariant,
and the first minors of its cubinvariant will be all separately zero. This theorem
may be made still more stringent; for by combining

                                    Ax2 + 2Bxy + Cy 2 ,

                                     αx2 + 2βxy + γy 2 ,
                                     βx2 + 2γxy + δy 2 ,
it becomes manifest that in the case supposed all the first minor determinants of

                                          A B C
                                          α β γ
                                          β γ δ

will be zero, showing in addition to the theorem last enunciated that also

                             A : B : C :: α : β : γ :: β : γ : δ.

It is curious and instructive to remark that this last set of equations, stringent
as they appear, and far more than enough to express a duplex condition, are not
sufficient to imply unequivocally the existence of three equal roots, unless we
have also AC − B 2 = 0; for suppose ϕ to take the form ax5 + f y 5 (b, c, d, e all
vanishing); then it will easily be seen that

                       α = 0, qquadβ = 0,           γ = 0,        δ = 0,

                           A = 0,        B = af,         C = 0.199
                                                                                                    p. 350
 199
    If we take L, M, N a system of fundamental invariants to ϕ, of which all the other invariants
                                               A B
of ϕ are rational integer functions, then L =          , and the simplest forms for M and N
                                               B C
are
                                                       α 2β γ 0
                             A B C
                                                        0 α 2β γ
                     M= α β γ               and N =                        ,
                                                        β 2γ       δ   0
                             β γ δ
                                                        0 β 2γ δ


                                              355
   Consequently we shall still have all the first minors of

                                             A B C
                                             α β γ
                                             β γ δ

zero, although there is not even so much as a pair of equal roots in ϕ; AC − B 2
however, it will be observed, is not zero in this supposition.
   The theory of Hessians, simple or bordered, may be regarded as one among
the infinite diversity of applications of the principle of the plexus. Let U, V, W ,
&c. be any number of concomitants having the common system of variables
x, y . . . z. Let χ represent
                                      d       d             d
                                x′      + y′    + · · · + z′ ,
                                     dx      dy             dz
and take
                             χ2 U + λχV + &c. + µχW = S;
then
                                       dS      dS      dS
                                           ,        ... ′
                                       dx′     dy ′    dz
forms a plexus; and this, combined with χV , &c. . . . χW , enables us to eliminate
dialytically x′ , y ′ , z ′ , λ . . . µ. The result is a Hessian of U , bordered with
                                       dV      dV     dV
                                          ,       ...
                                       dx      dy     dz
horizontally and vertically, and also with
                                      dW       dW      dW
                                         ,         ...     ,
                                      dx        dy      dz
&c. similarly dispersed; which Hessian, so bordered, is thus seen to be a
concomitant to U, V . . . W . The Hessian, as ordinarily bordered with ξ, η . . . ζ, is
derived by taking for V the universal concomitant

                                      xξ + yη + · · · + zζ,
where L and N are the discriminants of the quadratic and cubic covariants of ϕ respectively,
and a linear function of M, L2 is the discriminant of ϕ itself (L, M, N being of 4, 8, and 12
dimensions respectively in the coefficients of ϕ). For many purposes of the calculus of forms it
is desirable to have the command of cases for which any two out of these three invariants may
be made to vanish without the third vanishing; and it will be found that when ϕ is of the form
y 2 (cx3 + f y 3 ), L = 0, M = 0; when ϕ is of the form y(bx4 + f y 4 ), N = 0, L = 0; and when ϕ
is of the form ax5 + ey 5 , M = 0, N = 0; and of course when ϕ is of the form y 3 (dx2 + f y 2 ),
L = 0, M = 0, N = 0; it being obviously true in general, as remarked by Mr Cayley, that when
not less than half the roots of a function of two variables are equal, all its invariants must
vanish together.


                                               356
and for W (if there be a double border)

                                  xξ ′ + yη ′ + · · · + zζ ′ ,

and so forth.
   If V be taken identical with U , the resulting form, consisting of U bordered
with dU/dx, dU/dy . . . dU/dz, has been shown200 in my paper “On certain general
Properties of Homogeneous Functions,” in this Journal, to be equal to the product
of the simple Hessian of U and of U itself multiplied by a                        p. 351
   numerical factor. The theory of the bordered Hessian may be profitably
extended by taking

                            S = χ2r U + λχr V + · · · + µχr W,

and combining with χr V . . . χr W the plexus obtained by operating upon S with
the rth powers and products of
                                     d        d       d
                                        ,         ... ′,
                                    dx′      dy ′    dz

and eliminating dialytically the rth powers and products of x′ , y ′ . . . z ′ . Thus if

           U = ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4           and   V = (xξ + yη)2 ,

we obtain, by taking S = χ4 U + λχ2 V , and proceeding as indicated in the
preceding,
                             a b c ξ2
                              b c d ξη
                              c d e η2
                             ξ 2 ξη η 2
as a concomitant to U . So again, if

                             U = ax5 + 5bx4 y + · · · + f y 5 ,

we find
                            ax + by bx + cy cx + dy ξ 2
                            bx + cy cx + dy dx + ey ξη
                            cx + dy dx + ey ex + f y η 2
                               ξ2      ξη      η2
a concomitant to U .
   These extensions of the ordinary theory of Hessians will be found to be of
considerable practical importance in the treatment of forms, for which reason
they are here introduced.
 200
       p. 173 above.


                                             357
       Section VI.On the Partial Differential Equations to
                         Concomitants,
            Orthogonal and Plagiogonal Invariants, &c.
   In the 7th note of the Appendix to the three preceding sections201 I alluded
to the partial differential equations by which every invariant may be defined.
   This method may also be extended to concomitants generally. M. Aronhold,
as I collect from private information, was the first to think of the application of
this method to the subject; but it was Mr Cayley who communicated to me the
equations which define the invariants of functions of                               p. 352
   two variables.   202    The method by which I obtain these equations and prove
their sufficiency is my own, but I believe has been adopted by Mr Cayley in a
memoir about to appear in Crelle’s Journal. I have also recently been informed
of a paper about to appear in Liouville’s Journal from the pen of M. Eisenstein,
where it appears the same idea and mode of treatment have been made use of.
Mr Cayley’s communication to me was made in the early part of December last,
and my method (the result of a remark made long before) of obtaining these and
the more general equations, and of demonstrating their sufficiency, imparted a
few weeks subsequently—I believe between January and February of the present
year.
   The method which I employ, in fact, springs from the very conception of what
an invariant means, and does but throw this conception into a concise analytical
form.
   Suppose, to fix the ideas,
                                          1
               ϕ = axn + nbxn−1 y + n(n − 1)cxn−2 y 2 + · · · + ly n ,
                                          2
and let I(a, b, c . . . l) be any invariant to ϕ.
   Now suppose x to become x + ey, but y to remain unchanged; the modulus of
the transformation,
                                           1 e
                                                  ,
                                           0 1
being unity, I cannot alter in consequence of this substitution; but the effect of
this substitution is to convert ϕ into the form
                                   1
                αxn + nβxn−1 y + n(n − 1)γxn−2 y 2 + · · · + λy n ,
                                   2
 201
     p. 326 above.
 202
     It is extremely desirable to know whether M. Aronhold’s equations are the same in form as
those here subjoined. It is difficult to imagine what else they can be in substance. Should these
pages meet the eye of that distinguished mathematician he will confer a great obligation on the
author and be rendering a service to the theory by communicating with him on the subject; and
I take this opportunity of adding that I shall feel grateful for the communication of any ideas or
suggestions relating to this new Calculus from any quarter and in any of the ordinary mediums
of language—French, Italian, Latin or German, provided that it be in the Latin character.


                                              358
where

             α = a,        β = b + ae,          γ = c + 2be + ae2 ,        &c. &c.,

                                λ = l + · · · + nben−1 + aen .
Consequently, if we make

                      ∆b = ae,         ∆c = 2be + ae2 ,          &c. &c.,

we have by Taylor’s theorem, observing that ∆a = 0,

                    d       d          1       d   d      2
                                                                                 
         ∆I = ∆b       + ∆c + &c. I +       ∆b + ∆c + &c. I
                    db      dc        1·2      db  dc
                    1
                                   3
                              d
                         
                +          ∆b + &c. I + &c. = 0;
                  1·2·3      db
                                                                                                      p. 353
  and this being true for all the values of e, every separate coefficient of e in
∆I must be zero: hence we obtain n different equations by equating to zero the
coefficients of e, e2 . . . en respectively. The first of these equations will be
                                d    d    d
                                                            
                               a + 2b + 3c + &c. I = 0,
                                db   dc   dd
and it is obvious that this will imply all the rest; for, when e is taken indefinitely
small, I(a, b, c . . .) does not alter (when this equation is satisfied) by changing
a, b, c . . . into a′ , b′ , c′ . . .; consequently I(a′ , b′ , c′ , &c.) will not alter, when in
place of a′ , b′ , c′ we write a′′ , b′′ , c′′ , &c., obtained from a′ , b′ , c′ , &c., by the same
law as a′ , b′ , c′ , &c. from a, b, c, &c.
    Thus we may go on giving an indefinite number of increments, ey to x, without
changing the value of I. Consequently, if the equation above written be satisfied,
à priori all the rest must be so too. But there is not any difficulty in showing
the same thing by a direct method.203
    For we have
                                       d     d         d
                                                                  
                                    a + 2b + 3c + &c. I = 0,
                                      db     dc        dd
an identical equation. Hence
             d      d    d                          d      d    d
                                                                            
         a      + 2b + 3c + &c.                 a      + 2b + 3c + &c. I              = 0;
             db     dc   dd                         db     dc   dd
 203
    The method above given has the advantage however of being immediately applicable to
every species of concomitant, and we learn from it that concomitance, whether absolute or
conditional, is sufficiently determined when affirmed to exist for infinitesimal variations; it
cannot exist for infinitesimal variations without, by necessary implication, existing for finite
variations also; a most important consideration this in conducing to a true idea of the nature of
invariance and the other kinds of concomitance, and in cutting off all superfluous matter from
the statement of the conditions by which they are defined.


                                                359
hence
      d    d    d                       d    d    d                           d      d    d
                                                                    
     a + 2b + 3c + &c.                 a + 2b + 3c + &c.              I+ a       + 2b + 3c + &
      db   dc   dd                      db   dc   dd                          db     dc   dd
that is
     (                                                                    )
                                              2
         d    d    d         d    d    d
                                              
      2 a + 3b + 6c + &c. + a + 2b + 3c + &c.   I = 0;
         dc   dd   de        db   dc   dd

repeating the application of the symbolic operator
                                      d    d
                                                            
                                     a + 2b + &c. ,
                                      db   dc
                                                                                       p. 354
   we obtain
                                                                      
                        d       d        d
                                                        
           1 · 2 · 3 a + 4b + 10c + &c.
                                                          
                                                          
                                                          
                        dd      de      df                
                                                          
                                                          
                                                         
                           d       d           d   d
                                                       
                                                          
               + 1 · 2 a + 2b + &c.          a + 3b + &c.   I = 0,
                          db      dc          dc   dd     
                                                          
                                                          
                                             3
                      d      d       d
                                                         
                                                          
               + a + 2b + 3c + &c.
                                                          
                                                          
                                                          
                     db      dc      dd
                                                          

and so on; the numerical multipliers of the terms of the several series within the
parentheses forming the regular succession of figurate numbers

                                       1, 2, 3, &c.
                                       1, 3, 6, &c.
                                       1, 4, 10, &c.
It is easy to see that these equations correspond to the results of making the
coefficients of the successive powers of e equal to zero.
   I may remark, that the first instance as far as I know on record of this,
(as some may regard it rather bold) but in point of fact perfectly safe and
legitimate method of differentiating conjointly operator and operand, occurs in a
paper by myself in this Journal, Feb. 1851, “On certain General Properties of
Homogeneous Functions” [p. 165 above]; where I have applied it in operating
with
                                    d                  d
                                                              
                       (x1 − a1 e)     + (x2 − a2 e)      + &c.
                                   da1                da2
upon                                                          r
                                  d                  d
                   
                     (x1 − a1 e)      + (x2 − a2 e)      + &c. ω,
                                 da1                da2
which, as I have there noticed, gives the result
                                       r+1                              r
                              d                                d
                                                                       
               (x1 − a1 e)       + &c.     ω − re (x1 − a1 e)     + &c. ω.
                             da1                              da1

                                               360
The equation
                                      d    d
                                                        
                                     a + 2b + &c. I = 0
                                      db   dc
is evidently not enough to define I as an invariant; it merely serves to show that
I does not alter when in place of x we write x + ey, but this is true for any
function of the differences of the roots of the form multiplied by a suitable power
of a, namely that power which is just sufficient to cause the product to become
integer. But if we now, for convenience, write
                  1                       1
ϕ = axn +nbxn−1 y+ n(n−1)cxn−2 y 2 +· · ·+ n(n−1)c′ x2 y n−2 +nb′ xy n−1 +a′ y n ,
                  2                       2
                                                                                        p. 355
  and form the similar equation from the other side, namely

                           ′ d           ′ d
                                           d
                                                              
                                                    ′
                           a ′ + 2b ′ + 3c ′ + &c. I = 0,
                            db     dc     dd
these two equations together will suffice to define any invariant, as I shall proceed
to show—these are the two equations alluded to brought under my notice by Mr
Cayley. If they coexist, it follows from the method by which I have deduced them
that x may be changed into x + ey, or y into y + f x, without I being altered, e
and f having any values whatever: and it is obvious that these substitutions may
be performed, not merely alternatively but successively, because the equations
between the coefficients are identical equations, and depend only on the form of
I.
   Let now x become x + ey, and then y become y + f x; the result of these
substitutions is to convert

                                     x   into   x + ef x + ey,

and
                                         y   into   f x + y.
Finally, let x become x + qy; then x is converted into (1 + ef )(x + qy) + ey, and
y into y + f (x + qy), that is

                       x becomes (1 + ef )x + (eg + ef g)y,

and
                                 y becomes f x + (1 + f g)y.
The modulus of substitution it is evident, à priori, always remains unity, and
nothing would be gained by pushing the substitutions any further, as it is clear
that we may satisfy the equations

         1 + ef = p,         e + g + ef g = q,            f = p′ ,   1 + f g = q′,


                                                361
for all values of p, q, p′ , q ′ , which satisfy the equation

                                         pq ′ − p′ q = 1,

and for none other except such values; hence I remains unaltered for any unito-
modular linear transformation of x, y, and is therefore an invariant by definition.
   If ϕ be taken a function of three variables, x, y, z, and be thrown under the
form
           az n + (a1 x + b1 y)z n−1 + (a2 x2 + 2b2 xy + c2 y 2 )z n−2 + &c.,
and I be any invariant of ϕ, by supposing x to become x + ey, and giving
b1 , b2 , c2 , &c., the corresponding variations, and taking e indefinitely small, we
obtain
           d        d         d               d         d         d
                                                                              
       a1     + a2     + 2b2            + a3     + 2b3     + 3c3              + &c. I = 0,
          db1      db2       dc2             db3       dc3       dd3
                           d        d         d
                                                               
                       b1     + c2     + 2b2             + &c. &c. I = 0 :
                          da1      db2       da2
                                                                                                 p. 356
   and in like manner, by arranging ϕ according to the powers of y and of x, we
obtain two other pairs of equations: it is clear, however, that three equations (it
would seem any three out of the six) would suffice and imply the other three.
The method of demonstration will be the same as in the instance of two variables:
First, it can be shown by the method of successive accretions, that I remaining
invariable when x receives an indefinitely small increment ey, or y an indefinitely
small increment ez, or z an indefinitely small increment ex, it will also remain
invariable when these increments are taken of any finite magnitude. Secondly, by
eight successive transformations, admissible by virtue of the preceding conclusion,
x, y, z may be changed into any linear functions of x, y, z, consistent with the
modulus of transformation being unity. And in general for a function of m
variables, m partial differential equations similarly constructed (but not however
arbitrarily selected) will be necessary and sufficient to determine any invariant:
and it is clear that all the general properties of invariants must be contained in
and be capable of being deduced out of such equations.
   The same method enables us also to establish the partial differential equations
for any covariant, or indeed any concomitant whatever.
   Thus let
                           1
       ϕ = axn + nbxn−1 y + n(n − 1)cxn−2 y 2 + · · · + nb′ xy n−1 + a′ y n = 0,
                           2
and let K(a, b, c, &c.; x, y, x′ , y ′ , &c.; ξ, η, &c.) represent any concomitant, x, y; x′ , y ′
being cogredient, and ξ, η, &c. contragredient systems; when x, y become x+ey, y,
any such system x′ , y ′ becomes x′ + ey ′ , y ′ ; and any such system as ξ, η becomes



                                              362
ξ, η −eξ; and taking e indefinitely small, the second coefficients a, b, c, &c. become
a, b + ae, c + 2be, &c. as before; hence the equation to the concomitant becomes
                d    d             d      d              d
                                                                              
               a + 2b + · · · − y    − y′ ′ + · · · + ξ    − &c. K = 0204 ;
                db   dc           dx     dx             dη
and in like manner, by changing y into y + ex, results the corresponding equation

                 ′ d     ′ d           d    d            d
                                                                                  
                a ′ + 2b ′ + · · · − x − x′ ′ + · · · + η − &c. K = 0.
                 db     dc            dy   dy            dξ
These two equations define in a perfectly general manner every concomitant
(with any given number of cogredient and contragredient systems) to the form
ϕ; and the due number of pairs of similarly constituted equations will serve to
define the concomitant to a function of any given number of variables.205                p. 357
   In like manner we may proceed to form the equations corresponding to what
may be termed conditional concomitants, whether orthogonal or plagiogonal.
The concomitants previously considered may be termed absolute, the linear
transformations admissible being independent of any but the one general relation,
imposed merely for the purpose of convenience, namely of their modulus being
made unity. An orthogonal concomitant is a form which remains invariable,
not for arbitrary unito-modular, but for orthogonal transformation, that is for
linear substitutions of x, y . . . z, which leave unchanged x2 + y 2 + · · · + z 2 : in
like manner, a plagiogonal concomitant may be defined of a form which remains
invariable for all linear substitutions of x, y . . . z, which leave unaltered any given
quadratic function of x, y . . . z. Thus, let it be required to express the condition
of Q(a, b, c . . . x, y; ξ, η), being an orthogonal concomitant to the form

                         axn + nbxn−1 y + · · · + nb′ xy n−1 + a′ y n .

Let x become x + ey, e being indefinitely small, then y must become y − ex,
and the variations of a, b . . . b′ , a′ will be the sum of the variations produced by
taking separately x + ey for x and y − ex for y. Hence the one sole condition for
Q being of the required form becomes
         d      d             d     d                    d        d             d   d
                                                                                             
     a      + 2b + · · · − y    +ξ              − a′        + 2b′ ′ + · · · − x + η                   Q = 0,
         db     dc           dx    dη                   db′      dc            dy   dξ
or, as it may be written, θQ − ωQ = 0, where θQ = 0, ωQ = 0 are the two
equations expressing the conditions of Q, being an unconditional or absolute
concomitant; and so in general if ϕ be a function of m variables, we may obtain
 204
       For we have

   K(a, b + ae, c + 2be, &c.; x, y, &c.; ξ, η, &c.) = K(a, b, c, &c.; x, x + ey, &c.; ξ, η − eξ, &c.).

 205
     Vide Note (10) [p. 361 below].


                                                  363
2 m(m − 1) equations of the form L − M = 0 for the concomitant, of which
1

however (m − 1) only will be independent.
   Supposing, again, the substitutions to which x, y are subject to be conditioned
by lx2 + 2mxy + ny 2 remaining unalterable, or which is a more convenient
and only in appearance less general supposition by x2 + 2mxy + y 2 remaining
unalterable, the general type of an infinitesimal system of substitutions will
be rendered by supposing x, y to become (1 + me)x + ey, −ex + (1 − me)y,
respectively, for then x2 + 2mxy + y 2 becomes

            (1 − m2 e2 )x2 + {2m + (2m − 2m3 )e2 }xy + (1 − m2 e2 )y 2 ,

which differs from x2 + 2mxy + y 2 only by quantities of the second order of
smallness which may be neglected, and ξ and η will therefore become (1 − me)ξ −
eη, −ex + (1 + me)y, respectively: then, as to the coefficients of ϕ, in addition to
the variations which they undergo when m is zero, there will be the variations
consequent upon x assuming the increment mex, and y                                  p. 358
   the increment −mey: but by making x become x + mex, a, b, c, &c., b , a      ′  ′

assume respectively the variations

                      n · mea,     (n − 1)meb,      ··· ,       meb′ ,       0,

respectively; and by making y become y − mey, the corresponding variations
become
                0, −meb, · · · , −(n − 1)meb′ , −n · mea′ ,
respectively. Hence the equation becomes

                                θQ − ωQ + m(λQ − µQ) = 0,

where θ and ω have the same signification as before, and where λ denotes
                       d           d             d      d   d
                 na      + (n − 1)b + · · · + b′ ′ + x    −ξ ,
                      da           db           db     dx   dξ
and µ denotes
                          d      d              d      d    d
                      b      + 2c + · · · + na′ ′ − y    +η .
                          db     dc            da     dy   dη
If there be several systems of x, y or of ξ, η, or of both, the only difference in the
equation of condition will consist in putting
                        d                d               d                    d
                                                                           
                Σ y       ,       Σ x      ,    Σ x        ,       Σ y          ,
                       dx               dy              dx                   dy
                          d              d              d                     d
                                                                           
                Σ η          ,    Σ ξ      ,    Σ ξ        ,       Σ η          ,
                          dξ            dη              dξ                   dη
instead of the single quantities included within the sign of definite summation.

                                              364
   Fearing to encroach too much on the limited space of the Journal, I must
conclude for the present with showing how to integrate the general equation to
the orthogonal invariant of ϕ, the general function of x, y.
   Beginning with ϕ = ax2 + 2bxy + cy 2 , the equation becomes
                          d          d    d     d     d
                                                                    
                   −2b      + (a − c) + 2b + y    −x    Q = 0.
                         da          db   dc   dx    dy
Write now
                           da = −2b dθ,                dx = y dθ,
                           db = (a − c)dθ,             dy = −x dθ,
                           dc = +2b dθ;
we have then

                    λda + µdb + νdc = dθ{µa + 2(ν − λ)b − µc}.

Let
                     µ = κλ,          2(ν − λ) = κµ,           −µ = κν;
then
                                  d log(λa + µb + νc) = κdθ;
or
                                    λa + µb + νc = beκθ .
                                                                                 p. 359
     To find κ we have the determinant

                                       κ −1 0
                                       2 κ −2           = 0,
                                       0 1  κ

that is,
                                        κ3 + 4κ = 0,
and calling the three roots of this equation κ1 , κ2 , κ3 , we have

                           κ1 = 0,       κ2 = 2ι,        κ3 = −2ι;

accordingly we may put

                         κ = 0,       λ = 1,         µ = 0,     ν = 1,

or
                    κ = 2ι,          λ = 1,      µ = 2ι,        ν = −1,
or
                   κ = −2ι,          λ = 1,      µ = −2ι,           ν = −1.


                                               365
Again,
                                pdx + qdy = (py − qx)dθ;
and putting −q = ep, p = eq, so that px + qy = Eeeθ ,

                          e2 = −1,          e1 = ι,        e2 = −ι;

and we may put
                             e = ι,        p = 1,         q = −ι,
or
                            e = −ι,         p = 1,         q = +ι.
Consequently the complete integral of the given partial differential equation is
found by writing
                           a + c = l,          x − ιy = Eeιθ ,
                     a + 2ιb − c = l e ,
                                    ′  2ιθ     x + ιy = E ′ e−ιθ ,
                     a − 2ιb − c = l e
                                    ′′  −2ιθ .
By means of these five equations, after eliminating θ, we may obtain four
independent equations between a, b, c; x, y. Suppose

                   Q1 = 0,         Q2 = 0,          Q3 = 0,         Q4 = 0;

then Q = F (Q1 , Q2 , Q3 , Q4 ) is the complete integral required.
   Pursuing precisely the same method for the general case, it will be found that,
calling the degree of the given function n when n is even, the equation in κ to
be solved will be
                       κ(κ2 + 4)(κ2 + 9) · · · (κ2 + n2 ) = 0;
and when n is odd (say 2m + 1), the equation in κ to solve will be

                           (κ + 1)(κ2 + 9) · · · (κ2 + n2 ) = 0;
                                                                                              p. 360
   and performing the necessary reductions, and calling the roots of the equation,
arranged in order of magnitude, κ0 , κ2 . . . κn , respectively, it will be found that
the equations containing the integral become
                          = l1 eκ1 ιθ
                                            
                    L1                        
                                              
                    L2    = l2 eκ2 ιθ
                                              
                                                     (
                                                          x − ιy = Eeιθ
                                              
                                              
                    L3    = l3 e κ3 ιθ                                      ,
                                                          x + ιy = E ′ e−ιθ
                    ···   ···
                                              
                                              
                                              
                                              
                          = ln+1 e   κ     ιθ
                                              
                 Ln+1                  n+1
                                              

where l1 , l2 . . . ln+1 ; E, E ′ are arbitrary constants, and where L1 , L2 . . . Ln+1 are
the values assumed by the 1st, 2nd . . . (n + 1)th coefficients of the given function
ϕ, or
                           axn + nbxn−1 y + · · · + nb′ xy n−1 + a′ y n ,

                                              366
when it is transformed by writing x + ιy in place of x, and y + ιx in place of
y. ι is of course
           √      employed in the foregoing according to the usual notation to
represent −1. The same method applies to the general theory of plagiogonal
concomitants, where the linear substitutions are supposed such as to leave
lx2 + 2mxy + ny 2 unaltered in form, and the equations in θ which contain the
integral present themselves under a similar aspect. But a more full discussion
of these interesting integrals must be reserved until the ensuing number of the
Journal.

                                      Notes in Appendix.

    (9) The scale of covariants to a function of (x, y) obtained by the method
of unravelment [on p. 297 above], may be otherwise deduced in a form more
closely analogous to that of the corresponding theorems for the corresponding
invariantive scale [on p. 295 above], by a method which has the advantage of
exhibiting the scale equally well for the case of functions of the degree 4ι + 2
or 4ι + 4, the only difference being that in the latter case the coefficients of the
odd powers of λ will be found all to vanish, so that the degrees of the covariants
will rise by steps of 4 instead of by steps of 2, just conversely to what happens
in the invariantive scale; whereas in the invariantive scale alluded to the forms
containing odd powers of λ vanish when the degree of the function is of the form
4ι + 2, but do not vanish when it is of the form 4ι. This method in the form
here subjoined is a slight modification of one suggested to me by my friend Mr
Cayley.
    Let F be the given function of x, y of the degree 2n; take the systems
x , y ′ ; x1 , y1 cogredient with one another and with x, y. Then form the con-
  ′

comitant
                                 n
                ′ d        ′ d
            
      K= x            +y              F + λ(x′ y − y ′ x)n−1 (x′ y1 − y ′ x1 )(xy1 − yx1 ).
                dx          dy
                                                                                              p. 361
   Then (by what may be termed the Divellent method, which has been previously
applied by me in the Philosophical Magazine for Nov. 1851) calling θ0 , θ1 , θ2 . . . θn ,
the coefficients of
                       x′n , x′n−1 y ′ , . . . , y ′n in K,
we shall have
                           θ0 = A0 xn + B0 xn−1 y + · · · + L0 y n ,
                           θ1 = A1 xn + B1 xn−1 y + · · · + L1 y n ,
                             ···
                        θn = An xn + Bn xn−1 y + · · · + Ln y n ,
the coefficients being functions of the coefficients of f and of quadratic combina-



                                                 367
tions of x1 , y1 , affected with the multiplier λ; and the determinant
                                     A0 B0 · · ·          L0
                                     A1 B1 · · ·          L1
                                     ··· ···              ···
                                     An Bn · · ·          Ln
will give a function of λ in which the coefficients of the several powers of λ will
be all zero or covariants of F .
   The actual form of this determinant is not here given for want of space and
time, but will be exhibited hereafter. Precisely an analogous method applies to
obtain the scale to (x, y, z) given in Note (2) [p. 322 above]. Calling F = (x, y, z)4 ,
let the systems x′ , y ′ , z ′ ; x1 , y1 , z1 be taken cogredient with one another and with
x, y, z. Then, using R to express the determinant
                                       x′ y ′ z ′
                                       x y z ,
                                       x1 y1 z1
and making
                                                           2
                                 ′ d     ′ d         ′ d
                             
                         K= x      +y       +z          F + λR,
                               dx      dy       dz
and proceeding as above by the divellent method, we obtain the scale required.
   (10) [p. 356 above.] It is obvious that these defining equations ought to give
the means of discovering and verifying all the properties of concomitants; but it
is very difficult to see how in the present state of analysis many of the general
theorems that have been stated, readily admit of being deduced from them.
   The comparatively simple but eminently important theory of the evector
symbol does however admit of a very pretty verification by aid of these equations.
Thus, suppose θ any concomitant; suppose a contravariant to a function F of
x, y, say
                      axn + nbxn−1 y + · · · + nb′ xy n−1 + a′ y n .
                                                                                              p. 362
   Then θ must satisfy the two equations
                              d                       d
                                                             
                                                      ′
                         L+ξ    θ = 0,           L +η    θ = 0,
                             dη                       dξ
where
                            d      d               d
                           L=a + 2b + · · · + nb ′ ,
                            db     dc             da
                             d       d               d
                     L′ = a′ ′ + 2b′ ′ + · · · + nb .
                            db      dc              da
Now let ϕ = χ(θ) where
                           d          d           d              d
                 χ = ξn      + ξ n−1 η + ξ n−2 η 2 + · · · + η n ′ ;
                          da          db          dc            da

                                           368
then
             L(χθ) = χ(Lθ) − (χL)θ
                               n d      n−1 d              n−1 d
                                                                 
                   = χ(Lθ) − ξ     + 2ξ    η + · · · + nξη          θ,
                                db          dc                da′
             d            d      d
                                               
        ξ      (χθ) = χ ξ θ + ξ χ θ
            dη           dη     dη
                          d     n d      n−1 d              n−1 d
                                                                
                    =χ ξ θ + ξ      + 2ξ    η + · · · + nξη          θ.
                         dη      db          dc                da′
Hence
                           d                            d
                                                       
                      L+ξ    χ(θ) = χ              L+ξ    θ = χ(0) = 0.
                          dη                           dη
Similarly
                                                 d
                                                  
                                        L′ + η      χ(θ) = 0.
                                                 dξ
Hence if θ is an integral of the two conditioning equations, so also is χ(θ). In
like manner, if θ be a covariant or any other kind of concomitant of F , it may
be proved that its evectant χ(θ) is the same.
   (11) [p. 331 above.] Very much akin with the supposed equations is the
following most remarkable equation, which can be proved to exist. Let ϕ be
a function of x and y of the 5th degree. Let P and Q be the quadratic and
cubic covariants of ϕ. P is of two dimensions in the coefficients and also in the
variables, and Q of three dimensions in both; they are in fact the s and t (in
respect to x′ and y ′ ) of
                                     d       d 4
                                              
                                 x′    + y′      ϕ.
                                    dx      dy
Then, giving P and Q proper numerical factors, it will be found that

                                H2 ϕ + P Hϕ + Qϕ = 0.

I believe that a similar equation connects any function of x and y above the
3rd degree with its first and second Hessians. The proof will be given in a
subsequent Section, where also I shall give a complete proof, which occurred to
me immediately after sending the preceding note to the press, of the complete
Theory of the Respondent by means of the general equations of concomitance. p. 363
   P.S. Since the preceding was in type, I have ascertained the existence and
sufficiency of a general method for forming the polar reciprocal and probably
also the discriminant to functions of any degree of three variables by an explicit
process of permutation and differentiation. In particular I am enabled to give
the actual rule for constructing the polar reciprocal and the discriminant curves
of the 4th and 5th degrees. So far as regards the polar reciprocal of curves of
the 4th degree M. Hesse has already given a method of obtaining it, but mine is
entirely unlike to this, and rests upon certain extremely simple and universal

                                                 369
principles of the calculus of forms. The only thing necessary to be done in order
to carry on the process to curves of the 6th or higher degrees, is to ascertain
the relation of the discriminants of functions of two variables of those respective
degrees to such of the fundamental invariants as are of an inferior order to the
discriminant.
   The theory applies equally well to surfaces and to functions of any number of
variables, and may, I believe, without any serious difficulty be extended so as
to reduce to an explicit process the general problem of effecting the elimination
between functions of any degree and of any number of variables. The method
above adverted to will appear in a subsequent Section.

                      [Continued pp. 402 and 411 below.]




                                       370
                                          44.
      Sur une propriété nouvelle de l’équation qui sert à
       déterminer les inégalités séculaires des planètes
       [Nouvelles Annales de Mathématiques, XI. (1852), pp. 438–440.]


                                       [Extract.]
                                                                                           p. 364
   6. Soit le déterminant carré symétrique

                                a1,1 a1,2 · · · a1,n
                                a2,1 a2,2 · · · a2,n
                                                     ,                              (M)
                                ··· ···          ···
                                an,1 an,2 · · · an,n

dans lequel on a, d’après la définition,

                                      ai,e = ae,i .

Élevant le déterminant à la puissance p, on obtient le déterminant

                               A1,1 A1,2 · · · A1,n
                               A2,1 A2,2 · · · A2,n
                                                    ;                                (N)
                               ···  ···         ···
                               An,1 An,2 · · · An,n

et ce déterminant est symétrique aussi par rapport à la diagonale A1,1 , A2,2 . . . An,n .
   Retranchant de chaque terme de la diagonale symétrique du (M) la même
quantité λ, on obtient le déterminant

                         a1,1 − λ   a1,2   ···     a1,n
                           a2,1   a2,2 − λ · · ·   a2,n
                                                          .                          (P)
                            ···      ···            ···
                           an,1     an,2   · · · an,n − λ
                                                                                           p. 365
  Développant ce déterminant et ordonnant par rapport à λ, on obtient une
expression qui, étant égalée à zéro, donne l’équation

                     λn − f λn−1 + gλn−2 + · · · + (−1)n t = 0,                      (1)

équation qui a n racines réelles (voir t. x. p. 259).
   Retranchant de chaque terme de la diagonale symétrique du déterminant (N)
la quantité µ, et opérant comme ci-dessus, on parvient à l’équation

                    µn − F µn−1 + Gµn−2 + · · · + (−1)n T = 0,                       (2)

                                          371
équation qui a aussi n racines réelles. Les racines de cette équation sont les
racines de l’équation (1), élevées chacune à la puissance p.
   Démonstration. Représentons par

                                 ρ1 ,    ρ2 ,        ρ3 . . . ρp ,

les p racines de l’équation ρp − 1 = 0. Écrivons le déterminant

                     a1,1 − ρq λ     a1,2        ···     a1,n
                         a2,1    a2,2 − ρq λ     ···     a2,n
                                                              ,
                         ···         ···                 ···
                         an,1        ···     an,n − ρq λ

et faisons q égal successivement à tous les nombres de la suite 1, 2, 3 · · · p, on aura
p déterminants; le produit de tous ces déterminants reste évidemment le même
dans quelque ordre qu’on prenne ces déterminants, et, d’après les propriétés
connues des racines de l’unité, tous les termes en ρ qui ne seront pas élevés à
une puissance p disparaîtront, et λ accompagnant toujours ρ, il ne reste donc
que des λp , et le déterminant-produit sera

                      A1,1 − λp   A1,2     A1,3 · · · A1,n
                        A2,1    A2,2 − λ p ···     A2,n
                                                           ;                        (Q)
                         ···       ···              ···
                        An,1      An,2     · · · An,n − λp

où, faisant abstraction de λ, on a le déterminant (N). Ainsi

                                         µ = λp .

                                                                           C. Q. F. D.

   7. Application.
                                 n = 2,         et      p = 2;
déterminant
                                           a b
                                               ,                                   (M)
                                           b c
élevant ce déterminant au carré, on a

                                  a2 + b2 ab + bc
                                                  ;                                 (N)
                                  ab + bc b2 + c2
                                                                                           p. 366
   déterminant
                                        a−λ  b
                                                ,                                   (P)
                                         b  c−λ


                                            372
                           λ2 − (a + c)λ + ac − b2 = 0;                          (1)
déterminant
                            a2 + b2 − µ    ab + bc
                                                   ,
                              ab + bc   b + c2 − µ
                                         2


              µ2 − (a2 + c2 + 2b2 )µ + (ac − b2 )2 = 0,      où    µ = λ2 .      (2)
Faisons
                                  n = 2,      p = 3,
(M) ne change pas, et l’on a

                      a3 + 2ab2 + b2 c       a2 b + abc + b3 + bc2
                                                                   ;             (N)
                   a2 b + abc + b3 + b2 c       ab2 + 2b2 c + c3

le déterminant (P) et l’équation (1) restent les mêmes; mais l’équation (2) devient

                 µ2 − (a3 + c3 + 3ab2 + 3cb2 )µ + (ac − b2 )3 = 0,

où
                                         µ = λ3 ,
car, λ1 et λ2 étant les deux racines de l’équation (1), on a

            λ31 + λ32 = a3 + c3 + 3ab2 + 3cb2 ,        λ31 λ32 = (ac − b2 )3 .

8. M. Sylvester fait observer que son théorème est un cas particulier d’un théorème
plus général, démontré par M. Borchardt, pour des déterminants quelconques, et
qui devient le théorème démontré ci-dessus, lorsque le déterminant est symétrique
(Journal de Mathématiques, t. xii. p. 63, 1847).




                                           373
                                             45.
On a Remarkable Theorem in the Theory of Equal Roots and
                    Multiple Points
               [Philosophical Magazine, III. (1852), pp. 375–378.]
                                                                                             p. 367
   In order that the theorem which I propose to state may be the more easily
understood, and with the least ambiguity expressed, I shall commence with the
case of a homogeneous function of two variables only, x and y.
   Let
                            1
        ϕ = axn + nbxn−1 y + n(n − 1)cxn−2 y 2 + · · · + nb′ xy n−1 + a′ y n ,
                            2
and let the result of operating with the symbol
                          d         d                  d       d
                    xn      + xn−1 y + · · · + y n−1 x ′ + y n ′ ,
                         da         db                db      da
on any function of a, b, c . . . b′ , a′ be called the Evectant of such function, and the
result of repeating this process r times the rth Evectant.
   Understand by the multiplicity of the equation the number of equalities
between the roots that exist; so that a pair of equal roots will signify a multiplicity
1, two pairs of equal roots, or three equal roots a multiplicity 2; a pair of equal
roots and a set of three equal roots, a multiplicity 1 + 2 or 3, and so on. Now
suppose the total multiplicity of ϕ to be m: the first part of the proposition
consists in the assertion that the 1st, 2nd, 3rd . . . (m − 1)th Evectants of the
discriminant of ϕ, that is of the result of eliminating x and y between
                                         dϕ        dϕ
                                            ,
                                         dx        dy

(as well as the discriminant itself), will all vanish in whatever way the multiplicity
is distributed; the second part of the proposition about to be stated requires that
the mode should be taken into account of the manner in which the multiplicity
(m) is made up. Suppose, then, that there are r groups of roots, for one of which
the                                                                                    p. 368
   multiplicity is m1 , for the second m2 , &c., and for the rth mr , so that
m1 + m2 + · · · + mr = m. Then, I say, that the mth evectant of the determinant
of ϕ is of the form

                 (a1 x + b1 y)m1 n (a2 x + b2 y)m2 n · · · (ar x + br y)mr n ,

where a1 : b1 , a2 : b2 . . . ar : br are the ratios of x : y corresponding to the several
sets of equal roots.

                                             374
   This latter part of the theorem for the case of m = 1 was discovered inductively
by Mr Cayley, by considering the cases when ϕ is a cubic, or a biquadratic function.
I extended the theory to functions of any number of variables, and supplied a
demonstration, that is for the case of one pair of equal roots. Mr Salmon showed
that my demonstration could be applied to the case of two pairs of equal roots,
or two double points, &c., and very nearly at the same time I made the like
extension to the case of three equal roots, cusps, &c., and almost immediately
after I obtained a demonstration for the theorem in its most general form. This
demonstration reposes upon a very refined principle, which I had previously
discovered but have not yet published, in the Theory of Elimination.
   I have here anticipated a little in speaking of the theorem as applicable to
curves and other loci.
   Suppose ϕ(x, y, z) = 0 to be the equation to a curve expressed homogeneously.
   Let
                                          1                                      1
ϕ(x, y, z) = axn +(na′ xn−1 y+nb′ xn−1 z)+ n(n−1)a′′ xn−2 y 2 +n(n−1)b′′ xn−2 yz+ n(n−1)c′
                                          2                                      2
and understand by the evectant of any quantity the result of operating upon it
with the symbol
                       d          d          d            d
                 xn      + xn−1 y ′ + xn−1 z ′ + xn−2 y 2 ′′ + &c.
                      da         da         db           da
Suppose, now, the curve to have double points, the (r − 1)th evectant (and of
course all the inferior evectants) of the discriminant of ϕ (meaning thereby the
result of eliminating x, y, z between dϕ/dx, dϕ/dy, dϕ/dz) will all vanish, and
the rth evectant will be of the form

        (a1 x + b1 y + c1 z)n × (a2 x + b2 y + c2 z)n · · · × (ar x + br y + cr z)n ,

where a1 : b1 : c1 , a2 : b2 : c2 . . . ar : br : cr are the ratios of the coordinates at the
respective double points. If there be cusps the multiplicity of each                          p. 369
   such will be 2; and calling the total multiplicity m, to every cusp will cor-
respond a factor of the 2nth power in the mth evectant; and so on in general
for various degrees of multiplicity at the singular points respectively. The like
theorem extends to conical and other singular points of surfaces; so that there
exists a method, when a locus is given having any degree of multiplicity, of at
once detecting the amount and distribution of this multiplicity, and the positions
of the one or more singular points. In conclusion I may state, that precisely
analogous results (mutatis mutandis) obtain, when, in place of a single function
having multiplicity, we take the more general supposition of any number of
homogeneous functions being subject to the condition of pluri-simultaneity, that
is being capable of being made to vanish by each of several different systems of
values for the ratios between the variables. Multiplicity in a single function is, in

                                            375
fact, nothing more nor less than pluri-simultaneity existing between the functions
derived from it by differentiating with respect to each of the given variables
successively. But as I purpose to give these theorems and their demonstration,
which I have already imparted to my mathematical correspondents, in a paper
destined for reading before the Royal Society, I need not further enlarge upon
them on the present occasion.
   P.S. In the above statement I have spoken only of cusps of curves which are the
precise and unambiguous analogues of three coincident points in point-systems,
in order to avoid the necessity of entering into any disquisition as to the species of
singularity in curves or other loci corresponding to higher degrees of multiplicity
in point-systems, a subject which has not hitherto been completely made out. I
may here also add a remark, which gives a still higher interest to the theory, which
is (to confine ourselves, for the sake of brevity, to functions of two variables),
that if any root of x : y, say a : b, occur 1 + µ times, the total multiplicity of the
equation being supposed m, and its degree n, then taking ν any integer number
not exceeding µ, the (m + ν)th evectant of the discriminant will contain the
factor (ax + by)(µ−ν)n . So that, for instance, if there be but a single group of
equal roots, and they be 1 + µ in number, every evectant up to the (µ − 1)th
inclusive will vanish, and from the µth to the (2µ − ν)th will contain a power of
(ax + by)n .




                                         376
                                                46.
            Observations on a New Theory of Multiplicity
                  [Philosophical Magazine, III. (1852), pp. 460–467.]
                                                                                                         p. 370
   In the Postscript to my paper in the last number of the Magazine, I mis-stated,
or to speak more correctly, I understated the law of Evection applicable to
functions having any given amount of distributive multiplicity. The law may be
stated more perfectly, and at the same time more concisely, as follows. Every
point represented by the coordinates α1 , β1 . . . γ1 , for which the multiplicity is
m1 , will give rise in every evectant 206 of the discriminant of the function to a
factor (α1 x + β1 y + · · · + γ1 z)m1 n , n being supposed to be the degree of the
function. Hence if there be r such points, for which the several multiplicities are
m1 , m2 . . . mr , every evectant must contain (m1 + m2 + · · · + mr )n linear factors;
and as the uth evectant is of the degree un, it follows that all the evectants
below the (m1 + m2 + · · · + mr )th evectant must vanish completely, and this
Evectant itself be contained as a factor in all above it.207                            p. 371
   as already remarked, this simply means that there are r distinct groups of
equal roots, such groups containing 1 + m1 , 1 + m2 . . . 1 + mr roots respectively.
So for curves and higher loci, the total distributive multiplicity is the sum of
the multiplicities at the several multiple points. But the true theory of the
higher degrees of multiplicity separately considered at any point remains yet
to be elaborated, and will be found to involve the consideration of the theory
of elimination from a point of view under which it has never hitherto been
contemplated.
   Confining our attention for the present to curves, we have a clear notion of the
multiplicity 1: this is what exists at an ordinary double point. As well known,
  206
      Frequent use being made in what follows of the word Evectant, I repeat that the evectant
of any expression connected with the coefficients of a given function (supposed to be expressed
in the more usual manner with letters for the coefficients affected with the proper binomial or
polynomial numerical multipliers) means the result of operating upon such expression with a
symbol formed from the given function by suppressing all the binomial or polynomial numerical
parts of the coefficients to be suppressed, and writing in place of the literal parts of the
coefficients a, b, c, &c. the symbols of differentiation d/da, d/db, d/dc, &c.; in all that follows it
is the successive evectants of the discriminant alone which come under consideration. I need
hardly repeat, that the discriminant of a function is the result of the process of elimination
(clear from extraneous factors) performed between the partial differential quotients of the
function in respect to the several variables which it contains, or to speak more accurately, is the
characteristic of their coevanescibility. The constitution of the quotients obtained by dividing
all the other evectants of the discriminant by the first non-evanescent one, presents many
remarkable features which remain yet to be fully studied out, and promise a wide extension of
the existing theory.
  207
      When a function of only two variables is in question, there is no difficulty in understanding
what property of the function it is which is indicated by the allegation of the existence of
multiplicities m1 , m2 . . . mr ;


                                                377
its analytical character may be expressed by saying that the function of x, y, z,
which characterizes the curve, is capable, when proper linear transformations
are made, of being expanded under the form of a series descending according to
the powers of z, such that the constant coefficient of the highest power of z, and
the linear function of x, y, which is the coefficient of the next descending power
of z, may both disappear. Again, when the multiplicity is 2, the third coefficient,
which is a quadratic function of x and y, will become a perfect square. This is
the case of a cusp, which, as I have said, is the precise analogue to that of three
equal roots for a function of two variables. Before proceeding to consider what it
is which constitutes a multiplicity 3 for a curve, it will be well to pause for a
moment to fix the geometrical characters of the ordinary double point and the
cusp.
   If we agree to understand by a first polar to a curve the curve of one degree
lower which passes through all the points in which the curve is met by tangents
drawn from an arbitrary point taken anywhere in its own plane, we readily
perceive that at an ordinary double point all the infinite number of first polars
which can be drawn to the curve will intersect one another at the double point.
Again, at a cusp all these polars will not only all intersect, they will moreover
all touch one another at the cusp. Now we may proceed to inquire as to the
meaning of a multiplicity of the third degree, which, strange to say, I believe has
never yet been distinctly assigned by geometricians.
   This is not the case of a so-called triple point, that is a point where three
branches of the curve intersect. Supposing x = 0, y = 0, to represent such a
point, the characteristic of the curve must be reducible to the form

                       (gx3 + hx2 y + kxy 2 + ly 3 )z n−3 + &c.,

which, as is well known, involves the existence of four conditions. This, however,
would not in itself be at all conclusive against the multiplicity at a triple point
being only of the third degree; for it can readily be shown that there may exist
singular points of any degree of singularity (as measured by the number of
conditions necessary to be satisfied in order that such                                 p. 372
   singularity may come into existence), but for which the multiplicity may be as
low as we please; as, for instance, if at a double point (which is not a cusp) there
be a point of inflexion on one branch or on both, or a point of undulation, or any
other singularity whatever, still provided there be no cusps, the multiplicity will
stick at the first degree and never exceed it; for only the discriminant itself will
vanish on these suppositions, but no evectant of the discriminant. The reason,
on the contrary, why a so-called triple point must be said to have a multiplicity
of the degree 4, and not merely of the degree 3, springs from the fact that the
1st, 2nd and 3rd evectants of the discriminant all vanish at such a point.
   It is clear, then, that there ought to exist a species of multiplicity for which the
1st and 2nd evectants vanish, but not the 3rd. In fact, as at a double point the

                                          378
first polars all merely intersect, but at a cusp have all a contact with one another
of the first degree, so we ought to expect that there should exist a species of
multiple point such that all the first polars should have with each other a contact
of the second degree (or if we like so to say, the same curvature) at that point.
When the curve has a triple point, all its first polars will have that point upon
them as a double point; and it is not at the first glance, easy à priori to say what
is the nature of the contact between two curves which intersect at a point which
is a double point to each of them: we know upon settled analytical principles,
that when one curve having a double point is crossed there by another curve not
having a double point, that the two must be said to have with one another, a
contact of the 1st degree; and we now learn from our theory of evection, that if
each have a double point at the meeting-point, the degree of the contact must
from principles of analogy be considered to be of the 3rd degree.208 Now, then,
we come to the question of deciding definitely what is a multiple point for which
the degree of multiplicity is 3. It is, adopting either test, whether of first polar
contact or of evection, a cusp situated or having its nidus, so to say, at a point
of inflexion. In other words, x = 0, y = 0 will be a point whose multiplicity is
intermediate between that of the cusp and that of a so-called triple point, when
the characteristic of the curve admits of being written under the form

                   z n−2 x2 + z n−3 (gx3 + hx2 y + ixy 2 ) + z n−4 &c.;

or in other words, when over and above the vanishing of the constant and linear
coefficients, and the quadratic coefficient being a perfect square, as in the case of
an ordinary cusp, this square has a factor in common with the next (the cubic)
coefficient; or again, in other words, a curve has a point                            p. 373
   for which the multiplicity is 3 when its characteristic function admits of being
expanded according to the powers of one of the variables, in such a manner that
the first coefficient and the second (the linear) coefficient vanish, and that the
discriminant of the third and the resultant of the third and fourth are both
at the same time zero. This being the case, it may be shown that the first
polars will all have with each other a contact of the second degree; and moreover,
that all the evectants of the discriminant will have as a common factor a linear
function of the variables, raised to a power whose index is three times that of
the characteristic function. As, then, there is but one kind of ordinary double
point, and but one kind of point with multiplicity 2, so there is one, and only
one, kind of point with a multiplicity 3. A cusp is a peculiar double point; a
flex-cusp (as for the moment I call the point last above discussed) is a peculiar
cusp. This law of unambiguity, however, appears to stop at the third degree. A
so-called triple point (which ought in fact to be called a quintuple point) is a
 208
    This may easily be verified by direct analytical means; as also the more general proposition,
that two curves meeting at a point where there are m branches of the one and n branches of
the other, must be considered to have mn coincident points in common, that is, if we like so to
express it, to have a contact of the degree mn − 1.


                                              379
point for which the multiplicity, as shown above, is of the fourth degree; but it
is not the only point of that degree of multiplicity. Without assuming to have
exhausted every possible supposition upon which such a degree of multiplicity
may be brought into existence, it will be sufficient to take as an example a curve
whose characteristic is capable of assuming the form
  z n−2 x2 + z n−3 (gx3 + hx2 y) + z n−4 (kx4 + lx3 y + mx2 y 2 + nxy 3 ) + z n−5 &c.
It may readily be demonstrated that the first polars of this curve have all with
one another at the point x, y a contact of a degree exceeding the 2nd, that is of
at least the 3rd degree (and, I believe, in general not higher). Now the point
x, y is evidently not a triple-branched point, but a cusp with three additional
degrees of singularity; so that we have evidence of the existence of a point whose
degree of singularity is 5, and whose multiplicity is at least 4, but which is in
no sense a modified triple point. It is probably true (but to demonstrate this
requires a further advance to be made than has yet been realized in the theory of
the constitution of discriminants) that a cusp may be so modified by the nidus
at which it is posited, as, without ever passing into a triple point, to be capable
of furnishing any amount of multiplicity whatever, curiously in this contrasting
with an ordinary double point, no amount whatever of extraordinary singularity
imparted to which, or so to speak, to its nidus, can ever heighten its multiplicity
so as to make it surpass the first degree without first converting it into a cusp. I
may illustrate the nature of a flex-cusp by what happens to a curve of the third
degree. When it breaks up into a conic and a right line, there are two ordinary
double points; for the existence of these double points, as for the existence of a
cusp, two conditions are required. When, however, the right line and conic touch
one another (a casus omissus this in the works of the special geometers), the
characters of the cusp and the point of inflexion are combined at the point           p. 374
   of contact; the multiplicity is of the third degree, and the singularity also of a
degree not exceeding this; three conditions only being necessary to be satisfied
in order that a given cubic may degenerate into such a form; and it will be found
that the discriminant and the first and second evectants thereof vanish for this
case, and that the third evectant of the discriminant will be a perfect 9th power;
whereas in order that the cubic may have a so-called triple point, that is may
degenerate into a trident of diverging rays, four conditions must be satisfied, and
it will be found that when this is the case, the first, second, and third evectants
of the discriminant will all vanish, and the fourth will be a perfect 12th power of
a linear function of the variables. I may mention, by the way, at this place, that
the law of a discriminant and the successive evectants up to the mth inclusive,
all vanishing, may be expressed otherwise (not in identical, but in equivalent or
equipollent terms), by saying that the discriminant and all its derivatives of a
degree not exceeding the mth will all vanish—understanding by a derivative of
the discriminant any function obtained from the discriminant by differentiating
it any specified number of times with respect to the constants of the function to

                                         380
which it belongs, the same constants being repeated or not indifferently.209 And
very surprising it must be allowed to be, stated as a bare analytical fact, that
(m + 1) conditions imposed upon the coefficients of a function of any number
of variables and of any degree should suffice to make the inordinately greater
number of functions which swarm among the derivatives of the mth and inferior
degrees of the discriminant each and all simultaneously vanish.
   Without pushing these observations too far for the patience of the general
reader, it may be remarked by way of setting foot with our new theory upon
the almost unvisited region of the singularities of surfaces, that by the light of
analogy we may proceed with a safe and firm step as far as multiplicity of the
third degree inclusive.
   The function characteristic of the surface being supposed to be expressed
in terms of the four variables x, y, z, t, and expanded according to descending
powers of t, then when x, y, z is an ordinary double point of the first degree of
multiplicity, the constant and the linear coefficient disappear; when the point
has a multiplicity 2, the discriminant of the quadratic coefficient will be zero,
that is this coefficient will be expressible by means of due linear transformations
under the form of x2 + y 2 ; and when the multiplicity is to be of the degree 3,
the cubic coefficient will, at the same time that the quadratic coefficient is put
under the form x2 + y 2 , itself (for the same system of x and y) assume the form
of a cubic function of x, y, z, in which the highest power of z, that is z 3 , will not
appear; or in other words (restoring to x, y, z their                                   p. 375
   generality), not only will the first derivatives of the quadratic function be
nullifiable simultaneously with each other, but likewise at the same time with
the cubic function itself. These three cases will be for surfaces, the analogues so
far, but only so far as regards the degree of the multiplicity, to the double point,
cusp, and flex-cusp of curves.210 The analogue to the so-called triple point of the
curves will be a point whose degree of singularity, depending upon the vanishing
of the six constants in the third coefficient (which is a quadratic function of
x, y, z) at the same time as the three constants in the linear factor, would seem
to be but 6 more than for a double point, that is in all 1 + 6 or 7, but whose
multiplicity, as inferred from the nature of the contact of its first polars, which
will be of the 7th order, would appear to be 8 (a seeming incongruity which I
 209
     Or, to speak more simply, the discriminant and its successive differentials up to the mth
exclusive must all vanish simultaneously.
 210
     At an ordinary conical point of a surface for which the multiplicity is 1, every section of the
surface is a curve with a double point. When the multiplicity is 2, the cone of contact becomes
a pair of planes, through the intersection of which any other plane that can be drawn cuts
the surface in a section having an ordinary cusp of multiplicity 2, but which themselves cut
the surface in sections, having so-called triple points, so that for these two principal sections
(which is rather surprising) the multiplicity suddenly jumps up from 2 to 4. All other things
remaining unaltered when the multiplicity of the conical point is 3, the cusp belonging to any
section of the surface drawn through any intersection of the two tangent planes passes from an
ordinary cusp to a flex-cusp.


                                               381
am not at present in a condition to explain);211 so that there will apparently be
4 steps of multiplicity to interpolate between this case and the case analogous
(sub modo) to the flex-cusp, last considered. Whether these intervening degrees
correspond to singularities of an unambiguous kind, no one is at present in a
condition to offer an opinion. I will conclude with a remark, the result of my
experience in this kind of inquiry as far as I have yet gone in it, namely that
it would be most erroneous to regard it as a branch of isolated and merely
curious or fantastic speculation. Every singularity in a locus corresponds to the
imposition of certain conditions upon the form of its characteristic; by aid of the
theory of evection we are able to connect the existence of these conditions with
certain consequences happening to the form of the discriminant, and thereby it
becomes possible, upon known principles of analysis, to infer particulars relating
to the constitution of the discriminant itself in its absolutely general form, very
much upon the same principle as when the values of a function for particular
values of its variable or variables are known, the general form of the function
thereby itself, to some corresponding extent, becomes known. Thus, for instance,
I have by the theory of evection in its most simple application, been led to a
representation of the discriminant                                                  p. 376
   of a function of two variables under a form very different and very much more
complete and fecund in consequences than has ever been supposed, or than I
had myself previously imagined, to be possible.
   According to the opinion expressed by an analyst of the French school, of
pre-eminent force and sagacity, it is through this theory of multiplicity, here
for the first time indicated, that we may hope to be able to bridge over for the
purposes of the highest transcendental analysis, the immense chasm which at
present separates our knowledge of the intimate constitution of functions of two
from that of three, or any greater number of variables.
   It is, as I take pleasure in repeating, to a hint from Mr Cayley,212 who
habitually discourses pearls and rubies, that I am indebted for the precious and
 211
      So, too, at a so-called quadruple point in a curve, the degree of the contact of the 1st polars
is 8, and therefore the multiplicity of the curve at such point is 9; but the number of constants
which vanish for this case (namely all those of the cubic coefficient in x, y) over and above what
vanish for the case of a so-called triple point is only 4, which is a unit less than the difference
between the measures of the multiplicities at the respective points; and this difference continues
to increase as we pass on to so-called quintuple and higher multiple points in the curves.
  212
      Mr Cayley’s theorem stood thus:—If

                             axn + nbxn−1 y + · · · + nb′ xy n−1 + a′ y n

have two equal roots, and ϖ be its discriminant, then will
                                      d           d           d
                               n                                  o
                                yn      − y n−1 x    &c. = xn ′       ϖ
                                     da           db         da
be a perfect nth power. It will easily be seen that this theorem is convertible into a theorem of
evection by interchanging in the result x and y with y and −x.



                                                382
pregnant observation on the form assumed by the first discriminantal evectant of
a binary function with a pair of equal roots, out of which, combined with some
antecedent reflections of my own, this new theory of multiplicity has taken its
rise. The idea of the process of evection, and the discovery of its fundamental
property of generating what, in my calculus of forms (Cambridge and Dublin
Mathematical Journal), I have called contravariants, is due to my friend M.
Hermite. The polar reciprocals of curves and other loci are contravariants and, as
I have recently succeeded in showing, for curves at least, evectants, but of course
not discriminantal evectants; and I am already able to give the actual explicit
rule for the formation of the polar reciprocal of curves as high as the 5th degree,
which with a little labour and consideration can be carried on to the 6th, and in
fact to curves of any degree n when once we are acquainted with any mode of
determining all such independent invariants of a function of two variables as are
of dimensions not exceeding 2(n − 1) in respect of the coefficients.
   By the special geometers (by whom I mean those who, unvisited by a higher
inspiration, continue to regard and to cultivate geometry as the science of mere
sensible space) this problem has only been accomplished, and that but recently,
for curves whose degrees do not exceed the 4th. Mr Salmon has made the
happy and brilliant (and by the calculus of forms instantaneously demonstrable)
discovery, communicated to me in the course of a most instructive and suggestive
correspondence, that a certain readily ascertainable                                  p. 377
   evectant of every discriminant of any function whatever is an exact power of
its polar reciprocal.213
   I believe that it may be shown, that, with the sole exception of odd-degreed
functions of two variables, the polar reciprocal itself (as distinguished from a
power thereof) of every function is an evectant, not (of course) of the discriminant,
but of some determinable inferior invariant.
   P.S. The terms pluri-simultaneous and pluri-simultaneity, used or suggested
by me in my last paper in the Magazine, may be advantageously replaced by the
more euphonious and regularly formed words consimultaneous, consimultaneity.
Multiplicity and all its attributes and consequences are included as particular
cases in the general conception and theory of consimultaneity, that is of con-
simultaneous equations, or, which is the same thing, of consimultevanescent
functions.




 213
     Namely, for a function of degree n, and variability (that is, having a number of variables)
p, the (n − 1)p − 1th evect of the discriminant is the (n − 1)th power of the polar reciprocal.


                                             383
                                             47.
A Demonstration of the Theorem that Every Homogeneous
  Quadratic Polynomial is Reducible by Real Orthogonal
Substitutions to the Form of a Sum of Positive and Negative
                         Squares
                 [Philosophical Magazine, IV. (1852), pp. 138–142]
                                                                                                  p. 378
   It is well known that the reduction of any quadratic polynomial
                      (1, 1)x2 + 2(1, 2)xy + (2, 2)y 2 + · · · + (n, n)t2
to the form a1 ξ 2 + a2 η 2 + · · · + an θ2 , where ξ, η . . . θ are linear functions of
x, y . . . t, such that x2 + y 2 + · · · + t2 remains identical with ξ 2 + η 2 + · · · + θ2
(which identity is the characteristic test of orthogonal transformation), depends
upon the solution of the equation
                        (1, 1) + λ   (1, 2)      ···     (1, n)
                          (2, 1)   (2, 2) + λ    ···     (2, n)
                                                                     = 0.
                            ···        ···       ···       ···
                          (n, 1)     (n, 2)      · · · (n, n) + λ
The roots of this equation give a1 , a2 . . . an ; and if they are real, it is easily shown
that the connexions between x, y . . . t; ξ, η . . . θ, are also real. M. Cauchy has
somewhere given a proof of the theorem214 , that the roots of λ in the above
equation must necessarily always be real; but the annexed demonstration is, I
believe, new; and being very simple, and reposing upon a theorem of interest
in itself, and capable no doubt of many other applications, will, I think, be
interesting to the mathematical readers of this Magazine.                                   p. 379
   Let
                        (1, 1) + λ   (1, 2)         ···          (1, n)
                          (2, 1)   (2, 2) + λ       ···          (2, n)
            f (λ) =       (3, 1)     (3, 2)   (3, 3) + λ · · ·   (3, n)   .
                            ···        ···          ···            ···
                          (n, 1)     (n, 2)         ···        (n, n) + λ
It is easily proved that f (λ) × f (−λ)
                          [1, 1] − λ2    [1, 2]      ···      [1, n]
                             [2, 1]   [2, 2] − λ2    ···      [2, n]
                    =                                                  ,
                               ···         ···       ···        ···
                             [n, 1]      [n, 2]      · · · [n, n] − λ2
 214
    Jacobi and M. Borchardt have also given demonstrations; that of the latter consists in
showing that Sturm’s functions for ascertaining the total number of real roots expressed by my
formulæ (many years ago given in this Magazine) are all, in the case of f (λ), representable as
the sums of squares, and are therefore essentially positive.


                                             384
where
                [ι, e] = (ι, 1)(1, e) + (ι, 2)(2, e) + · · · + (ι, n)(n, e).
If, now, for all values of r and s, (r, s) = (s, r), that is, if f (0) becomes
the complete determinant to a symmetrical matrix, then every term [r, s] in
the derived matrix becomes a sum of squares, and is essentially positive, and
(−1)n f (λ) × f (−λ) assumes the form

                     (λ2 )n − F (λ2 )n−1 + G(λ2 )n−2 + · · · ± L,

where F, G, . . . L will evidently be all positive; for it may be shown that F will
be the sum of the squares of the separate terms, that is, of the last minor
determinants of the given matrix, G the sum of the squares of the last but one
minors, and so on, L being the square of the complete determinant. For instance,
if
                                    a+λ       γ        β
                           f (λ) =    γ     b+λ        α    ,
                                      β       α      c+λ
                     −f (λ) × f (−λ) = λ6 − F λ4 + Gλ2 − H,
where
                F = a2 + b2 + c2 + 2α2 + 2β 2 + 2γ 2 ,
                G = (ab − γ 2 )2 + (bc − α2 )2 + (ac − β 2 )2
                     + 2(aα − βγ)2 + 2(bβ − γα)2 + 2(cγ − αβ)2 ,
                                    2
                        a γ β
                H=      γ b α           .
                        β α c
Hence it follows immediately
                       √     that f (λ) = 0 cannot have imaginary roots; for, if
possible, let λ = p + q −1, and write

            a + p = a′ ,      b + p = b′ ,         c + p = c′ ,    λ + p = λ′ ,
                                                                                         p. 380
   f (λ) becomes
                               a′ + λ′     γ       β
                                  γ    b ′ + λ′    α   ,
                                  β        α    c +λ
                                                 ′   ′

or say ϕ(λ′ ), and the equation ϕ(λ′ ) × ϕ(−λ′ ) = 0 will be of the form

                            λ′6 − F ′ λ′4 + G′ λ′2 − H ′ = 0,

where F ′ , G′ , H ′ are all essentially positive. Hence, by Descartes’ rule, no value
of λ′2 can be negative, that is, (λ − p)2 cannot be of the form −q 2 ; that is to say,
it is impossible for any of the roots of f (λ) = 0 to be imaginary, or, as was to be
demonstrated, all the roots are real.

                                             385
   I may take this occasion to remark, that by whatever linear substitutions,
orthogonal or otherwise, a given polynomial be reduced to the form ΣA1 ξ 2 , the
number of positive and negative coefficients is invariable: this is easily proved.
If now we proceed to reduce the form (expressed under the umbral notation)
(a1 x1 + a2 x2 + · · · + an xn )2 to the form

                            A1 ξ12 + A2 ξ22 + · · · + An−1 ξn−1
                                                            2
                                                                + An ξn2 ,

by first driving out the mixed terms in which x1 enters, then those in which x2
enters, and so forth until eventually only xn of the original variables is left, it
may readily be shown that
                   !                        !         !                     !           !
                a1                     a1 a2   a1                    a1 a2 a3   a1 a2
       A1 =        ,     A2 =                ÷    ,           A3 =            ÷       ,            ...
                a1                     a1 a2   a1                    a1 a2 a3   a1 a2
                                                          !                     !
                                              a1 a2 . . . an   a1 a2 . . . an−1
                        ...        An =                      ÷                  .
                                              a1 a2 . . . an   a1 a2 . . . an−1
It follows, therefore, that in whatever order we arrange the umbræ a1 a2 . . . an ,
the number of variations and of continuations of sign in the series
                                          !             !                  !
                                        a1         a1 a2     a1 a2 . . . an
                              1,           ,             ...                ,
                                        a1         a1 a2     a1 a2 . . . an

will be invariable, and in fact will be the same as the number of positive and
negative roots in the generating function in λ above treated of, that is, since all
the roots are real, will be the same as the number of variations and continuations
in the series formed by the coefficients of the several powers of λ, that is
                                          !               !                  !
                                     a1            a1 a2     a1 a2 . . . an
                          1,       Σ    ,        Σ       ...                .
                                     a1            a1 a2     a1 a2 . . . an

The first part of this theorem admits of an easy direct demonstration; for by my
theory of compound determinants, given in this Magazine 215 , we know that
                 a1 a2 ...ar−1 ar                        !                         !
                 a1 a2 ...ar−1 ar         a1 a2 . . . ar−1   a1 a2 . . . ar−1 ar ar+1
                  a1 a2 ...ar−1  =                        ×                          .
                  a1 a2 ...ar−1           a1 a2 . . . ar−1   a1 a2 . . . ar−1 ar ar+1
                                                                                                         p. 381
   The first member of this equation is equivalent to
                                   !                            !                         !2
             a1 a2 . . . ar−1 ar         a1 a2 . . . ar−1 ar+1    a1 a2 . . . ar−1 ar
                                       ×                       −                               .
             a1 a2 . . . ar−1 ar         a1 a2 . . . ar−1 ar+1   a1 a2 . . . ar−1 ar+1
 215
       Cf. pp. 241, 252 above.




                                                      386
Hence it follows, that if the two factors on the right-hand side of the equation
have the same sign,
                                                      !                                            !
                          a1 a2 . . . ar−1 ar                         a1 a2 . . . ar−1 ar+1
                                                           and
                          a1 a2 . . . ar−1 ar                         a1 a2 . . . ar−1 ar+1

have also the same sign inter se, and consequently the two triads
      "                        #          "                           #           "                                #
           a1 a2 . . . ar−1                    a1 a2 . . . ar−1 ar                    a1 a2 . . . ar−1 ar ar+1
                                      ,                                   ,                                            ,
           a1 a2 . . . ar−1                    a1 a2 . . . ar−1 ar                    a1 a2 . . . ar−1 ar ar+1

and
      "                       #           "                            #          "                                #
          a1 a2 . . . ar−1                    a1 a2 . . . ar−1 ar+1                     a1 a2 . . . ar−1 ar+1 ar
                                  ,                                           ,                                            ,
          a1 a2 . . . ar−1                    a1 a2 . . . ar−1 ar+1                     a1 a2 . . . ar−1 ar+1 ar

will in all cases present the same number of changes and continuations, which
proves that the contiguous umbræ, ar , ar+1 , may be interchanged without affect-
ing the number of variations and continuations in the entire series; but, as is
well known, any one order of elements is always convertible into any other order
by means of successive interchanges of contiguous elements, which demonstrates
that, in whatever order the elements a1 , a2 . . . an be arranged, the number of
continuations and variations in
                                               !              !                              !
                                              a1          a1 a2     a1 a2 . . . an
                              1,                 ,              ...                ,
                                              a1          a1 a2     a1 a2 . . . an

is invariable. But that the same thing is true (as we know it to be), for the
relation between any one of these unsymmetrical series and the symmetrical
series (resulting from the method of orthogonal transformation)
                                          !                  !                                     !
                                a1                     a1 a2                          a1 a2 . . . an
                     1,       Σ    ,                 Σ       ,       ...                             ,
                                a1                     a1 a2                          a1 a2 . . . an

is by no means so easily demonstrable in the general case by a direct method,
and the attention of algebraists is invited to supply such direct method of
demonstration. My knowledge of the fact of this equivalence is, as I have stated,
deduced from that remarkable but simple law to which I have adverted, which
affirms the invariability of the number of the positive and negative signs between
all linearly equivalent functions of the form Σ ± cr xr (subject, of course, to the
condition that the equivalence is expressible by means of equations into which
only real quantities enter); a law to which my view of the physical meaning of
quantity of matter inclines me, upon the ground of analogy, to give the name of
the Law of Inertia for Quadratic Forms, as expressing the fact of the existence
of an invariable number inseparably attached to such forms.

                                                            387
                                              48.
On Staudt’s Theorems concerning the Contents of Polygons
 and Polyhedrons, with a Note on a New and Resembling
                   Class of Theorems
                 [Philosophical Magazine, IV. (1852), pp. 335–345]
                                                                                                    p. 382
    The beautiful and important geometrical theorems of Staudt are, I believe,
little, if at all, known to English mathematicians. They originally appeared in
Crelle’s Journal for the year 1843, and have been recently reproduced in M.
Terquem’s Nouvelles Annales for the August Number of the present year.
    These theorems may be summed up, in a word, as intended to show the
possibility and method of expressing the product of any two polygons or any two
polyhedrons as entire functions of the squares of the distances of the angular
points of the two figures from one another. The well-known expression for the
square of the area of a triangle in terms of the sides (in which, when expanded,
only even powers of the lengths of the sides appear) is but a particular case
of Staudt’s theorem for polygons, for it may be considered as the case of two
equal and similar triangles whose angular points coincide. So in like manner, as
observed by Staudt, a similar expression in terms of its sides may be found for
the square of a pyramid. This expression had, however, been previously given
(although, by a strange negligence, not named for what it was) by Mr Cayley in
the Cambridge Mathematical Journal for the year 1841,216 in his paper on the
relations between the mutual distances to one another of four points in a plane
and five points in space; the singularly ingenious (and as singularly undisclosed)
principle of that paper consisting in obtaining an expression for the volume of a
pyramid in terms of its sides, and equating this, or rather its square, to zero as
the conditions of the four angular points lying in the same plane.                   p. 383
    The analogous condition for five points in space is virtually deduced by going
out into rational space of four dimensions, and equating to zero the expression
obtained for the volume of a plupyramid; meaning thereby the figure which
stands in the same relation to space of four as a pyramid to space of three
dimensions. Mr Cayley’s method, if it had been pursued a step further, would
have led him to a complete anticipation of the principal part of Staudt’s discovery.
The method here given is not substantially different from Mr Cayley’s, but is
made to rest upon a more general principle of transformation than that which
he has employed. As to Staudt’s own method, it is as clumsy and circuitous
as his results are simple and beautiful. Geometry, trigonometry and statics,
are laid under contribution to demonstrate relations which will be seen to flow
as immediate and obvious consequences from the most elementary principles
  216
      Query, Is not this expression for the volume of a pyramid in terms of its sides to be found
in some previous writer? It can hardly have escaped inquiry.


                                              388
in the algorithm of determinants. Perhaps, however, M. Staudt’s method is
as good as could be found in the absence of the application of the method of
determinants, the powers of which, even so recently as ten years ago, were not
so well understood or so freely applied as at the present day.
   The following new but simple theorem, of which I shall have occasion to
make use, will be found to be a very useful addition to the ordinary method
for the multiplication of determinants. “If the determinants represented by two
square matrices are to be multiplied together, any number of columns may be
cut off from the one matrix, and a corresponding number of columns from the
other. Each of the lines in either one of the matrices so reduced in width as
aforesaid being then multiplied by each line of the other, and the results of the
multiplication arranged as a square matrix and bordered with the two respective
sets of columns cut off arranged symmetrically (the one set parallel to the new
columns, the other set parallel to the new lines), the complete determinant
represented by the new matrix so bordered (abstraction made of the algebraical
sign) will be the product of the two original determinants.”
   Thus                               !        !
                                   ab       αβ
                                        ×
                                   cd       γδ
may be put under any one of the three following forms:

                                     aα + bβ aγ + bδ
                                                     ,
                                     cα + dβ cγ + dδ

or
                                                    2   2 a b
                         aα aγ b
                                                    2   2 c d 217
                         cα cγ d            or                .
                                                    α   β 0 0
                          β  δ 0
                                                    γ   δ 0 0
                                                                                                    p. 384
   And in general for two matrices of n2 terms each, this rule of multiplication
will give (n + 1) distinct forms representing their products.
   Thus, as a further example,

                              a b c                 α β γ
                              a′ b′ c′        ×     α′ β ′ γ ′ ,
                              a′′ b′′ c′′           α′′ β ′′ γ ′′
 217
    Any quantities might be substituted instead of 2 in the places occupied by the figure in the
above determinant, as such terms do not influence the result; this figure is probably, however,
the proper quantity arising from the application of the rule, because (as all who have calculated
with determinants are aware) the value of the determinant represented by a matrix of no places
is not zero but unity.




                                              389
besides the first and last forms, will be representable by the two intermediate
forms
                     aα + bβ      aα′ + bβ ′     aα′′ + bβ ′′      c
                     aα+bβ aα +bβ
                      ′      ′     ′ ′    ′  ′   aα +bβ
                                                  ′ ′′      ′  ′′  c′
                 − ′′
                    a α + b β a α + b β a α + b β c′′
                             ′′   ′′ ′    ′′   ′ ′′ ′′      ′′  ′′

                         γ             γ′              γ ′′        0
and
                           aα aα′ aα′′ b c
                           a′ α a′ α′ a′ α′′ b′ c′
                         + a′′ α a′′ α′ a′′ α′′ b′′ c′′ .
                            β     β′     β ′′   0 0
                            γ     γ  ′   γ  ′′  0 0
To arrive, for instance, at the latter of these two forms, we have only to write
the two given matrices under the respective forms
                    a b c             0   0          α     0   0 β γ
                    a′ b′ c′          0   0          α′    0   0 β′ γ′
                    a′′ b′′ c′′       0   0          α′′   0   0 β ′′ γ ′′
                    0 0 0             1   0          0     1   0 0 0
                    0 0 0             0   1          0     0   1 0 0
and then apply the ordinary rule of multiplication. So, again, to arrive at the
first of the above written two forms, we must write the two given matrices under
the respective forms
                  a b c           0                    α β           0 γ
                  a′ b′ c′        0                    α′ β ′        0 γ′
                                              and    −
                  a′′ b′′ c′′     0                    α′′ β ′′      0 γ ′′
                  0 0 0           1                    0    0        1 0
and proceed as before.
   This rule is interesting as exhibiting, as above shown, a complete scale whereby
we may descend from the ordinary mode of representing the product of two
determinants to the form, also known, where the two original deter-                 p. 385
   minants are made to occupy opposite quadrants of a square whose places in
one of the remaining quadrants are left vacant, and shows us that under one
aspect at least this latter form may be regarded as a matrix bordered by the two
given matrices.
   A second but obvious theorem requiring preliminary notice is the following,
namely that the value of the determinant to the matrix
                            a1,1 a1,2           . . . a1,n 1
                            a2,1 a2,2           . . . a2,n 1
                            ··· ···                    ··· ···
                            an,1 an,2           . . . an,n 1
                             1    1             ...     1   0

                                               390
is the same as the value of the determinant to the matrix
                              A1,1 A1,2        . . . A1,n 1
                              A2,1 A2,2        . . . A2,n 1
                              ···  ···                ··· ···
                              An,1 An,2        . . . An,n 1
                               1    1          ...     1   0

where in general
                                   Ar,s = ar,s + hr + ks ,
h1 , h2 . . . hn and k1 , k2 . . . kn being any two perfectly arbitrary series of quantities.
This simple transformation is of course derived by adding to the respective
columns in the first matrix the last column (consisting of units) multiplied
respectively by h1 , h2 . . . hn , 0; and to the respective lines, the last line (consisting
of units) multiplied respectively by k1 , k2 . . . kn , 0.
    Suppose, now, that we have two tetrahedrons whose volumes are represented
respectively by one-sixth of the respective determinants

                         x1   y1    z1   1         ξ1   η1    ζ1   1
                         x2   y2    z2   1         ξ2   η2    ζ2   1
                                           ,                         ,
                         x3   y3    z3   1         ξ3   η3    ζ3   1
                         x4   y4    z4   1         ξ4   η4    ζ4   1

xr , yr , zr representing the orthogonal coordinates of the point r in one tetrahedron,
and ξr , ηr , ζr the same for the point r in the other.
    By the first theorem their product may be represented (striking off the last
column only from each matrix) by the matrix

                          Σx1 ξ1    Σx1 ξ2     Σx1 ξ3   Σx1 ξ4     1
                          Σx2 ξ1    Σx2 ξ2     Σx2 ξ3   Σx2 ξ4     1
                          Σx3 ξ1    Σx3 ξ2     Σx3 ξ3   Σx3 ξ4     1 ,
                          Σx4 ξ1    Σx4 ξ2     Σx4 ξ3   Σx4 ξ4     1
                            1         1          1        1        0
                                                                                                p. 386
   where, in general, any such term as Σxr ξs represents

                                    xr ξs + yr ηs + zr ζs .

Again, by virtue of the second theorem, adding
                        1            1            1            1
                       − Σx21 ,     − Σx22 ,     − Σx23 ,     − Σx24
                        2            2            2            2
to the respective lines, and
                        1            1            1            1
                       − Σξ12 ,     − Σξ22 ,     − Σξ32 ,     − Σξ42
                        2            2            2            2

                                             391
to the respective columns, the above matrix becomes (after a change of signs
not affecting the result) the − 288
                                 1
                                    th of

              Σ(x1 − ξ1 )2    Σ(x1 − ξ2 )2     Σ(x1 − ξ3 )2    Σ(x1 − ξ4 )2     1
              Σ(x2 − ξ1 )2    Σ(x2 − ξ2 )2     Σ(x2 − ξ3 )2    Σ(x2 − ξ4 )2     1
              Σ(x3 − ξ1 )2    Σ(x3 − ξ2 )2     Σ(x3 − ξ3 )2    Σ(x3 − ξ4 )2     1 .
              Σ(x4 − ξ1 )2    Σ(x4 − ξ2 )2     Σ(x4 − ξ3 )2    Σ(x4 − ξ4 )2     1
                   1               1                1               1           0

Or calling the angular points of the one tetrahedron a, b, c, d, and of the other
p, q, r, s, 8 × 36, that is 288 times, their product is representable by −1× the
determinant
                            (ap)2 (aq)2 (ar)2 (as)2 1
                            (bp)2 (bq)2 (br)2 (bs)2 1
                            (cp)2 (cq)2 (cr)2 (cs)2 1 ,
                            (dp)2 (dq)2 (dr)2 (ds)2 1
                              1     1       1     1     0
and of course if p, q, r, s coincide respectively with a, b, c, d, 576 times the square
of the tetrahedron abcd will be represented under Mr Cayley’s form,

                             0   (ab)2 (ac)2 (ad)2            1
                           (ba)2   0   (bc)2 (bd)2            1
                           (ca) (cb)
                               2     2   0   (cd)2            1 , 218
                           (da) (db) (dc)
                               2     2     2   0              1
                             1     1     1     1              0

four out of the sixteen distances vanishing, and the remaining twelve reducing to
six pairs of equal distances. The demonstration of Staudt’s                       p. 387
   theorem for triangles is obtained in precisely the same way by throwing the
product of the two determinants

                             x1 y1 1                    ξ1 η 1 1
                             x2 y2 1          and       ξ2 η 2 1
                             x3 y3 1                    ξ3 η 3 1

under the form of − 41 th of

                      Σ(x1 − ξ1 )2 Σ(x1 − ξ2 )2 Σ(x1 − ξ3 )2            1
                      Σ(x2 − ξ1 )2 Σ(x2 − ξ2 )2 Σ(x2 − ξ3 )2            1
                                                                          .
                      Σ(x3 − ξ1 )2 Σ(x3 − ξ2 )2 Σ(x3 − ξ3 )2            1
                           1            1            1                  0
 218
     The corresponding quantity to the above determinant for the case of the triangle (hereafter
given) is identical with the Norm to the sum of the sides. I have succeeded in finding the Factor
(of ten dimensions in respect of the edges), which, multiplied by the above Determinant itself,
expresses the Norm to the sum of the Faces, that is, the superficial area of the Tetrahedron.


                                              392
When the two triangles coincide, calling their angular points a, b, c the above
written determinant becomes
                                0   (ab)2 (ac)2      1
                              (ba)2   0   (bc)2      1
                                                       ,
                              (ca)2 (cb)2   0        1
                                1     1     1        0
or
         (ab)4 + (ac)4 + (bc)4 − 2(ab)2 (ac)2 − 2(ab)2 (bc)2 − 2(ac)2 (bc)2 ,
the negative of which is the well-known form expressing the square of four times
the area of the triangle abc.
   There is another and more general theorem of Staudt for two triangles not in
the same plane, which may be obtained with equal facility. In fact, if we start
from the determinant
                             (aα)2 (aβ)2 (aγ)2        1
                             (bα)2 (bβ)2 (bγ)2        1
                                                        ,
                             (cα)2 (cβ)2 (cγ)2        1
                               1     1     1          0

and add to each column respectively the last column multiplied by εξ12 , εξ22 , εξ32
respectively, we arrive at the form

                   (aα)2 + εξ12 (aβ)2 + εξ22 (aγ)2 + εξ32      1
                   (bα)2 + εξ12 (bβ)2 + εξ22 (bγ)2 + εξ32      1
                                                                 ,
                   (cα)2 + εξ12 (cβ)2 + εξ22 (cγ)2 + εξ32      1
                        1            1            1            0

and considering ξ1 , η1 ; ξ2 , η2 ; ξ3 , η3 as the coordinates of α, β, γ, the   p. 388
   projections upon the plane of abc of a triangle ABC, whose plane intersects
the former plane in the axis of y, and makes with that plane an angle whose
tangent is ε, it is easily seen that this determinant is term for term identical
with the determinant
                            (aA)2 (aB)2 (aC)2         1
                            (bA)2 (bB)2 (bC)2         1
                                                        ,
                            (cA)2 (cB)2 (cC)2         1
                              1     1     1           0

which therefore expresses −16 times the product of the triangles abc and αβγ,
that is abc × ABC× cosine of the angle between the two. A similar method,
if we ascend from sensible to rational geometry, may be given for expressing
in terms of the distances the product of any two pyramids (in a hyperspace)



                                         393
by the cosine of the angle included between the two infinite spaces219 in which
they respectively lie. To pass from the cases which have been considered of two
triangles to two polygons, or of two tetrahedrons to two polyhedrons, generally
presents no difficulty; and for Professor Staudt’s method of doing so, which is
simple and ingenious, and does not admit of material improvement, the reader is
referred to the memoir in Crelle’s Journal or Terquem’s Annales already adverted
to. It is, however, to be remarked (and this does not appear to be sufficiently
noticed in the memoirs referred to), that whilst the expression for the product of
any two polygons in terms of the distances given by Staudt’s theorem is unique,
that for the product of two polyhedrons given by the same is not so, but will
admit of as many varieties of representation as there are units in the product of
the numbers respectively expressing the number of ways in which each polygonal
face of each polyhedron admits of being mapped out into triangles. I cannot
help conjecturing (and it is to be wished that Professor Staudt or some other
geometrician would consider this point) that in every case there exists, linearly
derivable from Staudt’s optional formulæ (but not coincident with any one of
them), some unique and best, because most symmetrical, formula for expressing
the product of two polyhedrons in terms of the distances of the angular points
of the one from those of the other. In conclusion I may observe, that there is
a theorem for distances measured on a given straight line, which, although not
mentioned by Staudt, belongs to precisely the same class as his theorems for
areas in a plane and volumes in space; namely a theorem which expresses twice
the rectangle of any two such distances under the form of an aggregate of four
squares, two taken positively and two                                              p. 389
   negatively; that is to say, if A, B, C, D be any four points on a right line

                      2AB × CD = AD2 + BC 2 − AC 2 − BD2 .

I know not whether this theorem be new, but it is one which evidently must be
of considerable utility to the practical geometer.

                                Note on the above.

   The fundamental theorem in determinants, published by me in the Philo-
sophical Magazine in the course of last year220 , leads immediately to a class of
theorems strongly resembling, and doubtless intimately connected with, those of
Staudt.
 219
      In rational or universal geometry, that which is commonly termed infinite space (as if
it were something absolute and unique, and to which, by the conditions of our being, the
representative power of the understanding is limited), is regarded as a single homaloid related
to a plane, precisely in the same way as a plane is to a right line. Universal geometry brings
home to the mind with an irresistible force of conviction the truth of the Kantian doctrine of
locality.
  220
      See pp. 249, 253 above.


                                             394
  Thus for triangles we have by this fundamental theorem

       x1 x2 x3        ξ1 ξ2 ξ3                x1 ξ1 ξ2      ξ3 x2 x3
       y1 y2 y3      × η1 η2 η3        =       y1 η1 η2    × η3 y2 y3
       1 1 1           1 1 1                   1 1 1         1 1 1
                                                x1 ξ2 ξ3      ξ1 x2 x3
                                           +    y1 η2 η3    × η1 y2 y3
                                                1 1 1         1 1 1
                                             x1 ξ3 ξ1         ξ2 x2 x3
                                           + y1 η3 η1       × η2 y2 y3 .
                                             1 1 1            1 1 1

and consequently, if ABC, DEF be any two triangles,

        ABC × DEF = ADE × F BC + AEF × DBC + AF D × BCE.

This may be considered a theorem relating to two ternary systems of points in a
plane. The analogous and similarly obtainable theorem for two binary systems
of points in the same right line is

                      AB × CD = AC × DB − AD × CB.

As in applying this last theorem to obtain correct numerical results we must give
the same algebraical sign to any two lengths denoted by the two arrangements
XY, ZT , according as the direction from X to Y is the same as that from Z to
T , or contrary to it, so in the theorem for the products of triangles, the areas
denoted by any two ternary arrangements XY Z, T U V must be taken with the
like or the contrary sign, according as the direction of the rotation XY Z is
consentient with or contrary to that of T U V ; so that three of the six possible
arrangements of XY Z may be used indifferently for one another, but the other
three would imply a change of sign. If we                                           p. 390
    analyse what we mean by fixing the direction of the rotation of XY Z, and
reduce this form of speech to its simplest terms, we easily see that it amounts to
ascertaining on which side of B, C lies, that is whether to its right or left, to a
spectator stationed at A on a given side of the plane ABC.
    Let us now pass to the corresponding theorems for two tetrahedrons put
respectively under the forms

                      x1 x2 x3 x4              ξ1 ξ2 ξ3 ξ4
                      y1 y2 y3 y4              η1 η2 η3 η4
                                                           .
                      z1 z2 z 3 z4             ζ1 ζ2 ζ3 ζ4
                      1 1 1 1                  1 1 1 1



                                        395
We may represent this product in either of two ways by the application of our
fundamental theorem, namely as

                   x1 ξ1 ξ2 ξ3         ξ4 x2 x3 x4
                   y1 η1 η2 η3         η y y3 y4
                                     × 4 2                  + &c.
                   z1 ζ1 ζ2 ζ3         ζ4 z 2 z3 z4
                   1 1 1 1             1 1 1 1
or as
                   x 1 x 2 ξ1 ξ2       ξ3 ξ4 x3 x4
                   y1 y2 η 1 η 2       η η y y4
                                     × 3 4 3                + &c.
                   z1 z2 ζ1 ζ2         ζ3 ζ4 z3 z4
                   1 1 1 1             1 1 1 1
there being four products to be added together in the first expression and six in
the latter; and the rule, if we wish that all the products may be additive, being
that on removing the sign of multiplication the determinant to the square matrix
formed by the Greek letters in situ shall always preserve the same sign. Hence we
derive two geometrical formulæ concerning the products of polyhedrons, namely

        (1)   ABCD × EF GH = ABCE × F GHD − ABCF × GHED
                                 + ABCG × HEF D − ABCH × F GED,
        (2)   ABCD × EF GH = ABEF × GHCD + ABGH × EF CD
                                 + ABEG × HF CD + ABHF × EGCD
                                 + ABEH × F GCD + ABF G × EHCD.

These formulæ give rise to an exceedingly interesting observation. In order that
they shall be numerically true, we must have a rule for fixing the sign to be
given to the solid content represented by any reading off of the four points of a
tetrahedron, that is we must have a rule for determining                            p. 391
   the sign of solid contents of figures situated anywhere in space analogous to
that which, as applied to linear distances reckoned on a given right line, is the
true foundation of the language of trigonometry, and the condition precedent for
the possibility of any system of analytical geometry such as exists, and which, not
altogether without surprise, I have observed in the pages of this Magazine one of
the learned contributors has thought it necessary to vindicate the propriety of
importing into his theory of quaternions.
   Various rules may be given for fixing the sign of a tetrahedron denoted by
a given order of four letters. One is the following: the content of ABCD is to
be taken positive or negative, according as to a spectator at A the rotation of
BCD is positive or negative. Another, again, is to consider AB and CD as
representing, say two electrical currents, and to suppose a spectator so placed
that the current AB shall pass through the longitudinal axis of his body from
the head towards the feet, and looking towards the other current CD; the sign

                                        396
of the solid content of the tetrahedron (and, indeed, also the effect, in a general
sense, of the action of the two currents upon one another) will depend upon the
circumstance of this latter current appearing to flow from the right to the left,
or contrariwise in respect of the spectator. Last and simplest mode of all, the
sign of the solid content of ABCD will depend upon the nature (in respect to
its being a right-handed or left-handed-screw) of any regular screw-line (whether
the common helix or one in which the inclination is always in the same direction)
terminating at B and C, and so taken that BA shall be the direction of the
tangent produced at B, and CD the direction of the tangent produced at C.
Inasmuch as of the twenty-four permutations of a quaternary arrangement a
defined twelve have one sign, and the other twelve the contrary sign, these various
definitions of the direction, or, as it may be termed, polarity, of a tetrahedron
corresponding to a given reading, whether as taken each in itself or compared one
with another, give rise to, or rather imply a considerable number of interesting
theorems included in our intuitions of space, and probably belonging to the,
in my belief, inexhaustible class of primary and indemonstrable truths of the
understanding.




                                       397
                                       49.
     On a Simple Geometrical Problem illustrating a
Conjectured Principle in the Theory of Geometrical Method
              [Philosophical Magazine, IV. (1852), pp. 366–369]
                                                                                      p. 392
    The following theorem deserves attention as illustrating a principle of geomet-
rical method which will be presently adverted to. It is curious, also, from the
fact of its solution being by no means so obvious and self-evident as one would
expect from the extreme simplicity of its enunciation. It appeared, and for the
first time, it is believed, at the University of Cambridge about a twelvemonth
back, where it excited considerable attention among some of the mathematicians
of the place. The proposition, as originally presented, was merely to prove that
if ABC be a triangle, and if AD and BE drawn bisecting the angles at A and
B and meeting the opposite sides in D and E be equal, then the triangle must
be isosceles. It is particularly

                                              C

                                        F
                                                      D
                K                                               G
                               A E     H
                                                            B

noticeable that all the geometrical demonstrations yet given of this theorem are
indirect. Thus the first and simplest (communicated to me by a promising young
geometrician, Mr B. L. Smith of Jesus College, Cambridge), was the following:—
Assume one of the angles at DAB to be greater than the corresponding angle
EBA; it can easily be shown that, upon this supposition, D will be higher up
from AB than E; so that if DF and EG be drawn parallel to AB, DF will be
above EG; it is then easily shown that DF = AF , EG = BG, and consequently
DF and AF are each respectively less than EG                                      p. 393
    and BG; and also DF A, which is the supplement of twice DAB, will be less
than EGB, which is the supplement of twice F BA; from which it is readily
inferred, by an easy corollary to a proposition of Euclid, that DA will be less
than F B, whereas it should be equal to it; so that neither of the half angles
at the base can be greater than the other, and the triangle is proved to be
isosceles. Another and independent demonstration by the writer of this article
is less simple, but has the advantage of lending itself at once to a considerable
generalization of the theorem as proposed. Assuming, as above, that DAB is
greater than EBA, it is easily seen that DE produced will cut BA at K on the

                                       398
side of it: also if AD and BE intersect in H, it is readily demonstrable, by a
suitably constructed apparatus of similar triangles, that

                              AH : BH :: CE : CD.

But as HBA is less than HAB, AH is less than BH, and therefore CE is less
than CD, and therefore CED is greater than CDE; that is to say, CAB less K
is greater than CBA plus K, and therefore DAB less K is greater than EBA,
that is ADE is greater than ABE, and therefore the perpendicular from A upon
DE is greater than that from E on AB, which is easily proved to be absurd.
Hence, as before, the triangle is proved to be isosceles. This proof, it is obvious,
remains good for all cases in which EB and DA, drawn on either side of the
base, divide the angles at the base proportionally, provided that these lines
remain equal, and make positive or negative angles with the base not less than
one-half of the respective corresponding angles which the sides of the triangle
are supposed to make with it. The analytical solution of the question, as might
be expected, extends the result still further. To obtain this, let

                     BAC = n · BAD,           ABC = n · ABE,

n for the present being any numerical quantity, positive or negative; calling
BAC = 2nα, ABC = 2nβ, we readily obtain, by comparison of the equal
dividing lines with the base of the triangle,
                         sin(2nα + 2β)   sin(2nβ + 2α)
                                       =               ,
                             sin 2nα         sin 2nβ
or
                            sin(2nα + 2β)   sin 2nα
                                          =         ;
                            sin(2nβ + 2α)   sin 2nβ
and by an obvious reduction,
                     tan(n − 1)(α − β)   tan(n + 1)(α + β)
                                       =                   .
                       tan n(α − β)        tan n(α + β)
When this equation is put under an integer form, it is of course satisfied by
making α = β; on any other supposition than α = β it evidently cannot be
satisfied by admissible values of the angles for any value of n between              p. 394
   +1 and +∞; for on that supposition, since (α − β) and (α + β) are each less
than 180
       2n , the first side of the equation will be necessarily a proper fraction and
positive; but the second side, either a positive improper fraction if (n + 1)(α + β)
be less, and a negative proper or a negative improper fraction if (n + 1)(α + β)
be greater than a right angle.
   If n be negative, let it equal −ν, then
                     tan(ν + 1)(α − β)   tan(ν − 1)(α + β)
                                       =                   ;
                       tan ν(α − β)        tan ν(α + β)

                                        399
and for the same reason as before, if ν lies between ∞ and 1, this equation
cannot be satisfied. Hence the theorem is proved to be true for all values of n,
except between +1 and −1. For these values it ceases to be true; in fact, for
such values for any given values of (α − β) there will be always, as it may be
easily proved, one or more values of (α + β); thus if n = 12 , the equation becomes
                                                   
                                              α+β
                                      tan 3     2
                                                        = −1;
                                         tan α+β
                                               2

and if n = − 12 ,                                  
                                             α−β
                                      tan 3   2
                                            α−β
                                                        = −1,
                                         tan 2
showing that α + β = 90 and α − β = ±90 in these respective cases will
afford a solution over and above the solution α = β, which is easily verified
geometrically.221 It would be an interesting inquiry (for those who have leisure
for such investigations) to determine for any given value of n between +1 and
−1 the superior and inferior limits to the number of admissible values of α + β
corresponding to any given value of α − β.222
   My reader will now be prepared to see why it is that all the geometrical
demonstrations given of this theorem, even in the simplest case of all, namely
when n = 2, are indirect, I believe I may venture to say necessarily indirect. It
is because the truth of the theorem depends on the necessary non-existence of
real roots (between prescribed limits) of the analytical equation expressing the
conditions of the question; and I believe that it may be safely taken as an axiom
in geometrical method, that whenever this is the case no other                     p. 395
   form of proof than that of the reductio ad absurdum is possible in the nature
of things. If this principle is erroneous, it must admit of an easy refutation in
particular instances.
   As an example, I throw out (not a challenge, but) an invitation to discover
a direct proof, if such exist, of the following geometrical theorem, as simple a
one as it is perhaps possible to imagine:—“To prove that if from the middle of
a circular arc two chords be drawn, and the remoter segments of these chords
cut off by the line joining the end of the arc be equal, the nearer segments will
also be equal.” The analytical proof depends upon the fact of the equation
x2 + ax = b2 (where a is the given length of each segment, and b the length of the
chord of half the given arc) having only one admissible root; and if the principle
assumed or presumed to be true be valid, no other form of pure geometrical
 221
     In the first of these cases, if the base of the triangle is supposed given, the locus of the
vertex is a right line and a circle; in the second case, a right line and an equilateral hyperbola.
 222
     When ±n lies between 2ι−1  1
                                    and 2ι+1
                                           1
                                               (ι being any positive integer), it is easily seen that
the superior limit must be at least as great as ι.



                                                  400
demonstration than the reductio ad absurdum should be applicable in this case.
For the converse case, where the nearer segments are given equal, the reducing
equation is a(a + x) = b2 , indicating nothing to the contrary of the possibility of
there being a direct solution, which accordingly is easily shown to exist. The
indirect form of demonstration, it may be mentioned, is sometimes liable to
be introduced in a manner to escape notice. As, for instance, if it should be
taken for granted in the course of an argument, that one triangle upon the
same base and the same side of it as another triangle, and having the same
vertical angle, must have its vertex lying on the same arc; this would seem to be
immediately true by virtue of the well-known theorem, that angles in the same
circular segment are equal, but in reality can only be inferred from it indirectly
by showing the impossibility of its lying outside or inside the arc in question. To
go one step further, I believe it to be the case, that granted to be true all those
fundamental propositions in geometry which are presupposed in the principles
upon which the language of analytical geometry is constructed, then that the
reductio ad absurdum not only is of necessity to be employed, but moreover in
propositions of an affirmative character never need be employed, except when
as above explained the analytical demonstration is founded on the impossibility
or inadmissibility of certain roots due to the degree of the equation implied in
the conditions of the question. If this surmise turn out to be correct, we are
furnished with a universal criterion for determining when the use of the indirect
method of geometrical proof should be considered valid and admissible and when
not.223




  223
      If report may be believed, intellects capable of extending the bounds of the planetary
system and lighting up new regions of the universe with the torch of analysis, have been baffled
by the difficulties of the elementary problem stated at the outset of this paper, in consequence,
it is to be presumed, of seeking a form of geometrical demonstration of which the question
from its nature does not admit. If this be so, no better evidence could be desired to evince the
importance of such a criterion as that suggested in the text.


                                              401
                                              50.
 On the Expressions for the Quotients which Appear in the
Application of Sturm’s Method to the Discovery of the Real
                   Roots of an Equation
            [Hull British Association Report (1853), Part II., pp. 1–3]
                                                                                             p. 396
    Many years ago I published expressions for the residues which appear in
the application of the process of common measure to f x and f ′ x, and which
constitute Sturm’s auxiliary functions. These expressions are complete functions
of the factors of f x and of differences of the roots of f x, and are therefore in effect
functions of the factors exclusively, since the difference between any two roots
may be expressed as the difference between two corresponding factors. Having
found that in the practical applications of Sturm’s theorem the quotients may be
employed with advantage to replace the use of the residues, I have been led to
consider their constitution; and having succeeded in expressing these quotients
(which are of course linear functions of x) under a similar form to that of the
residues, that is, as complete functions of the factors and differences of the roots
of f x, I have pleasure in submitting the result to the notice of the Mathematical
Section of the British Association.
    Let h1 , h2 , h3 . . . hn be the n roots of f x.
    Let ζ(a, b, c . . . l) in general denote the squared product of the differences of
a, b, c . . . l.
    Let Zi denote in general Σζ(hθ1 , hθ2 . . . hθi ), where θ1 , θ2 . . . θi indicate any
combination of i out of the n quantities a, b, c, . . . , l, with the convention that
Z0 = 1, Z1 = n; and let (i) denote 12 {1 + (−1)i }, being zero when i is odd, and
unity when i is even; then I find that the ith quotient Qi may be written under
the form
                  Qi = i P12 (x − h1 ) + i P22 (x − h2 ) + · · · + i Pn2 (x − hn ),
where in general
                     2    2            2
                                     Z(i)
               Zi−1 Zi−3 Zi−5
      i Pe =         2    2   · · ·  2
                Zi Zi−2  Zi−4       Z(i)+1
               × Σ{ζ(he , hθ2 . . . hθi−1 ) × (he − hθ1 )(he − hθ2 ) . . . (he − hθi−1 )}.
                                                                                             p. 397
                       ′
  If we suppose ff xx , by means of the common measure process, to be expanded
under the form of an improper continued fraction, the successive quotients will




                                              402
be the values of Q1 , Q2 . . . Qn above found, that is
                            f ′x                     1
                                 =                                         ;
                            fx                            1
                                     Q1 −
                                                              1
                                             Q2 −
                                                                       1
                                                     Q3 − · · · −
                                                                      Qn
the successive convergents of this fraction will be
              1           Q2                Q2 Q3 − 1                              f ′x
                 ,               ,                         ,               ...,         .
              Q1       Q1 Q2 − 1        Q1 Q2 Q3 − Q1 − Q3                         fx
The numerators and denominators of these convergents will consequently also
be functions of the factors exclusively. They are the quantities the sum of the
products of which multiplied respectively by f x and f ′ x produce (to constant
factors près) the residues. The denominators are expressible very simply in
terms of the factors and the differences of the roots; and their values under such
forms were published by me about the same time as the values of the residues
in the Philosophical Magazine; the expression for the numerators is much more
complicated, but is given in my paper, “The Syzygetic Relations,” &c., in the
Philosophical Transactions. [p. 429 below.]
   By comparing the expression for any quotient with the expressions for the
two residues from which it may be derived, we obtain the following remarkable
identity: Zi−1 × Zi , that is
       Σζ(h1 h2 . . . hi−1 ) × Σζ(h1 h2 . . . hi ) = i P12 + i P22 + i P32 + · · · + i Pn2 .
When the roots are all real, we have thus the product of one sum of squares by
the product of another sum of squares (the number in each sum depending upon
the arbitrary quantity i), brought under the form of a sum of a constant number
n of squares, which in itself is an interesting theorem.
   The expression above given for Qi leads to a remarkable relation between the
                                 ′
quotients and convergents to ff xx .
   Let it be supposed, as before, that
                        f ′x                         1
                             =                                                 ,
                        fx                                1
                                 Q1 x −
                                                                  1
                                          Q2 x −
                                                                       1
                                                     Q3 x − · · · −
                                                                      Qn x
and let the successive convergents to this continued fraction be
                      N1 (x)      N2 (x)       N3 (x)                 Nn (x)
                             ,           ,            ,       ...,           ,
                      D1 (x)      D2 (x)       D3 (x)                 Dn (x)

                                               403
where the numerators and denominators are not supposed to undergo any
reductions, but are retained in their crude forms as deduced from the law

                  Ni = Qi Ni−1 − Ni−2 ,              Di = Qi Di−1 − Di−2 .
                                                                                                     p. 398
  N1 (x) being 1, and D1 (x) being Q1 (x); then it may be deduced from the
published results above adverted to that
              2 Z2 · · · Z2
             Zi−1 i−3     (i)
  Di (x) =                         {ζ(hθ1 , hθ2 . . . hθi )(x − hθ1 )(x − hθ2 ) . . . (x − hθi )}.
             Zi2 Zi−2
                  2 · · · Z2
                           (i)+1

Hence
           Σ{ζ(he , hθ2 . . . hθi−1 ) × (he − hθ1 )(he − hθ2 ) . . . (he − hθi−1 )}
                      2 Z2 · · · Z2
                     Zi−1 i−3     (i−1)+1
                 =     2 Z2 · · · Z2             Di−1 (he );
                      Zi−2 i−4     (i−1)

and we have therefore
                                3    4    4            4
                                                     Z(i)
                               Zi−1 Zi−3 Zi−5
                      i Pe =         4    4   · · ·  4     Di−1 (he ),
                                Zi Zi−4  Zi−6       Z(i)+1

and consequently
                          6   8            8
                                         Z(i)
                        Zi−1 Zi−3
                 Qi =             · · ·        Σ{(Di−1 (he ))2 (x − he )},
                         Zi2 Zi−4
                              8          8
                                        Z(i)+1

which is the general equation connecting the form of each quotient with that
of the denominator to the immediately preceding unreduced convergent in the
                 ′
expansion of ff xx under the form of an improper continued fraction.
    If instead of the denominator of the unreduced convergents, the denominators
of the convergents reduced to their simplest forms be employed, the powers of
Z in the constant factor will undergo a diminution. The essential part of this
theorem admits of being stated in general terms as follows:—
    “If the quotient of an algebraical function of x by its first differential coefficient
be expressed under the form of a continued fraction whose successive partial
quotients are linear functions of x, any one of these quotients may be found (to
a constant factor près) by taking the sum of the products formed by multiplying
each factor (x − h) of the given function by the square of what the denominator
of the immediately antecedent convergent fraction becomes after substituting in
it for x the root corresponding to such factor.”
    P.S. Since the above was read before the British Association, the theory has
been extended by the author to comprise the general case of the expansion of any
two algebraical functions under the form of a continued fraction, and has been
incorporated into the paper in the Philosophical Transactions above referred to.

                                               404
                                                               51.
On a Theorem concerning the Combination of Determinants
    [Cambridge and Dublin Mathematical Journal, VIII. (1853), pp. 60–62]
                                                                                                                          p. 399
   Let 1 A represent the line of terms 1 a1 , 1 a2 , . . . 1 am ,
                                     1
                                         B     ”       ”       1
                                                                   b1 , 1 b2 , . . . 1 bm .

Let 1 A × 1 B represent Σ(1 ar × 1 br ), where of course there are m terms within
the symbol of summation.
   Again, let 2 A represent the line 2 a1 , 2 a2 , . . . 2 am ,
                                     2
                                         B     ”       ”       2
                                                                   b1 , 2 b2 , . . . 2 bm ,

and let
               1A           1B                                            1a       1a               1b      1b
                                                                               r        s             r       s
               2A     ×     2B           represent                  Σ     2a       2a         ×     2b      2b        ,
                                                                               r        s             r       s

             1a      1a
               r       s
             2a      2a           denoting the determinant (1 ar 2 as − 1 as 2 ar ),
               r       s

                            1b     1b
                              r      s
                            2b     2b              ”       ”        ” (1 br 2 bs − 1 bs 2 br ),
                              r      s

there being of course 12 m(m − 1) terms comprised within the sign of summation;
and so, in general, let
                             1A                1B
                             2A                2B
                             3A                3B
                                     ×                     ,       n being less than m,
                              ..               ..
                               .                .
                             nA               nB

                                                                                                                          p. 400
   (and where in general r A denotes r a1 , r a2 , . . . r am and r B denotes r b1 , r b2 , . . . r bm )
represent
                    1a       1a          ···       1a                      1b        1b           ···     1b
                      h1       h2                    hn                      h1        h2                   hn
                    2a       2a          ···       2a                      2b        2b           ···     2b
                      h1       h2                    hn                      h1        h2                   hn
              Σ                                                     ×                                             .
                     ···      ···               ···                        ···        ···                ···
                    na       na                n
                                         · · · ahn                        nb         nb                 n
                                                                                                  · · · bhn
                       h1       h2                                          h1         h2

Now let r be any integer less than m, and let
                                             m(m − 1) . . . (m − r + 1)
                                   µ=                                   ,
                                                   1 · 2···r

                                                               405
and, supposing θ1 , θ2 , . . . θr to be r numbers of the set 1, 2, . . . m, let G1 , G2 , . . . Gµ
denote the µ rectangular matrices of the forms
                                     θ1 A
                                     θ2 A
                                                respectively,
                                      ···
                                     θr A


and let H1 , H2 , . . . Hµ denote the µ rectangular matrices of the forms
                                     θ1 B
                                     θ2 B
                                                respectively.
                                      ···
                                     θr B


Now form the determinant
                          G1 × H1 G1 × H2 · · · G1 × Hµ
                          G2 × H1 G2 × H2 · · · G2 × Hµ
                                                        ,
                            ···     ···            ···
                          Gµ × H1 Gµ × H2 · · · Gµ × Hµ

then, if we give r the successive values 1, 2, 3 . . . m (in which last case the
determinant in question reduces to a single term), the values of the determinant
above written will be severally in the proportions of
                                            1
                             K, K m , K 2 m(m−1) , . . . K m , K;

that is to say, the logarithms of these several determinants will be as the
coefficients of the binomial expansion (1 + x)m .
   When we make r = m, and equate the determinant corresponding to this
value of r with that formed by making r = 1, the theorem becomes identical
with a theorem previously given by M. Cauchy, for the Product of Rectangular
Matrices.                                                                            p. 401
   It would be tedious to set forth the demonstration of the general theorem in
detail. Suffice it here to say that it is a direct corollary from the formula marked
(4) in my paper in the Philosophical Magazine for April 1851, entitled “On the
Relations between the Minor Determinants of Linearly Equivalent Quadratic
Functions224 ,” when that formula is particularized by making
                              (                                  )
                                  am+1 am+2 . . . am+n
                                  bm+1 bm+2 . . . bm+n
 224
       p. 249 above.




                                                406
represent a determinant all whose terms are zeros except those which lie in one
of the diagonals, these latter being all unities, which comes, in fact, to defining
that
                           am+e               am+e
                                  = 1,                = 0.
                           bm+e                bm+e
The important theorem here referred to is made almost unintelligible by an
unfortunate misprint of θ θm , 1 θm , 2 θm , µ θm , in place of θ θr , 1 θr , 2 θr , µ θr . I may
here take notice of another and still more inexplicable blunder in the same paper,
formula (3)225 , in the latter part of the equation belonging to which
                  (                                                           )
                      aθ1   aθ2    . . . aθm   aθm+1   aθm+2    . . . aθm+s
                      aϕ1   a ϕ2   . . . aϕm   aϕm+1   aϕm+2    . . . aϕm+s

is written in lieu of
   (                                                                                     )
         a1 a2 . . . am aθm+1           aθm+2    . . . aθm+s   an+1 an+2 . . . an+m
                                                                                             .
         a1 a2 . . . am aϕm+1           aϕm+2    . . . aϕm+s   an+1 an+2 . . . an+m




 225
       See pp. 246, 251 above.


                                                407
                                             52.
                      Note on the Calculus of Forms
    [Cambridge and Dublin Mathematical Journal, VIII. (1853), pp. 62–64]
                                                                                                   p. 402
   Accidental causes have prevented me from composing the additional sections
on the Calculus of Forms, which I had destined for the present Number of this
Journal. In the meanwhile the subject has not remained stationary. Among the
principal recent advances may be mentioned the following.
   1. The discovery of Combinants; that is to say, of concomitants to systems of
functions remaining invariable, not only when combinations of the variables are
substituted for the variables, but also when combinations of the functions are
substituted for the functions; and as a remarkable first-fruit of this new theory
of double invariability, the representation of the Resultant of any three quadratic
functions under the form of the square of a certain combinantive sextic invariant
added to another combinant which is itself a biquadratic function of 10 cubic
invariants. When the three quadratic functions are derived from the same cubic
function, this expression merges in M. Aronhold’s for the discriminant of the
cubic. The theory of combinants naturally leads to the theory of invariability for
non-linear substitutions, and I have already made a successful advance in this
new direction.
   2. The unexpected and surprising discovery of a quadratic covariant to any
homogeneous function in x, y, of the nth degree, containing (n − 1) variables
cogredient with
                               xn−2 , xn−3 y . . . y n−2 ,
and possessing the property of indicating the number of real and imaginary roots
in the given function. This covariant, on substituting for the (n − 1) variables
the combinations of the powers of x, y with which they are cogredient, becomes
the Hessian of the given function226 .                                           p. 403
   3. The demonstration due to M. Hermite of a law of reciprocity connecting
the degree or degrees of any function or system of functions with the order or
orders of the invariants belonging to the system. The theorem itself was first
propounded by me about a twelvemonth back, and communicated to Messrs
Cayley, Polignac, and Hermite, as serving to connect together certain phenomena
 226
     This covariant furnishes, if we please, functions symmetrical in respect to the two ends of
an equation for determining the number of its real and imaginary roots. The ordinary Sturmian
functions, it is well known, have not this symmetry. As another example of the successful
application of the new methods to subjects which have been long before the mathematical world
and supposed to be exhausted, I may notice that I obtain without an effort, by their aid, a much
more simple, practical, and complete solution of the question of the simultaneous transformation
of two quadratic functions, or the orthogonal transformation of one such function, than any
previously given, even by the great masters Cauchy and Jacobi, who have treated this question.


                                             408
which had presented themselves to me in the theory: unfortunately it appeared to
contradict another law too hastily assumed by myself and others as probably true,
and I consequently laid aside the consideration of this great law of reciprocality.
To M. Hermite, therefore, belongs the honour of reviving and establishing, to
myself whatever lower degree of credit may attach to suggesting and originating,
this theorem of numerical reciprocity, destined probably to become the corner-
stone of the first part of our new calculus; that part, I mean, which relates to
the generation and affinities of forms227 .
   4. I may notice that the Calculus of Forms may now with correctness be
termed the Calculus of Invariants, by virtue of the important observation that
every concomitant of a given form or system of forms may be regarded as an
invariant of the given system and of an absolute form or system of absolute forms
combined with the given form or system. As regards that particular branch of the
theory of invariants which relates to resultants, or, in other words, to the doctrine
of elimination, I may here state the theorem alluded to in a preceding Number
of the Journal, to wit that if R be the resultant of a system of n homogeneous
functions of n variables, written out in their complete and most general form (so
that by definition R = 0 is the condition that the equations got by making the n
given functions zero, shall be simultaneously satisfiable by one system of ratios),
then the condition that these equations may be satisfied by i distinct systems of
ratios between the n variables is δ i R = 0, the variation δ being taken in respect
to every constant entering into each of the n equations.




 227
     This theorem of numerical reciprocity promises to play as great a part in the Theory of Forms
as Legendre’s celebrated theorem of reciprocity in that of Numbers. Another demonstration of
it, which leaves nothing to be desired for beauty and simplicity, has been since discovered by
Mr Cayley, which ultimately rests upon that simple law (essentially although not on the face
of it a law of reciprocity) given by Euler, which affirms that the number of modes in which a
number admits of being partitioned is the same whether the condition imposed upon the mode
of partitionment be that no part shall exceed a given number, or that the number of parts
constituting any one partition shall not exceed the same number.


                                              409
                                       53.
On the Relation between the Volume of a Tetrahedron and
  the Product of the Sixteen Algebraical Values of its
                       Superficies
  [Cambridge and Dublin Mathematical Journal, VIII. (1853), pp. 171–178]
                                                                                      p. 404
   The area of a triangle is related (as is well known) in a very simple manner to
the eight algebraical values of its perimeter: If we call the values of the squared
sides of the triangle a, b, c, there will be nothing to distinguish the algebraical
affections of sign of the simple lengths so as to entitle one to a preference over
the other. The area of the triangle can only vanish by reason of the three
vertices coming into a straight line; hence, according
                                                    √    to the general doctrine of
                                              √          √
characteristics, we must have the Norm of a + b + c, containing as a factor
some root or power of the expressions for the area of the triangle. The Norm
                                                                        1    1    1
in question being representable as −N 2 where N is the Norm of a 2 ± b 2 ± c 2 ,
which is of four dimensions in the elements a, b, c, and undecomposable into
rational factors, we infer that to a numerical factor près the square of the
area must be identical with the Norm N , and thus, by a logical coup-de-main,
completely supersede all occasion for the ordinary geometrical demonstration
given of this proposition, which in its turn, with certain superadded definitions,
would admit of being adopted as the basis of an absolutely pure system of
Analytical Trigonometry that should borrow nothing from the methods and
results of sensuous or practical geometry. But into this speculation it is not my
present purpose to enter: what I propose to do is to extend a similar mode of
reasoning to space of three dimensions, and to point out a general theorem in
determinants which is involved as a consequence in the generalization of the
result of the inquiry when pushed forward into the regions of what may be
termed Absolute or Universal Rational Space.
   Let F, G, H, K be the four squared areas of the faces of a tetrahedron, and
V the volume; then, since V only becomes zero in the case of the four vertices
coming into the same plane, which is characterised by the equation
                             √     √       √     √
                               F + G+ H + K =0
                                                                                      p. 405
  subsisting, we infer that N the Norm of
                            √     √    √  √
                              F ± G± H ± K

must contain a power of V as a rational factor. V 2 is rational and of three
dimensions in the squared edges; the Norm above spoken of is of eight dimensions
in the same. Consequently there is a rational factor, say Q, remaining, which
is of five dimensions in the squared edges, and this factor I now proceed to

                                       410
determine, the other factor V 2 being, as is well known, a numerical product of
the determinant
                             0 ab2 ac2 ad2 1
                            ba2 0 bc2 bd2 1
                            ca2 cb2 0 cd2 1 ,
                            da2 db2 dc2 0 1
                             1    1     1     1 0
a, b, c, d being the four angular points of the tetrahedron. See London and
Edinburgh Philosophical Magazine, 1852. [p. 386 above.]
    The quantity Q possesses an interest of a geometrical character; for if we
call the radii of the eight spheres which can be inscribed in a tetrahedron
r1 , r2 , r3 , r4 , r5 , r6 , r7 , r8 , we evidently have r1 r2 r3 r4 r5 r6 r7 r8 × N = (3V )8 . Hence
(R), the product of the eight radii in question,

                                             38 V 8   38 V 6
                                      R=            =        .
                                              N        Q
Consequently Q is the quantity which characterises the fact of one or more of the
radii of the inscribed spheres becoming infinite. For the triangle there exists no
corresponding property; this we know à priori, and can explain also analytically
from the fact that if we call P the product of the radii of the four inscribable
circles, v the Norm of the perimeter, and A the area, we have

                                           P v = 24 A4 ,

and
                                      2 4 A4
                                        v=   = A2 ,
                                        P
which contains no denominator capable of becoming zero, so that as long as the
sides remain finite the curvature of the inscribed circles is incapable of vanishing.
   To determine N as a function of the edges, and then to discover by actual
division the value of VN2 , would be the direct but an excessively tedious and
almost impracticably difficult process. I have ever felt a preference for the à
priori method of discovering forms whose properties are known, and never yet
have met with an instance where analysis has denied to gentle                         p. 406
   solicitation conclusions which she would be loth to grant to the application
of force. The case before us offers no exception to the truth of this remark. Q
is a function of five dimensions in terms of the squared edges: let us begin by
finding the value of that part of Q in which at most a certain set of four of these
edges make their appearance, and to find which consequently the other two edges
may be supposed zero without affecting the result. We may make two distinct
hypotheses concerning these two edges; we may suppose that they are opposite,
that is non-intersecting edges, or that they are contiguous, that is intersecting
edges.

                                                411
   To meet the first hypothesis suppose ab = 0, cd = 0.
   For convenience sake, use F, G, H, K to denote 16 times the square of each
area, instead of the simple square of the areas. Call

          16(abc)2 = K,      16(abd)2 = H,    16(acd)2 = G,        16(bcd)2 = F.

Then
       −K = (ab)4 + (ac)4 + (bc)4 − 2(ab)2 (ac)2 − 2(ab)2 (bc)2 − 2(ac)2 (bc)2
             = ac4 + bc4 − 2(ac)2 (bc)2 .

Similarly,
                             −H = ad4 + bd4 − 2(ad)2 (bd)2 ,
                               −G = ca4 + da4 − 2ca2 da2 ,
                              −F = cb4 + db4 − 2cb2 db2 .
                       √      √   √      √
Hence one value of         F + G + H + K will be
       √
         −1{(ac2 − bc2 ) + (bd2 − ad2 ) + (da2 − ac2 ) + (bc2 − bd2 )} = 0.

Hence, on this first supposition, the Norm vanishes. But V 2 does not vanish
when ab = 0, cd = 0, for it becomes, saving a numerical factor,

                                  0   0 ac2 ad2        1
                                  0   0 bc2 bd2        1
                                 ca cb2 0
                                    2        0         1 ,
                                    2
                                 da db  2 0  0         1
                                  1   1   1  1         0

that is
                    (ac2 bd2 − ad2 bc2 )(cb2 + ad2 − ca2 − bd2 )
                       + (bc2 − ac2 )(ca2 db2 − cb2 da2 )
                       + (ad2 − bd2 )(ca2 db2 − cb2 da2 )
                     = 2(ac2 bd2 − ad2 bc2 )(ad2 + bc2 − ac2 − bd2 );
and consequently, since N vanishes but V 2 does not vanish, Q vanishes, showing
that there is no term in Q but what contains one at least of any                 p. 407
   two opposite edges as a factor; or, in other words, there is no term in Q of
which the product of the square of the product of all three sides of some one or
other of the four faces does not form a constituent part.
   Next, let us suppose ab = 0, ac = 0, then

                                  K = 16abc2 = −bc4 ,

                              H = 16abd2 = −(ad2 − bd2 )2 ,

                                            412
                             G = 16acd2 = −(ad2 − cd2 )2 ,
          F = 16bcd2 = −bc4 − bd4 − cd4 + 2bc2 bd2 + 2bc2 cd2 + 2bd2 cd2 .
Four of the factors of N will be therefore

                 {ι(bc2 + cd2 − bd2 ) ± F },    {ι(bc2 − cd2 + bd2 ) ± F },
             √
ι denoting       −1, and the product of these four factors will be

             {(bc2 + cd2 − bd2 )2 + F 2 } × {(bc2 − cd2 + bd2 )2 + F 2 },

which is equal to
                                     16bc4 · bd2 · cd2 ;
and similarly, the remaining part of the Norm will be

     {(2ad2 − bd2 − cd2 + bc2 )2 + F 2 } × {(2ad2 − bd2 − cd2 − bc2 )2 + F 2 },

that is

          {4ad4 − 4ad2 (bd2 + cd2 + bc2 ) + 4bc2 bd2 + 4bd2 cd2 + 4cd2 bc2 }

                     ×{4ad4 − 4ad2 (bd2 + cd2 − bc2 ) + 4bd2 cd2 }.
Again, since ac2 = 0 and bc2 = 0, V 2 becomes

                                 0   0   0 ad2             1
                                 0   0 bc2 bd2             1
                                 0 cb2 0 cd2               1 ,
                                da2 db2 dc2 0              1
                                 1   1   1  1              0

which is evidently equal to

                           0   0 ad2 1
                                             0 ad2 1
                           0 cb2 cd2 1
                   2bc2                − bc da2 0 1 ,
                                           4
                          da2 db2 0 1
                                             1  0
                           1   1  1

          = 2bc2 {2bc2 ad2 + ad4 − ad2 bd2 − cd2 ad2 + bd2 cd2 } − 2bc4 ad2
                     = 2bc2 {ad4 − ad2 (bd2 + cd2 − bc2 ) + bd2 cd2 }.
                                                                                      p. 408
  Hence, paying no attention to any mere numerical factor, we have found that
when ac = 0 and bc = 0, Q or VN2 becomes

     bc2 · bd2 · cd2 {ad4 − ad2 (bd2 + cd2 + bc2 ) + bc2 bd2 + bd2 cd2 + cd2 bc2 }.


                                            413
Hence, with the exception of the terms in which five out of the six edges enter,
the complete value of Q will be

   Σ(bc2 · bd2 · cd2 ){ad4 − ad2 (bd2 + cd2 + bc2 ) + bc2 bd2 + bd2 cd2 + cd2 bc2 },

or more fully expressed, and still abstracting from terms containing five edges,

     = Σbc2 · bd2 · cd2 {(ab4 + ac4 + ad4 )
       − (ab2 + ac2 + bc2 )(bd2 + bc2 + cd2 ) + bc2 · bd2 + bd2 cd2 + cd2 · bc2 }.

It remains only to determine the value of the numerical coefficient affecting each
of the six terms of the form

                               ab2 · ac2 · ad2 · bc2 · bd2 .

To find this, let

                      ab2 = ac2 = ad2 = bc2 = bd2 = cd2 = 1;

then evidently, since all the squared areas are equal, several of the factors of N
will become zero, but V 2 evidently does not become zero for a regular tetrahedron;
hence Q becomes zero: and if we call the numerical factor sought for λ, we must
have (observing that the Σ includes four parts corresponding to each of the four
faces)
                              4{3 − 9 + 3} + 6λ = 0,
therefore
                             −12 + 6λ = 0,        or λ = 2.
Hence the complete value of Q is

      Σab2 · bc2 · ca2 {(da4 + db4 + dc4 )
         − (da2 + db2 + dc2 )(ab2 + bc2 + ca2 ) + ab2 bc2 + bc2 ca2 + ca2 ab2 }
         + 2Σ(ab2 · bc2 · cd2 · da2 · ac2 );

or, which is the same quantity somewhat differently and more simply arranged,

 Q = Σ(ab2 · bc2 · ca2 ){da4 + db4 + dc4 + da2 · db2 + db2 · dc2 + dc2 · da2
      + (ab2 · bc2 + bc2 · ca2 + ca2 · ab2 ) − (da2 + db2 + dc2 )(ab2 + bc2 + ca2 )},

and this quantity equated to zero expresses the conditions of a radius of an  p. 409
   inscribed sphere becoming infinite. The direct method would have involved,
as the first step, the formation of the Norm of a numerator consisting of
                             √      √     √     √
                               F ± G ± H ± K,


                                           414
the value of which is

                ΣF 4 − 4ΣF 3 G + 6ΣF 2 G2 + 4ΣF 2 GH − 40F GHK,

and contains 4 + 6 + 12, that is 22 positive terms, and 12 + 1, that is 13 negative
terms, together 35 terms, each of which might be an aggregate of 64 or 1296
quantities, and thus involve in all the consideration of 45360 separate parts,
for each of the quantities F, G, H, K being a quadratic function of three of the
squared edges, will contain six terms. It is not uninteresting to notice that in
addition to the case already mentioned of two opposite edges being each zero, as
ab = 0, cd = 0, Q will also vanish for the case of ab = cd, bc = ad; that is for the
case of two intersecting edges being each equal in length to the edges respectively
opposite to them. This is evident from the fact that on the hypothesis supposed
the face acb = acd and the face bdc = bda; hence N = 0, and therefore, V not
vanishing, VN2 , that is Q, will vanish.
   We may moreover remark that since ab = 0 and cd = 0 does not make V
vanish, the perpendicular distance of ab from cd, which, multiplied by ab × cd,
gives six times the volumes, must on this supposition become infinite. When
three edges lying in the same plane all vanish simultaneously, Q vanishes, since
one edge at least in every face of the pyramid vanishes, and V also vanishes, as
is evident from the expression for V 2 , when ab = 0, ac = 0, bc = 0, becoming a
multiple of
                                0    0    0 ad2 1
                                0    0    0 bd2 1
                                0    0    0 cd2 1 ,
                              ad2 bd2 cd2 0 0
                                1    1    1     0 0
which is evidently zero.
   It appeared to me not unlikely, from the situation and look of Q (the charac-
teristic of one of the inscribed spheres becoming infinite), that it might admit
of being represented as a determinant, but I have not succeeded in throwing
it under that form. I have a strong suspicion that if we take Q′ a function
corresponding to a tetrahedron p  a′ b′ c′ d′ , in the same way as Q corresponds to
abcd, QQ′ , and not improbably (QQ′ ), will be found to be                                   p. 410
   (as we know from Staudt’s Theorem of (V · V )) a rational integral function
                                               p
                                                    2     ′2

of the squares of the distances of the points a, b, c, d from the points a′ , b′ , c′ , d′ .
   That N should divide out by V 2 is in itself an analytical theorem relating
to 6 arbitrary quantities ab2 , ac2 , ad2 , bc2 , bd2 , cd2 , which evidently admits of
extension to any triangular number 10, 15, &c. of arbitrary quantities. Thus we
may affirm, à priori, that the norm of
                          √     √        √         √        p
                            L ± M ± N ± P ± Q,


                                            415
where (for the sake of symmetry, retaining double letters, as AB, AC, &c., to
denote simple quantities)

            0 AB AC AD 1                          0 AB AC AE 1
           AB  0 BC BD 1                         AB  0 BC BE 1
    Q=     AC BC  0 CD 1 ,                 P =   AC BC  0 CE 1 ,
           AD BD CD  0 1                         AE BE CE  0 1
            1  1  1  1 0                          1  1  1  1 0

                       N = &c.,    M = &c.,      L = &c.,
will contain as a factor the determinant
                         0   AB    AC       AD   AE   1
                        AB    0    BC       BD   BE   1
                        AC   BC     0       CD   CE   1
                                                        ,
                        AD   BD    CD        0   DE   1
                        AE   BE    CE       DE    0   1
                         1    1     1        1    1   0

and a similar theorem may evidently be extended to the case of any n(n+1)
                                                                      2
arbitrary quantities whatever.




                                      416
                                             54.
       On the Calculus of Forms, otherwise the Theory of
                           Invariants
   [Cambridge and Dublin Mathematical Journal, VIII. (1853), pp. 256–269]


                            [Continued from p. 363 above.]
                           Section VII. On Combinants.
                                                                                                  p. 411
    Reasons of convenience have induced me to depart from the plan to which
I originally intended to adhere in the development of this theory, and I shall
hereafter, from time to time, continue to add sections on such parts of the subject
as may chance to be most present to my mind or most urgent upon my attention,
without waiting for the exact place which they ought to occupy in a more formal
treatise, and without having regard to the separation of the subject into the two
several divisions stated at the outset of the first section. The present section
will be devoted to a brief and partial exposition of the theory of Combinants,228
with a view to the application of this theory to the solution of the problem of
throwing the resultant of three general homogeneous quadratic functions under
its most simple form, being analogous to that given by Aronhold in the particular
case where the three functions are derived from the same cubic, and becoming
identical therewith when the coefficients are accommodated to this particular
supposition.229 I shall confine myself for the present to combinants relating to
systems of functions, all of the same degree.
    If ϕ1 , ϕ2 , . . . , ϕr be homogeneous functions of any number of variables, any
invariant or other concomitant of the system which remains unchanged, not
only for linear substitutions impressed upon the variables contained within
the functions, but also for linear combinations impressed upon the functions
themselves, is what I term a Combinant. A Combinant is thus an invariant or
other concomitant of a system in its corporate capacity (quâ system), being in
fact                                                                                 p. 412
    common to the whole family of forms designated by
                                λ1 ϕ1 + λ2 ϕ2 + · · · + λr ϕr ,
where λ1 , λ2 , . . . , λr are arbitrary constants. If the coefficients of ϕ1 , ϕ2 , . . . , ϕr
be supposed to be written out in r lines (the coefficients of corresponding terms
 228
    Discovered by the Author of this paper in the winter of 1852.
 229
    A similar method will subsequently be applied to the representation of the resultant of
two cubic equations as a function of Combinants bearing relations to the quadratic and cubic
invariants of a quartic function of x and y, precisely analogous to those which the Combinants
that enter into the solution above alluded to bear to the Aronholdian invariants of a cubic
function.


                                             417
occupying the same place in each line), so as to form a rectangular matrix, any
combinantive invariant will be a function of the determinants corresponding to
the several squares of r2 terms each that can be formed out of such matrix, or,
as they may be termed, the full determinants belonging to such rectangular
matrix. If we call any such combinant K, then, over and above the ordinary
partial differential equations which belong to it in its character of an invariant, it
will be necessary and sufficient, in order to establish its combinantive character,
that K shall be subject to satisfy (r − 1) pairs of equations of the form

                                ′ d      ′ d        ′ d
                                                                 
                               a        +b        +c        ···       K = 0,
                                   da        db        dc
                                 d     d     d
                                                                 
                               a ′ + b ′ + c ′ ···                    K = 0,
                                da    db    dc
where a, b, c . . . ; a′ , b′ , c′ . . ., are respectively lines in the matrix above referred
to.
    So any combinantive concomitant will be a function of the full determinants
of the matrix formed by the coefficients of the given system of forms and of the
variables, and will be subject to satisfy the additional differential equations just
above written.
    It will readily be understood furthermore, that an invariant or other concomi-
tant may be combinantive in respect to a certain number of forms of a system,
and not in respect of other forms therein; or more generally, may be combinantive
in respect of each, separately considered, of a series of groups into which a given
system may be considered to be subdivided, without being so in respect of the
several groups taken collectively.
    In the fourth section of my memoir [p. 429 below] on a “Theory of the
Conjugate Properties of two rational integral Algebraical Functions,” recently
presented to the Royal Society of London, the case actually arises of an invariant
of a system of three functions, which is combinantive in respect only to two of
them.
    For greater simplicity, let the attention for the present be kept fixed upon
combinants which are such in respect of a single group of functions, all of the
same degree in the variables. (It will of course have been perceived that when the
system is made up of several groups, there would be nothing gained by limiting
the groups to be all of the same degree inter se; it is sufficient that all of the
same group be of the same degree per se.)                                                     p. 413
    All such combinants will admit of an obvious and immediate classification.
Let us suppose that a combinant is proposed which is in its lowest terms, that is
to say, incapable of being expressed as a rational integral algebraical function
of combinants of an inferior order. Such a combinant may, notwithstanding
this, admit of being decomposed into non-combinantive invariants of inferior
dimensions to its own, and in such event will be termed a complex combinant;

                                                  418
or it may be indecomposable after this method, in which event it will be termed
a simple combinant. It will presently be shown, that the resultant of a system
of three quadratic functions is made up of a complex combinant of twelve
dimensions, and of the square of a simple combinant of six dimensions, expressible
as a biquadratic function of ten non-combinantive invariants, each of three
dimensions in the coefficients. There is an obvious mode of generating complex
combinants; according to which they admit of being viewed as invariants of
invariants. Supposing ϕ1 , ϕ2 , . . . , ϕr , to be the functions of the given system,
λ1 ϕ1 + λ2 ϕ2 + · · · + λr ϕr may conveniently be termed the conjunctive of the
system: if now one or more invariants or other concomitants be taken of this
conjunctive, there results a derivative function or system of functions of the
quantities λ1 , λ2 , . . . , λr , in which every term affecting any power or combination
of powers of the λ series is necessarily an invariant or concomitant of the given
system. If now an invariant or other concomitant be taken of the new system
in respect to λ1 , λ2 , . . . , λr , (the original variables (supposing them to enter)
being treated as constants), this secondarily derived invariant will be itself an
Invariant, or at all events a Concomitant in respect of the original system, and
being unaffected by linear substitutions impressed upon the λ system, is by
definition a combinant of such system. A similar method will obviously apply
if the original system be made up of various groups; each group will give rise
to a conjunctive, and one or more concomitants being taken of this system of
conjunctives and treated as in the case first supposed, (the only difference being,
that there will on the present supposition be several unrelated systems instead of
a single system of new variables, that is, several λ systems instead of one only)
the result, when all the λ systems have been invariantized out (that is, made to
disappear by any process for forming invariants), will be a combinant in respect
to each of the groups, severally considered, of the given system of functions.
    Here let it be permitted to me to make a momentary digression, in order to be
enabled to avoid for the future the inconvenience of using the phrase “invariant
or other concomitant,” and so to be enabled at one and the same time to simplify
the language and to give a more complete unity to the matter of the theory, by
showing how every concomitant may in fact be viewed as a simple invariant, so
that the calculus of forms may hereafter admit of being cited, as I propose to
cite it, under the name of the Theory of Invariants.                                           p. 414
    Thus, to begin with the case of simple contragredience and cogredience, if
ξ, η, ζ . . . are contragredient to x, y, z . . ., any form containing ξ, η, ζ . . ., which is
concomitantive to a given form or system of forms S, which contains x, y, z . . .,
may be regarded as concomitantive to the system S ′ , made up of S and the
superadded absolute form ξx + ηy + ζz + · · · , say S; where ξ, η, ζ . . . are treated
no longer as variables, but as constants. In like manner every system of variables
contragredient to x, y, z . . ., or to any other system of variables in S, will give
rise to a superadded form analogous to S, the totality of which may be termed


                                             419
S1 ; and thus the various systems ξ, η, ζ . . . will no longer exist as variables in
the derived form, but purely as constants. Again, if S contain any system of
variables ϕ, ψ, A, &c., contragredient to x, y, z, &c., the system of variables u, v, w,
&c., cogredient with x, y, z, &c., may be considered as constants belonging to
the superadded form ϕu + ψv + Aw · · · ; but if S do not contain any system
contragredient to x, y, z, &c., then u, v, w, &c. may be treated as constants
belonging to the superadded system of forms xv − yu, yw − zv, zu − xw, &c.;
and so in general any concomitant containing any sets of variables in simple
relation, whether of cogredience or contragredience, with any of the sets in the
given system S, may in all cases be treated as an invariant of the system S ′ ,
made up of S and a certain superadded system S1 , all the forms contained in
which are absolute, by which I mean, that they contain no literal coefficient. The
same conclusion may be extended to the case of concomitants containing sets
of variables in compound relation with the sets in the given system of forms S.
Thus, suppose u1 , u2 , . . . , un , to be in compound relation of cogredience with

                       xn−1 ,   xn−2 y,   xn−3 y 2 , . . . ,   y n−1 ;

u1 , u2 , . . . , un may be regarded as constants belonging to the superadded form
                                   1
     u1 y n−1 − (n − 1)u2 y n−2 x + (n − 1)(n − 2)u3 y n−3 x2 ∓ · · · ± un xn−1 ,
                                   2
say Ω. And thus universally we are enabled to affirm, that a concomitant
of whatever nature to a given system of forms, may be reduced to the form
of an invariant of a system made up of the given system and a certain other
superadded system of absolute forms: without, therefore, abandoning the use
of the terms concomitant, cogredience, contragredience, &c., which for many
purposes are highly convenient and save much circumlocution, we may regard
every concomitant as a disguised invariant, and under the name of the Theory of
Invariants comprise the totality of the theory of Concomitance. I have already
had occasion to make use of the superadded form Ω in discussing the theory
of the Bezoutiant (a quadratic form concomitant to two functions of the same
degree in x, y, which plays a most important part in the theory of the relations
of their real roots), in the memoir for the Royal Society previously adverted to.
   I now return to the question of applying the theory of combinants to the
decomposition of the resultant of three general quadratic functions of            p. 415
   x, y, z. It will of course be apparent that every resultant of any system of
n functions of the same degree of a single set of n variables is a combinantive
invariant of the system. This is an immediate and simple corollary to the theorem
given by me in this Journal, in May, 1851. Accordingly, in proceeding to analyse
the composition of the resultant of three quadratic functions, I may, besides
impressing linear combinations upon the variables, impress linear combinations
upon the functions themselves, in any way most conducive to simplicity and


                                          420
facility of expression and calculation; and whatever relations shall be proved to
exist between the resultant and other combinants for such specific representation,
must be universal, and hold good for the functions in their most general form.
   (1) The system, by means of linear substitutions impressed upon the variables
which enter into the functions, may be made to assume the form

                                    x2 + y 2 + z 2 ,

                                  ax2 + by 2 + cz 2 ,
                     lx2 + my 2 + nz 2 + 2pyz + 2qzx + 2rxy.
  (2) By means of linear combinations of the functions themselves the system
may evidently be made to take the form

                               (c − a)x2 + (c − b)y 2 ,

                               (a − b)y 2 + (a − c)z 2 ,
                            ky 2 + 2pyz + 2qzx + 2rxy;
and finally, by taking suitable multipliers of x, y, z in lieu of x, y, z, it may be
made to become
                                  ρ(x2 − y 2 ),
                                        σ(y 2 − z 2 ),
                            y 2 + 2f yz + 2gzx + 2hxy.
We have thus reduced the number of constants in the system from eighteen to
five; and as it will readily be seen that in any combinant of the system in its
reduced form ρ and σ can only enter as factors of the simple quantity (ρσ)i , for
all purposes of comparison of the combinants of the system of like dimensions
with one another, ρ and σ might admit of being treated as being each unity, and
accordingly, practically speaking, we have only to deal with three in place of
eighteen constants, a marvellous simplification, and which makes it obvious, à
priori, or at least affords a presumption almost amounting to and capable of
being reduced to certainty, that the number of fundamental combinants of the
system, of which all the rest must be explicit rational functions, will be exactly
four in number; which, for the canonical form hereinbefore written, on making ρ
and σ each unity, will correspond to

                1,   f 2 + g 2 + h2 ,     f 2 g 2 + g 2 h2 + h2 f 2 ,   f gh,
                                                                                       p. 416
  and will be of the 3rd, 6th, 12th, and 9th degrees respectively. The reason
why the squares of f, g, h, instead of the simple terms f, g, h, appear in the 2nd
and 3rd of these forms is, because, on changing x into −x, y into −y, or z into
−z, two of the quantities f, g, h will change their sign, but the forms representing

                                            421
the invariants of even degrees ought to remain absolutely unaltered for such
transformations. I shall in the course of the present section set forth the methods
for obtaining these four combinants, which, although of the regularly ascending
dimensions 3, 6, 9, 12, belong obviously to two different groups, the one of three
dimensions forming a class in itself, and the natural order of the three others
being that denoted by the sequence 6, 12, and 9, and not that which would
be denoted by the sequence 6, 9, 12, the combinant of the ninth degree being
properly to be regarded as in some sort an accidentally rational square root of a
combinant of 18 dimensions.
   Let now

    ρ(x2 − y 2 ) = U,      σ(y 2 − z 2 ) = W,      y 2 + 2f yz + 2gzx + 2hxy = V.

The resultant will be found by making

                                 x = ±y,         z = ±y,

when
                  x = +y, z = +y       :   V = (1 + 2f + 2g + 2h)y 2 ,
                  x = +y, z = −y       :   V = (1 − 2f − 2g + 2h)y 2 ,
                  x = −y, z = +y       :   V = (1 + 2f − 2g − 2h)y 2 ,
                  x = −y, z = −y       :   V = (1 − 2f + 2g − 2h)y 2 .
Hence the resultant R
= ρ4 σ 4 (1 + 2f + 2g + 2h)(1 − 2f − 2g + 2h)(1 + 2f − 2g − 2h)(1 − 2f + 2g − 2h)
= (ρσ)4 {(1 + 2h)2 − 4(f + g)2 }{(1 − 2h)2 − 4(f − g)2 }
= (ρσ)4 {(1 + 4h2 − 4f 2 − 4g 2 )2 − (4h − 8f g)2 }
= (ρσ)4 [1 − 8(f 2 + g 2 + h2 ) + 16{(f 4 + g 4 + h4 ) − 2(g 2 h2 + h2 f 2 + f 2 g 2 )} + 64f gh].

Let now
                                 K = λU + µV + νW,
K being what I term a linear conjunctive of U, V, W . The invariant of K, in
respect to x, y, z, will be the determinant

                               ρλ     hµ       gµ
                               hµ µ − ρλ + σν f µ ,
                               gµ     fµ      −σν
                                                                                             p. 417
   that is

= (2f gh − g 2 )µ3 + σ(h2 − g 2 )µ2 ν − ρ(f 2 − g 2 )µ2 λ − ρσµλν + ρ2 σλ2 ν − ρσ 2 λν 2 ;

or, multiplying by 6, we may write

          Ix,y,z K = 6dλµν + 3b3 µ2 ν + 3b1 µ2 λ + 3a3 λ2 ν + 3c1 λν 2 + b2 µ3 ,

                                           422
where
                    d = −ρσ,                          b2 = 12f gh − 6g 2 ,
                   b1 = −2ρ(f 2 − g 2 ),              b3 = 2σ(h2 − g 2 ),
                   a3 = ρ2 σ,                         c1 = −2ρσ 2 .
the notation being accommodated to that employed by Mr Salmon in The Higher
Plane Curves, λ, µ, ν in IK being correspondent to x, y, z in Mr Salmon’s form.
If now we employ Mr Salmon’s expression for the S (the biquadratic Aronholdian
of IK), observing that

                        a2 = 0,       c2 = 0,         a1 = 0,      c3 = 0,

we have the complex combinant

    Sλ,µ,ν Ix,y,z K = d4 − 2d2 (b1 c1 + a3 b3 ) + da3 b2 c1 − a3 c1 b1 b3 + b21 c21 + a23 b23
                   (                                                                    )
                       1 − 8(f 2 + h2 − 2g 2 ) + 4(12f gh − 6g 2 )2
        = ρ4 σ 4
                       −16(f 2 − g 2 )(h2 − g 2 ) + 16 (f 2 − g 2 )2 + (h2 − g 2 )2
                                                                                    

= ρ4 σ 4 [1 − 8(f 2 + g 2 + h2 ) + 16(f 4 + g 4 + h4 − h2 g 2 − g 2 f 2 − f 2 h2 ) + 48f gh].
Hence, calling the resultant R, we have
         −3R + 4Sλ,µ,ν Ix,y,z K = 1 − 8(f 2 + g 2 + h2 ) + 16(f 4 + g 4 + h4 )
                                        + 32(f 2 g 2 + g 2 h2 + h2 f 2 )
                                      = {1 − 4(f 2 + g 2 + h2 )}2 = P22 .
Let Ω be taken the polar reciprocal to the conjunctive

                                      −λU + µV + νW ;

and for greater simplicity, as we know, à priori, from the fundamental definition
of a combinant, which (save as to a factor) must remain unaltered by any linear
modification impressed upon the functions to which it appertains, that ρ and σ
can enter factorially only in any combinant, let ρ and σ be each taken equal to
unity in performing the intermediary operations.
   Then
                                −λ      hµ        gµ ξ
                                hµ λ + µ + ν f µ η
                         Ω=
                                gµ      fµ       −ν ζ
                                 ξ       η         ζ 0
                              ξ (ν + νµ + νλ + f 2 µ2 )
                             2 2                                      
                                                                      
                              +η 2 (−λν + g 2 µ2 )
                            
                                                                      
                                                                       
                            
                                                                      
                                                                       
                                                                      
                             +ζ 2 (λ2 + λµ + λν + h2 µ2 )
                                                                      
                                                                       
                        =                                                  .
                            
                             −2ηζ(f λµ + hgµ2 )               
                                                               
                              +2ξζ{g(µλ +      + (g         2} 
                                                              
                            
                            
                                          µν)      − f h)µ    
                                                               
                               −2ξη(hµν + f gµ )
                                                              
                                            2                 


                                                423
                                                                                          p. 418
   Upon Ω, which is a quadratic function in respect of each of the two unrelated
systems ξ, η, ζ; λ, µ, ν, and also in respect of the coefficients in (U, V, W ), we may
operate with the commutantive symbol
                                    d    d       d 
                                                   
                                                   
                                   dξ   dη      dζ 
                                                   
                                                   
                                    d    d       d 
                                                   
                                                   
                                                   
                                                   
                                   dξ   dη      dζ
                                                   
                                    d    d       d  ,
                                                   
                                                   
                                   dλ   dµ      dν 
                                                   
                                                   
                                    d    d       d 
                                                   
                                                   
                                                   
                                                   
                                   dλ   dµ      dν
which, for facility of reference, I shall term 8E.
  Considering the first line as stationary, we shall obtain, for the value of 8E(Ω),
216 commutantives, which may be expressed under the following forms:
                                   d     d         d
                                  dξ    dη        dζ
                                   d     d         d
                                                       ,
                                  dξ    dη        dζ
                                  d2    d2       d2 
                                  dλ2   dµ2      dν 2
                               d           d        d
                              dξ          dη        dζ
                               d           d        d
                           −                            ,
                              dξ          dη        dζ
                              d2        d d       d d
                              dλ2       dµ dν     dµ dν
                                d          d        d
                               dξ         dη        dζ
                                d          d        d
                           −                            ,
                               dξ         dη        dζ
                              d d         d2      d d
                              dλ dν       dµ2     dλ dν
                                d            d        d
                               dξ           dη       dζ
                                d            d        d
                           −                             ,
                               dξ           dη       dζ
                              d d         d d        2
                                                     d 
                              dλ dµ       dλ dµ     dν 2
                                                                                          p. 419




                                         424
                              d          d       d
                             dξ         dη       dζ
                              d          d       d
                         2                            .
                             dξ         dη       dζ
                            d d       d d      d d 
                            dλ dµ     dµ dν    dν dλ
In this expression the first lines may be considered stationary, the second lines
are subject to the usual process of commutation, which makes three of the six
permutations positive and three negative; and the third or bracketed lines are
subject to the simple process which makes all the permutations of the same sign.
In the three middle groups two of the terms in the final line are always identical;
it will therefore be more convenient to introduce the multiplier 2, and then to
consider each such line to represent the three distinct permutations, taken singly.
   Let now                   (                   )
                                  d2 d2 d2
                           1
                           8         ,     ,
                                    2 dη 2 dζ 2
                                                   Ω = (Ω),
                         (
                                dξ               )
                           d2 d d d d
                      1
                      8         ,        ,
                              2 dη dζ dη dζ
                                                   Ω = (Ω)′ ,
                         (
                           dξ                    )
                            d d d2 d d
                       1
                       8           , 2,            Ω = (Ω)′′ ,
                         (
                           dξ  dζ    dη    dξ dζ )
                            d d d d d2
                       1
                       8           ,       , 2 Ω = (Ω)′′′ ,
                           dη  dξ    dη dξ   dζ 
                         d d d d d d
                      
                                ,        ,         Ω = (Ω)1 .
                         dξ dη dη dζ dζ dξ
And let
                                d2  d2 d2 
                                   ,    ,        = L,
                               dλ2 dµ2 dν 2
                           d2 d d d d 
                               ,      ,          = L′ ,
                           dλ2 dµ dν dµ dν
                           d d d2 d d 
                                 ,    ,          = L′′ ,
                           dλ dν dµ2 dλ dν
                           d d d d d2 
                                 ,      ,        = L′′′ ,
                           dλ dµ dλ dµ dν 2
                         d d d d d d 
                               ,      ,          = L1 .
                         dλ dµ dµ dν dν dλ
Then, attending to the convention just previously explained, we shall have

E(Ω) = (L − 2L′ − 2L′′ − 2L′′′ + 2L1 ){(Ω) − 2(Ω)′ − 2(Ω)′′ − 2(Ω)′′′ + 2(Ω)1 };
                                                                                      p. 420




                                       425
  a symbolical product, any term in which such as L′ Ω′′ will mean
                                                   
                             d2 d d d d  
                            dλ2 , dµ dν , dµ dν 
                           
                                                   
                                                    
                                                   1
                             (                    )     Ω,
                              d d d2 d d            8
                            dξ dζ , dη 2 , dξ dζ
                           
                                                   
                                                    
                                                   
                                                    

and a similar interpretation must be extended to each of the 25 partial products;
we have then
                  L(Ω) = 8g 2 ,      −2L′ (Ω) = 0,          −2L′′′ (Ω) = 0,
                          −2L′′ (Ω) = −4g 2 ,         2L1 (Ω) = −2,
                                     ′
                            −2L(Ω) = 0,            −2L(Ω)′′′ = 0,
                             4L′ (Ω)′ = 0,         4L′′ (Ω)′′′ = 0,
                            4L′′ (Ω)′ = 0,         4L′′′ (Ω)′′′ = 0,
                          4L′′′ (Ω)′ = 8f 2 ,      4L′ (Ω)′′′ = 8h2 ,
         −2L(Ω)′′ = 0,       4L′ (Ω)′′ = 0,        4L′′ (Ω)′′ = 0,      4L′′′ (Ω)′′ = 0,
                           −4L1 (Ω)′ = 0,          −4L1 (Ω)′′′ = 0,
                                    −4L1 (Ω)′′ = 4g 2 ;

and, finally, the five terms comprised in

                                2L(Ω)1 , . . . ,     4L1 (Ω)1 ,

each = 0. All the above equations can be easily verified by direct inspection, it
being observed that 8(Ω) represents

         ν 2 + λν + µν + f 2 µ2 ,        −λν + g 2 µ2 ,     λ2 + µλ + νλ + h2 µ2 ;

that 8(Ω)′ represents

            ν 2 + µν + λν + f 2 µ2 ,        −f λµ − hgµ2 ,        −f λµ − hgµ2 ;

that 8(Ω)′′ represents

     −λν + g 2 µ2 ,   g(µλ + µν) + (g − f h)µ2 ,            g(µλ + µν) + (g − f h)µ2 ;

that 8(Ω)′′′ represents

            λ2 + µλ + νλ + h2 µ2 ,          −hµν − f gµ2 ,        −hµν − f gµ2 ;

and that (Ω)1 represents

          −f λµ − hgµ2 ,       g(µλ + µν) + (g − f h)µ2 ,              −hµν − f gµ2 .

                                              426
We have thus

      E(Ω) = 8g 2 − 4g 2 − 2 + 8f 2 + 8h2 + 4g 2 = 2{4f 2 + 4g 2 + 4h2 − 1}.

Hence
                                                      1
                               3R = 4Sλ,µ,ν Ix,y,z K − {EΩ}2 .                                  (A)
                                                      4
                                                                                                      p. 421
  If we restore to U, V, W their general values, and make

                  U = ax2 + by 2 + cz 2 + 2f yz + 2gzx + 2hxy,
                  V = a′ x2 + b′ y 2 + c′ z 2 + 2f ′ yz + 2g ′ zx + 2h′ xy,
                 W = a′′ x2 + b′′ y 2 + c′′ z 2 + 2f ′′ yz + 2g ′′ zx + 2h′′ xy,

and construct the cubic function
 ϑ = (ax + a′ y + a′′ z)(bx + b′ y + b′′ z)(cx + c′ y + c′′ z)
      − (ax + a′ y + a′′ z)(f x + f ′ y + f ′′ z)2 − (bx + b′ y + b′′ z)(gx + g ′ y + g ′′ z)2
      − (cx + c′ y + c′′ z)(hx + h′ y + h′′ z)2
      + 2(f x + f ′ y + f ′′ z)(gx + g ′ y + g ′′ z)(hx + h′ y + h′′ z),

that is
   Σ(abc − af 2 − bg 2 − ch2 + 2f gh)x3
    + Σ{a′ bc + ab′ c + abc′ − (a′ f 2 + 2af f ′ ) − (b′ g 2 + 2bgg ′ ) − (c′ h2 + 2chh′ )
                                         + 2f ′ gh + 2f g ′ h + 2f gh′ }x2 y
    + {a′ b′′ c + a′ bc′′ + a′′ b′ c + a′′ bc′ + ab′ c′′ + ab′′ c′
          − 2a′ f f ′′ − 2af ′ f ′′ − 2a′′ f f ′
          − 2b′ gg ′′ − 2bg ′ g ′′ − 2b′′ gg ′
          − 2c′ hh′′ − 2ch′ h′′ − 2c′′ hh′
          + 2f ′′ g ′ h + 2f ′ g ′′ h + 2f g ′ h′′ + 2f ′′ gh′ + 2f ′ gh′′ + 2f g ′′ h′ }xyz,

Sλ,µ,ν Ix,y,z K in the preceding equation becomes simply the Aronholdian S to ϑ,
which may be calculated by Mr Salmon’s formula previously quoted.
   Ω may be taken equal to the determinant

                ax + a′ y + a′′ z hx + h′ y + h′′ z gx + g ′ y + g ′′ z             ξ
                hx + h′ y + h′′ z bx + b′ y + b′′ z f x + f ′ y + f ′′ z            η
                                                                                      .
                gx + g ′ y + g ′′ z f x + f ′ y + f ′′ z cx + c′ y + c′′ z          ζ
                       ξ                    η                   ζ                   0

And the cubic commutant of this, obtained by affecting it with the commutantive



                                                   427
operator,
                                    d    d    d 
                                                  
                                                 
                                   dx   dy    dz 
                                                 
                                                 
                                    d    d    d 
                                                 
                                                 
                                                 
                                                 
                                   dx   dy    dz
                                                 
                                    d    d    d 
                                                 
                                   dξ   dη    dζ 
                                                 
                                                 
                                    d    d    d 
                                                 
                                                 
                                                 
                                   dξ   dη    dζ
                                                                                         p. 422
    will give 48E(Ω) if each of the four lines of the operator undergoes permutation,
or 8E(Ω), if one of the four lines is kept stationary. Thus it falls within the limits
of practical possibility to calculate explicitly, by the formula (A), the value of
the resultant. I give to the S of ϑ the appellation of the Hebrew letter shin, and
to the commutant of Ω the appellation of the Hebrew letter teth. These letters
are chosen with design; for I shall presently show that when the three given
quadratic functions are the differential derivatives of the same cubic function ψ,
the teth becomes the Aronholdian T to the cubic function, or, as we may write
it, T ψ, and the shin becomes the Aronholdian S of the Hessian thereto, that is
SHψ.
    Thus for the first time the true inward constitution of the resultant of three
quadratics is brought to light. The methods anteriorly given by me, and the one
subsequently added by M. Hesse for finding this resultant, adverted to in Section
II., lead, it is true, to the construction of the form, but throw no light upon the
essential mode of its composition.




                                        428
                                              55.
 Théorème sur les limites des racines réelles des équations
                        algébriques
           [Nouvelles Annales de Mathématiques, XII. (1853), pp. 286–287]
                                                                                        p. 423
   Soit
                                           f (x) = 0
une équation algébrique de degré n, et supposons qu’en opérant sur f (x) et f ′ (x)
comme dans le théorème de M. Sturm, on obtienne les n quotients

                    a1 x + b1 ,     a2 x + b2 ,     a3 x + b3 . . . an x + bn ;

il faut remarquer seulement qu’on obtient le nième quotient, an x + bn , en divisant
l’avant-dernier résidu par le dernier résidu.
    Formons la série de 2n quantités
                       ±2 − b1      ±2 − b2       ±2 − b3     ±2 − bn
                               ,            ,             ...         ;
                         a1           a2            a3          an
il n’y a aucune racine de l’équation

                                           f (x) = 0

entre la plus grande de ces quantités et +∞, ni entre la plus petite de ces
quantités et −∞.230




 230
       Prochainement, une démonstration de ce théorème généralisé. [p. 424 below.]


                                              429
                                                  56.
Nouvelle méthode pour trouver une limite supérieure et une
    limite inférieure des racines réelles d’une équation
                   algébrique quelconque
        [Nouvelles Annales de Mathématiques, XII. (1853), pp. 329–336]
                                                                                                                      p. 424
Art. (1).    Lemme. Soient

                                   C1 , C2 , C3 . . . Cr−1 , Cr

une suite de quantités positives, assujetties à cette loi
                                1                           1                   1
   C1 = µ1 ,    C2 = µ2 +          ,     C3 = µ3 +             . . . Ci = µi +      . . . Cr = µr ,
                                µ1                          µ2                 µi−1
où les µ sont des quantités positives quelconques.
  Si, dans la fraction continue

                                                   1
                                                                               ,
                                                        1
                               q1 +
                                                             1
                                       q2 +
                                                                    1
                                              q3 + · · · +
                                                                          1
                                                                 qr−1 +
                                                                          qr
les quantités q1 , q2 . . . étant des quantités positives ou négatives, on a les inégalités

         [q1 ] > C1 ,     [q2 ] > C2 ,    [q3 ] > C3 . . . [qr−1 ] > Cr−1 ,                  [qr ] > Cr

(les crochets indiquent la racine carrée positive du carré de la quantité que ces
crochets renferment), le dénominateur de la fraction continue aura même signe
que le produit q1 q2 q3 . . . qr−1 qr .
   Démonstration. Posons
                      1                                           1                                               1
q1 = m 1 ,     q2 +      = m2 ,           ··· ,         qi +          = mi ,                 ··· ,        qr +          = mr ;
                      m1                                         mi−1                                            mr−1
                                                                                                                      p. 425
   il est aisé de vérifier que les dénominateurs successifs de la fraction continue
sont
               m1 , m1 m2 , m1 m2 m3 , . . . , m1 m2 m3 . . . mr−1 mr .
m1 a même signe que q1 :
      1    1             1      1        1    1                         1                     1
         =    ,               <    ,        <    ,           [q2 ] >       ,       [q2 ] >       ,   etc.;
      q1   m1           [q1 ]   µ1       m1   µ1                        µ1                    m1

                                                  430
donc q2 a même signe que m2 , et aussi m1 m2 est de même signe que q1 q2 :
                             1                      1    1                   1
               m2 > µ2 +        ,     m2 > µ2 ,        <    ,      [q3 ] >      ;
                             µ1                     m2   µ2                  µ2
donc q3 a même signe que m3 ; ainsi m1 m2 m3 est de même signe que q1 q2 q3 ,
et, en continuant, on parvient à démontrer que m1 m2 m3 . . . mr−1 mr , c’est-à-
dire le dénominateur de la fraction continue, est de même signe que le produit
q1 q2 q3 . . . qr−1 qr .
Art. (2).     Théorème. Si f (x) est une fonction algébrique entière de degré
n, et si l’on prend arbitrairement une autre ϕ(x) algébrique et entière, et d’un
degré moindre que n, et qu’on développe la fraction ϕ(x)
                                                     f (x) en fraction continue

                        ϕ(x)                        1
                              =                                         ,
                        f (x)                            1
                                    X1 +
                                                               1
                                           X2 + · · · +
                                                                   1
                                                          Xr−1 +
                                                                   Xr
où X1 , X2 . . . Xr sont des fonctions rationnelles de x, et si l’on forme l’équation

         (θ)       (X12 − C12 )(X22 − C22 ) · · · (Xr−1
                                                    2      2
                                                        − Cr−1 )(Xr2 − Cr2 ) = 0,

la racine réelle supérieure de cette équation sera plus grande, et la racine
réelle inférieure de cette équation sera moindre qu’aucune des racines réelles de
l’équation
                                     f (x) = 0;
et si toutes les racines de l’équation (θ) sont imaginaires, l’équation

                                           f (x) = 0

aura aussi toutes ses racines imaginaires.
   Démonstration. Tous les quotients de la fraction continue qui suivent le
premier quotient, savoir: X2 , X3 . . . Xr , sont en général des fonctions linéaires
de x, et X1 sera aussi linéaire, si ϕ(x) est de degré n − 1; les cas particuliers ne
changent pas la marche de la démonstration; mais il faut remarquer que lorsque
f (x) et ϕ(x) ont des racines communes, le dernier quotient aura la forme [X]     0 ,
[X] étant l’avant-dernier terme, et alors, dans l’équation (θ), au lieu de Xr2 − Cr2 ,
on écrit simplement Xr2 .                                                              p. 426
   Soient L la plus grande racine et Λ la plus petite racine de l’équation (θ);
alors aucun facteur de (θ) ne peut devenir nul pour des valeurs de x comprises
entre +∞ et L, et entre Λ et −∞; donc on aura toujours

    [X1 ] > C1 ,       [X2 ] > C2 ,        ··· ,        [Xr−1 ] > Cr−1 ,       [Xr ] > Cr .

                                              431
Or f (x) est évidemment égal au dénominateur de la fraction continue multiplié
par un facteur constant. Donc, en vertu du lemme, le dénominateur de la fraction
continue est de même signe que le produit X1 X2 X3 . . . Xr−1 Xr pour les valeurs
de x comprises entre +∞ et L, et entre Λ et −∞; mais dans ces intervalles la
fonction générale Xi n’étant pas comprise entre +Ci et −Ci ne peut devenir
nulle, et, par conséquent, ne peut changer de signe; donc le dénominateur de
la fraction continue conserve le même signe pour toute valeur de x renfermée
entre ces intervalles, et de même f (x); L est donc une limite supérieure et Λ une
limite inférieure des racines de l’équation

                                       f (x) = 0.

Le nombre des racines réelles de l’équation (θ) est évidemment pair, zéro com-
pris; dans ce dernier cas, c’est-à-dire (θ) n’ayant aucune racine réelle, f (x) ne
changera donc pas de signe pour des valeurs de x comprises entre +∞ et −∞;
autrement toutes les racines de f (x) = 0 sont imaginaires. Le théorème est donc
complètement démontré.
Art. (3). Si ϕ(x) est de degré n − 1, la fraction continue renferme en général
(sauf les cas où quelques-uns des coefficients deviennent nuls), comme il a été dit
plus haut, n quotients linéaires de la forme

             a1 x − b1 ,     a2 x − b2 . . . an−1 x − bn−1 ,     an x − bn ;

donc, d’après le théorème, la plus grande et la plus petite des 2n quantités
                 b1 ± C1     b2 ± C2     bn−1 ± Cn−1           bn ± Cn
                         ,           ...             ,                 ,
                    a1          a2          an−1                  an

sont respectivement une limite supérieure et une limite inférieure des racines de
l’équation
                                  f (x) = 0.
Si l’on prend (r = n)

                      µ1 = µ2 = · · · = µn−1 = 1,          µn = 2,

on vient au théorème énoncé [p. 423].                                                 p. 427

Art. (4). Lors même que les quotients X1 , X2 , etc., ne sont pas linéaires, on
n’aura pourtant jamais à résoudre que des équations du premier degré. En effet,
soient les 2r équations de degré quelconque

                  X1 − C1 = 0,      X2 − C2 = 0 . . . Xr − Cr = 0,

                  X1 + C1 = 0,      X2 + C2 = 0 . . . Xr + Cr = 0.



                                          432
Il suffit de trouver une quantité l supérieure aux racines de ces équations, et
une quantité λ inférieure à ces mêmes racines, l et λ seront des limites pour
l’équation
                                    f (x) = 0.
Si donc une de ces équations est de degré p > 1, on applique à cette équation le
procédé ci-dessus, en choisissant une fonction ϕ(x) de degré p − 1, et, en agissant
ainsi, on arrivera par une sorte de trituration à n’avoir à traiter que des équations
du premier degré.
Art. (5).    On a
                                                  1
                                   Ci = µi +            ;
                                                 µi−1
plus la valeur de µi est petite, et plus on aura de chances à resserrer les limites
dans les deux fractions bi ±C
                           ai ; par contre, on aura un désavantage sous ce rapport
                              i


dans les deux fractions suivantes bi+1a±C
                                       i+1
                                           i+1
                                               , car Ci+1 = µi+1 + µ1i ; plus µi diminue,
et plus Ci+1 augmente. Cet inconvénient n’a pas lieu pour la dernière fraction;
on peut donc prendre µn = 0 et Cn = µn−1     1
                                                  .
Art. (6). Il est à remarquer que tous les raisonnements précédents subsistent
en renversant la suite des µ et l’écrivant ainsi:
                      1            1                        1
                          ,            + µr−1 ,       ...      + µ1 .
                     µr−1         µr−2                      µ2

Art. (7).      Il y a lieu à des recherches intéressantes sur la forme à donner à
ϕ(x), et sur les valeurs à donner aux quantités µ pour obtenir les limites les plus
resserrées, et je crois être parvenu à démontrer que la forme la plus avantageuse
est f ′ (x), précisément la forme que M. Sturm a adoptée.
Art. (8). Dans la réduction en fraction continue de ϕ(x)
                                                     f (x) , nous n’avons considéré
que des quotients binômes; mais on peut pousser les divisions plus loin et obtenir
des quantités de la forme
                                         c   d         l
                              ax + b +     +   + ··· + r;
                                         x x2         x
                                                                                            p. 428
   le reste correspondant sera de la forme
                                                                l
                         a′ xr+1 + b′ xr + c′ xr−1 + · · · +      .
                                                               xr
En opérant ainsi, le nombre de termes dans chaque reste ira en diminuant, comme
dans le procédé ordinaire, et le dernier reste sera de la forme Cxµ , µ étant un
entier positif ou négatif, et le dernier quotient de la forme P xp + Qxp−1 , p étant
un entier positif ou négatif; nommant les quotients ainsi obtenus q1 , q2 . . . qr , on
voit aisément qu’on aura
                                   f (x) = M x±i D,

                                           433
où M est une constante, i un nombre entier positif ou négatif dont la valeur
dépend de la manière dont on a opéré dans les divisions successives, et D est le
dénominateur de la fraction continue
                                             1
                                                                        .
                                                  1
                          q1 +
                                                      1
                                 q2 +
                                                             1
                                        q3 + · · · +
                                                                   1
                                                          qr−1 +
                                                                   qr
Donc, si l’on écrit, comme ci-dessus,

                   X = (q12 − C12 )(q22 − C22 ) · · · (qr2 − Cr2 ) = 0,

nommant L et Λ les racines extrêmes de cette équation, si zéro n’est pas compris
entre +∞ et L, ni entre Λ et −∞, la démonstration donnée ci-dessus subsiste
encore pour le cas général. Et lors même que zéro est compris entre ces limites,
L et Λ restent tout de même les limites pour les racines, abstraction faite de la
racine zéro.




                                            434
                                        57.
 On a Theory of the Syzygetic Relations of Two Rational
Integral Functions, comprising an Application to the Theory
of Sturm’s Functions, and that of the Greatest Algebraical
                     Common Measure
 [Philosophical Transactions of the Royal Society of London, CXLIII. (1853),
                           Part III., pp. 407–548]
                                                                                      p. 429


                                  Introduction.

     “How charming is divine philosophy!
     Not harsh and crabbed as dull fools suppose,
     But musical as is Apollo’s lute,
     And a perpetual feast of nectar’d sweets,
     Where no crude surfeit reigns!”—Comus.

   In the first section of the ensuing memoir, which is divided into five sections, I
consider the nature and properties of the residues which result from the ordinary
process of successive division (such as is employed for the purpose of finding the
greatest common measure) applied to f (x) and ϕ(x), two perfectly independent
rational integral functions of x. Every such residue, as will be evident from
considering the mode in which it arises, is a syzygetic function of the two
given functions; that is to say, each of the given functions being multiplied
by an appropriate other function of a given degree in x, the sum of the two
products will express a corresponding residue. These multipliers, in fact, are
the numerators and denominators to the successive convergents to ϕx    f x expressed
under the form of a continued fraction. If now we proceed à priori by means of
the given conditions as to                                                            p. 430
   the degree in x of the multipliers and of any residue, to determine such residue,
we find, as shown in Art. 2, that there are as many homogeneous equations to be
solved as there are constants to be determined; accordingly, with the exception of
one arbitrary factor which enters into the solution, the problem is definite; and if
it be further agreed that the quantities entering into the solution shall be of the
lowest possible dimensions in respect of the coefficients of f and ϕ, and also of
the lowest numerical denomination, then the problem (save as to the algebraical
sign of plus or minus) becomes absolutely determinate, and we can assign the
numbers of the dimensions for the respective residues and syzygetic multipliers.
The residues given by the method of successive division are easily seen not to
be of these lowest dimensions; accordingly there must enter into each of them
a certain unnecessary factor, which, however, as it cannot be properly called


                                         435
irrelevant, I distinguish by the name of the Allotrious Factor. The successive
residues, when divested of these allotrious factors, I term the Simplified Residues,
and in Arts. 3 and 4 I express the allotrious factor of each residue in terms of
the leading coefficients of the preceding simplified residues of f and ϕ. In Art. 5
I proceed to determine by a direct method these simplified residues in terms
of the coefficients of f and ϕ. Beginning with the case where f and ϕ are of
the same dimensions (m) in x, I observe that we may deduce, from f and ϕ,
m linearly independent functions of x each of the degree (m − 1) in x, all of
them syzygetic functions of f and ϕ (vanishing when these two simultaneously
vanish), and with coefficients which are made up of terms, each of which is the
product of one coefficient of f and one coefficient of ϕ. These, in fact, are the
very same m functions as are employed in the method which goes by the name
of Bezout’s abridged method to obtain the resultant to (that is, the result of the
elimination of x performed upon) f and ϕ. As these derived functions are of
frequent occurrence, I find it necessary to give them a name, and I term them
the m Bezoutics or Bezoutian Primaries; from these m primaries m Bezoutian
secondaries may be deduced by eliminating linearly between them in the order
in which they are generated,—first, the highest power of x between two, then
the two highest powers of x between three, and finally, all the powers of x
between them all: along with the system thus formed it is necessary to include
the first Bezoutian primary, and to consider it accordingly as being also the first
Bezoutian secondary; the last Bezoutian secondary is a constant identical with
the Resultant of f and ϕ. When the m times m coefficients of the Bezoutian
primaries are conceived as separated from the powers of x and arranged in a
square, I term such square the Bezoutic square. This square, as shown in Art. 7,
is symmetrical about one of its diagonals, and corresponds therefore (as every
symmetrical matrix must do) to a homogeneous quadratic function of m variables
of which it expresses the determinant. This quadratic function,                       p. 431
   which plays a great part in the last section and in the theory of real roots,
I term the Bezoutiant; it may be regarded as a species of generating function.
Returning to the Bezoutic system, I prove that the Bezoutian secondaries are
identical in form with the successive simplified residues. In Art. 6 I extend these
results to the case of f and ϕ being of different dimensions in x. In Art. 7 I give
a mechanical rule for the construction of the Bezoutic square. In Art. 8 I show
how the theory of f (x) and ϕ(x), where the latter is of an inferior degree to f ,
may be brought under the operation of the rule applicable to two functions of the
same degree at the expense of the introduction of a known and very simple factor,
which in fact will be a constant power of the leading coefficient in f (x). In Art. 9
I give another method of obtaining directly the simplified residues in all cases.
In Art. 10 I present the process of successive division under its most general
aspect. In Arts. 11 and 12 I demonstrate the identity of the algebraical sign of
the Bezoutian secondaries with that of the simplified residues, generated by a
process corresponding to the development of ϕx  f x under the form of an improper


                                         436
continued fraction (where the negative sign takes the place of the positive sign
which connects the several terms of an ordinary continued fraction). As the
simplified residue is obtained by driving out an allotrious factor, the signs of the
former will of course be governed by the signs accorded by previous convention
to the latter; the convention made is, that the allotrious factors shall be taken
with a sign which renders them always essentially positive when the coefficients
of the given functions are real. I close the section with remarking the relation
of the syzygetic factors and the residues to the convergents of the continued
fraction which expresses ϕx  f x , and of the continued fraction which is formed by
reversing the order of the quotients in the first named fraction.
    In the second section I proceed to express the residues and syzygetic multipliers
in terms of the roots and factors of the given functions; the method becoming as
it may be said endoscopic instead of being exoscopic,231 as in the first section. I
begin in Arts. 14 and 15 with obtaining in this                                       p. 432
    way, under the form of a sum or double sum of terms involving factors and roots
of f and ϕ, and certain arbitrary functions of the roots in each term, a general
representative, or to speak more precisely, a group of general representatives for
a conjunctive of any given degree in x to f and ϕ, that is, a rational integral
function of x, which is the sum of the products of f and ϕ multiplied respectively
by rational integral functions of x, so as to vanish of necessity when f and ϕ
simultaneously vanish. This variety of representatives refers not merely to the
appearance of arbitrary functions, but to an essential and precedent difference of
representation quite irrespective of such arbitrariness.
    In Arts. 16, 17, 18, 19, 20, 21, I show how the arbitrary form of function
entering into the several terms of any one (at pleasure) of the formulæ that
represent a conjunctive of any given degree may be assigned, so as to make such
conjunctive identical in form with a simplified residue of the same degree. The
form of arbitrary function so assigned, it may be noticed, is a fractional function
of the roots, so that the expression becomes a sum or double sum of fractions. I
first prove in Arts. 16, 17 that such sum is essentially integral, and I determine
the weight of its leading coefficient in respect of the roots of f and ϕ (this weight
being measured by the number of roots of f and ϕ conjointly, which appear
in any term of such coefficient). Now in the succeeding articles I revert to the
Bezoutic system of the first section, and beginning with the supposition of m
 231
     These words admit of an extensive and important application in analysis. Thus the methods
for resolving an equation (or to speak more accurately, for making one equation depend upon
another of a simpler form) furnished by Tschirnhausen and Mr Jerrard (although not so presented
by the latter) are essentially exoscopic; on the other hand, the methods of Lagrange and Abel
for effecting similar objects are endoscopic. So again, the memoir of Jacobi, “De Eliminatione,”
hereinafter referred to, takes the exoscopic, and the valuable “Nota ad Eliminationem pertinens”
of Professor Richelot in Crelle’s Journal, the endoscopic view of the subject. In the present
memoir (in which the two trains of thought arising out of these distinct views are brought into
mutual relation) the subject is treated (chiefly but not exclusively) under its endoscopic aspect
in the second, third and fourth sections, and exoscopically in the first and last sections.


                                              437
and n being equal, I demonstrate that the most general form of a conjunctive
of any degree in x will be a linear function of the Bezoutics, from which it is
easy to deduce that the simplified residues of any given degree in x are the
conjunctives whose weight in respect of the roots is a minimum; so that all
conjunctives having that weight must be identical (to a numerical factor près),
and any integral form of less weight apparently representing a conjunctive must
be nugatory, every term vanishing identically. These results are then extended
to the case of two functions of unlike degrees. The conclusion is, that the weight
of the forms assumed in Arts. 16 and 17 being equal to the minimum weight,
they must (unless they were to vanish, which is easily disproved) represent the
simplified residues, or which is the same thing, the Bezoutian secondaries.
   We thus obtain for each simplified residue a number of essentially distinct
forms of representation, but all of which must be identical to a numerical factor
près, a result which leads to remarkable algebraical theorems.
   The number of these different formulæ depends upon the degree of the residue;
there being only one for the last or constant residue, two for the last but one,
three for the last but two, and so on. The formulæ continue to have a meaning
when their degree in x exceeds that of f or ϕ; but then, as although always
representing conjunctives, they no longer represent                                    p. 433
   residues, this identity no longer continues to subsist. In Arts. 22, 23, 24, 25, I
enter into some developments connected with the general formulæ in question;
these, it may be observed, are all expressed by means of fractions containing
in the numerator and denominator products of differences; the differences in
the numerator products being taken between groups of roots of f and groups
of roots of ϕ; and in the denominator between roots of f inter se and roots of
ϕ inter se. A great enlargement is thus opened out to the ordinary theory of
partial fractions.
   In Art. 26 I find the numerical ratios between the different formulæ which
represent (to a numerical factor près) the same simplified residue, and in Arts. 27
and 28 I determine the relations of algebraical sign of these formulæ to the
simplified residues or Bezoutian secondaries. In Art. 29 I determine the syzygetic
multipliers corresponding to any given residue in terms of the factors and roots
of the given functions; but the expressions for these, which are closely analogous
to those for the residues, cease to be polymorphic. They are obtained separately
from the syzygetic equation, and it is worthy of notice, that to obtain the one
we use the first of the polymorphic expressions for the residue, and to obtain
the other the opposite extremity of the polymorphic scale. In the subsequent
articles of this section, by aid of certain general properties of continued fractions,
I establish a theorem of reciprocity between the series of residues and either
series of syzygetic multipliers.
   Section III. is devoted to a determination of the values of the preceding
formulæ for the residues and multipliers in the case applicable to M. Sturm’s


                                         438
theorem, where ϕx becomes the differential derivative of f x. It becomes of
importance to express the formulæ for this case in terms of their roots and
factors of f x alone, without the use of the roots and factors of f ′ x, which will of
course be functions of the former.
   By selecting a proper form out of the polymorphic scale, the fractional terms
of the series for each residue in this case become separately integral, and we
obtain my well-known formulæ for the simplified residues (Sturm’s reduced
auxiliary functions) in terms of the factors and the squared differences of partial
groups of roots. This is shown in Art. 35. In Art. 36 the multiplier of f ′ x in the
syzygetic equation is expressed by formulæ of equal simplicity, and in a certain
sense complementary to the former. This method, however, does not apply to
obtaining expressions for the multiplier of f x in the same equation in terms
of the roots and factors of f x; for the separate fractions whose sum represents
any one of these factors, it will be found, do not admit of being expressed as
integral functions of the roots and factors. To obviate this difficulty I look to
the syzygetic equation itself, which contains five quantities, namely, the given
function, its first differential derivative, the residue of a given degree, and the
two multipliers, all of                                                                p. 434
   which, except the multiplier of f x, are known, or have been previously
determined as rational integral functions of the roots and factors of f x. I use
this equation itself for determining the fifth quantity, the multiplier in question.
To perform the general operations by a direct method required for this would
be impossible; the difficulty is got over by finding, by means of the syzygetic
equation, the particular form that the result must assume when certain relations
of equality spring up between the roots of f x; and then, by aid of these particular
determinations, the general form is demonstratively inferred.
   This investigation extends over Arts. 38, 39, 40, 41, 42, 43. It turns out that
the expressions for the multipliers of f x are of much greater complexity than for
the multipliers of f ′ x or for the residues. Any such multiplier consists of a sum
of parts, each of which, as in the case of the residues and the factors of f ′ x, is
affected with a factor consisting of the squared differences of a group of roots; but
the other factor, instead of being simply (as for the residues and factors before
mentioned) a product of certain factors of f x, consists of the sum of a series of
products of sums of powers by products of combinations of factors of f x, each
of which series is affected with the curious anomaly of its last term becoming
augmented in a certain numerical ratio beyond what it should be in order to be
conformable to the regular flow of the preceding terms in the series.232
   The fourth section opens with the establishment of two propositions concerning
quadratic functions which are made use of in the sequel. Art. 44 contains the
proof of a law which, although of extreme simplicity, I do not remember to have
 232
    The syzygetic multipliers are identical with the numerators and denominators (expressed in
                                                                                               ′
their simplest form) of the successive convergents to the continued fraction which expresses ff xx .


                                               439
seen, and with which I have not found that analysts are familiar: I mean the
law of the constancy of signs (as regards the number of positive and negative
signs) in any sum of positive and negative squares into which a given quadratic
function admits of being transformed by substituting for the variables linear
functions of the variables with real coefficients. This constant number of positive
signs which attaches to a quadratic function under all its transformations, which
is a transcendental function of the coefficients invariable for real substitutions,
may be termed conveniently its inertia, until a better word be found. This
inertia it is shown in Art. 45, by aid of a theorem identical with one formerly
given by M. Cauchy, is measured by the number of combinations of sign in
the series of determinants of which the first is the complete determinant of the
function, the second, the determinant when one variable is made zero, the next,
the determinant when another variable as well as the first is made zero, and so
on, until all the variables are exhausted, and the determinant                        p. 435
   becomes positive unity. In Art. 46 I give some curious and interesting expres-
sions for the residues and syzygetic multipliers, under the form of determinants,
communicated to me by M. Hermite; and in Art. 47 I show how, by the aid of
the generating function which M. Hermite employs, and of the law of inertia
stated at the opening of the section, an instantaneous demonstration may be
given of the applicability of my formulæ for M. Sturm’s functions for discovering
the number of real roots of f x, without any reference to the rule of common
measure; and moreover, that these formulæ may be indefinitely varied, and give
the generating function, out of which they may be evolved, in its most general
form. Had the law of inertia been familiar to mathematicians, this constructive
and instantaneous method of finding formulæ for determining the number of real
roots within prescribed limits would, in all probability, have been discovered long
ago, as an obvious consequence of such law. I then proceed in Arts. 48 and 49,
to inquire as to the nature of the indications afforded by the successive simplified
residues to two general functions f and ϕ; and I find that the succession of signs
of these residues serves to determine the number of roots of f or ϕ comprised
between given limits, after all pairs of roots of either function contained within
the given limits and not separated by roots of the other function have been
removed, and the operation, if necessary, repeated toties quoties until no two
roots of either function are left unseparated by roots of the other; or in other
words, until every root finally retained in one function is followed by a root of
the other, or else by one of the assigned limits. The system of roots comprised
between given limits thus reduced I call the effective scale of intercalations; such
a scale may begin with a root of the numerator or of the denominator of ϕx  f x ; and
upon this and the relative magnitudes of the greatest root of ϕx and f x it will
depend whether in the series of residues (among which f x and ϕx are for this
purpose to be counted) changes will be lost or gained as x passes from positive
infinity to negative infinity. In Art. 50 I observe that the theory of real roots of
a single function given by M. Sturm’s theorem is a corollary to this theory of the

                                         440
intercalations of real roots of two functions, depending upon the well-known law,
that odd groups of the limiting function f ′ x lie between every two consecutive
real roots of f x. In Art. 51 I verify the law of reciprocity, already stated to exist
between the residues of f x and ϕx, by an à posteriori method founded on the
theory of intercalations. In Arts. 52, 53, 54, I obtain a remarkable rule, founded
upon the process of common measure, for finding a superior and inferior limit
in an infinite variety of ways to the roots of any given function. This method
stands in a singular relation of contrast to those previously known. All previous
methods (including those derived through Newton’s Rule) proceed upon the idea
of treating the function whose roots are to be limited as made up of the sum of
parts, each of which                                                                   p. 436
    retains a constant sign for all values of the variable external to the quantities
which are to be shown to limit the roots. My method, on the other hand,
proceeds upon the idea of treating the function as the product of factors retaining
a constant sign for such values of the variable. In Art. 55, the concluding article
of the fourth section, I point out a conceivable mode in which the theory of
intercalations may be extended to systems of three or more functions.
    In Section V. Arts. 56, 57, I show how the total number of effective intercala-
tions between the roots of two functions of the same degree is given by the inertia
of that quadratic form which we agreed to term the Bezoutiant to f and ϕ; and in
the following article (58) the result is extended to embrace the case contemplated
in M. Sturm’s theorem; that is to say, I show, that on replacing the function of
x by a homogeneous function of x and y, the Bezoutiant to the two functions,
which are respectively the differential derivatives of f with respect to x and with
respect to y, will serve to determine by its form or inertia the total number of
real roots and of equal roots in f (x). The subject is pursued in the following
Arts. 59, 60. The concluding portion of this section is devoted to a consideration
of the properties of the Bezoutiant under a purely morphological point of view;
for this purpose f and ϕ are treated as homogeneous functions of two variables
x, y, instead of being regarded as functions of x alone. In Arts. 61, 62, 63, it is
proved that the Bezoutiant is an invariantive function of the functions from which
it is derived; and in Art. 64 the important remark is added, that it is an invariant
of that particular class to which I have given the name of Combinants, which
have the property of remaining unaltered, not only for linear transformations
of the variables, but also for linear combinations of the functions containing
the variables, possessing thus a character of double invariability. In Arts. 65,
66, I consider the relation of the Bezoutiant to the differential determinant, so
called by Jacobi, but which for greater brevity I call the Jacobian. On proper
substitutions being made in the Bezoutiant for the m variables which it contains
(m being the degree in x, y of f and ϕ), the Bezoutiant becomes identical with
the Jacobian to f and ϕ; but as it is afterwards shown, this is not a property
peculiar to the Bezoutiant; in fact there exists a whole family of quadratic forms


                                         441
of m variables, lineo-linear (like the Bezoutiant) in respect of the coefficients in f
and ϕ, all of which enjoy the same property. The number of individuals of such
family must evidently be infinite, because any linear combination of any two
of them must possess a similar property; I have discovered, however, that the
number of independent forms of this kind is limited, being equal to the number
of odd integers not greater than the degree of the two functions f and ϕ. In
Arts. 67 and 68, I give the means of constructing the scale of forms, which I term
the constituent or funda-                                                              p. 437
    mental scale, of which all others of the kind are merely numerico-linear
combinations. This scale does not directly include the Bezoutiant within it, and
it becomes an object of interest to determine the numbers which connect the
Bezoutiant with the fundamental forms; this calculation I have carried on (in
Arts. 69, 70, 71) from m = 1 to m = 6 inclusive, and added an easy method
of continuing indefinitely. In this method the numbers in the linear equation
corresponding to any value of m are determined successively, and each made
subject to a verification before the next is determined, there being always pairs
of equations which ought to bring out the same result for each coefficient.
    In the next and concluding Art. 72, I remark upon the different directions in
which a generalization may be sought of the subject-matter of the ideas involved
in M. Sturm’s theorem, and of which the most promising is, in my opinion,
that which leads through the theory of intercalations. Some of the theorems
given by me in this paper have been enunciated by me many years ago, but the
demonstrations have not been published, nor have they ever before been put
together and embodied in that compact and organic order in which they are
arranged in this memoir,—the fruit of much thought and patient toil, which I
have now the honour of presenting to the Royal Society.
    P.S. In a supplemental part to the third section I have given expressions in
terms of the roots of ϕx and f x for the quotients which arise in developing
ϕx
f x under the form of a continued fraction, and some remarkable properties
concerning these quotients. In a supplemental part to the fourth section I have
given an extended theory of my new method of finding limits to the real roots of
any algebraical equation. This method, so extended, possesses a marked feature
of distinction from all preceding methods used for the same purpose, inasmuch
as it admits in every case of the limits being brought up into actual coincidence
with the extreme roots, whereas in other methods a wide and arbitrary interval
is in general necessarily left between the roots and the limits.                       p. 438


                                     Section I.

 On the complete and simplified residues generated in the process of developing
under the form of a continued fraction, an ordinary rational algebraical fraction.

Art. (1).    Let P and Q be two rational integral functions of x, and suppose

                                         442
that the process of continued successive division leads to the equations

                              P − M0 Q + R1 = 0
                                                                  
                                                                  
                                                                  
                                                                  
                             Q − M1 R1 + R2 = 0
                                                                  
                                                                  
                                                                  
                                                                  
                                                                  
                            R1 − M2 R2 + R3 = 0                                     (1)
                                                                     
                                                    · · · · · · · · ·
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                    ·········
                                                                     

so that
                            Q                   1
                              =                                     .               (2)
                            P                         1
                                  M0 −
                                                        1
                                         M1 −
                                                     M2 − · · ·
which is what I propose to call an improper continued fraction, differing from
a proper only in the circumstance of the successive terms being connected by
negative instead of positive signs.
   M0 , M1 , M2 , &c., R1 , R2 , R3 , &c. are, of course, functions of x: the latter we
may agree to call the 1st, 2nd, 3rd, &c. residues (in order to avoid the use of the
longer term “residues with the signs changed”); and by way of distinction from
what they become when certain factors are rejected, we may call R1 , R2 , R3 , &c.
the complete residues. Each such complete residue will in general be of the form
Nι ρι
 Dι , Nι and Dι being integral functions of the coefficients only of P and Q, but
ρι an integral function of these coefficients, and of x; ρι may then be termed the
ιth simplified residue, and D   Nι
                                  ι
                                    the ιth allotrious factor. Suppose P to be of m
and Q of n dimensions in x, and m − n = e, the process of continued division
may be so conducted, that all the residues may contain only integer powers of x;
and we may upon this supposition make M0 of e dimensions, and M1 , M2 , M3 ,
&c. each of one dimension only in x; so that R1 , R2 , R3 , . . . will be respectively
of (n − 1), (n − 2), (n − 3), &c. dimensions in x.                                      p. 439
   P and Q are supposed to be perfectly unrelated, and each the most general
function that can be formed of the same degree. From (1) we obtain

                R1 = M0 Q − P
                                                                         
                                                                         
                                                                         
                                                                         
                R2 = M1 R1 − Q
                                                                         
                                                                         
                                                                         
                                                                         
                                                                         
                    = (M0 M1 − 1)Q − M1 P                                           (3)
                                                           
                R3 = (M0 M1 M2 + M0 + M2 )Q − (M1 M2 − 1)P 
                                                           
                                                           
                                                           
                                                           
                &c. = &c.
                                                           
                                                           

and in general we shall have

                                  Rι = Qι Q + Pι P,                                 (4)


                                          443
where it is evident that Qι will be of e + (ι − 1), and Pι of (ι − 1) dimensions in
x.
Art. (2). Hence it follows that the ratios Pι : Qι : Rι may be ascertained by
the direct application of the method of indeterminate coefficients, for Qι will
contain e + ι, and Pι will contain ι disposable constants, making e + 2ι disposable
constants in all. Again, Qι Q and Pι P will each rise to the degree n + e + ι − 1 in
x; but their sum Rι is to be only of n − ι dimensions in x. Hence we have to make
(n + e + ι − 1) − (n − ι), that is e + 2ι − 1 quantities (which are linear in respect
to the given coefficients in P and Q, as well as in respect to the new disposable
constants in Pι and Qι ) all vanish, that is to say, there will be e + 2ι − 1 linear
homogeneous equations to be satisfied by means of e + 2ι disposable quantities;
the ratios of these latter are, therefore, determinate, so that we may write
                                                         
                                       Pι = λι (Pι ) 
                                                     
                                                     
                                       Qι = λι (Qι ) ;                                      (5)
                                                  
                                       R = λ (R ) 
                                                  
                                         ι       ι   ι

and when (Pι ), (Qι ), (Rι ) are taken prime to one another, it is obvious that (Rι )
will be in all of e + 2ι dimensions in the given coefficients, that is of ι in respect
of the coefficients of P , and of e + ι in respect of those of Q; λι will correspond
to what I have previously called the allotrious factor; being in fact foreign to
the value of Rι as determined by means of the equation (4), and arising only
from the particular method employed to obtain it through the medium of the
system (1): it becomes a matter of some interest and importance to determine
the values of this allotrious factor for different values of ι.233                     p. 440

Art. (3).     This may be done by the following method, which is extremely
simple, and would admit of a considerable extension in its applications, were
it not beside my immediate purpose to digress from the objects set out in the
title to the memoir, by entering upon an investigation of the special or singular
cases which may arise in the process of forming the continued fraction, when
 233
    These are identical with what I termed quotients of succession in the London and Edinburgh
Philosophical Magazine (December, 1839) [p. 43 above]; but by an easily explicable error of
inadvertence, the quantities Q1 , Q2 , &c. therein set out are not as they are therein stated to
be, the quotients of succession or allotrious factors themselves, but the ratios of each such to
the one preceding, if in the series; so that
                                              λ2                 λ3
                        Q1 is λ1 ,    Q2 is      ,       Q3 is      ,   &c.
                                              λ1                 λ2
This error is corrected by my distinguished friend M. Sturm (Liouville’s Journal, t. viii. 1842,
Sur un théorème d’Algèbre de M. Sylvester), who appears, however, to have overlooked that
I was obviously well acquainted with the existence and nature of these factors, and their
essential character, of being perfect squares in the case contemplated in his memoir and my own.
MM. Borchardt, Terquem, and other writers, in quoting my formulæ for M. Sturm’s auxiliary
functions, have thus been led into the error of alluding to them as completed by M. Sturm.


                                               444
one or more of the leading coefficients in any of the residues vanish; such an
inquiry would require a more general character to be imparted to the values of
the quotients and residues than I shall for my present purposes care to suppose.
   Let us begin with supposing e = 1, and write

                        f = axn + bxn−1 + cxn−2 + &c.
                                                                   )
                                                                       .           (6)
                        ϕ = αxn−1 + βxn−2 + γxn−3 + &c.

Let ψ be the first residue of ϕf , and ω of ψϕ , and therefore of αϕ2 ψ , so that ω is
the second residue of ϕf .
   Let ω = λ(ω), ω being entirely integer, and λ a function of the coefficients in
f and ϕ. If we make λ = N   D , N and D being integer functions, D will evidently
be I , where I denotes the first coefficient in the simplified residue α2 ψ, and is
     2

evidently of two dimensions in α, β, &c., and of one in a, b, &c.; Dω is therefore
of 2 × 2 + 1, that is five dimensions in α, β, &c., and of two dimensions in a, b,
&c.; but ω (by virtue of what has been observed of the equations in system
(5)) is of three dimensions in α, β, &c., and of two in a, b, &c. Hence N is of
two dimensions in α, β, &c., and of none in a, b, &c. This enables us at once to
perceive that N = α2 .
   For ψ is of the form f − (px + q)ϕ, and ω is of the form ϕ − (p′ x + q ′ )ψ;

                                 ψ = f − (px + q)ϕ
                                                           )
                                                               .                   (7)
                                 ω = ϕ − (p′ x + q ′ )ψ
                                                                                         p. 441
   but N = 0 makes ω vanish, and therefore, upon this supposition, f and ϕ
would appear to have a common algebraical factor ψ, that is to say, N vanishing
would appear to imply that the resultant of f and ϕ must vanish, so that N
would appear to be contained as a factor in this general resultant, which latter is,
however, clearly indecomposable into factors—a seeming paradox—the solution
of which must be sought for in the fact, that the equation N = 0 is incompatible
with the existence of the usual equations (7) connecting f, ϕ, ψ and ω; but this
failure of the existence of the equations (7) (bearing in mind that N has been
shown to be a function only of the set of coefficients a, β, &c.), can only happen
by reason of a vanishing whenever N vanishes; a must therefore be a root of N ,
or which is the same thing, N a power of a and hence N = a2 .
   The same result may be obtained à posteriori by actually performing the
successive divisions; if the coefficients of any dividend be a, b, c, d, &c., and of
the divisor α, β, γ, δ, &c., the first remainder, forming the second divisor, will be
easily seen to have for its coefficients—
                   1     a b c       1    a b d       1    a b e
                         0 αβ    ,        0 αβ    ,        0 αβ        &c.
                   a2    αβ γ        a2   αβ δ        a2   αβ ε




                                            445
                                                                           a b c
Hence the coefficients in the next remainder (making                       0 αβ    = m) will be each
                                                                           αβ γ
of the form of the compound determinant,—
                                       α             β            γ
                                                   a b c         a b d
                           1           0           0 αβ          0 αγ
                                                   αβ γ          αβ δ      .
                           m2
                                   a b c           a b d         a b e
                                   0 αβ            0 αγ          0 αδ
                                   αβ γ            αβ δ          αβ ε

The compound determinant above written will be the first coefficient in the
remainder under consideration; the subsequent coefficients will be represented by
writing f, ϕ; g, γ, &c. respectively in lieu of e, ε. Omitting the common multiplier
 1
m2
   , the determinant above written is equal to                                       p. 442

                                                                          
                               a b c       a b e         a b d     a b d
                       α       0 αβ        0 αδ     −    0 αγ      0 αγ
                               αβ γ        αβ ε          αβ δ      αβ δ
                                                                          
                                a b c              a b d          a b c
                           +    0 αβ           β   0 αγ    −γ     0 αβ         .
                                αβ γ               αβ δ           αβ γ
The last written pair of terms are together equal to
                     a b c
                     0 αβ      × {−dβa2 + cγa2 + aα(βδ − γ 2 )},
                     αβ γ

which is of the form a2 A − a2 β 2 (βδ − γ 2 )α; and the sum of the first written
pair is of the form a2 B + (aβ 2 aβδ − aγβ aγβ)α. Hence the entire determinant
is of the form a2 (A + B), showing that a2 will enter as a factor into this and
every subsequent coefficient in the second remainder, as previously demonstrated
above.
   It may, moreover, be noticed, that this remainder, when a2 has been expelled,
will for general values of the coefficients be numerically as well as literally in its
lowest terms, as evinced by the fact that there exist terms (for example aα2 γε)
having +1 for their numerical part. The same explicit method might be applied
to show, that if the first divisor were e degrees instead of being only one degree
in x lower than the first dividend, ae+1 would be contained in every term of the
second residue: the difficulty, however, of the proof by this method augments
with the value of e; but the same result springs as an immediate consequence
from the method first given, which remains good mutatis mutandis for the general
case, as may easily be verified by the reader. Applying now this result to the
functions P and Q, supposed to be of the respective degrees n and n − e in
x, and calling the coefficients of the leading terms in the successive simplified
residues a1 , a2 , a3 , &c., and denoting by a the leading coefficient in Q, and as
before denoting the successive allotrious factors by λ1 , λ2 , &c., it will readily be
seen that
                     1              1             1             1
            λ1 = e+1 , λ2 λ1 = 2 , λ3 λ2 = 2 , λ4 λ3 = 2 , &c.,
                   a                a1           a2             a3

                                                   446
that is
                  1                 ae+1                   a21                      ae+1 a22
          λ1 =        ,      λ2 =        ,        λ3 =            ,        λ4 =              ,
                 ae+1                a21                 ae+1 a22                    a21 a23
and in general
                                             1
                                                                      
                                          a2 a2 · · · a22m−1 
                             λ2m+1 = e+1 1 2 3 2            ,
                                                             
                                     a     a2 a4 · · · a22m 
                                                             
                                                                                                 (8)
                                         a2 a2 · · · a22m−2 
                               λ2m = ae+1 22 42             .
                                                             
                                                       2
                                                             
                                         a1 a3 · · · a2m−1 
                                                                                                       p. 443
Art. (4).      Strictly speaking, we have not yet fully demonstrated that the
complete allotrious factors are represented by the values above given for λ, but
only that these latter are contained as factors in the allotrious factors; we must
further prove that there exist no other such factors. This may be shown as follows:
it is obvious from the nature of the process that the complete residues will always
remain of one dimension in respect of the given coefficients, that is, first of
one dimension in the set a, b, c, &c., and of zero dimensions in α, β, γ, &c.; then
conversely, of one dimension in α, β, γ, &c., and of zero dimensions in a, b, c, &c.,
and so on, the residues being evidently required to conform in their dimensions to
those of the first dividend and the first divisor alternately. These coefficients then
are always of unit dimensions in respect to the given coefficients; whereas it has
been shown (Art. 2) that the simplified residues in respect to these coefficients
are successively of the dimensions 2 + e, 4 + e, 6 + e, &c..
    Let the complete residue corresponding to λ2m be M λ2m a2m , that is
                                    ae+1 a22 a24       a22m−2
                               M                 · · ·        a2m ,
                                     a21 a23 a25       a22m−1
or say M L; in passing from a2q to a2q+1 the dimensions rise 2 units for all values
of q except zero, and when q = 0 the dimensions increase per saltum from 1 to
2 + e; hence the total dimensions of L in the joint coefficients will be

                      {(e + 1) − 2(e + 2)} − 4(m − 1) + 4m + e = 1,

and therefore M is of zero dimensions, and λ2m is the complete allotrious factor.
In like manner if the complete residue corresponding to λ2m+1 be M λ2m+1 a2m+1 ,
that is
                              1 a2 a2      a2
                         M e+1 12 32 · · · 2m−1  a2m+1 ,
                            a     a2 a4     a22m
or say M L, the dimensions of L will be

                   −(e + 1) − 4m + {e + 2(2m + 1)},                   that is, 1,

and hence, as in the preceding case, M is of zero dimensions, and λ2m+1 is the
complete allotrious factor.

                                                 447
Art. (5).     I proceed to show how the simplified residues may be most con-
veniently obtained by a direct process, identical with that which comes into
operation in applying to the two given functions of x the method familiarly
known under the name of Bezout’s abridged method of elimination. Let us call
the two given functions U and V , and commence with the case where U and
V are of equal dimensions (n) in x. The simplified ιth residue will then be a
function of n − ι dimensions in x, and of ι dimensions in respect of each given
set of coefficients, and may be taken equal to Vι U + Uι V , where Vι and Uι are
each of (ι − 1) dimensions in x.                                                 p. 444
   Let
                      U = a0 xn + a1 xn−1 + a2 xn−2 + · · · + an ,
                     V = b0 xn + b1 xn−1 + b2 xn−2 + · · · + bn ;
we may write in general, m being taken any positive integer not exceeding n,
U = (a0 xm + a1 xm−1 + · · · + am )xn−m + (am+1 xn−m−1 + am+2 xn−m−2 + · · · + an ),
V = (b0 xm + b1 xm−1 + · · · + bm )xn−m + (bm+1 xn−m−1 + bm+2 xn−m−2 + · · · + bn ).
Hence
        (b0 xm + b1 xm−1 + · · · + bm )U − (a0 xm + a1 xm−1 + · · · + am )V
                                                                                      (9)
              = m K1 x2n−1 + m K2 x2n−2 + m K3 x2n−3 + · · · + m Kn ,

where if we use (r, s) to denote ar bs − as br for all values of r and s, we have

              m K1 = (0, m + 1),      m K2 = (0, m + 2) + (1, m + 1),

                   m K3 = (0, m + 3) + (1, m + 2) + (2, m + 1),

and in general m Ki = Σ(r, s), the values of r and s admissible within the sign of
summation being subject to the two conditions, one the equality r + s = m + i,
the other the inequality r less than i. By giving to m all the different values
from 0 to m − 1 in succession, and calling

           b0 xm + b1 xm−1 + · · · + bm ,         a0 xm + a1 xm−1 + · · · + am

respectively Qm and Pm , we have

             Q0 U − P0 V = K1 x2n−1 + K2 x2n−2 + · · · + Kn ,
                                                                                 
                                                                                 
                                                                                 
                                                                                 
             Q1 U − P1 V = 1 K1 x2n−1 + 1 K2 x2n−2 + · · · + 1 Kn ,
                                                                                 
                                                                                 
                                                                                 
                                                                                 
                                                                                 
                                                                                 
             Q2 U − P2 V = 2 K1 x2n−1 + 2 K2 x2n−2 + · · · + 2 Kn ,                  (10)
                                                                              
                                                                              
                           ············                                       
                                                                              
                                                                              
                                                                              
                                                                              
        Qn−1 U − Pn−1 V = n−1 K1 x2n−1 + n−1 K2 x2n−2 + · · · + n−1
                                                                              
                                                                           K .
                                                                             n

The right-hand members of these n equations I shall henceforth term the Be-
zoutians to U and V .

                                            448
    The determinant formed by arranging in a square the n sets of coefficients
of the n Bezoutians, and which I shall term the Bezoutian matrix, gives, as is
well known, the Resultant (meaning thereby the Result in its simplest form of
eliminating the variables out) of U and V .
    Eliminating dialytically, first xn−1 between the first and second, then xn−1 and
xn−2 between the first, second and third, and so on, and finally, all the powers of
x between the first, second, third, . . . nth of these Bezoutians, and repeating the
first of them, we obtain a derived set of n equations, the right-hand members of
which I shall term the secondary Bezoutians to U and V , this secondary system
of equations being                                                                   p. 445



              Q0 U − P0 V = K1 x2n−1 + K2 x2n−2 + K3 x2n−3 + · · · + Kn ,
       (1 K1 Q0 − K1 Q1 )U − (1 K1 P0 − K1 P1 )V
                              = L1 x2n−2 + L2 x2n−3 + · · · + Ln−1 ,
                             {(1 K12 K2 − 2 K11 K2 )Q0 + (2 K1 K2 − K12 K2 )Q1
                                   + (K11 K2 − 1 K1 K2 )Q2 }U
                              − {(1 K12 K2 − 2 K11 K2 )P0 + (2 K1 K2 − K12 K2 )P1
                                   + (K11 K2 − 1 K1 K2 )P2 }V
                              = M1 x2n−3 + M2 x2n−4 + · · · + Mn−2 ,
                             &c. = &c.                                                      (11)

And we can now already without difficulty establish the important proposition,
that the successive simplified residues to VU , expanded under the form of an
improper continued fraction, abstracting from the algebraical sign (the correctness
of which also will be established subsequently), will be represented by the n
successive Secondary Bezoutians to the system U, V .
   For if we write the system of equations (11) under the general form

                       Dι U − Hι V = Aι xn−ι + Bι xn−ι−1 + &c.,

the degree of Dι and Hι in x will be that of Qι−1 and Pι−1 , that is ι − 1; and
the dimensions of Aι , Bι , &c., in respect of each set of coefficients is evidently
ι; consequently, by virtue of Art. 2, Aι xn−ι + Bι xn−ι−1 + &c., which is the ιth
Bezoutian, will (saving at least a numerical factor of a magnitude and algebraical
sign to be determined, but which, when proper conventions are made, will be
subsequently proved to be +1) represent the ιth simplified residue to VU ,234 as
was to be shown.
 234
     V is supposed to be taken as the first divisor, and the term residue is used, as hitherto in
this paper, throughout in the sense appertaining to the expansion conducted, so as to lead to
an improper continued fraction, in that sense, in fact, in which it would, more strictly speaking,
be entitled to the appellation of excess rather than that of residue.


                                              449
Art. (6). More generally, suppose U and V to be respectively of n + e and n
dimensions in x.
  Let

   U = a0 xn+e + a1 xn+e−1 + a2 xn+e−2 + &c.,        V = b0 xn + b1 xn−1 + &c.

Making

      U = (a0 xe+m + a1 xe+m−1 + &c. + ae+m )xn−m
           + (ae+m+1 xn−m−1 + &c. + an+e ),
      V = (b0 xm + b1 xm−1 + · · · + bm )xn−m + (bm+1 xn−m−1 + &c. + bn ),

we obtain the equation

          Qm U − Pe+m V = m K1 xn+e−1 + m K2 xn+e−2 + &c. + m Kn+e ,           (12)
                                                                                       p. 446
  where

          Qm = (b0 xm + · · · + bm ),    Pe+m = (a0 xe+m + · · · + ae+m );

            m K1 = a0 bm+1 ,

            m K2 = a0 bm+2 + a1 bm+1 ,    ··· ,
            m Ke = a0 bm+e + a1 bm+e−1 + &c. + ae bm ,

          m Ke+1 = a0 bm+e+1 + &c. + ae+1 bm − ae+m+1 b0 ,        &c. = &c.
By giving to m every integer value from 0 to (n − 1) inclusive, we thus obtain
n equations of the form of (12), each of the degree n + e − 1 in x, and of one
dimension in regard to each set of coefficients.
   In addition to these equations we have the e equations of the form

                     xµ V = b0 xn+µ + b1 xn+µ−1 + &c. + bn xµ ,                (13)

in which µ may be made to assume every value from 0 to (e − 1) inclusive, and
the right-hand side of the equation for all such values of µ will remain of a degree
in x not exceeding n + e − 1, the degree of the equations of the system above
described. There will thus be e equations in which only the (b) set of coefficients
appear, and n equations containing in every term one coefficient out of each of
the two sets.
   The total number of equations is of course n + e. Between the e equations of
the second system (13) and the r occurring first in order of the first system (12),
we may eliminate dialytically the e + r − 1 highest powers of x, and there will
thus arise an equation of the form

               θr−1 U − ωe+r−1 V = Lxn−r + L′ xn−r−1 + &c. + (L),              (14)

                                         450
where θr−1 and ωe+r−1 are respectively of the degrees r − 1 and e + r − 1 in x,
and L, L′ . . . (L) are of r dimensions in the (a) set, and of (e + r) dimensions in
the (b) set of coefficients, and consequently Lxn−r + L′ xn−r−1 + · · · + (L) must
satisfy the conditions necessary and sufficient to prove its being (to a numerical
factor près) a simplified residue to (U, V ).
   Thus suppose

         U = a0 x4 + a1 x3 + a2 x2 + a3 x + a4 ,         V = b0 x2 + b1 x + b2 .

Then, corresponding to the system of which equation (13) is the type, we have

               V = b0 x2 + b1 x + b2 ,       xV = b0 x3 + b1 x2 + b2 x.

Again, to form the system of which equation (12) is the type, we write

                b0 U − (a0 x2 + a1 x + a2 )V = b0 (a3 x + a4 ) − (a0 x2 + a1 x + a2 )(b1 x + b2 )
                                                 = −a0 b1 x3 − (a0 b2 + a1 b1 )x2
                                                   + (b0 a3 − a1 b2 − a2 b1 )x + (b0 a4 − a2 b2 ),
(b0 x + b1 )U − (a0 x + a1 x + a2 x + a3 )V = (b0 x + b1 )a4 − (a0 x3 + a1 x2 + a2 x + a3 )b2
                    3         2

                                                 = −a0 b2 x3 − a1 b2 x2
                                                   + (b0 a4 − a2 b2 )x + (b1 a4 − b2 a3 ).
                                                                                             p. 447
   Combining the two equations of the first system with the first of the second
system, we obtain the first simplified residue Lx + L′ , where

                             0          b0               b1
                 −L =        b0         b1               b2
                            a0 b1 a0 b2 + a1 b1 a1 b2 + a2 b1 − b0 a3

and
                               0          b0            b2
                        ′
                     L =       b0         b1            0       .
                              a0 b1 a0 b2 + a1 b1 a2 b2 − b0 a4
By again combining the two equations of the first system with both of the second
system, we have the determinant

                  0          b0                b1                b2
                  b0         b1                b2                0
          R=
                 a0 b1 a0 b2 + a1 b1 a1 b2 + a2 b1 − b1 a3 a2 b2 − b0 a4
                 a0 b2     a1 b2         a2 b2 − b0 a4     a0 b2 − a4 b1

which is the last simplified residue, or in other terms, the resultant to the system
U, V .


                                           451
Art. (7). It is most important to observe that the Bezoutian matrix to two
functions of the same degree (n) is a symmetrical matrix, the terms similarly
disposed in respect to one of the diagonals being equal.
   Thus retaining the notation of Art. 5, so that

             (0, 1) = aβ − bα, (1, 2) = bγ − cβ, (2, 3) = cδ − dγ,
             (0, 2) = aγ − cα, (1, 3) = bδ − dβ, &c.
             (0, 3) = aδ − dα, &c.
             &c.

when n = 1 the Bezoutian matrix consists of a single term (0, 1); when n = 2, it
becomes
                               (0, 1) (0, 2)
                               (0, 2) (1, 2);
when n = 3, it becomes

                           (0, 1)  (0, 2)  (0, 3)
                                    (0, 3)
                           (0, 2)  +  (1, 3)
                                          
                                    (1, 2)
                           (0, 3)   (1, 3)   (2, 3);
                                                                                   p. 448
  when n = 4, it becomes

                    (0, 1)  (0, 2)     (0, 3)      (0, 4)
                             (0, 3)       (0, 4)
                    (0, 2)  +             +  (1, 4)
                                              
                                        
                             (1, 2)      (1, 3) 
                             (0, 4)       (1, 4)
                                       

                    (0, 3)  +          +  (2, 4)
                                              
                             (1, 3)       (2, 3)
                    (0, 4)   (1, 4)       (2, 4)   (3, 4);




                                      452
when n = 5, it becomes

                (0, 1)  (0, 2)   (0, 3)                (0, 4)        (0, 5)
                         (0, 3)     (0, 4)                    (0, 5)
                (0, 2)  +   +                               +         (1, 5)
                                                                
                                                          
                         (1, 2)     (1, 3)                    (1, 4)
                                    (0, 5)
                                          

                         (0, 4)    +                      (1, 5)
                                                                    
                                          
                (0, 3)  +   (1, 4)                     +             (2, 5)
                                                              
                         (1, 3)    +                      (2, 4)
                                          

                                  (2, 3) 
                         (0, 5)     (1, 5)                  (2, 5)
                                                                     

                (0, 4)  +   +                          +  (3, 5)
                                                              
                         (1, 4)     (2, 4)                  (3, 4)
                (0, 5)   (1, 5)     (2, 5)                  (3, 5)   (4, 5),

and so forth. Every such square it is apparent may be conceived as a sort of
sloped pyramid, formed by the successive superposition of square layers, which
layers possess not merely a simple symmetry about a diagonal (such as is proper
to a multiplication table), but the higher symmetry (such as exists in an addition
table), evinced in all the terms in any line of terms parallel to the diagonal
transverse to the axis of symmetry being alike.235 Thus for n = 5, the three
layers or stages in question will be seen to be, the first—

                           (0, 1)   (0, 2)   (0, 3)    (0, 4) (0, 5)
                           (0, 2)   (0, 3)   (0, 4)    (0, 5) (1, 5)
                           (0, 3)   (0, 4)   (0, 5)    (1, 5) (2, 5)
                           (0, 4)   (0, 5)   (1, 5)    (2, 5) (3, 5)
                           (0, 5)   (1, 5)   (2, 5)    (3, 5) (4, 5);
                                                                                                   p. 449
   the second—
                                    (1, 2) (1, 3) (1, 4)
                                    (1, 3) (1, 4) (2, 4)
                                    (1, 4) (2, 4) (3, 4);
and the third—
                                             (2, 3).
In general, when n is odd, say 2p + 1, the pyramid will end with a single term
(p, (p + 1)), and when even, as 2p, with a square of four terms,

                            ((p − 2), (p − 1)),        ((p − 2), p),
 235
     A square arrangement having this kind of symmetry, namely, such as obtains in the so-called
Pythagorean addition table as distinguished from that which obtains in the multiplication table,
may be universally called Persymmetric.


                                              453
                             ((p − 2), p),   ((p − 1), p).
Each stage may be considered as consisting of three parts, a diagonal set of equal
terms transverse to the axis of symmetry, and two triangular wings, one to the
left, and the other to the right of this diagonal; the terms in each such diagonal
for the respective stages will be

                 (0, n),   (1, n − 1),   (2, (n − 2)) . . . (p, (p + 1)),

p being n2 − 1 when n is even, and n−1  2 when n is odd.
   If we change the order of the coefficients in each of the two given functions, it
will be seen that the only effect will be to make the left and right triangular wings
to change places, the diagonals in each stage remaining unaltered. The mode
of forming these triangles is an operation of the most simple and mechanical
nature, too obvious to need to be further insisted on here.
Art. (8). When we are dealing with two functions of unequal degrees, n and
n + e, we can still form a square matrix with the coefficients of the two systems
of e and n equations respectively, but this will no longer be symmetrical about a
diagonal; it is obvious, however, that if we treat the function of the lower degree,
as if it were of the same degree as the other function, which we may do by filling
up the vacant places with terms affected with zero coefficients, the symmetry will
be recovered; and it is somewhat important (as will appear hereafter) to compare
the values of the Bezoutian secondaries as obtained, first in their simplest form
by treating each of the two functions as complete in itself, and secondly, as
they come out, when that of the functions which is of the lower degree is looked
upon as a defective form of a function of the same degree as the other. A single
example will suffice to make the nature of the relation between the two sets of
results apparent.
   Take

      f x = ax4 + bx3 + cx2 + dx + e,          ϕx = 0x4 + 0x3 + γx2 + δx + ε.
                                                                                        p. 450
   The general method of Art. 7 then gives for the Bezoutian matrix
                       0    aγ           aδ              aε
                           aδ        aε
                      aγ  +      +         bε
                                     

                         
                           bγ  
                                     bδ   
                           aε        bε
                      aδ  +       +     cε − eγ
                                       
                           bδ     cδ − dγ
                      aε   bε     cε − eγ   dε − eδ.
We shall not affect the value either of the complete determinant, or of any of the
minor determinants appertaining to the above matrix, by subtracting the first

                                         454
line of terms, each increased in the ratio of b : a, from the second line of terms
respectively; the matrix so modified becomes

                          0      aγ      aδ        aε
                         aγ      aδ     aε       0
                                         bε
                         aδ aε + bδ     +     cε − eγ
                                             
                                      cδ − dγ
                         aε    bε     cε − eγ   dε − eδ.

Again, adopting the method of Art. 6, we should obtain the matrix

                          0       γ     δ         ε
                          γ       δ    ε        0
                                        bε
                         δ aε − bδ     +     cε − eγ
                                            
                                     cδ − dγ
                         aε   bε     cε − eγ   dε − eδ.

Hence it is apparent that the secondary Bezoutians obtained by the symmetrizing
method will differ from those obtained by the unsymmetrical method by a
constant factor a2 ; and so in general it may readily be shown that the secondary
Bezoutians, by the use of the symmetrizing method, will each become affected
with a constant irrelevant factor aω , where ω is the difference of the degrees
of the two functions, and a the leading coefficient of the higher one of the two.
When a is taken unity, the Bezoutian secondaries, as obtained by either method,
will of course be identical.
Art. (9). There is another method236 of obtaining the simplified residues to
any two functions U and V of the degrees n and n + e respectively, which,     p. 451
  although less elegant, ought not to be passed over in silence. This method
consists in forming the identical equations (of which for greater brevity the
 236
     Originally given by myself in the London and Edinburgh Philosophical Magazine, as long
ago as 1839 or 1840 [p. 54 above]; and some years subsequently in unconsciousness of that fact,
reproduced by my friend Mr Cayley, to whom the method is sometimes erroneously ascribed,
and who arrived at the same equations by an entirely different circle of reasoning.




                                             455
right-hand members are suppressed)

                                         V = &c.
                                        xV = &c.
                                          ..
                                           .
                                    xe−1 V     = &c.
                                         U     = &c.
                                      xe V     = &c.
                                       xU      = &c.
                                    xe+1 V     = &c.
                                      x2 U     = &c.
                                    xe+2 V     = &c.
                                       &c.     = &c.
                                    xn−1 U     = &c.
                                  xe+n−1 V     = &c.

If we equate the right-hand members of (e + 2ι) of the above equations to zero,
and then eliminate dialytically the several powers of x from xn+e+ι−1 to xn+ι+1
(both inclusive), the result of this process will evidently be of (e + ι) dimensions
in respect of the coefficients in V , of ι dimensions in respect of the coefficients in
U and of the degree xn−ι in x; it will also be of the form

            (A + Bx + · · · + Lxι−1 )U + (F + Gx + · · · + Qxe+ι−1 )V,

and by virtue of Art. 2, must consequently be the ιth simplified residue to the
system U, V .
Art. (10). The most general view of the subject of expansion by the method of
continued division, consists in treating the process as having reference solely to
the two systems of coefficients in U and V , which themselves are to be regarded
in the light of generating functions. To carry out this conception, we ought to
write
                   U = a0 + a1 y + a2 y 2 + a3 y 3 + &c. ad inf.,
                    V = b0 + b1 y + b2 y 2 + b3 y 3 + &c. ad inf.,
and might then suppose the process of successive division applied to U and V ,
so as to obtain the successive equations
                              U − M1 V + R1 = 0,
                             V − M2 R1 + R2 = 0,
                            R1 − M3 R2 + R3 = 0,
                                               &c. &c.,
                                                                                          p. 452
  M1 , M2 , M3 , &c. being each severally of any degree whatever in y, and in
general the degree of y in Mι being any given arbitrary function ϕ(ι) of ι. The

                                         456
values of the coefficients of the residues R1 , R2 , R3 . . ., or of these forms simplified
by the rejection of detachable factors, become then the distinct object of the
inquiry, and will, of course, depend only upon the coefficients in U and V and
the nature of the arbitrary continuous or discontinuous function ϕ(ι), which
regulates the number of steps through which each successive process of division
is to be pursued. Following out this idea in a particular case, if we again reduce
our two initial functions to the forms previously employed, and write

            U = a0 xn + a1 xn−1 + &c.,           V = b0 xn + b1 xn−1 + &c.;

and if, instead of making, according to the more usual course of proceeding,
the divisions proceed first through one step and ever after through two steps
at a time, which is tantamount to making ϕ1 = 1, ϕ(1 + ω) = 2, we push each
division through one step only at a time, and no more (so that in fact ϕ(ι) is
always 1), we shall have

                                 U − m1 V + R1 = 0,
                              V − m2 xR1 + R2 = 0,
                               R1 − m3 R2 + R3 = 0,
                             R2 − m4 xR3 + R4 = 0,
                                                  &c. &c.,

m1 , m2 , m3 , &c. being functions of the coefficients only of U and V ; and it is not
without interest to observe (which is capable of an easy demonstration) that
the simplified residues contained in R1 , R2 , &c., found according to this mode of
development, will be the successive dialytic resultants obtained by eliminating
the (ι − 1)th highest powers of x between the ι first of the system of annexed
equations (supposed to be expressed in terms of x)

                                        U    =    0,
                                        V    =    0,
                                      xU     =    0,
                                      xV     =    0,
                                     x2 U    =    0,
                                     x2 V    =    0,
                                      &c.    =    &c.,
                                   xn−1 U    =    0,
                                   xn−1 V    =    0.
                                                                                              p. 453
   If we combine together 2i + 1 of the above equations, the highest power of
x entering on the left-hand side will be xn+i , and we shall be able to eliminate
2i of these factors, leaving xn−i the highest power remaining uneliminated. If
we take 2i, that is i pairs of the equations, the highest power of x appearing
in any of them will be xn+i−1 , and we shall be able to eliminate between them

                                           457
so as still to leave xn+i−1−(2i−1) , that is xn−i as before, the highest power of x
remaining uneliminated; and it will be readily seen that such of the simplified
residues corresponding to this mode of development as occupy the odd places in
the series of such residues, will be identical with the successive simplified residues
resulting from the ordinary mode of developing VU under the form of a continued
fraction.
Art. (11). It has been shown that the simplified residues of f x and ϕx resulting
from the process of continued division are identical in point of form with the
secondary Bezoutians of these functions, but it remains to assign the numerical
relations between any such residue and the corresponding secondary.
   To determine this numerical relation, it will of course be sufficient to compare
the magnitude of the coefficient of any one power of x in the one, with that of
the same power in the other; and for this purpose I shall make choice of the
leading coefficients in each. In what follows, and throughout this paper, it will
always be understood that in calculating the determinant corresponding to any
square the product of the terms situated in the diagonal descending from left
to right will always be taken with the positive sign, which convention will serve
to determine the sign of all the other products entering into such determinant.
Now adopting the umbral notation for determinants,237 we have, by virtue of a
much more general theorem for compound determinants, the following identical
equation:
                                 a1 a2 a3 . . . am−1
                                                      a1 a2 a3 . . . am+1
                                                          
                                                     ×
                                 α1 α2 α3 . . . αm−1   α1 α2 α3 . . . αm+1
    a1 a2 a3 . . . am−1 am    a1 a2 . . . am−1 am+1    a1 a2 a3 . . . am−1 am    a1 a2 . . . am−1 am+1 
=                            ×                         −                          ×                         ,
    α1 α2 α3 . . . αm−1 αm     α1 α2 . . . αm−1 αm+1     α1 α2 . . . αm−1 αm+1       α1 α2 . . . αm−1 αm
and consequently
                            a1 a2 a3 . . . am−1    a1 a2 a3 . . . am−1 am am+1 
                                                  ×
                            α1 α2 α3 . . . αm−1     α1 α2 α3 . . . αm−1 am am+1
                 a1 a2 a3 . . . am−1 am    a1 a2 . . . am−1 am+1     a1 a2 . . . am−1 am 2
             =                            ×                         −                         ,
                 α1 α2 α3 . . . αm−1 am     α1 α2 . . . αm−1 am+1     α1 α2 . . . αm−1 am+1
                                                                                                                p. 454
    and consequently when
                                                                !
                                       a1 a2 . . . am−1 am
                                                                    = 0,
                                      α1 α2 . . . αm−1 am+1
                                         !                                            !
                      a1 a2 . . . am−1                    a1 a2 . . . am−1 am am+1
                                              and
                      α1 α2 . . . αm−1                    α1 α2 . . . αm−1 am am+1
will have different algebraical signs, it being of course understood that all the
quantities entering into the determinants thus umbrally represented above are
supposed to be real quantities. This theorem, translated into the ordinary
language of determinants, may be stated as follows:—Begin with any square of
terms whether symmetrical or otherwise, say of r lines and r columns: let this
square be bordered laterally and longitudinally by the same number r of new
quantities symmetrically disposed in respect to one of the diagonals, the term
 237
       See London and Edinburgh Philosophical Magazine, April 1851 [p. 242 above].


                                                    458
common to the superadded line and column being filled up with any quantity
whatever; we thus obtain a square of (r + 1) lines and columns; let this be
again bordered laterally and longitudinally by (r + 1) quantities symmetrically
disposed above the same diagonal as that last selected, the place in which this
new line and column meet being also filled up with any arbitrary quantity; and
proceeding in this manner, let the determinants corresponding to the square
matrices thus formed be called Dr , Dr+1 , Dr+2 . . .: this series of quantities will
possess the property, that no term in it can vanish without the terms on either
side of that so vanishing having contrary signs. Thus if we begin with a square
consisting of one single term, we may suppose that by accretions formed after
the above rule it has been developed into the square (M) below written, and
which of course may be indefinitely extended:
                                  a l m p s
                                  l b n q t
                                  m n c r u                                      (M)
                                  p q r d v
                                  s t u v e
Here D0 , D1 , D2 , D3 , D4 , D5 will represent the progression
                                                                  a l m p s
                                          a l m p
                         a l m                                    l b n q t
             a l                          l b n q
1,     a,        ,       l b n ,                  ,               m n c r u ;
             l b                          m n c r
                         m n c                                    p q r d v
                                          p q r d
                                                                  s t u v e
                                                                            (II) p. 455
     so if we use the matrix
                                  a l m p s
                                  l′ b n q t
                                  m n c r u
                                  p q r d v
                                  s t u v e
the determinants D1 , D2 , D3 , D4 , representing
                                                     a l m p
                                   a l m
                       a l                           l′ b n q
                a,          ,      l′ b n ,                   ,
                       l′ b                          m n c r
                                   m n c
                                                     p q r d

will possess the property in question; the line and column l, b; l′ , b not being
identical, the first determinant D0 representing unity must not be included in
the progression.
   We shall have occasion to use this theorem as applicable to the case of a matrix
symmetrical throughout, and we may term the progression (II), above written, a

                                         459
progression of the successive principal determinants about the axis of symmetry
of the square matrix (M), and so in general. Now it is obvious that the leading
coefficients of the successive Bezoutian secondaries are the successive principal
determinants about the axis of symmetry of the Bezoutian squares; they will
therefore have the property which has been demonstrated of such progressions;
to wit, if the first of them vanishes, the second will have a sign contrary to that
of +1; if the second vanishes, the third will have a sign contrary to that of the
first, and so on.
Art. (12). Now let f x and ϕx be any two algebraical functions of x with the
leading coefficients in each, for greater simplicity, supposed positive: and in the
course of developing ϕx f x under the form of an improper continued fraction by
the common process of successive division, let any two consecutive residues (the
word residue being used in the same conventional sense as employed throughout)
be
                             Axt + Bxt−1 + Cxt−2 + &c.
                         B ′ xt−1 + C ′ xt−2 + D′ xt−3 + &c.
The residue next following, obtained by actually performing the division and
duly changing the sign of the remainder, will be
                      AD′                 AC ′     C′
                                                    
                          −C −                 − B           xt−2 + &c.,
                       B′                 B′       B′
                                                                                       p. 456
  which is of the form
                           1
                               {B ′ M − AC ′2 }xt−2 + &c.
                          B ′2
Thus the leading coefficients in the complete unreduced residues will be
                                           1
                          A,    B′,            {B ′ M − AC ′2 },
                                          B ′2
and when reduced by the expulsion of the allotrious factor will become

                               A,   B′,      B ′ M − AC ′2 ,

and consequently, when B ′ the leading coefficient of one of the simplified residues
vanishes, the leading coefficients of the residues immediately preceding and
following that one will have contrary signs.
   First, let f x and ϕx be of the same degree. As regards the numerical ratio of
each Bezoutian secondary to the corresponding simplified residue, it has been
already observed that there are always unit coefficients in the latter of these,
and the same is obviously true of the former; hence if we call the progression of
the leading coefficients of the simplified residues

                               R1 , R2 , R3 , R4 , &c.,

                                            460
and that of the leading coefficients of the Bezoutian secondaries

                                B1 , B2 , B3 , B4 , &c.,

we have

        B1 = ±R1 ,       B2 = ±R2 ,        B3 = ±R3 ,               B4 = ±R4 ,   &c.

It may be proved by actual trial that B1 = R1 and B2 = R2 . Moreover, since
the signs are invariable, and do not depend upon the values of the coefficients,
we may suppose B2 = 0 (which may always be satisfied by real values of the
quantities of which B2 is a function); we shall also, therefore, have R2 = 0, and
consequently B3 has the opposite sign to that of B1 , and R3 the opposite sign
to that of R1 , which is equal to B1 : hence when B2 = 0, B3 and R3 are equal,
and consequently are always equal; in like manner we can prove that R4 and B4
have the same sign when R3 and B3 vanish, and consequently are always equal,
and so on ad libitum, which proves that the series B1 , B2 , . . . Bn is identical with
the series R1 , R2 , . . . Rn , and consequently that the Bezoutian secondaries are
identical in form, magnitude and algebraical sign with the simplified residues.
   Secondly, when f x and ϕx are not of the same degree, it has been shown that
the secondaries formed from the non-symmetrical matrix corresponding to this
case will be the same as those formed from the symmetrical matrix corresponding
to f x and Φx (where Φx is ϕx treated by aid of evanescent terms as of the same
degree as f x), with the exception merely of a constant multiplier (a power of
the leading coefficient of f x) being introduced into each secondary. By aid of
this observation, the proposition                                                       p. 457
   established for the case of two functions of the same degree may be readily
seen to be capable of being extended, from the case of f and ϕ being of equal
dimensions in x, to the general case of their dimensions being any whatever.
Art. (13). Before closing this section, it may be well to call attention to the
nature of the relation which connects the successive residues of f x and ϕx with
these functions themselves, and with the improper continued fractional form into
which ϕx
       f x is supposed to be developed in the process of obtaining these residues.
   If ϕx be of n degrees, and f x of n + e degrees in x, we shall have

                           ϕx                   1
                              =                                     ,
                           fx                       1
                                  Q1 −
                                                        1
                                         q2 −
                                                                1
                                                q3 − · · · −
                                                               qn
where Q1 may be supposed to be a function of x of the degree e, and q2 , q3 , . . . qn ,
are all linear functions of x; the total number of the quotients Q1 , q2 , . . . qn being

                                          461
of course n when the process of continued division is supposed to be carried out
until the last residue is zero. Upon this supposition the last but one residue is a
constant, the preceding one a function of x of the first degree, the one preceding
that a function of x of the second degree, and so on.
   Let us call the residue of the degree ι in x, Rι ; it will readily be seen that the
successive complete residues arranged in an ascending order will be

     R0 ,   R0 qn ,   R0 (qn−1 qn − 1),      R0 (qn−2 qn−1 qn − qn−2 − qn ),   &c.,

being in the ratios
                                       1                   1
                1,    qn ,   qn−1 −      ,   qn−2 −               ,   &c.
                                      qn               qn−1 − q1n

Again, we shall have in general

                                  Λ ι f − L ι ϕ = Rι ,                                (15)

Λι being an integral function of x of the degree n − ι − 1, and Lι an integral
function of x of the degree (n + e) − ι − 1; and it is easy to see that the successive
convergents to the continued fraction

                                              1
                                                   1
                                 Q1 −
                                                      1
                                          q2 −
                                                  q3 − &c.
have their respective numerators and denominators identical with those of the
fractions
                        Λn−1     Λn−2    Λn−3
                             ,        ,       , &c.
                        Ln−1     Ln−2    Ln−3
                                                                                             p. 458
   Adopting the language which I have frequently employed elsewhere, I call St
a syzygetic function, or more briefly a conjunctive of f and ϕ, and Λt and Lt
may be termed the syzygetic factors to St so considered. If we divide each term
of the equation (15) by the allotrious factor (M ), we have
                                 Λt    Lt
                                    f−    ϕ = Rt ,
                                 M     M
where Rt is the tth simplified residue to (f, ϕ); and if we call Λ
                                                                 M = τt , and M = tt ,
                                                                  t           Lt

so as to obtain the equation

                                   τt f − tt ϕ = Rt ,                                 (16)

we see that τttt , the fraction formed by the component factors to any simplified
residue of (f, ϕ), will be identical in value (although no longer in its separate

                                             462
terms) with one of the corresponding convergents to ϕf , exhibited under the form
of an improper continued fraction. I shall in the next section show how, not only
the successive simplified residues, but also the component syzygetic factors of
each of them, and consequently the successive convergents, may be expressed in
terms of the roots of the two given functions.
   Since the preceding section was composed the valuable memoir of the lamented
Jacobi, entitled “De Eliminatione Variabilis è duabus Equationibus Algebraicis,”
Crelle, Vol. xvi., has fallen under my notice. That memoir is restricted to the
consideration of two equations of the same degree, and the principal results in
this section as regards the Bezoutic square and the allotrious factors applicable
to that case will be found contained therein. The mode of treatment however is
sufficiently dissimilar to justify this section being preserved unaltered under its
original form.

                                        Section II.
  On the general solution in terms of the roots, of any two given algebraical
  functions of x, of the syzygetic equation, which connects them with a third
   function, whose degree in x is given, but whose form is to be determined.

Art. (14).    Let f and ϕ be two given functions in x of the degrees m and n
respectively in x, and for the sake of greater simplicity let the coefficients of the
highest power of x in f and ϕ be each taken unity, and let it be proposed to
solve the syzygetic equation
                                   τt f − tt ϕ + St = 0,                               (17)
                                                                                              p. 459
   where St is given only in the number of its dimensions in x, which I suppose to
be t; but the forms of τt , tt , St are all to be determined in terms of h1 , h2 . . . hm ,
the roots of f , and η1 , η2 . . . ηn the roots of ϕ.
   I shall begin with finding St ; and before giving a more general representation
of St , I propose now to demonstrate that we may make
               St = Σ{Pq1 ,q2 ...qt × (x − hq1 )(x − hq2 ) . . . (x − hqt )},          (18)
where Pq1 ,q2 ...qt is used to denote

           (hqt+1 − η1 )(hqt+1 − η2 ) . . . (hqt+1 − ηn ) 
                                                         
        
                                                         
         ×(hqt+2 − η1 )(hqt+2 − η2 ) . . . (hqt+2 − ηn ) 
        
                                                         
                                                          
                                                         
            ×(hqt+3 − η1 )(hqt+3 − η2 ) . . . (hqt+3 − ηn ) R(hq1 , hq2 . . . hqt ),
                           ············
        
                                                          
                                                           
        
                                                          
                                                           
             ×(hqm − η1 )(hqm − η2 ) . . . (hqm − ηn )
        
                                                          
                                                           

R(hq1 , hq2 . . . hqt ) denoting any rational symmetrical function whatever of the
quantities preceded by the symbol R, and q1 , q2 . . . qt , qt+1 . . . qm being any per-
mutation of the m indices 1, 2 . . . m.

                                            463
   Suppose f = 0 and ϕ = 0, then x is equal to one of the series of roots

                                            h1 , h2 . . . hm ,

and also to one of the series of roots

                                            η1 , η2 . . . ηn .

Suppose then that
                                            x = ha = ηω ,
and consider any term of St .
  If in any such term a is found in the series q1 , q2 . . . qt , then

                           (x − hq1 )(x − hq2 ) . . . (x − hqt ) = 0.

But if not, then x must be found in the complementary series

                                      hqt+1 , hqt+2 . . . hqm ,

and consequently Pq1 ,q2 ...qt will contain a factor ha − ηω and Pq1 ,q2 ...qt = 0; in
every case therefore

                    Pq1 ,q2 ...qt × (x − hq1 )(x − hq2 ) . . . (x − hqt ) = 0.

Therefore St as expressed in equation (18) is a syzygetic function of f and ϕ;
and we have found a function of the tth degree in x, and of course expressible by
calculating the symmetric functions as a function only of x and of the coefficients
of f and ϕ, which will satisfy the equation

                                      τt f − tt ϕ + St = 0.
                                                                                                     p. 460
   It will be remembered that by virtue of Art. 2 we know à priori that all the
values of St satisfying this equation are identical, save as to an allotrious factor,
which is a function only of the coefficients in f and ϕ.
   It is clear that we may interchange the h and η, m and n, and thus another
representation of a value of St satisfying the equation (17) will be

                                  (ηqt+1 − h1 )(ηqt+1 − h2 ) . . . (ηqt+1 − hm ) 
                                                                                
                                
                                                                                
                                 (η
                                
                                         − h )(η      − h ) . . . (η      −h ) 
                                                                                 
                                     qt+2        1     qt+2      2       qt+2    m
St = ΣR(ηq1 , ηq2 . . . ηqt )                                                            (x−ηq1 )(x−ηq2 ) . . . (x
                                
                                                ············                        
                                                                                     
                                     (ηqn − h1 )(ηqn − h2 ) . . . (ηqn − hm )
                                
                                                                                    
                                                                                     


Art. (15).     If we employ in general the condensed notation
                                     "                           #
                                         l m n ...p
                                                                     ,
                                         λ µ ν ...ρ

                                                     464
to denote the product of the differences resulting from the subtraction of each
of the quantities λ, µ . . . ν in the lower line from all of those in the upper line
l, m, n . . . p, the two values above given for St may be written under the respective
forms
                                "                                  #
                                     hqt+1 , hqt+2 . . . hqm
   ΣR(hq1 , hq2 . . . hqt )                                            (x − hq1 )(x − hq2 ) . . . (x − hqt ),
                                         η1 , η2 . . . ηn

and
                                 "                                #
                                     ηξt+1 , ηξt+2 . . . ηξn
      ΣR(ηξ1 , ηξ2 . . . ηξt )                                        (x − ηξ1 )(x − ηξ2 ) . . . (x − ηξt ),
                                        h1 , h2 . . . hm

in each of which equations disjunctively and in some order of relation each with
each
                         q1 , q2 , q3 . . . qm = 1, 2, 3 . . . m,
and
                                      ξ1 , ξ2 , ξ3 . . . ξn = 1, 2, 3 . . . n.
These two forms are only the two extremities of a scale of forms all equally well
adapted to express St ; for let v and v ′ be any two integers so taken as to satisfy
the equation
                                    v + v ′ = t,
and let R(· · ·; · · ·), where the dots denote any quantities whatever, be used to
denote a rational function which remains unaltered in value when any two of the
quantities under either of the two bars are mutually interchanged, then we may
write
                                                                                                            
                                                                              hqv+1 , hqv+2 . . . hqm         
                        R(hq1 , hq2 . . . hqv ; ηξ1 , ηξ2 . . . ηξv′ ) ×       ηξv′ +1 , ηξv′ +2 . . . ηξn
           St = Σ                                                                                                  .   (19)
                     ×(x − h )(x − h ) . . . (x − h )(x − η )(x − η ) . . . (x − η ) 
                             q1      q2             qv      ξ1      ξ2             ξv ′

                                                                                                                              p. 461
  For if, as above, we suppose x = ha = ηω , any term of St in which q1 , q2 . . . qv
comprise among them a, or in which ξ1 , ξ2 . . . ξv′ comprise among them ω, will
vanish by virtue of the factors

          (x − hq1 )(x − hq2 ) . . . (x − hqv ) × (x − ηξ1 )(x − ηξ2 ) . . . (x − ηξv′ );

but if neither a nor ω is so comprised, then a must be one of the terms in the com-
plementary series qv+1 , qv+2 . . . qm , and ω one of the terms in the complementary
series ξv′ +1 , ξv′ +2 . . . ξn , and therefore one of the quantities hqv+1 , hqv+2 . . . hqm
will equal one of the quantities ηξv′ +1 , ηξv′ +2 . . . ηξn , and consequently the term
of St in question will vanish by virtue of the factor
                                         "                                       #
                                             hqv+1 , hqv+2 . . . hqm
                                             ηξv′ +1 , ηξv′ +2 . . . ηξn

                                                          465
vanishing. In either case therefore every term included within the sign of
summation vanishes when x = ha = ηω , that is, whenever f x = 0 and ϕx =
0. Hence St , as given by equation (19), will satisfy the syzygetic equation
τt f − tt ϕ + St = 0 for all values of v and v ′ which make v + v ′ = t, and for all
symmetrical forms of the function denoted by the symbol R(· · ·; · · ·).
Art. (16).         I shall now proceed to show how to assign the arbitrary func-
tion whose form is denoted by this symbol in such a manner as to make St
become identical with a simplified residue to f and ϕ. To this end I take for
R(hq1 , hq2 . . . hqv ; ηξ1 , ηξ2 . . . ηξv′ ) the value
                                                      "                                   #
                                                           hq1 , hq2 . . . hqv
                                                           ηξ1 , ηξ2 . . . ηξv′
                  R= "                                                #       "                                     #;               (20)
                                hq1 , hq2 . . . hqv                                  ηξ1 , ηξ2 . . . ηξv′
                                                                          ×
                              hqv+1 , hqv+2 . . . hqm                             ηξv′ +1 , ηξv′ +2 . . . ηξn

we shall then have
                                                                                                      
                                                                   hqv+1 , hqv+2 . . . hqm
                          h                           i
                              hq1 , hq2 . . . hqv
                                                          ×        ηξv′ +1 , ηξv′ +2 . . . ηξn
                              ηξ1 , ηξ2 . . . ηξv′
             St = Σ h                                                                                     
                            hq1 , hq2 . . . hqv
                                                           i                 η ξ1 , η ξ2 . . . η ξv ′                                    (21)
                                                               ×
                          hqv+1 , hqv+2 . . . hqm                         ηξv′ +1 , ηξv′ +2 . . . ηξn
                       × {(x − hq1 )(x − hq2 ) . . . (x − hqv )(x − ηξ1 )(x − ηξ2 ) . . . (x − ηξv′ )}.

I shall first show this sum of fractions is in substance an integral function of
the quantities h1 , h2 . . . hm ; η1 , η2 . . . ηn . For greater conciseness write in general
x − h = E, x − η = H; we have then, since
     h − η = H − E,                   hqr − hqs = Eqs − Eqr ,                                  ηξr − ηξs = Hξs − Hξr ,
                                                                                             
                                                      Hξv′ +1 , Hξv′ +2 . . . Hξn
              h                           i
                  Hξ1 , Hξ2 . . . Hξv′
                                              ×
                  Eq1 , Eq2 . . . Eqv                 Eqv+1 , Eqv+2 . . . Eqm
   St = Σ h                                                                                       Eq1 . . . Eqv Hξ1 . . . Hξv′ .       (22)
                                                          Hξv′ +1 , Hξv′ +2 . . . Hξn
                                              i
               Eqv+1 , Eqv+2 . . . Eqm
                                                  ×
                 Eq1 , Eq2 . . . Eqv                        Hξ1 , Hξ2 . . . Hξv′
                                                                                                                                                p. 462
   On reducing the fractions contained within the sign of summation to a common
denominator, St will take the form D·∆      N
                                               , where D will be the product of the
1
2 m(m   − 1) differences of E1 , E2 . . . Em  subtracted each from each, and ∆ the
corresponding product of the differences inter se of H1 , H2 . . . Hn . Hence, unless
the sum in question is an integral function of the E’s and H’s it will become
infinite when any two of the E series, or any two of the H series of quantities are
made equal. Suppose now E1 = E2 ; the terms in (22) which contain E1 − E2
in the denominator will evidently group themselves into pairs of the respective
forms,
                                                                                                                                    
                                                                                                         E2 , Eqv+2 . . . Eqm
                                                          h                                i
                                                                E1 , Eq3 . . . Eqv
    (E1 Eq3 . . . Eqv ) × (Hξ1 Hξ2 . . . Hξv′ ) ×                                              ×       Hξv′ +1 , Hξv′ +2 . . . Hξn
                                                               Hξ1 , Hξ2 . . . Hξv′
                                                                                                              
                                                                             Hξ1 , Hξ2 . . . Hξv′
                          h                                   i
                                E1 , Eq3 . . . Eqv
                                                                  ×
                               E2 , Eqv+2 . . . Eqm                        Hξv′ +1 , Hξv′ +2 . . . Hξn


                                                                   466
and
                                                                                                                                                                             
                                                                                                                                E1 , Eqv+2 . . . Eqm
                                                                          h                                      i
                                                                               E2 , Eq3 . . . Eqv
      (E2 Eq3 . . . Eqv ) × (Hξ1 Hξ2 . . . Hξv′ ) ×                                                                  ×        Hξv′ +1 , Hξv′ +2 . . . Hξn
                                                                              Hξ1 , Hξ2 . . . Hξv′
                                                                                                                                                                                ;
                                                                                            Hξ1 , Hξ2 . . . Hξv′
                                    h                                         i
                                         E2 , Eq3 . . . Eqv
                                                                                  ×
                                        E1 , Eqv+2 . . . Eqm                              Hξv′ +1 , Hξv′ +2 . . . Hξn


the sum of this pair of terms will be of the form
               "                          #       "                                  #                 "                          #       "                                  #
                           E1                                      E2                                              E2                                      E1
                                               ×                                                                                       ×
    
                                                                                                                                                                              
                                                                                                                                                                               
                                                                                                                                                                              
    
  P   E1           Hξ1 , Hξ2 . . . Hξv′                Hξv′ +1 , Hξv′ +2 . . . Hξn             E2          Hξ1 , Hξ2 . . . Hξv′                Hξv′ +1 , Hξv′ +2 . . . Hξn    
                                                                                                                                                                               
                                "                                   #                     +                              "                                  #                         ,
    E1 − E2
  Q                                               E1                                         E2 − E1                                      E2
                                                                                                                                                                                 
                                                                                                                                                                                  
                                                                                                                                                                                 
                                                                                                                                                                                  
                                   Eqv+1 , Eqv+2 . . . Eqm                                                                  Eqv+1 , Eqv+2 . . . Eqm                              



where Q, it may be observed, does not contain H1 − H2 , so that Q
                                                                P
                                                                  remains finite
when H1 = H2 .
  The above pair of terms together make up a sum of the form

                                    P    1    ϕ(E1 , E2 )ψE2 − ϕ(E2 , E1 )ψE1
                                                                              ,
                                    Q E1 − E2            ψE1 × ψE2
which, as the numerator of the third factor vanishes when E1 = E2 , remains
finite on that supposition. Hence the whole sum of terms in (22) which is          p. 463
   made up of such pairs of terms, and of other terms in which E1 − E2 does not
enter, remains finite when E1 − E2 = 0, and therefore generally when D = 0, and
similarly when H1 − H2 = 0, and therefore also when ∆ = 0; hence the expression
for St in (22) is an integral function of the E and H series of quantities, as was
to be proved.
Art. (17). Let us now proceed to determine the dimensions of the coefficient
of xt , the highest power of x in this value of St , when supposed to be expressed
under the form of an integral function (as it has been proved to be capable of
being expressed) of h1 , h2 . . . hm ; η1 , η2 . . . ηn ; x.
   This coefficient is the sum of fractions the numerators of each of which consist
of two factors, which are respectively of v ×v ′ and of (m−v)×(n−v ′ ) dimensions
in respect of the two sets of roots taken conjointly, and the denominators of two
factors respectively of v(m − v) and v ′ (n − v ′ ) dimensions in respect of the same.
   Consequently, the exponent of the total dimensions of the coefficient in question

                             = vv ′ + (m − v)(n − v ′ ) − v(m − v) − v ′ (n − v ′ )
                             = (m − v − v ′ ) × (n − v − v ′ )
                             = (m − t)(n − t),

and thus is seen to depend only on the degree t in x of St , and not upon the
mode of partitioning t into two parts v and v ′ , for the purpose of representing
St by means of formula (19).


                                                                                      467
Art. (18). I shall now demonstrate that every form in this scale (to a numerical
factor près) is identical with a simplified residue to f, ϕ, of the same degree t in
x. Any such simplified residue is, like St , a syzygetic function, or to use a briefer
form of speech a conjunctive of f, ϕ; and if we agree to understand by the “weight”
of any function of the coefficients of f and ϕ its joint dimensions in respect of
the roots of f and ϕ combined, I shall prove,—first, that any simplified residue
of f and ϕ of a given degree in x is that conjunctive, whose weight in respect of
the roots of f and ϕ is less than the weight of any other such conjunctive; and
second, that St , as determined above (in equation 22), is of the same weight as
the simplified residue, and can therefore only differ from it by some numerical
factor. For the purpose of comparison of weights, it will of course be sufficient to
confine our attention to the coefficients of the highest power in x (or any other,
the same for each) of the forms whose weights are to be compared.
   Suppose f to be of m dimensions, and ϕ to be of n dimensions in x; and let
m = n + e.                                                                             p. 464
   Suppose
                        Λf + Lϕ = Axt + Bxt−1 + · · · + K,
                           Λ = λ0 xq + λ1 xq−1 + · · · + λq ,                       (23)
                        L = l0 x   q+e
                                         + l1 x   q+e−1
                                                          + · · · + lq+e ,
the number of homogeneous equations to be satisfied by the q + 1 quantities
λ0 , λ1 . . . λq , and the q + e + 1 quantities l0 , l1 . . . lq+e will be m + q − t, and
therefore q + 1 and q + e + 1 taken together must be not less than m + q − t + 1,
that is 2q + e + 2 must be not less than q + m − t + 1, that is q not less than
m − t − e − 1; and if this inequality be satisfied 2q + e + 2 − (q + m − t + 1) + 1,
that is q + t + e − m + 2 will be the number of arbitrary constants entering into
the solution of equation (23).
    If q be greater than (n − 1), let q = (n − 1) + t; and let

                       (Λ) = λ0 xn−1 + λ1 xn−2 + · · · + λn−1 ,

                    (L) = l0 xn+e−1 + l1 xn+e−2 + · · · + le+n−1 ;
and let (Λ), (L) be so taken as to satisfy the equation

                      (Λ)f + (L)ϕ = Axt + Bxt−1 + · · · + K;

and make
                        Ξ = (Λ) + (f + gx + · · · + hxt−1 )ϕ,
                        X = (L) − (f + gx + · · · + hxt−1 )f,
f, g . . . h being arbitrary constants; then

               Ξf + Xϕ = (Λ)f + (L)ϕ = Axt + Bxt−1 + · · · + K.


                                             468
Now the total number of arbitrary constants in the system (Λ) and (L) will
be n − 1 + t + e − m + 2, that is t + 1; hence the total number of arbitrary
constants in Ξ and X will be t + 1 + t, that is q − n + t + 2, which is equal
to q + t + e − m + 2, the number of arbitrary constants in the most general
values of Λ and L. Hence (Λ = Ξ, L = X) is the general solution of the equation
Λf + Lϕ = Axt + Bxt−1 + · · · + K; and consequently the most general form
of Axt + Bxt−1 + · · · + K, which is evidently independent of the (t) arbitrary
quantities f, g . . . h, will contain the same number of arbitrary constants as enter
into the system (Λ) and (L), that is t + 1.
Art. (19). Let us now begin with the case of greater simplicity when m = n,
that is e = 0; and let us revert to the system of equations marked (10) in Section
I., in which U and V are to be replaced by f and ϕ.
    First, let t = n − 1, then t + 1, the number of arbitrary quantities, in the
conjunctive, is n.
    From the system of equations (10) we have, for all values of ρ1 , ρ2 , ρ3 . . . ρn ,

(ρ1 Q0 + ρ2 Q1 + · · · + ρn Qn−1 )f − (ρ1 P0 + ρ2 P1 + · · · + ρn Pn−1 )ϕ
                                    = (ρ1 K1 + ρ21 K1 + · · · + ρnn−1 K1 )xn−1 + &c.,
                                                                                            p. 465
   and consequently the most general value of Sn−1 in the equation

                             τn−1 f − tn−1 ϕ + Sn−1 = 0,

where
                         Sn−1 = Axn−1 + Bxn−2 + · · · + L,
will be obtained by making

                      τn−1 = ρ1 Q0 + ρ2 Q1 + cdots + ρn Qn−1 ,

                       tn−1 = −ρ1 P0 − ρ2 P1 − · · · − ρn Pn−1 ,
which solution contains n, that is the proper number of arbitrary constants.
   Again, if t = n − 2, t + 1 = n − 1, which will therefore be the number of
arbitrary constants in the most general value of Sn−2 in the equation

                             τn−2 f − tn−2 ϕ + Sn−2 = 0.

This most general value of Sn−2 is therefore found by making

                       τn−2 = ρ′1 Q0 + ρ′2 Q1 + · · · + ρ′n Qn−1 ,

                       tn−2 = −ρ′1 P0 − ρ′2 P1 − · · · − ρ′n Pn−1 ,
where ρ′1 , ρ′2 . . . ρ′n are no longer entirely independent, but subject to the equation

                        ρ′1 K1 + ρ′2 1 K1 + · · · + ρ′n n−1 K1 = 0,

                                           469
so as to leave (n − 1) constants arbitrary.
   We thus obtain

               Sn−2 = (ρ′1 K2 + ρ′2 1 K2 + · · · + ρ′n n−1 K2 )xn−2 + &c.

In like manner, and for the same reasons, the most general values of Sn−3 in
the equation
                         τn−3 f − tn−3 ϕ + Sn−3 = 0,
will be found by making

                        τn−3 = ρ′′1 Q0 + ρ′′2 Q1 + · · · + ρ′′n Qn−1 ,

                        tn−3 = −ρ′′1 P0 − ρ′′2 P1 − · · · − ρ′′n Pn−1 ,
where ρ′′1 , ρ′′2 . . . ρ′′n are subject to satisfying the two equations

                         ρ′′1 K1 + ρ′′2 1 K1 + · · · + ρ′′n n−1 K1 = 0,
                         ρ′′1 K2 + ρ′′2 1 K2 + · · · + ρ′′n n−1 K2 = 0,

so as to leave (n − 2) constants arbitrary; and we thus obtain

               Sn−3 = (ρ′′1 K3 + ρ′′2 1 K3 + · · · + ρ′′n n−1 K3 )xn−3 + &c.,

and so on, the number of independent arbitrary constants in S decreasing (as
it ought) each time by one unit as the degree of S descends, until finally, if
τ0 f − t0 ϕ + S0 = 0, S0 being a constant, the general value of S0 is found by
making
                    τ0 = (ρ1 )Q0 + (ρ2 )Q1 + · · · + (ρn )Qn−1 ,
                      t0 = −(ρ1 )P0 − (ρ2 )P1 − · · · − (ρn )Pn−1 ,
                                                                                   p. 466
   where (ρ1 ), (ρ2 ) . . . (ρn ) are subject to satisfy the (n − 1) equations

                                    (ρ1 )K1 + &c. = 0,
                                    (ρ1 )K2 + &c. = 0,
                                        ············
                                  (ρ1 )Kn−1 + &c. = 0,

which gives
                     S0 = Kn (ρ1 ) + 1 Kn (ρ2 ) + · · · + n−1 Kn (ρn ).
Now evidently the lowest weight in respect to the roots of U and V that can
be given to (ρ1 K1 + ρ21 K1 + · · · + ρnn−1 K1 )xn−1 + &c., when the multipliers
ρ1 , ρ2 . . . ρn are absolutely independent, is found by taking

                      ρ1 = 1,        ρ2 = 0,        ρ3 = 0 . . . ρn = 0,

                                             470
which makes the weight of the leading coefficient in Sn−1 , the same as that of
K1 , that is 1.
  Again, when one equation,
                             ρ′1 K1 + ρ′2 1 K1 + · · · + ρ′n n−1 K1 = 0,
exists between the (ρ)’s, the lowest weight will be found by making
              ρ′1 = 1 K1 ,      ρ′2 = −K1 ,        ρ′3 = 0,      ρ′4 = 0 . . . ρ′n = 0,
which makes the weight of the leading coefficient in Sn−2 depend on

                                        1 K1 K2 − K11 K2 ,

which is of the weight 1 + 3, that is 4, in respect of the roots of f and ϕ.
  Similarly, Sn−3 will have its lowest weight when its leading coefficient is the
determinant
                                 K1 K2 K3
                                1 K1 1 K2 1 K3 ,
                                2 K1 2 K2 2 K3

the weight of which is 1 + 3 + 5 = 9; and finally, the lowest weighted value of S0
is the determinant represented by the complete Bezoutian square; the weight in
general of Sn−t being 1 + 3 + · · · + (2t − 1), that is t2 , or which is the same thing
otherwise expressed, the weight of the leading coefficient of the lowest-weighted
conjunctive of f and ϕ of the degree t in x is (n − t)(m − t)238 . It will of course
have been seen in the foregoing demonstration, that the weight of r Ks [which
means Σ(ar bs − as br ), ar , as being the coefficients of xn−r , xn−s in f , and br , bs
of the same in ϕ] has been correctly taken to be r + s in respect of the roots of
f and ϕ conjoined.                                                                        p. 467

Art. (20). If now we proceed in like manner with the general case of m = n + e,
it may be shown, in precisely the same way as in the preceding article, that the
most general value of any conjunctive of f and ϕ will be a linear function of e
functions,
                       xn + a1 xn−1 + a2 xn−2 + · · · + an ,
                      xn+1 + a1 xn + a2 xn−1 + · · · + an x,
                     xn+2 + a1 xn+1 + a2 xn + · · · + an x2 ,
                                ··············· ,
                      x m−1 + a1 xm−2 + &c. + an xe−1 ,
and of the n functions,
                       K1 xn−1 +           K2 xn−2 +  ···+           Kn ,
                             n−1 +               n−2 +···+
                      1 K1 x              1 K2 x                    1 Kn ,
                          &c.                          &c.
                              n−1 +             n−2 + · · · +
                     n−1 1 x
                         K                 K
                                        n−1 2 x                    n−1 Kn ,
 238
       n and m are supposed equal and t = n − i.


                                                471
and that consequently, if the degree of such conjunctive in x be (n − i), it will
be of the lowest weight when it is a linear function of the entire e upper set of
functions, and i of the lower set; and consequently, the coefficient of the highest
power of x in such conjunctive will be the determinant
                     K1        K2       K3         ···        Ki         · · · Ki+e
                    1 K1       1 K2     1 K3       ···        1 Ki      · · · 1 Ki+e
                    2 K1       2 K2     2 K3       ···        2 Ki      · · · 2 Ki+e
                     ···        ···      ···                  ···             ···
                   i−1 K1     i−1 K2   i−1 K3        ···     i−1 Ki   · · · i−1 Ki+e
                                                                                       ,
                      1         a1       a2     · · · ai−1    ai       · · · ai+e−1
                      0          1       a1     · · · ai−1    ai       · · · ai+e−2
                      0          0        1     · · · ai−2    ai       · · · ai+e−3
                     ···        ···      ···                  ···             ···
                      0          0        0        ···         1            · · · ai

the weight of which is evidently that of

                            K1 × K2 × K3 × · · · × Ki−1 × (ai )e ,

that is
                               1 + 3 + 5 + · · · + (2i − 1) + ei,
that is i2 + ei, or i(e + i), which is (n − t)(m − t) if t = n − i.                 p. 468
   Hence the weight of the leading coefficient in the lowest-weighted conjunctive
of f and ϕ of the degree t in x is (m − t)(n − t), m being the degree of f and n
of ϕ.
   From this we infer that any conjunctive of f and ϕ of the degree t, of which
the leading coefficient is of the weight (m − t)(n − t), all the coefficients being
of course understood to be integral functions of the roots of f and ϕ, must, to
a numerical factor près, be equivalent to any other of the same weight; and
furthermore, any supposed function of x of the tth degree which possesses
the property characteristic of a conjunctive of vanishing when f and ϕ vanish
simultaneously, but of which the weight of the leading coefficient would be less
than (m − t)(n − t), must be a mere nugatory form and have all its terms
identically zero239 .
Art. (21). We have previously shown, Art. 16, that S, as defined by equation
(21), is an integral function of the roots f and ϕ, and vanishes when f and ϕ
vanish. Moreover, its weight in the roots has been proved to be (m − t)(n − t),
and consequently, if by way of distinguishing the several forms of St we name
that one where t in the equation above cited is supposed to be divided into
two parts, v and v ′ , Sv,v′ , we have for all values of v and v ′ , such that v + v ′
is not greater than n, Sv,v′ to a constant numerical factor près identical with
 239
    And more generally it admits of being demonstrated by precisely the same course of
reasoning, that the number of arbitrary parameters in a conjunctive of the degree t, and of the
weight (m − t)(n − t) + e in the roots, cannot (abstraction being supposed to be made of an
arbitrary numerical multiplier) exceed the number e.



                                                472
the (v + v ′ )th simplified residue to (f, ϕ), so that the form of Sv,v′ depends only
upon the value of v + v ′ .
Art. (22). It must be well borne in mind that this permanency of the value of
Sv,v′ for different values of v has only been established for the case where t can
be the degree of a residue to f and ϕ, that is to say, when t is less than the lesser
of the two indices m and n. When t does not satisfy this condition of inequality,
the theorem ceases to be true. It is clear that when m = n and v + v ′ = m = n,
Sv,v′ , which always remains a conjunctive of f and ϕ, can only be a numerical
linear function of f and ϕ; and I have ascertained when m = n on giving to
v and v ′ the respective values successively (0, n), (1, n − 1), (2, (n − 2)) . . . (n, 0)
that
                                                             (n − 1)(n − 2)
S0,n = f,       S1,n−1 = (n−1)f +ϕ,            S2,n−2 =                     f +(n−1)ϕ, . . .
                                                                  1·2
                       Sn−1,1 = f + (n − 1)ϕ,               Sn,0 = ϕ.
Thus, by way of a simple example, let

                        f = x2 + ax + b = (x − h1 )(x − h2 ),

                        ϕ = x2 + αx + β = (x − k1 )(x − k2 ),
                                                                                               p. 469


                              "           #   "            #
                                   h1 h2            ···
                              
                                    ···            k1 k2
   S0,2 = (x − h1 )(x − h2 )                            = (x − h1 )(x − h2 ) = f,
                                                       
                             "            #×"         #
                                   ···            ··· 
                                   h1 h2           k1 k2
                                                        "        #
                                                            h1
                                                            k1
                  S1,1 = Σ(x − h1 )(x − k1 ) "          #        "        #
                                                   h1                k1
                                                            ×
                                                   h2                k2
                             x − h1 x − k1
                        =Σ                   {(h1 − k1 )(h2 − k2 )},
                             h1 − h2 k1 − k2




                                           473
that is
                   x − h1       1
                            
          S1,1 = Σ                   {(x − k1 )(h1 − k1 )(h2 − k2 )
                   h1 − h2 k1 − k2
                 −(x − k2 )(h1 − k2 )(h2 − k1 )}]
                   x − h1
               =Σ          {(h1 − h2 )x + [(k1 + k2 )h2 − (h1 h2 + k1 k2 )]}
                   h1 − h2
               = (x − h1 )x + (x − h2 )x − (k1 + k2 )x + (h1 h2 + k1 k2 )
               = x2 − (h1 + h2 )x + h1 h2 + x2 − (k1 + k2 )x + k1 k2
                                               

               = (x2 + ax + b) + (x2 + αx + β)
               = f + ϕ;

so we find also S2,0 = ϕ.
Art. (23). The expression Sv,v′ , which is universally a conjunctive of f and ϕ,
continues algebraically interpretable so long as v + v ′ has any value intermediate
between 0 and m + n; when v + v ′ = 0 we must of course have v = 0 and v ′ = 0,
and S0,0 becomes the resultant of f and ϕ; when v + v ′ = m + n we must also
have the unique solution v = m and v ′ = n, and Sm,n becomes necessarily f × ϕ,
which we thus see stands in a sort of antithetical relation to the resultant of f
and ϕ, say (f, ϕ). Nor is it without interest to remark that f × ϕ = 0 implies that
a factor of f or else of ϕ is zero; and (f, ϕ) = 0 implies that if a factor of the one
of the functions is zero, so also is a factor of the other, that is that a factor of
each or of neither is zero. As t increases from 0 to n or decreases from m + n to
m − 1, the number of solutions of the equation v + v ′ = t in the one case, and the
number of admissible solutions of the equation v + v ′ = t in the other case, which
is subject to the condition that v must not exceed n, continues to increase by a
unit at each step; there being thus n + 1 different forms Sv,v′ when v + v ′ = n,
and the same number when v + v ′ = m − 1. For all values of t intermediate
between n and (m − 1) (both taken exclusively) it is very remarkable that Sv,v′
will vanish, as I proceed to demonstrate.                                              p. 470

Art. (24). The weight of the coefficient of the highest power of Sv,v′ (v + v ′
being equal to t) is (m − t)(n − t), and consequently, when t is greater than
n, and less than m, Sv,v′ would contain fractional functions of the roots of f
and ϕ, if there were in it a power xt , but Sv,v′ has been proved to be always
an integer function of the roots. Hence the coefficient of xt will be zero, and so
more generally the first power of x in Sv,v′ of which the coefficient is not zero,
will be xt−ω , subject to the condition (since evidently the weight of the several
coefficients goes on increasing by units as the degree of the terms in x decreases
by the same) that ω be not less than (m − t)(t − n); let then ω = (m − t)(t − n),
Sv,v′ becomes of the form Axt−ω + Bxt−ω−1 + &c., where A is of zero dimensions;
but this is impossible if t − ω < n, for then Axt−ω + &c. is a conjunctive of weight
lower than the lowest-weighted simplified residue of the degree t − ω. Hence ω

                                         474
is not greater than t − n, that is (m − t)(t − n) is not greater than t − n, that
is m − t cannot be greater than 1, that is t when intermediate between m and
n cannot be less than m − 1, otherwise Sv,v′ will vanish identically. Moreover,
when t = m − 1, ω = t − n, and t − ω = n, and accordingly Sv,m−1−v is not
merely, as we might know, à priori an algebraical, but more simply a numerical
multiple of ϕ for all values of v. The same is of course true also, m being greater
than n, for every form Sn,m−n , since this is always a conjunctive of f and ϕ
of which the former is of a degree higher than the S in question, so that the
multiplier of f in this conjunctive must be zero240 .
Art. (25). To enter into a further or more detailed examination of the values
assumed by Sv,v′ for the most general values of m, n, t, would be to transcend
the limits I have proposed to myself in drawing up the present memoir. What
we have established is, that to every form of Sv,t−v appertaining to a value of t
between 0 and n, there is a sort of conjugate form for which t lies between m + n
and m; that for t = m − 1 or t = n, Sv,t−v becomes a numerical multiplier of
ϕ; and that when t lies in the intermediate region between n and m − 1, Sv,t−v
vanishes for all values of v. I pause only for a moment to put together for the
purpose of comparison the forms corresponding to t and to m + n − t. By Art. 16,
making t = v + v ′ ,
                                                                                                             
                                                                                                                 hqv+1 , hqv+2 . . . hqm
                                                                          h                          i
                                                                              hq1 , hq2 . . . hqv
                                                                                                         ×       ηξv′ +1 , ηξv′ +2 . . . ηξn
                                                                              ηξ1 , ηξ2 . . . ηξv′
St = Σ(x−hq1 )(x−hq2 ) . . . (x−hqv )(x−ηξ1 )(x−ηξ2 ) . . . (x−ηξv′ ) h                                          
                                                                                                                        ηξ1 , ηξ2 . . . ηξv′
                                                                                                         i
                                                                            hq1 , hq2 . . . hqv
                                                                                                             ×
                                                                          hqv+1 , hqv+2 . . . hqm                    ηξv′ +1 , ηξv′ +2 . . . ηξn

                                                                                                                                 p. 471
   The conjugate form for which t′ = m + n − t and m − v, n − v ′ , vv ′ take the
places of v, v ′ and (m − v)(n − v ′ ), will be got by taking
                                                                                                                                 
                                                                                                                                     hqv+1 , hqv
                                                                                          h                              i
                                                                                              hq1 , hq2 . . . hqv
                                                                                                                             ×       ηξv′ +1 , ηξv
                                                                                              η ξ1 , η ξ2 . . . η ξv ′
St′ = Σ(x−hqv+1 )(x−hqv+2 ) . . . (x−hqm )(x−ηξv′ +1 )(x−ηξv′ +2 ) . . . (x−ηξn ) h                                                  
                                                                                                                                            ηξ1 , η
                                                                                                                             i
                                                                                            hq1 , hq2 . . . hqv
                                                                                                                                 ×
                                                                                          hqv+1 , hqv+2 . . . hqm                        ηξv′ +1 , η


which it will be perceived are identical, term for term, in the fractional constant
factor, and differ only in the linear functions of x, which in St and in St′ are
complementary to one another. Our proper business is only with those forms for
which t < n.
Art. (26). It will presently be seen to be necessary to ascertain the numerical
relations between S0,t and S1,t−1 when t < n, and this naturally brings under
our notice the inquiry into the numerical relations which exist between the entire
series of forms Sv,t−v for a given value of t, corresponding to all values of v
between 0 and t inclusive.
 240
     It thus appears that if the indices m and n do not differ by at least 3 units, S will have
an actual quantitative existence for all values of t between 0 and m + n; or in other words,
the failure in the quantitative existence of the forms St only begins to show itself when this
difference is 3; thus if m = n + 3, Sn exists, and Sn+2 exists, but Sn+1 = 0.


                                                    475
   In order to avoid a somewhat oppressive complication of symbols, I shall take
a particular numerical example, that is m = 7, n = 6, t = 4, and compare the
values of S0,4 , S1,3 , S2,2 , S3,1 , S4,0 , all of which we know to be identical [to a
numerical factor près] with one another and with the second simplified residue
to f and ϕ, that being of the fourth degree in x; our object in the subjoined
investigation is to determine the numerical ratios of these several forms of S to
one another.
   First, let v = 0, v ′ = 4. The leading coefficient S0,4 is
                                     "                                #
                                                η5 η6
                                         h1 h2 h3 h4 h5 h6 h7
                                 Σ         "                    #         ,
                                                  η5 η6
                                               η1 η2 η3 η4

which we know à priori (it should be observed) to be essentially an integral
function of the h and the η system. In this, the term containing η63 will be
evidently                   "                      #
                                       η5
                              h1 h2 h3 h4 h5 h6 h7
                          Σ     "              #     ,                   (A)
                                       η5
                                  η1 η2 η3 η4
the η system to which the latter summation relates being now reduced to consist
of η1 , η2 , η3 , η4 , η5 . In this expression, again, the coefficient of η53 is evidently 1.
Hence, therefore, the leading coefficient in S0,4 contains the term η53 η63 .                 p. 472
   Secondly, let v = 1, v ′ = 3. The leading coefficient in S1,3 becomes
                            "              #       "                           #
                                η1 η2 η3                   η4 η5 η6
                                               ×
                                  h1                   h2 h3 h4 h5 h6 h7
                          Σ"                            #       "              #.
                                h2 h3 h4 h5 h6 h7                   η4 η5 η6
                                                            ×
                                       h1                           η1 η2 η3

In this, the factor affecting η63 will be
                            "              #       "                           #
                                η1 η2 η3                     η4 η5
                                               ×
                                  h1                   h2 h3 h4 h5 h6 h7
                          Σ"                            #       "              #,
                                h2 h3 h4 h5 h6 h7                    η4 η5
                                                            ×
                                       h1                           η1 η2 η3

η6 being now understood to be eliminated out of the η system included within




                                                   476
the above summation. Again, in this latter sum the factor affecting η53 will be
                           "               #        "                            #
                               η1 η2 η3                        η4
                                               ×
                                 h1                     h2 h3 h4 h5 h6 h7
                         Σ"                              #       "               #.   (B)
                               h2 h3 h4 h5 h6 h7                        η4
                                                             ×
                                      h1                             η1 η2 η3

η5 and η6 being now both eliminated out of the η system. This last sum can of
course only represent a numerical quantity.
   So in like manner, again, if v = 2, v ′ = 2, the coefficient of η63 η53 in S2,2 will
be similarly reducible to the form
                               "           #        "                        #
                                   η1 η2                    η3 η4
                                               ×
                                   h1 h2                h3 h4 h5 h6 h7
                           Σ"                            #       "           #.       (C)
                                   h3 h4 h5 h6 h7                    η3 η4
                                                             ×
                                       h1 h2                         η1 η2

So, again, when v = 3, v ′ = 1, the coefficient of η63 η53 in S3,1 will be
                               "                #       "                    #
                                      η1                      η2 η3 η4
                                                    ×
                                   h1 h2 h3                  h4 h5 h6 h7
                           Σ"                       #        "               #;       (D)
                                   h4 h5 h6 h7                   η2 η3 η4
                                                        ×
                                    h1 h2 h3                        η1

and finally, the coefficient of η63 η53 in S0,4 will be
                                           "                     #
                                               η1 η2 η3 η4
                                                h5 h6 h7
                                      Σ"                         #,                   (E)
                                                h5 h6 h7
                                               h1 h2 h3 h4

out of all which sums it is to be remembered that η5 and η6 are supposed excluded
from appearing. All these several coefficients being numbers in disguise, we may
determine them by giving any values at pleasure to the terms in the h and η
system.                                                                           p. 473
   Let now η1 = h1 , η2 = h2 , η3 = h3 , η4 = h4 , then in (B) it will readily be
seen that all the terms included within the sign of summation vanish identically,




                                                    477
except the following, namely,
                                        h                  i        h                           i
                                            η1 η2 η3                           η4
                                                                ×
                                              h4                        h1 h2 h3 h5 h6 h7
                                        h                                i       h              i,
                                          h1 h2 h3 h5 h6 h7             η4
                                                               ×
                                        h        h4i h               η1 η2 η3 i
                                          η1 η2 η4                 η3
                                                      ×
                                            h3              h1 h2 h4 h5 h6 h7
                                        h                                i       h              i,
                                          h1 h2 h4 h5 h6 h7             η3
                                                               ×
                                        h        h3i h               η1 η2 η4 i
                                          η1 η3 η4                 η2
                                                      ×
                                            h2              h1 h3 h4 h5 h6 h7
                                        h                                i       h              i,
                                          h1 h3 h4 h5 h6 h7             η2
                                                               ×
                                        h        h2i h               η1 η3 η4 i
                                          η2 η3 η4                 η1
                                                      ×
                                            h1              h2 h3 h4 h5 h6 h7
                                        h                                i       h              i.
                                            h2 h3 h4 h5 h6 h7                           η1
                                                                             ×
                                                   h1                                η2 η3 η4

In each of these expressions the first factor of the numerator is identical in value
(by reason of the equations h1 = η1 , h2 = η2 , h3 = η3 , h4 = η4 ) with (−)3 × the
second factor of the denominator, and the second factor of the numerator with
(−)6 × the first factor of the denominator; hence the coefficient of η53 η63 in S1,3 is
−4.
   In like manner the only effective terms of S2,2 will be
               h            i       h                           i         h             i       h                        i
                    η1 η2                   η3 η4                             η3 η4                     η1 η2
                                ×                                                           ×
                    h3 h4               h1 h2 h5 h6 h7                        h1 h2                 h3 h4 h5 h6 h7
                h                           i       h           i,        h                          i       h           i,
                 h1 h2 h5 h6 h7          η3 η4                              h3 h4 h5 h6 h7          η1 η2
                                   ×                                                          ×
               h     h3 i
                        h4 h             η1 η2 i                          h     h1 i
                                                                                   h2 h             η3 η4 i
                 η1 η3              η2 η4                                   η2 η4              η1 η3
                          ×                                                          ×
                 h2 h4          h1 h3 h5 h6 h7                              h1 h3          h2 h4 h5 h6 h7
                h                           i       h           i,        h                          i       h           i,
                 h1 h3 h5 h6 h7          η2 η4                              h2 h4 h5 h6 h7          η1 η3
                                   ×                                                          ×
               h     h2 i
                        h4 h             η1 η3 i                          h     h1 i
                                                                                   h3 h             η2 η4 i
                 η1 η4              η2 η3                                   η2 η3              η1 η4
                          ×                                                          ×
                 h2 h3          h1 h4 h5 h6 h7                              h1 h4          h2 h3 h5 h6 h7
                h                           i       h           i,        h                          i       h           i.
                    h1 h4 h5 h6 h7                      η2 η3                 h2 h3 h5 h6 h7                     η1 η4
                                                ×                                                        ×
                        h2 h3                           η1 η4                     h1 h4                          η2 η3

Any other term will necessarily contain in the numerator a factor, whose symbol-
ical representation will contain one of the quantities η1 , η2 , η3 , η4 , in the upper
line, and one of the quantities h1 , h2 , h3 , h4 , having the same subscript           p. 474
   index, in the lower line, and which will therefore vanish; the number of effective
terms being evidently the number of ways in which four things can be combined
                                                                     2       2
2 and 2 together, and the value of each term is evidently (−)2 (−1)2 1, so that
the entire value of the coefficient of η53 η63 in S2,2 is +6.
   Precisely in the same manner, we shall find that the leading coefficient in S3,1
will contain the term −4η53 η63 , the (−1) resulting from the operation (−1)13 (−)34 ,
and in S4,0 the term +η53 η63 , the +1 resulting from the operation (−1)43 . Hence
it appears that S0,4 ; S1,3 ; S2,2 ; S3,1 ; S4,0 are to one another in the ratios of
1; −4; 6; −4; 1; and so in general for any values of m, n, t (t being less than m

                                                                    478
and less than n) it will be found that

                            S0,t ,    S1,t−1 ,       S2,t−2 . . . St,0

will be in the ratios of the numbers
                                     t(t − 1)                      t(t − 1)(t − 2)
1; quad(−1)m−t t;     (−1)2(m−t)              ;      (−1)3(m−t)                    ;            ...;   (−1)t(m−t) .
                                       1·2                             1·2·3

Art. (27).      The method employed in the preceding investigation will enable
us to affix the proper sign and numerical factor to S0,t , or in general to Sv,t−v ,
in order that it may represent the Bezoutian secondary of the degree t in x.
This latter has been already identified with the simplified residue obtained by
expanding ϕx f x under the form of an improper continued fraction. For this purpose,
it will be sufficient to compare a single term of any such S with the corresponding
one in the Symmorphic Bezoutian secondary. Let us first suppose that m = n, f
and ϕ being of the same degree. A glance at the form of the Bezoutian square
will show that if we form the Bezoutian secondary of the degree (n − i) in x,
                                                                            (i−1)i
the coefficient of its leading term will contain the term (−) 2 ; (0, i)i ; (0, i) as
usual denoting the product of the coefficient of xn in f by the coefficient of xn−i
in ϕ, less the product of the coefficient of xn in ϕ by that of xn−i in f ; and as
we suppose the first coefficients in f and ϕ to be each 1, if we term the other
coefficients last spoken of ai and αi respectively, this said coefficient of the leading
                                                                        (i−1)i
term of the ith Bezoutian secondary will contain the term (−) 2 (αi − ai )i ,
                          (i−1)i                  i(i+1)
and consequently (−1) 2 αii and (−) 2 aii .
  Now by the like reasoning to that employed in the preceding article, the
coefficient of the leading term in Sm−i,0 , that is
                                                             "                         #
                                                                 hq1 , hq2 . . . hqi
                                                                  η1 , η2 . . . ηm
           Σ(x − hqi+1 )(x − hqi+2 ) . . . (x − hqm ) "                                    #,
                                                                hq1 , hq2 . . . hqi
                                                              hqi+1 , hqi+2 . . . hqm
                                                                                                           p. 475
   will contain the quantity Σ(h1 h2 h3 . . . hi )i , and therefore will contain a term
{Σ(h1 h2 h3 . . . hi )}i , that is (−)i aii , which is equal to (−)i aii , since (i − 1)i is
always even. Hence
                          (i−1)i
          Sm−i,0 = (−)       2     × the corresponding Bezoutian secondary.

The above applies to the case where we have supposed m = n.
Art. (28). When this equality does not exist we may proceed as follows. Prefix
to ϕx, the first coefficient of which is still supposed to be 1, a term εxm , where ε
is positive and indefinitely small, and let ϕx so augmented be called Φ(x). Then

                                             479
if η1 , η2 . . . ηn are the roots of ϕx, η1 , η2 . . . ηn , together with the (m − n) values
          1
of    1
      ε
           m−n
                 , will be the roots of Φ(x).
   But it has already been proved that when (as here supposed) the first coefficient
of f x is 1, the Bezoutian secondaries to f and ϕ will be identical with those to f
and Φ respectively; at least it has been proved that these latter, when ε = 0, but
the form of Φ is preserved, become identical with the former, and consequently
the same is true when ε is taken indefinitely small. Now if we call the (m − n)
roots of Φ which do not belong to ϕ, ηn+1 , ηn+2 . . . ηm , and make
                                                                        "                             #
                                                                            hq1 , hq2 . . . hqi
                                                                             η1 , η2 . . . ηm
      Ψm−i,0 = Σ(x − hqi+1 )(x − hqi+2 ) . . . (x − hqm ) "                                               #,
                                                                          hq1 , hq2 . . . hqi
                                                                        hqi+1 , hqi+2 . . . hqm

we have                                                   "                              #
                                                               hq1 , hq2 . . . hqi
                    Ψm−i,0 = ΣP (hq1 , hq2 . . . hqi )                                       ,
                                                              ηn+1 , ηn+2 . . . ηm
where
                                                                              "                                #
                                                                                  hq1 , hq2 . . . hqi
                                                                                   η1 , η2 . . . ηn
P (hq1 , hq2 . . . hqi ) = (x − hqi+1 )(x − hqi+2 ) . . . (x − hqm ) "                                             #.
                                                                                 hq1 , hq2 . . . hqi
                                                                               hqi+1 , hqi+2 . . . hqm

But since ηn+1 , ηn+2 . . . ηm are infinite in value,

                                                                                          1
             "                           #                                                i
                   hq1 , hq2 . . . hqi
                                             = {(−ηn+1 )(−ηn+2 ) . . . (−ηm )}       i
                                                                                                  .
                  ηn+1 , ηn+2 . . . ηm                                                       ε

Hence
                                  1                                     1
                                  i                                   i
                     Ψm−i,0 =            ΣP (hq1 , hq2 . . . hqi ) =           Sn−i,0 ,
                                   ε                                    ε
and
                                         Sn−i,0 = εi Ψm−i,0 .
But by what has been shown antecedently, taking account of the fact of the        p. 476
  leading coefficient of Φ being ε in place of 1, which introduces the factor ε ,
                                                                               i

we have                                     (i−1)i
                            εi Ψm−i,0 = (−) 2 Bi′ ,
where Bi′ is the Bezoutian secondary of the (m − i − 1)th degree in x to f and ϕ;
but Bi′ has been proved = Bi , the Bezoutian secondary of the same degree to f
and ϕ; hence
                                            (i−1)i
                              Sm−i,0 = (−) 2 Bi .

                                                    480
Art. (29).    If now we return to the syzygetic equation,

                                      τ f − tϕ + S = 0,

S may be treated as known, having in fact been completely determined as a
function of the roots, as well in its most general form, as also so as to represent the
simplified residues to f and ϕ in the preceding articles; it remains to determine
the values of τ and t as functions of the roots corresponding to any allowable
form of S, but I shall confine the investigation to the case where S is the
lowest-weighted conjunctive or, which is the same thing, a simplified residue
to f and ϕ of any given degree in x; each value of τt will then represent one of
the convergents to ϕf when expanded under the form of a continued fraction. If
S be of the ιth degree in x, τ is of the degree (n − ι − 1) and t of the degree
(m − ι − 1). This being supposed, and calling n − ι − 1 = ν, m − ι − 1 = µ, I say
that t will be represented by G and τ by Γ, where
                                                                 "                             #
                                                                         hq1 , hq2 . . . hqµ
                                                                          η1 , η2 . . . ηn
        G = (−)ι Σ(x − hq1 )(x − hq2 ) · · · (x − hqµ ) "                                          #,
                                                                   hq1 , hq2 . . . hqµ
                                                                 hµ+1 , hµ+2 . . . hqm

and τ is an analogous form Γ; h1 , h2 . . . hm , as heretofore, being the roots of
f , and η1 , η2 . . . ηn of ϕ. To fix the ideas and make the demonstration more
immediately seizable, give m and n specific values; thus let m = 5, n = 4, ι = 2,
so that µ = 5 − 2 − 1 = 2. Put S under the form Sι,0 , so that S in the case
before us                                         "             #
                                                    hq3 hq4 hq5
                                                    η1 η2 η3 η4
                          = Σ(x − hq1 )(x − hq2 ) "             #.
                                                    hq3 hq4 hq5
                                                     hq1 hq2
Now make x = h1 , then f = 0, and S becomes
                                                      "                     #
                                                          hq3 hq4 hq5
                                                          η1 η2 η3 η4
                        Σ(h1 − hq1 )(h1 − hq2 ) "                           #,
                                                          hq3 hq4 hq5
                                                           hq1 hq2
                                                                                                        p. 477
   that is                        "           #"                 #
                                       h1           h1 h2 h3
                                      h4 h5        η1 η2 η3 η4
                              Σ          "                #          ,
                                              h1 h2 h3
                                               h4 h5

                                               481
h1 being kept constant in the above sum, but h2 , h3 , h4 , h5 being partitionable
in all the six possible ways into two groups, as into h4 h5 ; h2 h3 in the term above
expressed. This sum is evidently identical with
            "                 #                                                      "                 #
                 h1 h2 h3                                                                  h2 h3
                                                       "                     #
                η1 η2 η3 η4                                    h1                        η1 η2 η3 η4
          Σ "                 # ,    that is                                     ×Σ "              #       .
                 h1 h2 h3                                  η1 η2 η3 η4                     h2 h3
                  h4 h5                                                                    h4 h5

Again, ϕ becomes                              "                  #
                                                      h1
                                                                     .
                                                  η1 η2 η3 η4

Hence t = S
          ϕ becomes                           "                     #
                                                     h2 h3
                                                   η1 η2 η3 η4
                                          Σ "                   #        .
                                                       h2 h3
                                                       h4 h5

But, when x = h1 , (−)
                    G
                      ι becomes

                                                       "                     #
                                                             h2 h3
                                    "              #
                                         h1                η1 η2 η3 η4
                                                           "             #       ,
                                        h2 h3                  h2 h3
                                                               h4 h5

that is                       "           #
                                   h1
                                              "                     #
                                  h2 h3             h2 h3
                              "           #                              = (−1)ι t.
                                  h2 h3           η1 η2 η3 η4
                                  h4 h5
Thus when x = h1 , t = G. In like manner, when x = h2 , or h3 , or h4 , or h5 , t
always = G; but t and G are both functions of x of the same, namely of only two,
dimensions in x. Hence t is identical with G. So in general it may be proved
that whenever x = h1 , or h2 , or h3 . . . or hm , t and G, which are each of only
(m − 1 − ι) dimensions in x, are equal. Hence universally t = G, as was to be
shown. To find τ we must avail ourselves of the symmorphic, or as we may better
say (it being at the opposite extremity of the scale of forms), the antimorphic,
value of S represented by S0,ι , taking care to preserve S strictly identical under
both forms of representation, in point of sign as well as quantity. That is to say,
we must make                                                                        p. 478




                                                       482
                                                                  "                                      #
                                                                          h1 , h2 . . . hm
                                                                       ηqν+1 , ηqν+2 . . . ηqn
     S0,ι = (−)ι(m−ι) Σ(x − ηq1 )(x − ηq2 ) · · · (x − ηqν ) "                                           #
                                                                       ηqν+1 , ηqν+2 . . . ηqn
                                                                         ηq1 , ηq2 . . . ηqν
                                                        "                                       #
                                                              ηqν+1 , ηqν+2 . . . ηqn
                                                                 h1 , h2 . . . hm
           = (−)ω Σ(x − ηq1 )(x − ηq2 ) · · · (x − ηqν ) "                                      #,
                                                              ηqν+1 , ηqν+2 . . . ηqn
                                                                ηq1 , ηq2 . . . ηqν

where
                                ω = ι(m − ι) + m(n − ι),
so that
                           (−)ω = (−)ι−1+mn−mι = (−)mn−ι ;
and consequently the same reasoning as was applied to t to prove t = G, will
serve to show that −τ = Γ, where
                                                               "                            #
                                                                      ηξ1 , ηξ2 . . . ηξν
                                                                       h1 , h2 . . . hm
        Γ = (−)mn Σ(x − ηξ1 )(x − ηξ2 ) · · · (x − ηξν ) "                                          #,
                                                                 ηξ1 , ηξ2 . . . ηξν
                                                               ηξν+1 , ηξν+2 . . . ηξn
or                                                            "                             #
                                                                    h1 , h2 . . . hm
                                                                   ηξ1 , ηξ2 . . . ηξν
          τ = (−)ω Σ(x − ηξ1 )(x − ηξ2 ) · · · (x − ηξν ) "                                     #,
                                                                ηξ1 , ηξ2 . . . ηξν
                                                              ηξν+1 , ηξν+2 . . . ηξn
where

            ω = mn − 1 − mν = mn − 1 − m(n − ι − 1) = mι + m − 1.

Art. (30). I have not succeeded in throwing t and τ under any other than the
single forms for each above given, and it is remarkable that whilst apparently t
and τ admit only of this single representation, S admits of the variety of forms
included under the general symbol Sν−ι for a given value of ι; and it ought to be
remarked that these forms, although the most perfectly symmetrical and exactly
balanced representations, and for that reason possibly the most commodious
for the ascertainment of the allotrious factor belonging to them respectively, by
no means exhaust the almost infinite variety of modes by which the simplified
residues, that is, the hekistobarytic, or if we like so to call them, the prime

                                             483
conjunctives, admit of being represented as functions of the roots of the given
functions; for if in Art. 16, instead of writing
                                       "                          #
                                           hq1 , hq2 . . . hqv
                                           ηξ1 , ηξ2 . . . ηξv′
            R= "                               #          "                                 #,
                      hq1 , hq2 . . . hqv                        ηξ1 , ηξ2 . . . ηξv′
                                                   imes
                    hqv+1 , hqv+2 . . . hqm                   ηξv′ +1 , ηξv′ +2 . . . ηξn
                                                                                                 p. 479
   we had made
                             P (hq1 , hq2 . . . hqv ; ηξ1 , ηξ2 . . . ηξv′ )
            R= "                               #          "                                 #,
                      hq1 , hq2 . . . hqv                        ηξ1 , ηξ2 . . . ηξv′
                                                   imes
                    hqv+1 , hqv+2 . . . hqm                   ηξv′ +1 , ηξv′ +2 . . . ηξn

where P represents any function symmetrical in respect of hq1 , hq2 . . . hqv , and
also in respect of ηξ1 , ηξ2 . . . ηξv′ (the interchanges, that is to say, between one h
and another h, or between one η and another η, leaving P unaltered), it might be
shown that the value of Sv,v′ resulting from the introduction of this more general
value of R would (as for the particular value assumed) always be expressible as
an integral function of the roots; and consequently, if P be taken of the same
dimensions in the roots as the numerator of R previously assumed, that is vv ′ ,
Sv,v′ would continue to be (unless indeed it vanish) identical (to some numerical
factor près) with the corresponding simplified residue. If, on the other hand, P
be taken of less than vv ′ dimensions in h and η, we know à priori that Sv,v′
must vanish, as otherwise we should have a conjunctive of a weight less than the
minimum weight. When P is of the proper amount of weight vv ′ , it is I think
probable that another condition as to the distribution of the weight will be found
to be necessary in order that Sv,v′ may not vanish, namely, that the highest
power of any single h in P shall not exceed v, nor the highest power of any
single η exceed v ′ . But as I have not had leisure to enter upon the inquiry, the
verification or disproval of this supposed law, and more generally the evolution of
the allotrious numerical factor introduced into Sv,v′ by assigning any particular
form to P satisfying the necessary conditions of amount and distribution of
weight, must be reserved, amongst other points connected with the theory of the
remarkable forms (19) Art. 15, as a subject for future investigation.
Art. (31). A property of continued fractions, which, if known, I have not met
with in any treatise on the subject (but which has been already cursorily alluded
to in these pages), gives rise to a remarkable property of reciprocity connecting
τ and t severally with S in the syzygetic equation τ f − tϕ + S = 0.




                                               484
  Let the successive convergents to the ordinary continued fraction

                                            1
                                                 1
                         q1 +
                                                     1
                                q2 +
                                                            1
                                        q3 + · · · +
                                                                    1
                                                         qi−1 +
                                                                    qi
be called
                           l1      l2     li−1              li
                              ,       ...      ,               ,
                           m1      m2     mi−1              mi
respectively; it is well known that

                           mi−1 li − mi li−1 = (−)i−1 1;
                                                                                    p. 480
   but I believe that it has not been observed that this is only the extreme case
of a much more general equation, namely

                         mi−ρ li − mi li−ρ = (−)i−ρ µρ−1 ,

where µ1 , µ2 . . . µi denote respectively the denominators to the convergents to
the continued fractions formed with the quotients taken in a reverse order, that
is, the continued fraction

                                            1
                                                                         .
                                                1
                        qi +
                                                     1
                               qi−1 +
                                                                1
                                        qi−2 + · · · +
                                                                    1
                                                           q2 +
                                                                    q1
This is easily proved when ρ = 1; µ0 is of course (as usual) to be considered 1.
So more simply for the improper continued fraction,

                        li                       1
                           =                                             ,
                        mi                           1
                                q1 −
                                                            1
                                        q2 − · · · −
                                                                    1
                                                         qi−1 −
                                                                    qi
of which the convergents are supposed to be
                           l1      l2     li−1              li
                              ,       ...      ,               ,
                           m1      m2     mi−1              mi

                                           485
and the reverse fraction
                                                      1
                                                                              ,
                                                          1
                                      qi −
                                                                     1
                                             qi−1 − · · · −
                                                                         1
                                                              q2 −
                                                                         q1
of which the convergents are supposed to be
                                             λ1       λ2      λi
                                                ,        ,       ,
                                             µ1       µ2      µi
we have the more simple equation

                                  li mi−ρ − li−ρ mi + µρ−1 = 0.

And it is well known, or at all events easily demonstrable, that

li−1                      1                            mi−1                                   1
     =                                            ,         =                                                           .
  li                          1                         mi                                        1
         qi −                                                        qi −
                                  1                                                                   1
                qi−1 −                                                            qi−1 −
                                             1                                                                1
                         qi−2 − · · · −                                                    qi−2 − · · · −
                                             q2                                                                    1
                                                                                                            q2 −
                                                                                                                   q1

Art. (32).    If now we use subscript indices to denote the degree in x of the
quantities to which they are affixed, we have the general syzygetic equation

                           Kτn−ι−1 fm − Ktm−ι−1 ϕn + KSι = 0,

where K, a constant (which I have given the means of determining in the first
section), being rightly assumed, Kτn−ι−1 , Ktm−ι−1 become the numerator and
denominator respectively of one of the convergents to ϕf , expressed as         p. 481
   an improper continued fraction, and KSι becomes the denominator to one
of the convergents to τn−1                                   τn−1
                         f , or, which is the same thing, to ϕ . Conversely, it
is obvious that if we adopt as our primitive functions cfm and tm−1 , c being
the value of K when ι = 0, we shall obtain as the general form of our syzygetic
equation, bearing in mind that (m − 1) now replaces n,

                     cKτm−ι−1 fm − K ′ Sm−ι−1 tm−1 + K ′ tι = 0;

and similarly, if we adopt as our primitive functions τn−1 and cϕn , we obtain for
our general syzygetic equation, observing that (n − 1) now replaces m,

                         K ′ Sn−ι−1 τn−1 − cK ′ tn−ι−1 ϕn + K ′ τι = 0;

                                                      486
so that (making abstraction of the constant factors and looking merely to the
forms of the several functions which enter into the equations) we see that on
the first hypothesis, namely of tm−1 being substituted for ϕn , the conjunctives
of each degree in x change places with the second conjunctive factors, that is
the original multipliers of ϕ of the same degree in x, and vice versâ; and in the
second hypothesis, where τn−1 takes the place of fm , the conjunctives of each
degree in x change places with the first conjunctive factors, that is the original
multipliers of f of the same degree in x, and vice versâ; tm−1 and τn−1 being
respectively multipliers of ϕ and f , such that the difference of the respective
products is independent of x. These results ought to be capable of being verified
by aid of our general formulæ for t, τ, S, and as this verification will serve to
exhibit in a clearer light the nature of the reciprocity between the conjunctives
and the conjunctive factors, it may be not uninteresting to set it out.
Art. (33). As usual, let h1 , h2 . . . hm be the roots of f x, and η1 , η2 . . . ηn the
roots of ϕx; the last conjunctive factor to ϕ, which is of the degree (m − 1) in x,
will be represented, neglecting powers of (−), by tm−1 , where
                                                                   "                           #
                                                                       hq1 , hq2 . . . hqm−1
                                                                          η1 , η2 . . . ηn
           tm−1 = Σ(x − hq1 )(x − hq2 ) · · · (x − hqm−1 ) "                                   #.
                                                                               hqm
                                                                       hq1 , hq2 . . . hqm−1

If now we for greater simplicity make tm−1 = t(x), and call the roots of t,
η1′ , η2′ . . . ηm−1
                 ′   , any such quantity as

                                                                                                   ϕ(hq1 )ϕ(hq2 ) ·
"                          #
             hqm
                               = t(hqm ) = (hqm −hq1 )(hqm −hq2 ) · · · (hqm −hqm−1 )
    η1′ , η2′ . . . ηm−1
                     ′
                                                                                            (hqm − hq1 )(hqm − hq2

                                                                         1
                               = ϕ(hq1 )ϕ(hq2 ) · · · ϕ(hqm−1 ) = R            ,
                                                                       ϕ(hqm )
                                                                                                        p. 482
   R denoting a constant independent of the root hqm selected, in fact the
resultant of the two functions f x and ϕx, that is to say,

                                       ϕ(h1 )ϕ(h2 )ϕ(h3 ) · · · ϕ(hm ).

But by our general formula the simplified residue to f x and t(x) of the ιth degree
in x will be represented by
                                                      "                            #
                                                      
                                                        hqι+1  , hqι+2   . . . hqm
                                                                                     
                                                                                     
                                                             ′ , η′ . . . η′
                                                                                    
          ′
                                                      
                                                          η 1    2         m−1
                                                                                     
                                                                                     
         Sι,0 = Σ(x − hq1 )(x − hq2 ) · · · (x − hqι ) "                            # ;
                                                      
                                                      
                                                           hq1 , hq2 . . . hqι      
                                                                                     
                                                                                     
                                                      
                                                       h                            
                                                                ,h  qι+1  ...h
                                                                             qι+2
                                                                                     
                                                                                       qm



                                                    487
therefore
                                                                                                       
                                             
                                                                                                       
                                                                                                        
                                                                                                       
                                                      ϕ(hqι+1 )−1 ϕ(hqι+2 )−1 · · · ϕ(hqm           ) 
                                                                                                     −1
                                             
                                                                                                       
S′ι,0 = Σ(x−hq1 )(x−hq2 ) · · · (x−hqι )× Rm−ι                   "                              #
                                             
                                             
                                                                      hq1 , hq2 . . . hqι             
                                                                                                       
                                                                                                       
                                                                                                      
                                                                    hqι+1 , hqι+2 . . . hqm           

                                                        ϕ(hq )ϕ(hq2 ) · · · ϕ(hqι )
        = Rm−ι−1 Σ(x − hq1 )(x − hq2 ) · · · (x − hqι ) " 1                       #,
                                                            hq1 , hq2 . . . hqι
                                                          hqι+1 , hqι+2 . . . hqm
or
                                      S′ι = Rm−ι−1 tι ,
the relation which was to be obtained. So conversely, in precisely the same
manner, calling t′ι the conjunctive factor of the degree ι in x to t(x) in the
syzygetic equation which connects f x and t(x) with a corresponding simplified
residue, we have
                                                             "                          #
                                                                 hq1 , hq2 . . . hqι
                                                                 η1′ , η2′ . . . ηm−1
                                                                                  ′
            t′ι = Σ(x − hq1 )(x − hq2 ) · · · (x − hqι ) "                                  #
                                                               hq1 , hq2 . . . hqι
                                                             hqι+1 , hqι+2 . . . hqm

                                              ϕ(hq )ϕ(hqι+2 ) · · · ϕ(hqm )
 = Rι−1 Σ(x − hq1 )(x − hq2 ) · · · (x − hqι ) " ι+1                      # = Rι−1 Sι ,
                                                    hq1 , hq2 . . . hqι
                                                  hqι+1 , hqι+2 . . . hqm
the conjugate equation to the one previously obtained241 .
   And evidently the same reasoning serves to establish the reciprocity, or rather
reciprocal convertibility, between the S series and the τ series, when in lieu of
the original primitives f x and ϕx we take as our primitives τ (x) and ϕx, τ (x)
being the function which satisfies the equation

                                τ (x)f x − t(x)ϕx + S = 0.
                                                                                                        p. 483
Art. (34).     It may be remarked that if n = m − 1, the last syzygetic equation
being thus
                             tm−2 ϕm−1 − τm−2 fm + S0 = 0,
when tm−2 and fm are taken as the primitives, the corresponding equation will
be of the form
                      S′m−1 tm−2 − τm−2
                                    ′
                                        fm + S′0 = 0;
 241
    M. Hermite, by a peculiar method, first discovered one of these two conjugate relations of
reciprocity, applicable to the case of Sturm’s theorem, where ϕx = f ′ x, and I am indebted to
him for bringing the subject under my notice.


                                             488
these two equations must therefore be identical, and consequently tm−2 = ϕm−1
(to a numerical factor près), so that tm−2 and ϕm−1 are reciprocal forms. This
is also obvious from the consideration that tm−2 must, by the general law of
reciprocity (established above), be a residue to (fm , ϕm−1 ), which the latter
function itself may be considered to be. Or the same fact may be presented
under a more general point of view, by observing that when two functions are
respectively the last conjunctive factors one to the other, the last conjunctive
factor to either of them is the other itself; and it is an immediate corollary that,
if ϕ is the differential derivative of f , the last conjunctive factor to f is the
differential derivative of f itself.

                                             Section III.

    On the application of the Theorems in the preceding Section to the expression
in terms of the roots of any auxiliary functions of Sturm’s auxiliary functions,
and the other functions which connect them with the primitive function and its
first differential derivative.
Art. (35).     The formulæ in the preceding Section had reference to the case
of two absolutely independent functions and their respective systems of roots;
when the functions become so related that the roots of the one system become
explicitly or implicitly functions of the roots of the other system, the formulæ
will become susceptible in some of these latter also, and, in some cases the terms
of which the sums are taken will vanish by reason of numerator factors becoming
zero, in other cases terms will require to be evaluated by means of ultimate
ratios; but, as I shall proceed to show, in the case here treated of, which now p. 484
   is the differential derivative of the other. When f and ϕ are thus related, so
          df
that ϕ = dx  , calling as before h1 , h2 . . . hm the roots of f , and η1 , η2 . . . ηm−1 the
roots of ϕ, we shall have in general
         "                          #
                     hqi+1
                                        = (hqi+1 − η1 )(hqi+1 − η2 ) · · · (hqi+1 − ηm−1 )
               η1 , η2 . . . ηm−1
               "                                           #       "                         # "                         #
                                   hqi+1                                     hqi+1                     hqi+1
= fh′ q    =                                                   =                             ×                               .
       i+1         hq1 , hq2 . . . hqi , hqi+2 . . . hqm               hq1 , hq2 . . . hqi     hqi+2 , hqi+3 . . . hqm




                                                     489
Consequently
"                               #       "                           #           "                            #
    hqi+1 , hqi+2 . . . hqm                       hqi+1                                       hqi+2
                                    =                                   imes                                     × ···
      η1 , η2 . . . ηm−1                    η1 , η2 . . . ηm−1                          η1 , η2 . . . ηm−1
                                            "                           #
                                                       hqm
                                        ×
                                                η1 , η2 . . . ηm−1
                                        "                           #           "                                 #
                                                  hqi+1                                     hqi+1
                                    =                                   imes
                                            hq1 , hq2 . . . hqi                     hqi+2 , hqi+3 . . . hqm
                                            "                           #           "                                 #
                                                      hqi+2                                       hqi+2
                                        ×                                   imes
                                                hq1 , hq2 . . . hqi                       hqi+1 , hqi+3 . . . hqm
                                                      "                         #             "                               #
                                                                 hqm                                        hqm
                                        × ··· ×                                     imes                                          .
                                                          hq1 , hq2 . . . hqi                     hqi+1 , hqi+2 . . . hqm−1

Hence
"                               #
    hqi+1 , hqi+2 . . . hqm
                                            m
                                                  "                                           #
      η1 , η2 . . . ηm−1                                             hqr                                 1
                                # =                                                               = (−) 2 (m−i)(m−i−1) ζ(hqi+1 , hqi+
                                            Y
"
    hqi+1 , hqi+2 . . . hqm             r=i+1
                                                      hqi+1 , . . . , h
                                                                      dqr , . . . , hqm
      hq1 , hq2 . . . hqi

the ζ denoting the operation of taking the product of the squares of the differences
of the quantities which this symbol governs. Hence the Bezoutian secondary to
f and f ′ of the (m − i − 1)th degree in x, namely
                                                                                    "                                 #
                                                                                         hqi+1 , hqi+2 . . . hqm
               i(i−1)                                                                      η1 , η2 . . . ηm−1
         (−)      2     Σ(x − hqi+1 )(x − hqi+2 ) · · · (x − hqm ) "                                                  #,
                                                                                           hq1 , hq2 . . . hqi
                                                                                         hqi+1 , hqi+2 . . . hqm

becomes

(−)i(m−i) Σζ(hq1 , hq2 . . . hqi )(x−hqi+1 )(x−hqi+2 ) · · · (x−hqm ) = Σζ(hq1 , hq2 . . . hqi )(x−hqi+1 )
                                                                                                                              p. 485
   since (−)i(m−i) = 1; this gives the well-known formulæ (enunciated242 by me
in the London and Edinburgh Philosophical Magazine for 1839) for expressing
M. Sturm’s auxiliary functions in terms of the roots of the primitive, and which
I therein stated were immediately deducible from the general formulæ (also
enunciated in the same paper) applicable to any two functions. These more
general formulæ appear to have completely escaped the notice of M. Sturm and
others, who have used the special formulæ applicable to the case of one function
becoming the first differential derivative of the other.
 242
       p. 45 above.


                                                             490
Art. (36).   In precisely the same manner, if we form as usual the ordinary
syzygetic equation
                             tf ′ x − τ f x + S = 0,
we may find the different values of t given by the complementary formulæ; and
using ti to denote the multiplier of the degree i in x, that is appertaining to the
residue of the degree (m − i − 1) in x, we have
                        "                         #
                            hq1 , hq2 . . . hqi
                            η1 , η2 . . . ηm−1
             ti = Σ "                                 # (x − hq1 )(x − hq2 ) · · · (x − hqi )
                          hq1 , hq2 . . . hqi
                        hqi+1 , hqi+2 . . . hqm
               = Σζ(hq1 , hq2 . . . hqi )(x − hq1 )(x − hq2 ) · · · (x − hqi ).

Art. (37).     Thus, if we make i = m − 1,

      f ′ x = tm−1 = Σζ(hq1 , hq2 . . . hqm−1 )(x − hq1 )(x − hq2 ) · · · (x − hqm−1 ).

It is evident from the form of f ′ x that it possesses relative to f x, the same
property as f ′ x, I mean the property that when x is indefinitely near to a real
                                                                                   ′
root of f x, and is passing from the inferior to the superior side of such root, ff xx
      ′
like ff xx will pass from being negative to being positive, or in other words, f1′ x
and f ′ x have always the same sign in the immediate vicinity to a real root of
f x. Hence it follows that f1′ x might be used instead of f ′ x, to produce, by the
Sturmian process of common measure, a series of auxiliary functions, which with
f x and f1′ x would form a rhizoristic series, that is a series for determining (as in
the manner of M. Sturm’s ordinary auxiliaries) the number of real roots of f x
comprised within given limits. The rhizoristic series generated by this process
will, it is easily seen, be (to a constant factor près) the denominators (reckoning
                                                                                    ′
+1 as the denominator in the zero place) of the successive convergents to ff xx
thrown under the form                                                                  p. 486
   of a continued fraction
                                                       1
                                                                          ;
                                                            1
                                    q1 −
                                                                 1
                                           q2 − · · · −
                                                                      1
                                                            qn−1 −
                                                                     qn

M. Sturm’s own rhizoristic series, on the contrary, will be (to a constant factor
                                                                    ′
près) the denominators of the convergents to the inverse fraction ff xx , which will



                                                      491
be of the form                                                     
                                                           
                                                           
                                                           
                                                           
                                                           
                                           1
                                                           
                                                            ;
                                                           
                             K
                                               1           
                               qn −
                                                           
                                                          1 
                                                            
                              
                                     qn−1 − · · · −
                                                           
                                                           1
                                                           
                              
                                                        q2 −
                                                               q1
accordingly these two rhizoristic series will be equivalent as regards the number
of changes and of combinations of sign (afforded by each) corresponding to any
given value of x, of which of course the q’s are linear functions. This result agrees
with what has been demonstrated by me243 by a more general method (in the
London and Edinburgh Philosophical Magazine, June and July 1853), where it
has been proved, by means of a very simple theorem of determinants, that the
two series
           1      1  1        1   1  1                   1   1   1         1
              ,     − ,         −   − ,       ...,         −   −   − ··· − ,
           q1     q1 q2       q1 q2 q3                   q1 q2 q3         qn
and
  1         1   1          1   1    1                      1   1    1         1
    ,         −    ,         −   −     ,         ...,        −   −     − ··· − ,
 qn        qn qn−1        qn qn−1 qn−2                    qn qn−1 qn−2        q1

always contain (for real values of q1 , q2 , q3 . . . qn ) the same number of positive
and negative signs.
Art. (38). Having now determined the general values of S and t in the equation
tf ′ x − τ f x + S = 0 as explicit integral functions of the roots of f x, the more
difficult task remains to assign to τ its value similarly expressed. This cannot
readily be effected by means of substitutions in the general formulæ, the method
we adopted for finding t and S; but all the other quantities except τ in the
syzygetic equation being integral functions of the roots, it is evident that τ
also must be an integral function of the same, and to obtain it we may use the
expression
                                        tf ′ x + S
                                    τ=             .
                                             fx
To obtain the general form of τ by direct calculation from this formula would
however be found to be impracticable; the mode I adopt therefore to discover the
general expression for τ corresponding to different values of S, is to ascertain its
value on the hypothesis of particular relations existing between the roots of f x,
and then from the particular values of τ thus obtained to infer demonstratively
 243
       See below pp. 616 and 621.


                                           492
its general form, as will be seen below. The demonstration of τ is unavoidably
somewhat long, τ being in fact represented by a double sum of partial symmetrical
functions.
   Using the subscript indices of each function as the syzygetic equation to denote
its degree in x, we have in general

                            tm−i−1 f ′ x − τm−i−2 f x + Si = 0,
                                                                                                   p. 487
   where if we make

                   h1 − x = k1 , quadh2 − x = k2 , . . . , hm − x = km ,

so that
                                      hi − hω = ki − kω ,
and therefore
                          ζ(hθ1 , hθ2 . . . hθρ ) = ζ(kθ1 , kθ2 . . . kθρ ),
we have in effect found

                        Si = Σkq1 kq2 · · · kqi ζ(kqi+1 , kqi+2 . . . kqm )

and
                 tm−i−1 = ±Σkq1 kq2 · · · kqm−i−1 ζ(kq1 , kq2 . . . kqm−i−1 );
we have also f ′ (x) = (−)m−1 Σkq1 kq2 · · · kqm−1 .
  Let us commence with the case where i = 0; we have then

                                    S0 = ζ(k1 , k2 . . . km ),

                     tm−1 = Σkq1 kq2 · · · kqm−1 ζ(kq1 , kq2 . . . kqm−1 );
we have thus

(−)m τm−2 k1 k2 · · · km = ζ(k1 , k2 . . . km )−Σkq1 kq2 · · · kqm−1 ×Σkq1 kq2 · · · kqm−1 ζ(kq1 , kq2 . . . kq

It may easily be verified that the negative sign interposed between the two parts
of the right-hand member of the equation has been correctly taken, for
                                                      2(m−1) 2(m−2)
           ζ(k1 , k2 . . . km ) contains a term k1          k2             4
                                                                    · · · km−2  2
                                                                               km−1 ,

                Σkq1 kq2 · · · kqm−1 contains a term k1 k2 · · · km−2 km−1 ,
and
                           Σkq1 kq2 · · · kqm−1 ζ(kq1 , kq2 . . . kqm−1 )
                                                                               2(m−1) 2(m−2)
contains a term k12m−3 k22m−5 · · · km−2
                                     3   km−1 , and thus the term k1  k2              4
                                                                               · · · km−2  2
                                                                                          km−1 ,
which does not contain km , will (as it ought to do) disappear from the right-hand
side of the equation.

                                                493
   Now suppose
                                               k1 = k2 ,
then
                                        ζ(k1 , k2 . . . km ) = 0,
and also
                                     ζ(kq1 , kq2 . . . kqm−1 ) = 0,
except when one or the other of the two disjunctive equations

                                q1 , q2 , q3 . . . qm−1 = 1, 3, 4 . . . m,

                                q1 , q2 , q3 . . . qm−1 = 2, 3, 4 . . . m,
is satisfied (by a disjunctive equation, meaning an equation which affirms the
equality of one set of quantities with another set the same in number, each with
each, but in some unassigned order).                                             p. 488
   Hence

          Σkq1 kq2 · · · kqm−1 ζ(kq1 , kq2 . . . kqm−1 ) = 2k1 k3 · · · km ζ(k1 , k3 . . . km ).

Hence when k1 = k2 , (−)m τm−2 becomes
                              2
                                 Σkq1 kq2 · · · kqm−1 ζ(k1 , k3 . . . km ),
                              k1
that is
                  2ζ(k1 , k3 . . . km ){k1 Σkr3 kr4 · · · krm−1 + 2k3 k4 · · · km },
the Σ referring to r3 , r4 . . . rm supposed to be disjunctively equal to 3, 4 . . . m.
   Now τm−2 is of (m − 2) dimensions in x, and whenever more than one equality
exists between the h’s, S0 and tm−1 both vanish (in fact every term in each
vanishes separately), and therefore τm−2 , which

                                             S0 + tm−1 f ′ x
                                         =                   ,
                                              k1 k2 · · · km
will vanish. Hence (−)m τm−2 must be always of the form

                     Σζ(hq1 , hq2 . . . hqm−1 ) × Ψ(kq1 , kq2 . . . kqm−1 , kqm ),

Ψ denoting some integral function of (m − 2) dimensions in respect of the system
of quantities kq1 , kq2 . . . kqm . The result above obtained enables us to assign the
value of
                                     Ψ(k1 , k3 . . . km , k2 ),
when k1 = k2 , namely

                            k1 Σ(kr3 , kr4 . . . krm−1 ) + 2k3 k4 · · · km .

                                                  494
Now for a moment suppose, selecting (m − 1) terms k1 , k3 , k4 . . . km out of the
m terms of the k series, that

 Ω(k1 , k3 , k4 . . . km , k2 ) = k2m−2 − k2m−3 S1 (k1 , k3 . . . km ) + k2m−4 S2 (k1 , k3 . . . km )

                 ± · · · + k2 Sm−3 (k1 , k3 . . . km ) ± 2Sm−2 (k1 , k3 . . . km ),
where S1 means that the quantities which it governs are to be simply added
together, S2 denotes that their binary, S3 that their ternary, and in general Sr
that their r-ary products are to be added together.
   When k1 = k2 , Ω becomes

k1m−2 −k1m−3 {k1 +S1 (k3 , k4 . . . km )}+k1m−4 {k1 S1 (k3 , k4 . . . km )+S2 (k3 , k4 . . . km )}

                 −k1m−5 {k1 S2 (k3 , k4 . . . km ) + S3 (k3 , k4 . . . km )} ± · · ·
    ±k1 {k1 Sm−4 (k3 , k4 . . . km ) + Sm−3 (k3 , k4 . . . km )} ± 2Sm−2 (k3 , k4 . . . km ),
which evidently equals

                   ±{2Sm−2 (k3 , k4 . . . km ) + k1 Sm−3 (k3 , k4 . . . km )},

that is
                        ±{k1 Σ(kr3 , kr4 . . . krm−1 ) + 2k3 k4 · · · km }.
                                                                                                        p. 489
   Hence when k1 = k2 , Ψ = Ω, and

          (−)m τm−2 = Σζ(hq1 , hq2 . . . hqm−1 )imesΩ(kq1 , kq2 . . . kqm−1 , kqm );

and so in like manner, when h1 is equal to any one of the (m − 1) quantities
k2 , k3 . . . km , the form of τm−2 above written will have been correctly assumed.
But τm−2 may be treated as a function of (m − 2) dimensions in k1 , and
consequently any form of (m−2) dimensions in k1 , which fits it for (m−1) different
values of k1 , must be its general form, and accordingly we have universally,

(−)m τm−2 = Σζ(hq1 , hq2 . . . hqm−1 ) × {(x − hqm )m−2 ± (x − hqm )m−3 S1 (x − hq1 , x − hq2 . . .
                + (x − hqm )m−4 S2 (x − hq1 , x − hq2 . . . x − hqm−1 ) + &c.
                ∓ (x − hqm )Sm−3 (x − hq1 , x − hq2 . . . x − hqm−1 )
                ± 2Sm−2 (x − hq1 , x − hq2 . . . x − hqm−1 )}.

Art. (39). With a view to better paving our way to the general form of τ for
all values of i, let us pass over the case of i = 1 and go at once to the equation

                                tm−3 f ′ x − τm−4 f x + S2 = 0;

and to better fix our ideas let m = 7, so that the equation becomes

                                    t4 f ′ x − τ3 f x + S2 = 0;

                                                495
we have then, preserving the same relation as before, that is, using h to denote
any root of f x, and k to denote h − x, the equation
±k1 k2 k3 k4 k5 k6 k7 τ3 = Σkq1 kq2 ζ(kq3 kq4 kq5 kq6 kq7 )−Σkq1 kq2 kq3 kq4 kq5 kq6 ×Σ{kq1 kq2 kq3 kq4 ζ(kq1
now τ3 will vanish whenever more than three relations of equality exist between
the k’s, for then each term in both of the two sums in the right-hand member of
the equation above written will separately vanish; and of course three relations
of equality between the same are sufficient to make all the terms in the first of
these sums vanish. This relationship between the different k’s corresponding to a
multiplicity 3 may arise in different ways; the multiplicity 3 may be divided into
3 units corresponding to 3 pairs of equal roots, or into 2 and 1 corresponding
one set of 3 equal roots, and a second set of 2 equal roots, or may be taken en
bloc, which corresponds to the case of one set of 4 equal roots. I shall make the
first of these suppositions, which will sufficiently well answer our purpose in the
case before us.
    Thus I shall suppose
                           k1 = k4 ,       k2 = k5 ,          k3 = k6 ,
then, as above remarked,
                                   ζ(kq3 kq4 kq5 kq6 kq7 ) = 0
                                                                                                 p. 490
   for all values of q3 , q4 , q5 , q6 , q7 , and therefore
                              Σkq1 kq2 ζ(kq3 kq4 kq5 kq6 kq7 ) = 0;
also Σkq1 kq2 kq3 kq4 kq5 kq6 becomes
                      k1 k2 k3 {k1 k2 k3 + 2k7 (k1 k2 + k1 k3 + k2 k3 )},
and ζ(kq1 kq2 kq3 kq4 ) vanishes, except for the cases where q1 , q2 , q3 , q4 represent
respectively, q1 the index 1 or 4, q2 the index 2 or 5, q3 the index 3 or 6, and q4
the index 7.
   Hence
              Σkq1 kq2 kq3 kq4 ζ(kq1 kq2 kq3 kq4 ) = 23 k1 k2 k3 k7 ζ(k1 k2 k3 k7 ),
and consequently τ3 becomes
               ±8ζ(k1 k2 k3 k7 ) × {k1 k2 k3 + 2k7 (k1 k2 + k1 k3 + k2 k3 )}.
Hence we are able to predict that the general expression for our τ in the case
before us will be
 τ3 = ∓Σζ(kq1 kq2 kq3 kq7 )
       × {(kq34 + kq35 + kq36 ) − (kq24 + kq25 + kq26 )(kq1 + kq2 + kq3 + kq7 )
         + (kq4 + kq5 + kq6 )(kq1 kq2 + kq1 kq3 + kq1 kq7 + kq2 kq3 + kq2 kq7 + kq3 kq7 )
         − 4(kq1 kq2 kq3 + kq1 kq2 kq7 + kq1 kq3 kq7 + kq2 kq3 kq7 )}.

                                              496
For in the first place, the fact that the τ vanishes when more than three relations
of equality exist between the k’s, proves that we may assume τ3 of the form

                    Σζ(kq1 kq2 kq3 kq7 ) × ϕ{kq1 kq2 kq3 kq7 ; kq4 kq5 kq6 },

the semicolon (;) separating the k’s into two groups, in respect of each of which
severally ϕ is a symmetrical form. But if in the expression last above written for
τ3 we make
                        k1 = k4 , quadk2 = k5 , k3 = k6 ,
it becomes

       ∓8ζ(k1 k2 k3 k7 ) × {(k13 + k23 + k33 ) − (k12 + k22 + k32 )(k1 + k2 + k3 + k7 )

+(k1 +k2 +k3 )(k1 k2 +k1 k3 +k1 k7 +k2 k3 +k2 k7 +k3 k7 )−4(k1 k2 k3 +k1 k2 k7 +k1 k3 k7 +k2 k3 k7 )}.
Now in general if
                              σr = ar1 + ar2 + ar3 + · · · + art ,
and
                                   Sr = Σ(a1 a2 a3 . . . ar ),
then
                        σr − σr−1 S1 + σr−2 S2 ± · · · ± rSr = 0.
Consequently the sum of the terms constituting the second factor in the above
expression
              = (3 − 4)k1 k2 k3 + (2 − 4)k7 (k1 k2 + k1 k3 + k2 k3 ).
                                                                                            p. 491
   Hence the above expression becomes

                 ±8ζ(k1 k2 k3 k7 ){k1 k2 k3 + 2(k1 k2 + k1 k3 + k2 k3 )k7 }.

Thus, then, whenever k1 , k2 , k3 are respectively equal to any three of the quantities
k4 , k5 , k6 , k7 , which may take place in twenty-four different ways (twenty-four
being the number of permutations of four things), our τ3 will have been correctly
assumed; but ζ(kq1 kq2 kq3 kq7 ) being replaceable by ζ(hq1 hq2 hq3 hq7 ), the τ3 may
be treated as a cubic function in k1 , k2 , k3 , and arranged according to the powers
of k1 , k2 , k3 will contain only twenty terms; hence, since the assumed form is
verified for more than twenty, that is, for twenty-four values of h1 , h2 , h3 , it
follows that the assumed form is universally identical with the form of τ , which
was to be determined.
Art. (40). Now, again, in order to facilitate the conception of the general proof,
let us suppose f x to be of only five dimensions in x, i still remaining 3; it will
no longer be possible when we suppose a multiplicity three to prevail among the
roots, to conceive this multiplicity to be distributed into three parts, for that
would require the existence of three pairs of roots, there being only five. But we

                                              497
may, if we please, make h1 = h2 = h3 , and h4 = h5 , or else h1 = h2 = h3 = h4 ,
or in any other mode conceive the multiplicity to be divided into two parts, 2
and 1 respectively, or to be taken collectively en bloc. As a mode of proceeding
the more remote from that last employed, I shall choose the latter supposition.
Then we obtain (τ now becoming τ5−2−2 , that is τ1 )

              k1 k2 k3 k4 k5 τ1 = ±Σkq1 kq2 kq3 kq4 × Σkq1 kq2 ζ(kq1 kq2 ),

and ζ(kq1 kq2 ) will vanish, except in the case where q1 represents the indices 1 or
2 or 3 or 4, and q2 the index 5; also

                           Σkq1 kq2 kq3 kq4 = k14 + 4k13 k5 .

Hence our equation becomes

                       k14 k5 τ = (k14 + 4k13 k5 )4k1 k5 ζ(k1 k5 ),

and τ becomes
                               −4ζ(k1 k5 )(k1 + 4k5 ).
If, now, we assume for the general value of τ in the case before us

                τ = Σζ(kq1 kq5 ){(kq2 + kq3 + kq4 ) − 4(kq1 + kq5 )},

when k1 = k2 = k3 = k4 , τ becomes

                          ±4ζ(k1 k5 ){3k1 − (4k1 + k5 )},

that is
                               ±4ζ(k1 k5 )(k1 + 4k5 ).
Hence then for the two systems of values of h1 , h2 , h3 , namely

                               h1 = h4    h1 = h5
                               h2 = h4 or h2 = h5
                               h3 = h4    h3 = h5 ,
                                                                                       p. 492
    the form of τ will have been correctly assumed. But since the derived form is
a linear function of h1 , h2 , h3 , this is not enough to identify the assumed with
the general form, since for such verification four systems of values must be taken,
four being the number of terms in a function of three variables of the first degree.
If, however, we had adopted a separation of the multiplicity three into two parts,
and had started with supposing k1 = k2 = k3 , k4 = k5 , we should have found
that τ would have become

                              = 6ζ(k1 , k5 )(2k1 + 3k5 ).


                                          498
Moreover, when these equalities subsist,

             k1 k2 k3 k4 + k1 k2 k3 k5 + k1 k2 k4 k5 + k1 k3 k4 k5 + k2 k3 k4 k5

becomes 2k13 k5 + 3k12 k52 , and the common factor k12 k4 disappears in the course of
the operations for finding τ , and eventually we have to show (in order to support
the universality of the previously assumed form for τ ) that

                            kq2 + kq3 + kq4 − 4(kq1 + kq5 )

becomes −2kq4 − 3k5 when

                     kq2 = kq3 = kq1 = k1 ,            kq4 = kq5 = k5 ,

which is evidently true. Hence then τ will have been correctly assumed for the
following cases,

                    k1 = k2 = k5 = k3 ,            k1 = k2 = k5 = k4 ;

and also for the cases
                                                              
                              k1 = k2 = k3 and k5 = k4 
                                                       
                              k1 = k5 = k3 and k2 = k4
                              k2 = k5 = k3 and k1 = k4 
                                                       

                                                              
                             k1 = k2 = k4 and k5 = k3 
                                                      
                             k1 = k5 = k4 and k2 = k3 ,
                             k2 = k5 = k4 and k1 = k3 
                                                      

that is, for eight cases in all, whereas four only would have sufficed. Hence, ex
abundantiâ demonstrationis, the form assumed for τ1 is in the case before us the
general form.
Art. (41). We may now easily write down the general form which τ assumes
for all values of i and prove its correctness. If the roots be

                                    h1 , h2 , h3 . . . hm ,

and
                          tm−i−1 f ′ x − τm−i−2 f x + Si = 0,
                                                                                        p. 493
  we shall have
±τm−i−2 = Σ{ζ(hq1 hq2 hq3 . . . hqm−i−1 ) × [σm−i−2 − σm−i−3 S1 + σm−i−4 S2 + &c.
            + (−)m−i−3 σ1 Sm−i−3 + (−)m−i−2 (σ0 + 1)Sm−i−2 ]},

where σr denotes in general the sum of the rth powers of the (i + 1) quantities

                      (x − hqm−i ), (x − hqm−i+1 ), . . . (x − hqm ),

                                             499
and Sr denotes in general the sum of the products of the complementary (m−i−1)
quantities
                      (x − hq1 ), (x − hq2 ) . . . (x − hqm−i−1 )
combined r and r together. It will of course also be understood that σ0 = i + 1,
so that σ0 + 1 = i + 2.
Art. (42). To prove the correctness of this general determination of the form of
τm−i−2 , let us suppose in general that i+1 relations of equality spring up between
the m quantities h1 , h2 . . . hm ; we shall then easily obtain (N representing a
certain numerical multiplier)

                                                   Σkθ1 kθ2 . . . kθm−i−1
              ±Q = N ζ(k1 , k2 . . . km−i−1 )    µ1 −1 µ2 −1        µm−i−1 −1 ,
                                                µ1 k 1       · · · µm−i−1

k1 , k2 . . . km−i−1 being what the k system becomes when repetitions are excluded,
and being respectively supposed to occur µ1 , µ2 . . . µm−i−1 times respectively, so
that
                             µ1 + µ2 + · · · + µm−i−1 = m;
the fractional part of the right-hand member of the equation immediately above
written will be readily seen to be equivalent to

                              Σµθm−i−1 kθ1 kθ2 · · · kθm−i−2 .

To establish the correctness of the assumed form, we must be able, as in the
particular cases previously selected, to prove two things; the one, and the
more difficult thing to be proved is, that when the series of distinct quantities
k1 , k2 , k3 . . . km become converted into µ1 groups of k1 ; µ2 groups of k2 , . . . µm−i−1
groups of km−i−1 , then that

                               Σµθ1 kθ2 kθ3 kθ4 · · · kθm−i−1 ,

or in other terms
                                                          m−i−1
                        Σ ± kθ1 kθ2 kθ3 · · · kθm−i−1 ×           (µθ ),
                                                           X

                                                           π=1

becomes identical with

             σm−i−2 − σm−i−3 S1 + &c. + (−1)m−i−2 (σ0 + 1)Sm−i−2 .

The other step to be made, and with which I shall commence, consists in showing
that the number of terms in the expression last above written, considered as a
function of (m − i − 2)th degree of (i + 1) variables, is never greater than the
entire number of ways in which (i + 1) quantities out of m quantities may be


                                            500
equated to the remaining (m − i − 1) quantities, namely each of the first set
respectively to all the same, or all different, or some the                         p. 494
   same and some different; in short, in any manner each of the i + 1 quantities
with some one or another (without restriction against repetitions) of the m − i − 1
remaining quantities. This latter number being in fact the number of ways in
which (m − i − 1) quantities may be combined (i + 1) together with repetitions
admissible, by a well-known arithmetical theorem, is (m − i − 1)i+1 , and the first
number is
                            (i + 1)(i + 2) · · · (m − 2)
                                                         ,
                               1 · 2 · · · (m − i − 2)
which is always less than the other. It remains then only to prove the remaining
step of the demonstration.244
Art. (43).     To fix the ideas let m = 10, i = 5, and consider the expression

(k53 + k63 + k73 + k83 + k93 + k10
                                3
                                   ) − (k52 + k62 + k72 + k82 + k92 + k10
                                                                       2
                                                                          )(k1 + k2 + k3 + k4 )
   + (k5 + k6 + k7 + k8 + k9 + k10 )(k1 k2 + k1 k3 + k1 k4 + k2 k3 + k2 k4 + k3 k4 )
   − 7(k1 k2 k3 + k1 k2 k4 + k1 k3 k4 + k2 k3 k4 ).

Now suppose the six quantities k5 , k6 , k7 , k8 , k9 , k10 to become respectively equal
each to some one or another of the four quantities k1 , k2 , k3 , k4 , as for instance, I
shall suppose

                 k5 = k6 = k7 = k1 ,         k8 = k9 = k2 ,        k10 = k3 .

Then
                         µ1 = 4,     µ2 = 3,     µ3 = 2,     µ4 = 1,
 244
    If this first step of the demonstration appear unsatisfactory or subject to doubt, it may be
dispensed with, and the result obtained in the succeeding article (the demonstration of which is
wholly unexceptionable) being assumed, it may be proved that the formula there obtained on a
particular hypothesis must be universally true, in precisely the same way and by aid of the
same Lemma in and by aid of which the formula obtained in the Supplement to this section for
                                ′
the simplified quotients to ff xx upon a like particular hypothesis is shown to be of universal
application, that is, by showing that otherwise a function of 2i − 1 variables would contain a
function of 2i variables as a factor.




                                             501
and the formula of Art. 41 becomes
(3k13 + 2k23 + k33 ) − (3k12 + 2k22 + k32 )(k1 + k2 + k3 + k4 )
   + (3k1 + 2k2 + k3 )(k1 k2 + k1 k3 + k1 k4 + k2 k3 + k2 k4 + k3 k4 )
   − 7(k1 k2 k3 + k1 k2 k4 + k1 k3 k4 + k2 k3 k4 )
= 3[{k13 − k12 (k2 + k3 + k4 ) + k1 } + k1 {(k2 k3 + k2 k4 + k3 k4 ) + k4 (k2 + k3 + k4 )}]
   + 2{k23 − k22 (k1 + k3 + k4 ) + k2 } + k2 {(k1 k3 + k1 k4 + k3 k4 ) + (k2 k1 + k3 ) + k4 }
   + {k33 − k32 (k1 + k2 + k4 ) + k3 } + k3 {(k1 k2 + k1 k4 + k2 k4 ) + k3 (k1 + k2 + k4 )}
   − (k2 k3 k4 + k1 k3 k4 + k1 k2 k4 + k1 k2 k3 )
= −k1 k2 k3 − 2k1 k2 k4 − 3k1 k3 k4 − 4k2 k3 k4
                 µ1 µ2 µ3 µ4
                                      
= −k1 k2 k3 k4      +     +      +       .
                 k1    k2   k3      k4
                                                                                                    p. 495
    In the above investigation the quantities which with their repetitions make
up the k’s system, are k4 , k1 , k2 , k3 , appearing respectively 1, 2, 3, 4 times, that
is to say repeated 0, 1, 2, 3 times; 7 is one more than the sum of the repetitions
0 + 1 + 2 + 3, and the numbers 1, 2, 3, 4 arise from subtracting from 7 the sums
1+2+3; 0+2+3; 0+1+3; 0+1+2; respectively, so that the remainders 1, 2, 3, 4
denote respectively one more than the number of repetitions of k4 , k1 , k2 , k3 , that
is, are the number of appearances of k4 , k1 , k2 , k3 ; and thus with a slight degree
of attention to the preceding process the reader may easily satisfy himself that
the preceding demonstration (although not so expressed) is in essence universal,
and the form of τ as an explicit function of x and of the roots of f x is thus
completely established for all values of m and of i.

                           Supplement to Section III.
 On the Quotients resulting from the process of continuous division ordinarily
  applied to two Algebraical Functions in order to determine their greatest
                              Common Measure.

Art. (a). 245 We have now succeeded in exhibiting the forms of the numerators
                        ′
and denominators of ff xx developed into a continued fraction in terms of the
differences of the roots and factors of f x. It remains to exhibit the quotients
themselves of this continued fraction under a similar form.
   Lemma. An equation being supposed of an arbitrary degree n, there exists no
function of n and of less than 2i of the coefficients 246 , which vanishes for all
 245
      The articles in this and subsequent sections to which Latin, Greek and Hebrew letters
are prefixed, although in strict connexion with the context, are supplementary in the sense of
having been supplied since the date when the paper was presented for reading to the Royal
Society. All the articles marked with numbers (from 1 to 72), and the Introduction, appeared
in the memoir as originally presented to the Society, June 16, 1853.
  246
      In the proposition thus enunciated the coefficient of the highest power of x is supposed to
be a numerical quantity.


                                              502
values of n whenever the n roots reduce in any manner to i distinct groups of
equal roots; or in other words, any function of n and the first 2i − 1 coefficients
of an equation of the nth degree, which vanishes for all values of n in every case
where the roots retain only i distinct names, must be identically zero.
   To render the statement of the proof more simple, let i be taken equal to 3.
And let the roots be supposed to reduce to p roots a, q roots b, and                p. 496
   r roots c. And let sr in general denote the sum of the rth powers of the roots.
Then we have evidently
                                   p + q + r = s0 ,
                                pa + qb + rc = s1 ,
                             pa2 + qb2 + rc2 = s2 ,
                             pa3 + qb3 + rc3 = s3 ,
                             pa4 + qb4 + rc4 = s4 ,
                             &c. &c., ad inf initum.
Eliminating p, q, r between the first, second, third and fourth equations, we
obtain
                              1 1 1 s0
                              a b c s1
                                               = 0.
                             a2 b2 c2 s2
                             a3 b3 c3 s3
In like manner eliminating ap, bq, cr between the second, third, fourth and fifth
equations, we have
                               1 1 1 s1
                              a b c s2
                                               = 0;
                              a2 b2 c2 s3
                              a3 b3 c3 s4
and so in general we have for all values of e,
                              1 1 1     se
                              a b c se+1
                                                     = 0;
                              a2 b2 c2 se+2
                              a3 b3 c3 se+3
whence it may immediately be deduced, that, upon the given supposition of there
being only three groups of distinct roots, we must have the following infinite
system of coexisting equations satisfied, namely,
                   s0 t + s1 u + s2 v + s3 w =   0     say L0 = 0,
                   s1 t + s2 u + s3 v + s4 w =   0     ” L1 = 0,
                   s2 t + s3 u + s4 v + s5 w =   0     ” L2 = 0,
                   s3 t + s4 u + s5 v + s6 w =   0     ” L3 = 0,
                   s4 t + s5 u + s6 v + s7 w =   0     ” L4 = 0,
                               &c. &c. &c.           &c.;

                                        503
                                                                                                p. 497
   and conversely, when this infinite system of equations is satisfied the roots
must reduce themselves to three groups of equal roots.
   Let now ϕ be any function of s0 , s1 , s2 . . . which vanishes when this is the case.
Then ϕ must necessarily contain as a factor some derivee of the infinite system
of equations above written, that is, some function of s0 , s1 , s2 . . . which vanishes
when these equations are satisfied, that is, some conjunctive of the quantities
L0 , L1 , L2 , L3 , . . .; but it is obviously impossible in any such conjunctive to exclude
s6 from appearing, unless by introducing some other s with an index higher than
6, and consequently ϕ cannot be merely a function of s0 , s1 , s2 , s3 , s4 , s5 , nor
consequently of n and the first five coefficients; or if such, it is identically zero.
And so in general any function of n and only 2i − 1 of the coefficients which
vanishes when the roots reduce to i groups of equal roots, must be identically
zero; as was to be proved.
Art. (b).     It ought to be observed that the preceding reasoning depends
essentially upon the circumstance of n being left arbitrary. If n were given the
proposition would no longer be true. In fact, on that supposition, the n roots
reducing to i distinct roots would imply the existence of n − i conditions between
the n roots; and consequently n − i independent equations would subsist between
the n coefficients, and functions could be formed of i only of the coefficients,
which would satisfy the prescribed condition of vanishing when the roots resolved
themselves into i groups of distinct identities.
Art. (c).    Let Dr1 ,r2 ...ri be used in general to denote the determinant
                            sr1 sr1 +1 sr1 +2 · · · sr1 +i−1
                            sr2 sr2 +1 sr2 +2 · · · sr2 +i−1
                                                             ;
                            ··· ···     ···          ···
                            sri sri +1 sri +2 · · · sri +i−1
then the simplified ith Sturmian residue Ri may be expressed under the form
    D1,2,3...i xn−i−1 − D2,3...i+1 xn−i−2 + D3,4...i+2 xn−i−3 · · · ± Dn−i,n−i−1...n ,
which is easily identifiable with the known expression for such residue.
   Now obviously the necessary and sufficient condition in order that the n roots
may consist of only repetitions of i distinct roots is, that Ri shall be identically
zero, that is to say, we must have
                D1,2...i = 0,      D2,3...i+1 = 0 . . . Dn−i,n−i−1...n = 0.
But the reasoning of the preceding article shows that although these equations
are necessary and sufficient, they are but a selected system of equations of an
infinite number of similar equations which subsist,247 and that, in fact,       p. 498
 247
     But quære whether any other sufficient system can be found of equations so few in number
as this system.


                                            504
   whatever be the value of n, we may take r1 , r2 . . . ri perfectly arbitrary and
as great as we please, and the equation

                                         Dr1 ,r2 ...ri = 0

must exist by virtue of the existence of the n − i equations last above written.
Art. (d). I now return to the question of expressing the successive quotients
     ′
of ff xx as functions of the differences of the roots and factors; that they must
be capable of being so expressed is an obvious consequence of the fact that the
numerators and denominators of the convergents have been put under that form,
since, if
                             Ni−2        Ni−1       Ni
                                   ,          ,        ,
                             Di−2        Di−1       Di
are any three consecutive convergents of the continued fraction
                                                1
                                                1            ,
                                       Q1 − Q −···− 1
                                                2      Qi


we must have
                                  Di−2 Ni − Ni−2 Di = Qi .
It would not, however, be easy to perform the multiplications indicated in the
above equation, so as to obtain Qi under its reduced form as a linear function of
x. I proceed therefore to find Qi constructively in the following manner.
   Let Ri−2 , Ri−1 , Ri be three consecutive residues, f ′ x counting as the residue
                                      −Ri                          p′
in the zero place, then Qi = Ri−2                             p
                                  Ri−1 , and is of the form q x + q ′ .
   Now in general if we denote the n roots of f x, where the coefficient of xn is
supposed unity, by h1 , h2 . . . hn , and if we use Zi to denote Σζ(hθ1 , hθ2 . . . hθi ),
with the convention that Z1 = n, Z0 = 1, we have, employing (i) to denote
2 {(−1) + 1},
1       i

          2 Z2 · · · Z2
         Zi−1 i−3     (i)
  Ri =                         Σ{ζ(hθ1 , hθ2 . . . hθi+1 )(x − hθi+2 )(x − hθi+3 ) . . . (x − hθn )},
         Zi2 Zi−2
              2 · · · Z2
                       (i)+1
          2 Z2 · · · Z2
         Zi−2 i−4     (i)+1
Ri−1 =     2 Z2 · · · Z2         Σ{ζ(hθ1 , hθ2 . . . hθi )(x − hθi+1 )(x − hθi+2 ) . . . (x − hθn )},
          Zi−1 i−3     (i)
           2 Z2 · · · Z2
          Zi−3 i−5     (i)
Ri−2 =    2 Z2 · · · Z2          Σ{ζ(hθ1 , hθ2 . . . hθi−1 )(x − hθi )(x − hθi+1 ) . . . (x − hθn )}.
         Zi−2 i−4     (i)+2
                                                                                                  p. 499
   The part of Ri−1 within the sign of summation is

      Zi xn−i − Σ(hθi+1 + hθi+2 + · · · + hθn )ζ(hθ1 , hθ2 . . . hθi )xn−i−1 + &c.,

say
                                 Zi xn−i − Zi′ xn−i−1 + &c.,

                                               505
and the part of Ri−2 within the sign of summation is
                                              ′
                               Zi−1 xn−i+1 − Zi−1 xn−i + &c.,

and
                       ′ xn−i
      Zi−1 xn−i+1 − Zi−1
Zi2                           = Zi−1 Zi x+(Zi−1 Zi′ −Zi Zi−1
                                                         ′
                                                             )+an algebraic fraction.
        Zi xn−i − Zi′ xn−i−1
Hence                                                                        )−1
                           2    2         2            ( 2    2 · · · Z2
                      1 Zi−3 Zi−5 · · · Z(i)            Zi−2 Zi−4      (i)+1
                 Qi = 2 2     2 · · · Z2                   2 Z2 · · · Z2
                     Zi Zi−2 Zi−4       (i)+1             Zi−1 i−3     (i)
                        × {Zi−1 Zi x + (Zi−1 Zi′ − Zi Zi−1
                                                       ′
                                                           )}
                         2     4 Z4 · · · Z4
                             Zi−3
                       Zi−1         i−5       (i)
                     =     2 4    4         4     Ti ,
                        Zi Zi−2 Zi−4 · · · Z(i)+1
Ti denoting
                               Zi−1 Zi x + (Zi−1 Zi′ − Zi Zi−1
                                                           ′
                                                               ).

Art. (e). If the process of obtaining the successive quotients and residues be
considered, it will easily be seen that each step in the process imports two new
coefficients into the quotients, the first quotient containing no literal quotient
in the part multiplying x and containing the first literal coefficient in the other
part, the second quotient containing two literal coefficients in the one part and
three in the other, and in general the ith quotient containing 2i − 2 of the letters
in the one part and 2i − 1 of them in the other. Hence Ti being made equal to
Li x + Mi , Li contains 2i − 2 and Mi contains 2i − 1 of the literal coefficients of
f x.
   Moreover, we have Zi of the form
                                               Pi−2 − mPi
                                         Ti2              ,
                                                  Pi−1
where
                        Pi−1 = Σζ(hθ1 , hθ2 . . . hθi )ηθi+1 ηθi+2 . . . ηθn ,
                        Pi−2 = Σζ(hθ1 , hθ2 . . . hθi−1 )ηθi ηθi+1 . . . ηθn ,
and Pi , which is the ith simplified residue, vanishes when the n roots in any
manner become reduced to only i distinct groups.
  I proceed to show that if we make

          Ai x + Bi = Ui = A2i,1 (x − h1 ) + A2i,2 (x − h2 ) + · · · + A2i,n (x − hn ),

where in general

       Ai,e represents Σζ(hθ1 , hθ2 . . . hθi−1 )(he − hθ1 )(he − hθ2 ) . . . (he − hθi−1 ),

                                                 506
then will
                                          Ti = Ui .
                                                                                           p. 500
   It will be observed that Ai,e is identical with what the simplified denominator of
the (i−1)th convergent becomes when we write he in place of x, and consequently,
when arranged according to the powers of he , will be of the form

                               c1 hi−1
                                   e   + c2 hi−2
                                             e   + · · · + ci ,

where c1 , c2 . . . ci are functions of the coefficients, but containing no more of them
than enter into Qi−1 , that is, containing only 2i − 2 of them.
  Now Ai is made up of terms, each consisting of some binary product of

                                        c1 , c2 . . . ci ,

combined with some term of the series

                               Σh2i−2 ,      Σh2i−3 . . . Σh0 ;

and any one of this latter set of terms expressed as a function of the coefficients
of f x contains at most 2i − 2 of them.
   Hence only 2i − 2 of the coefficients enter into Ai , and in like manner only
2i − 1 of them into Bi .
   The number of letters, therefore, in Ai and in Bi is the same as in Li and in
Mi , namely 2i − 2 and 2i − 1 respectively.
   Now let the roots consist of only i distinct groups of equal roots, so that
                                                        Pi−2
                                  Ti becomes Zi2             .
                                                        Pi−1
I shall show that in whatever way the equal roots are supposed to be grouped
upon this supposition, there will result the equation

                                          Ti = Ui ,

where
                                                               Pi−2
                            Ti = [Σζ(ηθ1 , ηθ2 . . . ηθi )]2        ,
                                                               Pi−1
                   Pi−2 = Σ{ηθi ηθi+1 . . . ηθn ζ(ηθ1 , ηθ2 . . . ηθi−1 )},
                   Pi−1 = Σ{ηθi+1 ηθi+2 . . . ηθn ζ(ηθ1 , ηθ2 . . . ηθi )},
and
                           Hi = A21 η1 + A22 η2 + · · · + A2n ηn ,
Ae meaning

            Σ{(ηe − ηθ1 )(ηe − ηθ2 ) . . . (ηe − ηθi−1 )ζ(ηθ1 , ηθ2 . . . ηθi−1 )},

                                              507
and ηω meaning x − hω .
    Let the n factors be constituted of m1 factors η1 , m2 factors η2 . . . mi factors
ηi . Then
                   Zi = µζ(η1 , η2 . . . ηi ), µ = m1 m2 · · · mi ,
                      Pi−1 = µζ(η1 , η2 . . . ηi )η1m1 −1 η2m2 −1 . . . ηimi −1 ,
                                                                                                      p. 501
   and
                    Pi−2 = µ1 ζ(η2 , η3 . . . ηi )η1m1 η2m2 −1 . . . ηimi −1
                               + µ2 ζ(η1 , η3 . . . ηi )η1m1 −1 η2m2 . . . ηimi −1
                               + &c. &c.
                               + µi ζ(η1 , η2 . . . ηi−1 )η1m1 −1 η2m2 −1 . . . ηimi ,
where
                                     µ                    µ             µ
                            µ1 =        ,         µ2 =       . . . µi =    .
                                     m1                   m2            mi
Hence
                            η1 ζ(η2 , η3 . . . ηi ) η2 ζ(η1 , η3 . . . ηi )         ηi ζ(η1 , η2 . . . ηi−1 )
                                                                                                          
Ti = µ ζ(η1 , η2 . . . ηi )
         2
                                                   +                        + ··· +                           .
                                    m1                      m2                                mi
Again, in Ui the term containing η1 will be

                 m1 η1 [Σ(η1 − η2 )(η1 − η3 ) . . . (η1 − ηi )ζ(η2 , η3 . . . ηi )]2

 = m1 η1 × (m2 m3 . . . mi )2 × (η1 − η2 )2 (η1 − η3 )2 . . . (η1 − ηi )2 {ζ(η2 , η3 . . . ηi )}2
                               µ2
                          =       η1 × ζ(η1 , η2 . . . ηi )ζ(η2 , η3 . . . ηi ).
                               m1
Hence
                                         η1 ζ(η2 η3 . . . ηi ) η2 ζ(η1 η3 . . . ηi )
                                                                                        
      Ui = µ2 ζ(η1 , η2 . . . ηi )                            +                      + &c. = Ti .
                                                m1                    m2
Hence, therefore, Ui − Ti vanishes whenever the roots of f x contain only i distinct
groups of equal roots, and it has been shown that Ui and Ti each contain only
2i − 1 of the coefficients of f x, so that Ui − Ti is a function only of n and these
2i − 1 letters, and consequently, by virtue of the Lemma in Art. (a), Ui − Ti
is universally zero, that is, Ui is identical with Ti , as was to be proved. In the
same manner, as observed in a preceding note [p. 494], the expression given in
the antecedent articles for the numerator of the ith convergents, having been
verified for the case of the roots consisting of only i distinct groups, could have
been at once inferred to be generally true by aid of the Lemma above quoted.
Art. (f).     Since the coefficient of x in Ti is Zi−1 × Zi , we deduce the unexpected
relation

             Σζ(h1 , h2 . . . hi−1 ) × Σζ(h1 , h2 . . . hi ) = P12 + P22 + · · · + Pn2 ,

                                                    508
where

        Pe = Σ{(he − hθ1 )(he − hθ2 ) . . . (he − hθi−1 )ζ(hθ1 , hθ2 . . . hθi−1 )}.
                                                            ′
So that every simplified Sturmian quotient to ff xx , when the n roots of f x are
real, will be the sum of n squares. But the equation is otherwise remarkable, in
exhibiting the product of the sum of

                                n(n − 1) . . . (n − i + 2)
                                    1 · 2 · · · (i − 1)

squares by another sum of

                                n(n − 1) . . . (n − i + 1)
                                       1 · 2···i
squares under the form of the sum of n squares.                                p. 502
    If we denote the ith simplified denominator to the Sturmian convergents to
f ′x
 f x by Di x, and if we call the ith simplified quotient Xi x, we have

                                     1
                            Xi x =     Σ(Di−1 he )2 (x − he ).
                                     n
If we construct the numerators and denominators of the convergents to
                                               1
                                                   1        ,
                                  Q1 − Q −          1
                                           2
                                               Q3 −···− 1
                                                       Qi


according to the general rule for continued fractions, as functions of Q1 , Q2 , Q3 ,
&c., so that calling the denominators ∆1 , ∆2 , ∆3 . . . ∆i ,

             ∆1 = Q,         ∆2 = Q1 Q2 − 1 . . . ∆i = Qi ∆i−1 − ∆i−2 ,

we have
                                      2 Z2 · · · Z2
                                     Zi−2 i−4     (i−1)
                         ∆i−1 x =      2 Z2 · · · Z2            Di−1 x,
                                      Zi−1 i−3     (i)

∆i−1 x being in fact the multiplier of f ′ x in the equation which connects f x and
f ′ x with the (i − 1)th complete residue, and consequently, retaining Q(x) to
designate the complete ith quotient, we have
                           2     4 Z4 · · · Z4
                               Zi−3
                         Zi−1        i−5      (i)
              Qi (x) =                            Σ{Di−1 he }2 (x − he )
                          Zi2 Zi−2
                               4 Z4 · · · Z4
                                    i−4     (i)+1
                           6     8 Z8 · · · Z8
                               Zi−3
                         Zi−1        i−5      (i)
                     =                            Σ{∆i−1 he }2 (x − he ),
                          Zi2 Zi−2
                               8 Z8 · · · Z8
                                    i−4     (i)+1


                                           509
which equation gives the connexion between the form of any quotient and that
of the immediately preceding convergent denominator of the continued fraction
                  ′
which expresses ff xx .
Art. (g).      I have found that the coefficients of the n factors of f x in the
expression above given for the quotients possess the property that the sum of their
square roots taken with the proper signs is zero for each quotient except the first
(the coefficients for the first being all units), that is Di h1 + Di h2 + · · · + Di hn = 0
for all values of i except i = 1. Moreover I find that the determinant formed
by the n sets of the n coefficients of the factors of f x in the complete set of n
quotients is identically zero, that is, the determinant represented by the square
matrix
                 1           1           1                  ···       1
              (D1 h1 )2   (D1 h2 )2   (D1 h3 )2             ···    (D1 hn )2
              (D2 h1 )2   (D2 h2 )2   (D2 h3 )2             ···    (D2 hn )2     = 0.
                ···         ···         ···                          ···
             (Dn−1 h1 )2 (Dn−1 h2 )2 (Dn−1 h3 )2            · · · (Dn−1 hn )2
                                                                                              p. 503
Art. (h). It should be observed that Ui is the form of the simplified quotients
for all the quotients except the nth (that is, the last), for which the simplified
form is not Un , but Un ÷ ζ(h1 , h2 . . . hn ), which arises from the circumstance of
the last divisor, which is the final Sturmian residue, not containing x; it being
evidently the case that the division of a rational function of x by another one
degree lower, introduces into the integral part of the quotient the square of the
leading coefficient of the divisor, subject to the exception that when the divisor
is of the degree zero, the simple power enters in lieu of the square. The general
formula gives for the reduced nth quotient the expression

            Σ{(h1 − h2 ), (h1 − h3 ) . . . (h1 − hn )ζ(h2 , h3 . . . hn )}2 (x − h1 ),

which equals
                        ζ(h1 , h2 . . . hn )Σζ(h2 , h3 . . . hn )(x − h1 ).
Rejecting the first factor, we have

                                 Σζ(h2 , h3 . . . hn )(x − h1 ),

which is equal to the penultimate residue, which residue is (as it evidently ought
to be) identical with the simplified last quotient.
                                                                                         ′
Art. (i).     We have thus succeeded in giving a perfect representation of ff xx , that
is, of
                               1      1               1
                                   +       + ··· +        ,
                             x − h1 x − h2         x − hn



                                               510
under the form of a continued fraction of the form
                                           1
                                                                                ,
                          m1 (x − e1 ) − m (x−e )−···−
                                                   1
                                                                       1
                                               2       2           mn (x−en )


where m1 , m2 . . . mn ; e1 , e2 . . . en are all determinate and known functions of
h1 , h2 . . . hn .
    We may by means of this identity, differentiating any number of times with
respect to x both sides of the equation, obtain analogous expressions for the
series
                          1                 1                   1
                                    +               + ··· +            .
                     (x − h1 )   t      (x − h2 ) t         (x − hn )t
But to do this we must be in possession of a rule for the differentiation of
continued fractions whose quotients are linear functions of the variable. I subjoin
here the first step only toward such investigation.
   Let the denominator of
                                       1
                                           1    ,
                                 q1 − q −···− 1
                                                2          qn
                                                                                                    p. 504
   where q1 , q2 . . . qn are any n arbitrary quantities, be denoted by [q1 , q2 , q3 . . . qn ],
so that the entire fraction will be equal to
                                        [q2 , q3 . . . qn ]
                                                               .
                                      [q1 , q2 , q3 . . . qn ]
Any such quantity as [qi , qi+1 . . . qn ] may be termed a Cumulant, of which
qi , qi+1 . . . qn may be severally termed the elements or Components, and the
complete arrangement of the elements may be termed the Type. The cumulant
corresponding to any type remains unaffected by the order of the elements in the
type being reversed, as is evident from any cumulant being in fact representable
under the form of a symmetrical determinant, thus, for example, the cumulant
[q1 , q2 , q3 , q4 ] may be represented by the determinant

                                      q1 1 0 0
                                      1 q2 1 0
                                               ,
                                      0 1 q3 1
                                      0 0 1 q4

and [q4 , q3 , q2 , q1 ] will in like manner be represented by the determinant

                                      q4 1 0 0
                                      1 q3 1 0
                                               ,
                                      0 1 q2 1
                                      0 0 1 q1

which is equal to the former.

                                               511
Art. (j). Let it be proposed in general to find the first differential coefficient
in respect to x of the fraction
                                          [qi , qi+1 . . . qn ]
                                                                  = Fi ,
                                         [q1 , q2 , q3 . . . qn ]
where each q is a function of one or more variables.
  I find that the variation of Fi may be expressed as follows:
                −δFi = {δ[q1 , q2 . . . qi−2 , qn ] + δ[q1 , q2 . . . qi−2 , qn−1 ]qn2
                           + δ[q1 , q2 , q3 . . . qi−2 , qn−2 ][qn , qn−1 ]2 + · · ·
                           + δ[q1 , q2 , q3 . . . qi−2 , qi−1 ][qn , qn−1 , qn−2 . . . qi ]2 }
                           ÷ [q1 , q2 , q3 . . . qn ]2 .
                                                                                                                p. 505
Art. (k). Suppose i = 2, and q1 = a1 x + b1 , q2 = a2 x + b2 . . . qn = an x + bn ,
we shall have by virtue of the above equation,
                                                                                        
                                                                                        
                                                                       1
                                                                                        
                          d                          d                                  
                            F2 ,     that is                                 1
                         dx                         dx 
                                                        q1 −                   1
                                                                                         
                                                                  q2 −                  
                                                                           q3 −···− q1
                                                                                         
                                                                                     n

               1
=−                         {an 12 +an−1 qn2 +an−2 [qn , qn−1 ]2 +&c.+a1 [qn , qn−1 , qn−2 . . . q2 ]2 }.
      [q1 , q2 . . . qn ]2
If we call F2 = ϕx
                f x every such quantity as [qn , qn−1 . . . qi ] represents to a constant
factor près the (i − 1)th simplified residue (ϕx counting as the first of them) to
ϕx
f x , and making certain obvious but somewhat tedious reductions, and rejecting
the common factor − (f 1x)2 , we obtain the expression

              C0 R12    R22   R32            Rn2
                     +      +      + ··· +         = ϕxf ′ x − ϕ′ xf x,
               C1      C1 C2 C2 C3         Cn−1 Cn
where R1 , R2 . . . Rn represent ϕx and the successive simplified residues to f x, ϕx,
while Ci means the coefficient of the highest power of x in Ri , and C0 the first
coefficient in f x.248
Art. (l).   If we take gx of the same degree as f x, and for greater simplicity
make the first coefficients in f x and gx each of them unity,                      p. 506
                                          gx
   the successive simplified residues to f x will be identical with the simplified
residues to
                                     −f x + gx
                                        gx
 248
     This result may be obtained directly as follows:—Let f x, ϕx and the (m − 1) complete
Sturmian residues be called ρ0 , ρ1 , ρ2 . . . ρn ; let the n complete quotients be called q1 , q2 . . . qn ,
and let the allotrious factors to the residues ρ2 , ρ3 . . . ρn be called µ2 , µ3 . . . µn ; then
                    ρ0 = q 1 ρ1 − ρ2 ,    ρ1 = q2 ρ2 − ρ3 ,     ρ2 = q3 ρ3 − ρ4 ,        &c.;


                                                      512
(including amongst them the quantity gx − f x itself), and, since

                   {f x − gx}g ′ x − {f x − gx}′ gx = g ′ xf x − f ′ xgx,

the right-hand side of the equation above written, when the residues, instead of
referring to f and ϕ, are made to refer to f and g, taken of the same degree in
x, becomes equal to f ′ xgx − f xg ′ x; and if we now agree to consider f and g as
homogeneous functions each of the nth degree in x and 1, the equation becomes

R12    R22   R32            Rn2              d                   d
    +      +      + ··· +         = g(x, 1) f (x, 1) − f (x, 1) g(x, 1)
C1    C1 C2 C2 C3         Cn−1 Cn           dx                  dx
                                    1      d     d       d         1    d d      d
                                                                        
                                  =     x g+ g             f −        x f+ f
                                    n     dx     d1     dx         n   dx d1    d
                                    1 df dg      df dg       1
                                                      
                                  =            −         = J(f, g),
                                    n dx d1 d1 dx           n
where J indicates the Jacobian of the given functions f and g in respect to
the variables x and 1, meaning thereby the so-called Functional Determinant
of Jacobi to f and g in respect of x and 1, which equation also obviously must
continue to hold good when we restore to the coefficients of xn in f and g their
general values.
   It may happen that for particular relations between the coefficients of f and
g certain of the residues may be wanting, which will be the case when any of
the secondary Bezoutics have their first or successive terms affected with the
coefficient zero; the equation connecting the residues with the Jacobian will then
change its form (as some of the quantities C1 , C2 . . . Cn will become zero); but

hence
                        ρ1 δρ0 − ρ0 δρ1 = ρ21 δq1 + (ρ2 δρ1 − ρ0 δρ2 )
                                          = ρ21 δq1 + ρ22 δq2 + (ρ3 δρ2 − ρ2 δρ3 )
                                          = &c.
                                          = ρ21 δq1 + ρ22 δq2 + ρ23 δq3 + · · · + ρ2n δqn ;
but we have in general ρi = µi Ri , hence
                                                     Ci−1 µi+1
                                             δqi =             δx,
                                                      C i µi
and
                                                   Ci−1
                                       ρ2i δqi =        µi−1 µi Ri2 δx;
                                                    Ci
but it may be easily seen that
                                                             1
                                               µi−1 µi =     2
                                                                 ,
                                                            Ci−1
except when i = 1, for which case µi−1 µi = 1, hence
                           1                                            C0 2
            ρ2i δqi =           Ri2 δx,     when i > 1,       and =        R1 δx     when i = 1,
                        Ci−1 Ci                                         C1
which proves the theorem in the text.


                                                      513
I do not propose to enter for the present into the theory of these failing, or as
they may more properly be termed, Singular cases in the theory of elimination.
Art. (m). The series last obtained for J(f, g) leads to a result of much interest
in the theory, and of which great use is made in the concluding section of this
memoir, namely the identification of the Jacobian (abstraction made of the
numerical factor n) with what the Bezoutiant becomes when in place of the n
variables in it, u1 , u2 . . . un , we write xn−1 , xn−2 . . . x, 1. Thus suppose f and g
to be each of the third degree, and let
             Ax2 + Hx + G,            Hx2 + Bx + F,                    Gx2 + F x + C,
                                                                                                p. 507
   be the three primary Bezoutics; if we make
                            x2 = u,        x = v,             1 = w,
these may be written under the form
   Au + Hv + Gw = L,             Hu + Bv + F w = M,                        Gu + F v + Cw = N,
and if the Bezoutiant be called I, we have
                              dI                 dI                      dI
                        L=       ,      M=          ,          N=           .
                              du                 dv                      dw
The simplified residues to f and g are L, (L, M ), (L, M, N ), where (L, M ) means
the result of eliminating u between L and M , and (L, M, N ) the result of
eliminating u and v between L, M, N ; and by a theorem (virtually implied in
the direct method249 of reducing a quadratic function to the form of a sum of
squares), if we call the leading coefficients of these quantities C1 , C2 , C3 , we have
                           L2   (L, M )2 (L, M, N )2
                              +         +            = I.
                           C1    C1 C2      C2 C3
Hence, when n = 3, 13 J(f, g) = I when in I, u, v, w are turned into x2 , x, 1;
and so in general for any values of n, the Bezoutiant correspondingly modified,
becomes n1 J(f, g), as was to be shown.250
                                                                                ′
Art. (n).    The expressions obtained for the quotients to ff xx may be generalized
and extended to the quotients to ϕx
                                 f x , where ϕx and f x are two functions of x of
any degrees m and n, whose roots are respectively k1 , k2 . . . km , and h1 , h2 . . . hn .
If we suppose
                     ϕx                    1
                         =                     1            ,
                     fx    Q(x) − q (x)−         1
                                           2                           1
                                                     q3 (x)−···−
                                                                 q   m+1 (x)

 249
    Namely, that of M. Cauchy, adverted to in Section IV. Arts. 44–45. [p. 511 below.]
 250
    Compare Jacobi, De Eliminatione, § 2. The general expression for the allotrious factor, I
may here incidentally mention, is given under the head Theorem α, § 16, which comes quite at
the end of the same paper.


                                               514
where Q(x) is of n − m dimensions, and q2 (x), q3 (x) . . . qm+1 (x) each of one
dimension in x, it may be proved that on writing

                                             1                     Ni (x)
                                                               =          ,
                                             1
                               Q(x) − q (x)−···−   1               Di (x)
                                       2         q (x)  i
                                                                                                             p. 508
    we shall have
                         m 
                                            f kθ
                                                              
                                (Ni kθ )2          (x − kθ ) = Cqi+1 (x),                             (A)
                         X

                         θ=1
                                            ϕ′ k θ

                         n 
                                        ϕhθ
                                                              
                               (Di hθ )2 ′ (x − hθ )              = C ′ qi+1 (x),                      (B)
                         X

                         θ=1
                                        f hθ
where
                                             C + C ′ = 0,                                              (E)
Cqi+1 (x) being the (i + 1)th simplified quotient. When Q(x) is a linear function
of x, in finding q1 x from the formula (B), we must take D0 x = 1. The proof of
this theorem being generally true, may easily be shown to depend upon its being
true in the special case,251 when m = µ + i, and n = µ + i′ (m being supposed
less than n), and h1 , h2 . . . hn become l1 , l2 . . . lµ , h1 , h2 . . . hi , while k1 , k2 . . . km
become l1 , l2 . . . lµ , k1 , k2 . . . ki ; and the truth of the theorem for this special case
(if for instance we wish to prove the formula (B)) depends upon the expression
                                              !                                     !
                    h1 h2 . . . hi′ −1                        h1 h2 . . . hi
                    k1 k2 . . . km                            k1 k2 . . . km
                                              ! ×                                       !
                    hi hi+1 . . . hn                        hi′ +1 hi′ +2 . . . hn
                    hi hi+1 . . . hn                        hi′ +1 hi′ +2 . . . hn

being identical with the expression
                              !                                                                                       !

       h1 h2 . . . hi′ −1                                                          
                                                                                                          hi
                                                                                   

       k1 k2 . . . km                                                              
                                                                                                    k1 , k2 . . . km
                               × (hi − h1 )(hi − h2 ) . . . (hi − hi′ −1 ) ×


       hi hi+1 . . . hn                                                    
                                                                            
                                                                                                            hi
                                                                           
                                                                                          h1 , h2 . . . hi−1 , hi+1 . . .

as it may readily be shown to be. And the formula (A) may be verified in
precisely the same manner. There is no difficulty in finding the values of C
and C ′ , which are products of powers, some positive and some negative, of the
  251
      By virtue of the Lemma, that when ϕx and f x are two algebraical functions, no function of
the coefficients vanishing identically when i roots of f x coincide with i roots of ϕx respectively
can be formed, in which there are fewer of the coefficients of f and ϕ respectively than appear
in the leading coefficient of the (n − i + 1)th residue of ϕf .


                                                  515
leading coefficients in the simplified residues, and recognizing that they satisfy
the equation (E); when ϕx is of one degree below f x this equation is of the form
C + C ′ = 0.
Art. (o). When ϕx = f ′ x, this expression for the (i + 1)th simplified quotient
becomes Σ(Di h)2 (x − h), as previously found; the correlative expression will be

                                                   fk
                                     −Σ(Ni k)2            (x − k),
                                                   f ′′ k
                                                                                                          p. 509
   k being any root of f ′ x = 0, which is equal to the former expression. The
general expressions above given for the simplified quantities are of course integral
functions of h and k, although given under the form of the sums of fractions, by
virtue of the well-known theorem that Σ fϑh′ h , where ϑ is an integral function of h,

and the summation comprises all the roots (h) of f h = 0, is always integral.
Art. (p).      It will be found that for all values of i greater than unity
                                       m
                                                       f kθ
                                            (Ni kθ )          = 0,
                                       X

                                       θ=1
                                                       ϕ′ k θ

and that                               n
                                                       ϕhθ
                                           (Di hθ )           = 0.
                                       X

                                      θ=1
                                                       f ′ hθ
The theorem of Art. (n) is in effect a theorem of cumulants of the form

                               [Q1 (x), q2 (x) . . . qi (x) . . . qn (x)],

where the elements are all independent of one another, and

       f x = [Q1 (x), q2 (x), q3 (x) . . . qn (x)],          ϕx = [q2 (x), q3 (x) . . . qn (x)],

n being any number whatever greater than i; this makes the theorem still more
remarkable. The urgency of the press precludes my investigating for the present
the more general theorem which must be presumed to exist, whereby qi+1 can
be connected with [q1 , q2 , q3 . . . qi ], or [q2 , q3 . . . qi ], and with [q1 , q2 , q3 . . . qi+e ]
and [q2 , q3 . . . qi+e ], when each q represents a function of an arbitrary degree in
x. The theorem so generalized would comprehend the complete theory of the
quotients arising from the process of continued division, without exclusion of
the singular cases (at present supposed to be excluded) where one or several
consecutive principal coefficients in one or more of the residues, vanish.
Art. (q).    The complete statement of two twin theorems suggested by and
intimately connected with the biform representation of the quotients ϕx
                                                                     f x , given
in the preceding article, is too remarkable to be omitted.



                                                  516
                                                                         ′
   Suppose ϕx = f ′ x, and let the successive convergents to ff xx be called

                         1       t2 x             tn−1 x       tn x
                             ,        ,   ...            ,          ,
                        T1 x     T2 x             Tn−1 x       Tn x
where the subscript index to t or T indicates the degree in x. Then if we call the
roots of f x, h1 , h2 . . . hn , the theorem already cited in a preceding          p. 510
   article, concerning the denominators of the convergents, may be expressed as
follows:—               ′ 2           ′ 2              ′ 2
                          f h1          f h2               f hn
                          ϕh1            ϕh2       · · ·    ϕhn
                       (T1′ h1 )2      (T1′ h2 )2 · · · (T1′ hn )2
                       (T2′ h1 )2      (T2′ h2 )2 · · · (T2′ hn )2    = 0,
                            ···           ···                ···
                     (Tn−1
                         ′    h1 )2 (Tn−1
                                        ′    h2 )2 · · · (Tn−1
                                                           ′   hn ) 2
where it will be observed that the first line of terms consists exclusively of units,
since f ′ x = ϕx by hypothesis.
   Correlatively I have ascertained that preserving the same assumption that
                                       ′          ′′
ϕx = f ′ x, so that consequently ϕf kk means ff kk , the following theorem obtains,
namely that if k1 , k2 . . . kn−1 are the (n − 1) roots of ϕx,

                  ϕ k1 2           ϕ k2 2                    ϕ kn−1 2
                  ′              ′                    ′        
                   f k1             f k2         ···         f kn−1
                {t1 (k1 )}2      {t1 (k2 )}2     ···    {t1 (kn−1 )}2
                {t2 (k1 )}2      {t2 (k2 )}2     ···    {t2 (kn−1 )}2        = 0.
                   ···           ···                         ···
               {tn−2 (k1 )}2 {tn−2 (k2 )}2       · · · {tn−2 (kn−1 )}2

It may consequently be conjectured, when ϕ and f are independent functions
of x and respectively of the degree n − 1 and n, and ϕx
                                                     f x is expanded under the
form of a continued fraction, of which, as before,
                                   1       t1     tn−1
                                      ,       ...
                                   T1      T2      Tn
are the successive convergents, that we shall have analogous determinants to
the twin forms above given, each separately vanishing, these more general
determinants differing only from their model forms in respect of the uppermost
line of terms in the one of them, being each multiplied by certain functions of
h1 , h2 . . . hn respectively (all of which become units when ϕx = f ′ x), and in the
other of them by certain functions of k1 , k2 . . . kn .
    The exact form, however, of such functions, and even the possibility of such
form being found capable of making the determinants vanish, remains open for
further inquiry.                                                                      p. 511


                                     Section IV.

                                           517
 On some further Formulae connected with M. Sturm’s theorem, and on the
 Theory of Intercalations, whereof that theorem may be treated as a corollary.

Art. (44). As preparatory to some remarks about to be made on the formulæ
connected with M. Sturm’s theorem, it is necessary to premise two theorems of
great importance concerning quadratic functions, one of which, notwithstanding
extreme simplicity, is as far as I know very little (if at all) known, and the other
was given in part many years ago by M. Cauchy, but is also not generally known.
The former of these two theorems is as follows. If a quadratic homogeneous
function of any number of variables be (as it may be in an infinite variety of
ways) transformed into a function of a new set of variables, linearly connected
by real coefficients with the original set, in such a way that only positive and
negative squares of the new variables appear in the transformed expression, the
number of such positive and negative squares respectively will be constant for a
given function whatever be the linear transformations employed. This evidently
amounts to the proposition, that if we have 2n positive and negative squares of
homogeneous real linear functions of n variables identically equal to zero, the
number of positive squares and of negative squares must be equal to one another,
so that for example we cannot have

              u21 + u22 + · · · + u2n + u2n+1 − u2n+2 − u2n+3 − · · · − u22n

identically zero when n of the variables are linear functions of the remaining n;
and this is obviously the case, for if the equation could be identically satisfied
we might make

                    un+2 = u1 ,       un+3 = u2 , . . . u2n = un−1 ,

and we should then be able to find un+1 as a real numerical multiple of un ,
and consequently should have the equation u2n (1 + k 2 ) = 0, which is obviously
impossible; à fortiori we may prove that in the identical equation existing between
the sum of an even number of positive and of negative squares of real linear
functions of half the number of independent variables, there cannot be more
than a difference of two (as we have proved that there cannot be that difference)
between the number of positive and negative squares. Hence there must be as
many of one as of the other; and as a consequence, the number of positive squares
or of negative squares in the transform of a given quadratic function of any
number of variables effected by any set of real linear substitutions is constant,
being in fact some unknown transcendental function of the coefficients of the
given function. I quote this law (which I have enunciated before, but of which
I for the first time publish the proof) under the name of the law of inertia for
quadratic forms.                                                                    p. 512

Art. (45).   The other theorem is the following. If any quadratic function be


                                           518
represented in the umbral notation252 under the form of

                             (a1 x1 + a2 x2 + · · · + an xn )2 ,

where a1 , a2 . . . an are the umbræ of the coefficients, and x1 , x2 . . . xn the variables,
then by writing

        a1      a1      a1      a1              a1
           x1 +    x2 +    x3 +    x4 + · · · +    xn = y1 ,
        a1      a2      a3      a4              an
        a1 a2      a1 a2      a1 a2              a1 a2
              x2 +       x3 +       x4 + · · · +       x n = y2 ,
        a1 a2      a1 a3      a1 a4              a1 an
        a1 a2 a3      a1 a2 a3              a1 a2 a3
                 x3 +          x4 + · · · +          xn = y3 ,
        a1 a2 a3      a1 a2 a4              a1 a2 an
                       &c. &c. &c.,
        a1 a2 · · · an
                       xn = yn ,
        a1 a2 · · · an

(a1 x1 + a2 x2 + · · · + an xn )2 will assume the form

               a1 a2            a1 a2 a3                        a1 a2 · · · an−1 , an
  a1 2         a1 a2            a1 a2 a3                        a1 a2 · · · an−1 , an
     y +                y22 +                   y32 + · · · +                            yn2 ,
  a1 1           a1                a1 a2                           a1 a2 · · · an−1
                 a1                a1 a2                           a1 a2 · · · an−1

and consequently the number of positive squares in the reduced form of the given
function will always be the number of continuations or permanencies of sign of
the series

             a1         a1 a2           a1 a2 a3                    a1 a2 · · · an
       1;       ;             ;                  ··· ,                             ,
             a1         a1 a2           a1 a2 a3                    a1 a2 · · · an

the several terms of this progression being in fact the determinants of what the
given function becomes when we obliterate successively all the variables but one,
then all but that and another, then all but these two and a third, until finally the
last term is the determinant of the given function with all the variables retained.
This comes to saying that if we call the function (suppose of four variables) f ,
   For an explanation of the umbral notation, see London and Edinburgh Philosophical
 252

Magazine, April 1851, or thereabouts [p. 243 above].




                                            519
and write down the matrix
                             d2 f            d2 f          d2 f          d2 f
                                  ,                 ,             ,             ,
                             dx21           dx1 dx2       dx1 dx3       dx1 dx4
                             d2 f            d2 f          d2 f          d2 f
                                    ,             ,               ,             ,
                            dx2 dx1          dx22         dx2 dx3       dx2 dx4
                             d2 f            d2 f          d2 f          d2 f
                                    ,               ,           ,               ,
                            dx3 dx1         dx3 dx2        dx23         dx3 dx4
                             d2 f            d2 f          d2 f          d2 f
                                    ,               ,             ,           ,
                            dx4 dx1         dx4 dx2       dx4 dx3        dx24
                                                                                                                 p. 513
   (where all the terms are of course coefficients of the given function expressed
as above for greater symmetry of notation), the inertia of f will be measured
by the number of continuations of sign in the series formed of the successive
principal minor coaxal determinants (in writing which I shall use in general (r, s)
             2f
to denote dxdr dxs
                   ),

                                                                    (1, 1) (1, 2) (1, 3)
                                  (1, 1) (1, 2)
              1, (1, 1),                        ,                   (2, 1) (2, 2) (2, 3) ,
                                  (2, 1) (2, 2)
                                                                    (3, 1) (3, 2) (3, 3)
                                   (1, 1)     (1, 2)     (1, 3)     (1, 4)
                                   (2, 1)     (2, 2)     (2, 3)     (2, 4)
                                                                           ,
                                   (3, 1)     (3, 2)     (3, 3)     (3, 4)
                                   (4, 1)     (4, 2)     (4, 3)     (4, 4)
and in like manner in general.253
Art. (46).     Reverting now to the simplified Sturmian residues, since by the
theory set out in the first Section these differ from the unsimplified complete
residues required by the Sturmian method only in the circumstance of their
being divested of factors which are necessarily perfect squares and therefore
essentially positive, these simplified Sturmians may of course be substituted for
the complete Sturmians for the purposes of M. Sturm’s theorem. The leading
coefficients in these simplified Sturmians, reckoning f ′ (x) as one of them, will be

                    m2 Σζ(h1 , h2 ),          Σζ(h1 , h2 , h3 ) . . . ζ(h1 , h2 . . . hm ),
  253
      I have given a direct à posteriori demonstration in the London and Edinburgh Philosophical
Magazine, that the number of continuations of sign in any series formed like the above from a
symmetrical matrix, is unaffected by any permutations of the lines and columns thereof, which
leaves the symmetry subsisting, that is to say (using the umbral notation), if θ1 , θ2 , θ3 . . . θi
are disjunctively equal, each to each, in any arbitrary order to 1, 2, 3 . . . i, the number of
continuations of sign in the series

              aθ1          aθ1    aθ2          aθ1      aθ2   aθ3              aθ1   aθ2   ···   aθi
      1,             ,                  ,                            ...,                              ,
              aθ1          aθ1    aθ2          aθ1      aθ2   aθ3              aθ1   aθ2   ···   aθi

is irrespective of the order of the natural numbers 1, 2, 3 . . . i in the arrangement θ1 , θ2 , θ3 . . . θi .


                                                     520
which it is easily seen, as remarked long ago by Mr Cayley, are the successive
principal minor coaxal determinants of the matrix

                                 σ0  σ1 σ2 σ3 · · · σm−1
                                 σ1  σ2 σ3        · · · σm
                                 σ2  σ3        · · · σm+1
                                 ··· ···             ···
                                σm−1 σm       · · · σ2m−2 ,
                                                                                                        p. 514
   where in general σr = hr1 + hr2 + · · · + hrm , and of course σ0 = m. M. Hermite
has improved upon this remark by observing, what is immediately obvious, that
if we use σr to denote, not the quantity above written, but
                                 hr1    hr2            hrm
                                     +       + ··· +        ,
                               x − h1 x − h2         x − hm
the successive coaxal determinants of the above matrix will become respectively

                                1                   ζ(h1 , h2 )
                                                                      
                            Σ        ,        Σ                    ,
                              x − h1            (x − h1 )(x − h2 )

                       ζ(h1 , h2 , h3 )                 ζ(h1 , h2 . . . hm )
                                               
         Σ                                   ···                                    ;
                 (x − h1 )(x − h2 )(x − h3 )     (x − h1 )(x − h2 ) · · · (x − hm )
that is to say, these successive coaxal determinants, when multiplied up by f x,
will become respectively

Σ(x − h2 )(x − h3 ) · · · (x − hm ),          Σζ(h1 , h2 ){(x − h3 )(x − h4 ) · · · (x − hm )}, . . .

                                         Σζ(h1 , h2 . . . hm ),
that is to say, will represent the simplified Sturmian series given by my general
formulæ. M. Hermite further remarks, that the matrix formed after this rule will
evidently be that which represents the determinant of the quadratic function
(which may be treated as a generating function)
                            1
                      Σ          {u1 + h1 u2 + h21 u3 + · · · + hm−1
                                                                 1   um }2 ,
                          x − h1
in which, since only the squared differences of the terms in the (h) series finally
remain in the successive coaxal determinants, we may write (x − h1 ), (x −
h2 ) . . . (x − hm ) simultaneously in place of h1 , h2 . . . hm without affecting the
result; consequently the generating function above may be replaced by the
generating function
             1
       Σ          {u1 + (x − h1 )u2 + (x − h1 )2 u3 + · · · + (x − h1 )m−1 um }2 ,
           x − h1


                                                 521
the corresponding matrix to which becomes
                              1
                       Σ             θ0       θ1    ···    θm−2
                           x − h1
                            θ0       θ1       θ2    ···    θm−1
                            θ1       θ2             ···     θm
                            ···      ···                    ···
                           θm−2     θm−1 · · · θ2m−3 ,
                                                                                          p. 515
                                                   f ′x
   where θi denotes Σ(x − h)i , and Σ       x−hi = f x . Hence every simplified residue
                                             1

is of the form
                                                  0  θ0  θ1        ···   θr
             θ1  θ2 · · ·    θr
                                                 θ0  θ1            ···   θr+1
             θ 2 θ 3 · · · θ r+1
        f ′x                               + f x θ1                ···   θr+2 .
             ··· ···        ···
                                                 ··· ···                  ···
             θr θr+1 · · · θ2r−1
                                                 θr θr+1 · · ·     · · · θ2r+1

The residue in question will be of the degree m − r − 2 in x, and consequently
we have, according to the notation antecedently used for the syzygetic equations,

                                    θ1  θ2 · · ·    θr
                                    θ2  θ3 · · · θr+1
                           tr+1 =                       ,
                                    ··· ···        ···
                                    θr θr+1 · · · θ2r−1

                                 0  θ0  θ1          ···     θr
                                θ0  θ1              ···   θr+1
                       −τr =    θ1                  ···   θr+2 .
                                ··· ···                    ···
                                θr θr+1 · · ·       · · · θ2r+1
Elegant and valuable for certain purposes as are these formulæ for tr+1 and τr ,
they are affected with the disadvantage of being expressed by means of formulæ
of a much higher degree in the variable x than really appertains to them, the
paradox (if it may be termed such) being explained by the circumstance of the
coefficients of all the powers of x above the right degree being made up of terms
which mutually destroy one another; upon the face of the formulæ, tr+1 and τr ,
which are in fact only of the degrees r + 1 and r respectively in x, would appear
to be of the degree
                             1 + 3 + 5 + · · · + (2r − 1),
that is of the degree r2 .
Art. (47). I may add the important remark, which does not appear to have
occurred immediately to my friend M. Hermite when he communicated to me
the above most interesting results, that in fact, by virtue of the law of inertia

                                            522
for quadratic forms, we may dispense with any identification of the successive
coaxal determinants of the matrix to the generating function
                        1
                  Σ          {u1 + h1 u2 + h21 u3 + · · · + hm−1
                                                             1   um }2
                      ρ − h1
with my formulæ for the Sturmian functions, and prove ab initio in the most
simple manner, that the successive ascending coaxal determinants             p. 516
   (always of course supposed to be taken about the axis of symmetry) of the
matrix to the form above written, or to the more general form (which I shall
quote as (G), namely)

              Σ(ρ − h1 )q {ϕ1 (h1 )u1 + ϕ2 (h1 )u2 + · · · + ϕm (h1 )um }2 ,     (G)

(where ϕ1 , ϕ2 . . . ϕm are absolutely arbitrary integral forms of function with real
coefficients), will form a rhizoristic series in regard to f x (that is a series, the
difference between the number of the continuations of sign between the successive
terms of which corresponding to two different values of ρ will determine the
number of real roots of x lying between such two assumed values), provided only
that q be an odd positive or negative integer. Nothing can be easier than the
demonstration, for whenever ρ is greater than any one of the real roots as hi :
   Firstly, any pair of imaginary roots will give rise to two terms of the form
                   √            √                       √             √
        (L + M −1)q (v + w −1)2 and (L − M −1)q (v − w −1)2 ,

or more simply
          √                 √            √                  √
   (L + M −1)(v 2 − w2 + 2vw −1) + (L − M −1)(v 2 − w2 − 2vw −1),

where v and w are real linear functions of u1 , u2 . . . um . The sum of which couple
will be

     2{L(v 2 − w2 ) − 2M wv} = 2{(Lv − M w)2 − (Lw + M v)2 } = p2 − q 2 ;

so that each such couple combined will for every value of x give rise to one
positive and one negative square.
    Secondly, any real root of the series h1 , h2 . . . hm , when ρ is taken greater
than such root, will give rise to a positive square of a real linear function of
u1 , u2 . . . um .
    Thirdly, any real root of the same series, when ρ is beneath it in value (q
being odd), will give rise to the negative of the square of a real linear function of
the same.
    Hence the number of real roots between ρ taken equal to one value (a), and
ρ taken equal to any other value (b), will be denoted by the loss of an equal
number of positive squares in the reduced form of the expression (G) when ρ

                                           523
is taken a and when ρ is taken b; that is by virtue of Art. 45 will be denoted
by the difference of the number of permanencies of sign in the successive minor
determinants of the matrix corresponding to the quadratic form (G)254 (which
we have taken as our                                                               p. 517
   generating function) resulting from the substitution respectively of a and b in
place of ρ, which gives a theorem equivalent to that of M. Sturm, transformed
by my formulæ, when we choose to adopt the particular suppositions

       q = −1, qquadϕ1 h = 1,          ϕ2 h = h,        ϕ3 h = h2 , . . .   ϕm h = hm−1 .

This method of constructing a rhizoristic series to f x by a direct process is
deserving of particular attention, because it does not involve the use of the
notion of continuous variation, upon which all preceding proofs of Sturm’s
theorem proceed. It completes the cycle of the Sturmian ideas. Happily this
cycle was commenced from the other end, for it would have been difficult to have
suspected that the root-expressions for the terms in the rhizoristic series could
be identified with the residues, had the former been the first to be discovered,
and much of the theory of algebraical common measure laid open by means of
this identification would probably have remained unknown.
Art. (48). I proceed now to consider a theorem concerning the relative positions
of the real roots of two independent algebraical functions as indicated by the
succession of signs presented by their Bezoutian secondaries; this more general
theory of intercalations or relative interpositions will be seen to include within it
as a corollary the justly celebrated theorem of M. Sturm.
   Let the real roots of f x taken in descending order of magnitude be h1 , h2 . . . hp ,
and the real roots of ϕx taken in the like order η1 , η2 . . . ηq , so that

                          f x = (x − h1 )(x − h2 ) · · · (x − hp )H,

                          ϕx = (x − η1 )(x − η2 ) · · · (x − ηq )K,
H and K being functions of x incapable of changing their signs. Now, as in M.
Sturm’s method, let us inquire what takes place in respect to the sign of ϕ(x)f (x) ,
which I shall call the Indicatrix, as x descends the scale of real magnitude from
+∞ to −∞. If between +∞ and h1 , i real roots of ϕx are contained, it is obvious
 254
     The inertia of the quadratic form (G) is the measure of the number of real roots of f x
comprised between ∞ and ρ, and may be estimated in any manner that may be found most
convenient. If ρ be made infinity, and hi be taken equal to ϕiq−1 , and the inertia of the
corresponding value of (G) be estimated by means of the formulæ in ordinary use by geometers
for determining the nature of a surface of the second degree, the criteria of the number of
real roots in f x will be, or may be made to be, symmetrical in respect to the two ends of the
expression f x. This system of criteria, however, is not so good as that given by the Bezoutiant
to the two differential coefficients of f (x, 1) taken with regard to x and 1 respectively, which
will also possess the like character of symmetrical indifference, and be one less in number than
the former.


                                              524
that as x travels from +∞ to the superior brink of h1 , the Indicatrix will change
its sign from + to − and from − to + altogether i times, so that at the moment
when x is about to pass through h1 , it                                              p. 518
    will be positive if i is zero or even, and negative if i is odd; but the moment
after x has passed through the value h1 , the indicatrix will be negative on the
first supposition, and positive on the other supposition. Hence immediately after
the passage of x through h1 the indicatrix will have been once oftener negative
than positive on the one supposition, and as often negative as positive on the
other. Again, in like manner as x traverses the interval between h1 and the
inferior brink of h2 , if no η or an even number of η’s occupy this interval, the
sign which the indicatrix had at the beginning of this interval will have been
reversed once oftener than restored; but if there be an odd number of η’s so
interposed, the number of reversals and restorations will have been identical;
and so for each successive interval, reckoned from a value for x immediately
subsequent to one real root of f x, down to a value immediately subsequent to
the next less real root of the same; and it is evident that the effect upon the
sign of the indicatrix at the end of every such interval depends, not upon the
number of η’s grouped together in such interval, but upon the form of the group
as regards its being made up of an odd or even number of terms, the first interval
being of course understood to extend from +∞ to a value immediately inferior
to h1 , and the last from a value immediately inferior to hp to −∞. Hence as
regards the relation of the sign of the indicatrix at the beginning to the sign
at the end of every such interval, nothing will be altered by taking away any
even number of η’s that may be found therein. If we suppose this to be done,
we shall then have in some of the intervals one η occurring and in the other
intervals no η; that is to say, some of the h’s will be separated by single η’s, but
other h’s will come together. Again, by removing any even number of h’s not
separated by η’s (and thus removing an even number of intervals), it is clear
that as many changes of sign of the indicatrix will have been done away with
from + to − as from − to +, and no effect of the one kind of sign changes over
the other kind of changes will have been produced. By removing pairs of h’s in
this manner, it may happen that η’s will again be brought together, any even
number of which, not separated by h’s, may again be removed and then pairs of
h’s not separated by η’s in their turn, and so continually toties quoties until at
length we must arrive at a reduced system of h’s and η’s, where no two h’s and
no two η’s come together, or else all the h’s and all the η’s will have disappeared.
Let the scale of h’s and η’s thus simplified and reduced be called the effective
scale of intercalations. The number of h’s and the number of η’s in any such
scale will be equal, or will at most differ from one another by a unit, since at
each part of the scale, except at the end, every h is followed by an η and every η
by an h. If the scale begins and ends with an h, there will of course be one more
h than η; if it begin and end with an η, there will be one more η than h; if it
begin with an h or an η and end with an η or h, there will be as many of the

                                        525
one as of the other.                                                                    p. 519
   Firstly, suppose the effective intercalation scale to commence with an h; then
in passing from +∞ to just beyond the first h the sign of the indicatrix changes
from + to −; it changes again from − to + as it passes the first η, then again
from + to − as it passes the second h, and so on; that is to say, there will
be a change always in the same direction from + to − as x passes from being
just greater than to being just less than any h appearing in the effective scale.
Secondly, if the effective scale begin with η, the indicatrix will conversely be
negative after passing the first and every subsequent η, and change from − to +
in the act of passing through the first and every subsequent h. So that on either
supposition the changes of sign for the effective scale always take place in the
same direction, and the number of h’s in the effective scale will be measured by
the number of such changes, and consequently will be measured by the difference
between the number of times that the indicatrix ϕf changes its sign from + to
− as x passes through each in turn of the real roots of f x, and the number of
times that in passing through any such root it changes its sign from − to +; if
the former number be greater than the latter, the effective scale of interpositions
will begin with a root of f x; if it be less, the scale will begin with a root of ϕx.
   If instead of beginning with +∞ and ending with −∞ we begin and end with
any two limits, a and b respectively (making abstraction of all roots of f x or of
ϕx lying outside these limits, and forming the effective intercalation scale with
the roots comprised within these limits exclusively), we shall obviously obtain
a similar result, but with the condition that the changes from + to − will be
in excess if an even number of h’s and η’s combined be cut off by the superior
limit, and the effective scale begin with an h, or if an odd number of h’s and η’s
combined be so cut off and the scale begin with an η; and in defect if an odd
number of h’s and η’s combined be so cut off and the scale begin with an h, or
an even number be so cut off and the scale begin with an η.
   If, now, supposing f x to be of n, and ϕx of not more than n, say m dimensions,
we form the signaletic series f x, ϕx, B1 , B2 . . . Bm−1 (where the B1 , B2 . . . Bm−1
are the Bezoutian secondaries or simplified successive residues corresponding
to ϕx
    f x expanded under the form of an improper continued fraction), it may be
shown, in the same way as for Sturm’s theorem, that whenever ϕx    f x changes from
+ to − a change of sign will be gained in the series, and when from − to +
a change will be lost; and that no change can be gained or lost except as x
passes through the successive real roots of f x. Hence the difference between the
number of changes of sign in the above signaletic series when x is taken a, and
the number of the same when x is taken b, will indicate the number of roots         p. 520
   of f x remaining in the effective scale of interpositions formed between such of
the roots of f x and of ϕx as lie between a and b; calling the one number I(a) and
the other I(b), the sign of I(b) − I(a) depends not on the relative magnitudes of a
and b, but upon the manner in which the effective scale commences; if I(a) − I(b)


                                          526
is positive, the effective scale formed between the a and b will commence with a
root of f x; if negative, it will commence with a root of ϕx.
Art. (49). In forming the scale of effective interpositions, it is evidently not
necessary to go on reducing the h series and the η series separately and alternately;
the same result will be effected more expeditiously by eliding simultaneously any
even number of h’s that come together without being separated by an η, and any
even number of η’s that come together without being separated by an h, and,
repeating this process of simultaneous elision, as often as may be required, until
no two h’s or η’s come together. Thus, for instance, denoting the magnitudes of
the series of real roots of f and of ϕ by the distances of h and η points taken
along a right line from a fixed point therein, and supposing such series of roots
between the limits a and b to be

                       hhhηηηhηηhηηηhhηhηhhhhhhηηh,

our first reduction brings this scale to the form

                                   hηhhηηhηhh;

the next reduction brings it to the form

                                      hηηηhη;

and a third and final reduction brings it to the form

                                       hηhη;

and accordingly we shall find for such an arrangement of the h and η system

                                 I(b) − I(a) = +2.

Art. (50). If we suppose ϕx = dfdxx , by a well-known theorem of algebra, any
two consecutive roots of f x will contain between them an odd number of roots
of ϕx, and the number of real roots of ϕx greater than the greatest root of f x,
and the number of real roots of ϕx less than the least root of f x will each be
even. Hence the effective intercalation scale between any two limits a and b will
be formed by merely reducing the η groups to single units, and the number of
h’s in the scale so formed will be the total number of h’s between the limits a
and b. Moreover, since such scale commences always with a root of f x, or with
an even number of roots of f x followed by                                         p. 521
    a root of ϕx, if the number of h’s and η’s cut off be even, and with a root of
f x or an even number of roots of f x followed by a root of ϕx, if the number
  ′

so cut off be odd, it follows that for this case I(a) − I(b), a being the superior
limit, will be always positive, and will measure the total number of real roots of


                                        527
f x lying between a and b; this, then, is Sturm’s theorem, treated as a corollary
to the Theory of Intercalations.
Art. (51). If we write down the last syzygetic equation between f x of m and
ϕx of n dimensions, namely

                        Tn−1 (x)f x − tm−1 (x)ϕx + S0 = 0,

it has been shown that the succession of signs in the series formed with f x, ϕx
and their successive Bezoutian secondaries will contain the same number of
continuations and variations as the series formed with f x, tm−1 (x), and their
successive Bezoutian secondaries. This indicates that the effective scale of
interpositions for f x and ϕx will contain an equal number of roots of f x with
the effective scale for f x and tm−1 (x); the two scales however will not necessarily
be identical, because the roots of ϕx will not necessarily be in the same order
relative to the h’s in the one scale as those of tm−1 (x) relative to the h’s in the
other scale. This equality is perfectly well explained à posteriori by the form of
tm−1 (x), which by the formula in Section II. will be represented by
                          (                                          )
                                     ϕh1 ϕh2 · · · ϕhq−1
                      Σ                                                  .
                              (x − h1 )(x − h2 ) · · · (x − hq−1 )

Now, whenever x is indefinitely near to any one of the roots of f x, as hq , this
sum reduces to the simple expression
                                                                              1
           ϕh1 ϕh2 · · · ϕhq−1 · ϕhq+1 · · · ϕhm = {ϕh1 ϕh2 · · · ϕhm }          ,
                                                                             ϕhq

and consequently in the immediate neighbourhood of every real root of f x, ϕx
and tm−1 (x) will have always the same or always a contrary sign, according
as ϕh1 ϕh2 · · · ϕhm is positive or negative, which will depend upon the relative
disposition of the real roots in ϕx and f x; in either case the effective scale of
interpositions for f x with ϕx and for f x with tm−1 x must contain the same
number of h’s; but the difference will be, that if ϕh1 ϕh2 · · · ϕhm is positive an h
will occupy the first place in each scale, or the second place in each scale; but
if negative, then in one scale an h will occupy the first place, and in the other
scale the second place.
Art. (52). The same process of common measure or residues which serves to
furnish a rhizoristic series for f x or a syrrhizoristic series for f x and ϕx, will
serve also to furnish superior and inferior limits to the real roots of any proposed
equation. Thus suppose f x to be any rational integral function of                   p. 522
   x of the degree n and ϕx any other function of x, which I shall begin with
supposing to be of the degree (n − 1), and let the successive quotients resulting
from the process of finding the greatest common measure of f x, ϕx continued
until the last remainder is not a constant but zero, be supposed to be (as they

                                             528
may generally be taken, but subject to cases of exception, which will hereafter
be alluded to) n linear functions q1 , q2 . . . qn ; then we shall have
                             ϕx                       1
                                =                                         ,
                             fx                           1
                                  q +   1                     1
                                               q2 +
                                                                   1
                                                       qn−1 +
                                                                  qn
and therefore
                              ϕx = KN,                 f x = KD,
where N is the numerator and D the denominator of the continued fraction and
K is a constant; the value of this constant is immaterial but is in fact

                                            L0 L22 L24
                                        ±               &c.,
                                            L21 L23 L25
L0 , L1 , L2 , L3 , &c. being the leading coefficients of the last, the last but one, the
last but two, &c. of the Bezoutian secondaries to f x and ϕx. Accordingly,

        if n = 1, let D = q1 = µ1 ;
                                                           1
                                                                 
        if n = 2, let D = q2 q1 + 1 = µ1              q2 +            = µ 1 µ2 ;
                                                           µ1
                                                                               1
                                                                                   
        if n = 3, let D = q3 {q2 q1 + 1} + q1 = µ1 µ2                     q3 +          = µ1 µ2 µ3 ;
                                                                               µ2
                                  ·····················
and in general let
                                   D = µ 1 µ2 µ 3 · · · µ n ,
where
                                 1                            1                                   1
     µ 1 = q1 ,      µ2 = q2 +      ,         µ3 = q3 +          ,...          µn = qn +               .
                                 µ1                           µ2                               µn−1
Now suppose x to be so taken that

                     q1 does not lie between              +1 and − 1 
                                                                              
                                                                     
                                  ”                       +2 and − 2 
                                                                     
                     q2
                                                                     
                                                                     
                                                                     
                     q3           ”                       +2 and − 2 
                                                                     
                                                                     
                                                                     
                     q4           ”                       +2 and − 2                     (ω)
                    ···         ···                       ···
                                                                     
                                                                     
                                                                     
                                                                     
                                  ”                       2 and − 2 
                                                                     
                  qn−1
                                                                     
                                                                     
                                                                     
                                  ”                       1 and − 1 
                                                                     
                    qn

where it will be observed that the excluded region lies between +2 and −2 for
all the intermediate quotients, but between only +1 and −1 for the first      p. 523


                                                529
   and last quotients. Then µ1 is positively or negatively greater than 1, therefore
µ1 is a positive or negative fraction; but q2 is positively or negatively greater than
 1

2; therefore µ2 will be of the same sign as q2 , and also µ2 will be positively or
negatively greater than 1; therefore µ12 will be a positive or negative fraction; but
q3 is positively or negatively greater than 2; therefore µ3 will be of the same sign
as q3 , and also µ3 will be positively or negatively greater than 1; and proceeding
in this way, we find that all values of µi , from i = 1 to i = n − 1, will be of the
same sign as qi , and positively or negatively greater than 1. Finally, µn−1      1
                                                                                      will
be a fraction, and therefore, since qn is positively or negatively greater than 1,
µn = qn + µn−1 1
                  will have the same sign as qn (but of course is not necessarily
greater than 1, nor would that condition serve any purpose were it satisfied). We
infer consequently, that when the conditions (ω) are satisfied, µ1 , µ2 , µ3 . . . µn will
respectively have the same signs as q1 , q2 . . . qn ; and therefore D = µ1 µ2 µ3 · · · µn
has the same sign as q1 q2 q3 · · · qn .
   Now suppose

             q1 = a1 x + b1 ,      q2 = a2 x + b2 . . . ,    qn = an x + bn ,

and solve the 2n equations

             a1 x + b1 = +c1 , a2 x + b2 = +c2 . . . an x + bn = +cn ,
             a1 x + b1 = −c1 , a2 x + b2 = −c2 . . . an x + bn = −cn ,

where
              c1 = 1,      c2 = 2,       c3 = 2 . . . cn−1 = 2,     cn = 1.
Whenever in any one of the n pairs of equations above written the coefficient of
x is positive, the upper equation of the pair will bring out the greater value of
x; but when the coefficient is negative the lower equation will give the greater
value. Take the pair

                          ai x + bi = ci ,      ai x + bi = −ci .

If ai is positive ai x + bi will always be positive, and greater than ci , between
x = ∞ and x = the greater of the two values of x; if ai is negative, ai x + bi will
always be negative, and less (that is nearer to −∞) than −ci , for all values of x
between the same limits as before. So again it will be seen in like manner, that
whether ai be positive or negative, between x = −∞ and x = the lesser of the
two values of x corresponding to the above pair of equations, ai x + bi will always
retain the same sign, and will be greater than +ci , or less than −ci , according
as ai is negative or positive. If, then, we                                         p. 524
   take the greatest of the greaters of the n pairs of values of x, that is the
absolute greatest of the 2n values, and the least of the lessers, that is the
absolute least of the same, say L and Λ, then between L and Λ, q1 , q2 . . . qn
will each always retain an invariable sign, and will then fall without the limits

                                             530
±c1 , ±c2 , . . . ± cn−1 , ±cn , so that between +∞ and L and between Λ and −∞,
µ1 µ2 . . . µn , that is a constant multiple of f (x), will retain the same sign as
q1 q2 · · · qn , that is will never change its sign from the beginning to the end of
one interval, nor from the beginning to the end of the other; and consequently L
and Λ will be a superior and inferior limit respectively to the real roots of f x. It
will of course be observed that it is indifferent for the purposes of the foregoing
theorem, whether ϕx      f x be expanded under the form of a proper or an improper
fraction, that is whether we employ the ordinary or the Sturmian process of
successive division; for changing the signs of the residues will only have the effect
of changing qi into (−)qi , and the pair of equations (±)qi = ±ci remains the
same whether the + or the − sign be prefixed to qi . The result is, that if we
form the 2n quantities
            ±1 − b1       ±2 − b2      ±2 − b3     ±2 − bn−1        ±1 − bn
                    ,             ,            ...           ,              ,
              a1            a2           a3          an−1             an

the greatest of them will be a superior, and the least of them an inferior limit to
the roots of f x.255
   It may be remarked that if the successive dividends in the course of the process
be multiplied respectively by k1 , k2 . . . kn , ϕx
                                                 f x will take the form

                                   k1 k2 k3         kn
                                                 ··· ,
                                  q1 + q2 + q3 +    qn
and if we write

             a1 x + b1 = ±c1 ,        a2 x + b2 = ±c2 . . . an x + bn = ±cn

and make

             c1 = 1,       c2 = 1 + k2 ,      c3 = 1 + k3 . . . cn = 1 + kn ,

the same reasoning as above will show that the greatest and least of the 2n
quantities

          ±1 − b1       ±(1 + k2 ) − b2       ±(1 + kn ) − bn−1       ±1 − bn
                  ,                     ,...,                   ,             ,
            a1               a2                    an−1                 an

will be a superior and inferior limit to the roots of f x.
   For greater simplicity, again, consider k1 , k2 . . . kn to be all equal to unity; we
may make this addition to the theorem as above stated, namely calling                    p. 525

 255
    For a generalization and improved form of statement of this theorem see Supplement to
the present Section.




                                           531
   L1 , Λ1 ; L2 , Λ2 . . . Ln , Λn the greatest and least values of the terms contained
respectively in the series marked below 1, 2, 3 . . . n, namely
 ±1 − b1      ±2 − b2      ±2 − b3                    ±2 − bn−1        ±1 − bn
         ,            ,                  ···                    ,              , (1)
   a1           a2           a3                          an−1             an
              ±1 − b2      ±2 − b3                    ±2 − bn−1        ±1 − bn
                      ,                  ···                    ,              , (2)
                a2           a3                          an−1             an
                           ±1 − b3                    ±2 − bn−1        ±1 − bn
                                         ···                    ,              , (3)
                             a3                          an−1             an
                           ..........................................................
                                     ±1 − bn−1 ±1 − bn
                                               ,          , (n − 1)
                                        an−1         an
                                                                       ±1 − bn
                                                                               , (n)
                                                                          an
L1 , Λ1 ; L2 , Λ2 . . . Ln , Λn will be respectively superior and inferior limits to f x,
ϕx and their successive residues. As a corollary, we see, of course, that L and
Λ, the superior and inferior limits to the roots of the given function f x, must
always lie between +∞ and the greatest root, and between −∞ and the least
root, of the arbitrarily assumed function ϕx.
Art. (53). Let us now assume somewhat more generally that ϕx is any number
of degrees θ1 in x lower than f x, which will cause the first quotient qθ1 to be of
the degree θ1 in x; and let us further suppose that ϕx stands in such a relation
to f x that the following quotients, qθ2 , qθ3 . . . qθp , are of the degrees θ2 , θ3 . . . θp
in x (θ2 , θ3 . . . θp being supposed not necessarily units, as they would generally
be, but any positive integers whatever, as may happen in consequence of one or
more of the leading coefficients in any residue vanishing); then

                            ϕx     1     1     1          1
                               =                   ··· +     ,
                            fx   qθ1 + qθ2 + qθ3 +       qθp

where θ1 + θ2 + θ3 + · · · + θp = n; and consequently f x will be equal to the
denominator of the last convergent above written, multiplied by a constant, so
that we have now cf x = m1 m2 . . . mp , where

                                          1                                 1
            m1 = qθ1 ,      m2 = qθ2 +       ,...,        mp = qθp −1 +         .
                                          m1                               mp−1

And as in the case previously considered, so long as

           >1                   >2                    >2                  >1
                                                                              

    qθ1  or  ,          qθ2  or  ,         qθ3  or  , . . . , qθp  or  ,
                                                                        
          < −1                  < −2                 < −2                 < −1

f x will have the same sign as qθ1 qθ2 · · · qθp .                                               p. 526



                                            532
   Let now
                      qθ1 = ±c1 ,        qθ2 = ±c2 . . . qθp = ±cp ,
where
                    c1 = 1,       c2 = 2 . . . cp−1 = 2,       cp = 1.
Consider any pair of the above equations as qθ2i − c2i = 0.
    Firstly, suppose all the roots of this equation are impossible; qθ2i − c2i must be
positive for all values of x, and qθi can never lie between +ci and −ci ; moreover,
since upon the hypothesis made, qθi + ci and qθi − ci always retain the same sign,
namely, that of the coefficient of the highest power of qθi , it follows that qθi must
also always retain the same sign; for if we construct the two curves y = qθi + ci
and y = qθi − ci , these will both lie on the same side of the axis of x, and never
cut the axis, consequently the curve y = qθi , which lies between them, must also
lie on the same side as either of them, and never cut the axis.
    Hence, then, if the roots of the equation are all impossible, qθi will always
retain the same sign, and will never fall within the region bounded on two sides
by +ci and −ci .
    Secondly, suppose the equation to have one or more possible roots, and li to
be the greatest, and λi the least (which of course, if there is but one possible root,
will be identical). If the leading coefficient of qθi is positive, the greatest root
(l) of the equation qθi − ci = 0 will exceed the greatest root (l′ ) of the equation
qθi + ci = 0; for between x = ∞ and x = l′ , qθi must go through all values
intermediate between ∞ and −ci ; hence there must be a quality l intermediate
between l′ and +∞, which will make qθi = ci . In like manner, if the leading
coefficient of qθi is negative, it will be seen that the greatest root of qθi + ci = 0
will exceed that of qθi − ci = 0. Moreover, in the one case qθi will be always
positive and greater than ci , and in the other always negative and less than ci .
In every case, therefore, between +∞ and li , qθi retains the same sign, and does
not fall within the region bounded by +ci and −ci ; the same thing may be shown
to be true for all values of x between −∞ and λi . Hence, then, by the same
reasoning as that employed in the preceding article, we are enabled to affirm,
that if we form the equation
         (qθ21 − 1)(qθ22 − 4)(qθ23 − 4) · · · (qθ2p−1 − 4)(qθ2p − 1) = 0,   (ψ)
its greatest root will be a superior limit, and its least root an inferior limit to
the roots of the equation f x = 0, whatever be the value of the assumed function
ϕx; and if the above equation (ψ) has no real root, all the roots of f x will be
imaginary.
Art. (54).    In the preceding two articles it has been supposed that all the
quotients are taken integral functions of x; but the process of successive division
may be so conducted as to give rise to quotients of the form
                                                    d           l
                       axi + bxi−1 + · · · + c +      + · · · + i′ .
                                                    x          x

                                           533
                                                                                          p. 527
   Suppose then that we have in general
                              ϕx    1 1            1
                                 =           ··· + ,
                              fx   q1 + q2 +      qω

where q1 , q2 . . . qω are each of the general form above written (but of course i and
i′ being not necessarily the same for any two of the quotients), and suppose that
the sum of the degrees in x of q1 , q2 . . . qω is n + t, where t is essentially (as it
must be) positive. Then we shall find, as in the last article, that L and Λ being
called the greatest and least roots of

                      (q12 − 1)(q22 − 4) · · · (qω−1
                                                 2
                                                     − 4)(qω2 − 1),

D, the denominator of the last convergent to the continued fraction above written,
will never change its sign between +∞ and L, nor between Λ and −∞; but here
we shall have
                                 f x = Kxt × D.
Hence xt D will be invariable in sign within each of these two intervals.
   Firstly, let t be even; then f x will be invariable in sign, whatever L and Λ
may be for each such interval.
   Secondly, let t be odd; then if L is > 0 and Λ < 0, f x cannot change its
sign in either interval; but if L is < 0 or Λ > 0, f x will change its sign as x
passes through zero, but will be invariable for each of the three regions contained
between +∞ and L, L and 0, or 0 and Λ (as the case may be), and Λ and −∞;
so that universally L and Λ will be a superior and inferior limit to the roots of
f x, making abstraction of the roots (if any such there be in f x) whose value is
zero.
Art. (55). I shall close this section with offering (for what it is worth) a bare
suggestion as to the mode in which the theory of Intercalations may hereafter be
found to admit of being extended from a system of two general functions of x,
to a system of three general functions of x, y, four general functions of x, y, z,
and in general to a system of e general functions of e − 1 variables, or which is
the same thing, of e homogeneous functions of e variables. In the case of two
functions of x, f x and ϕx, f x = 0 and ϕx = 0 may be considered to represent
two systems of points in a right line; and the theory relates in this case to the
relative positions of these two “Kenothemes” or point systems; and of course
using x and y to denote the distances of any point in a line from two fixed points
therein respectively, instead of f x and ϕx, we may employ two homogeneous
functions of x and y, as f (x, y) and ϕ(x, y), to denote these two systems of points.
So, similarly, if we have three functions of two variables, f (x, y), g(x, y), h(x, y),
which I shall suppose to be of the same degree, we may consider the mutual
relations of the Monothemes, that is to say, the three plane curves, denoted            p. 528



                                          534
   by the equations f (x, y) = 0, g(x, y) = 0, h(x, y) = 0. Now every two of these
will intersect one another in a system of points, which we may call (f, g) for the
intersections of f and g, (g, h) for those of g and h, and (h, f ) for those of h
and f . If we take any two of these systems of intersections, as (f, g) and (g, h),
they will both lie upon one of the given curves (g). And by reading off the two
systems of points (f, g) and (g, h), arranged according to the order upon which
they are disposed upon the curve g, we may, by following the course of such
curve, form a scale of effective intercalations for these two systems, and in like
manner for the two systems (g, h) and (h, f ); (h, f ) and (f, g). Now I believe
that it will be found that when f, g, h represent any algebraical curves consisting
of a single continuous line, either extending to infinity in both directions, or
returning to itself (and I have fully satisfied myself of the truth of this for the
case of ellipses), each effective scale of intercalation will contain the same number
of pairs of points; if, however, the curves consist of more than one branch, as
if hyperbolæ be considered, such is no longer necessarily the case; from these
facts, conjoined with the light thrown upon the subject by its relation to the
theory of combinants explained in the succeeding section, I am induced to infer
the probability of the truth of the following law (which, for avoidance of further
uncertainty, I confine to the case of functions of the same degree), namely, that
if f, g, h be three homogeneous functions of x, y, z of the same degree, and if
U, V, W be any three linear functions of f, g, h, and if U = 0, V = 0, W = 0 be
treated as the equations to three cones, and if we form an effective scale of the
intercalations of the lines of intersection of U and W , and V and W , according
to the order in which they are disposed upon W (which seems to require that
the lines shall be continuous, in order to admit of a fixed order of reading off the
intersections of any two of them upon the third); then, whatever value may have
been given to the coefficients in the linear functions, the number of elements
remaining in any such scale will (as I conjecture) be constant, and some theory
(to be discovered) for three functions, analogous to that of Bezoutian residues for
two functions, will serve to determine the number of the elements so remaining.
And so, in like manner, but with a difficulty increasing at each step (as at the
next step we should have to pass into quasi-space of four dimensions), a theory
of intercalations may be conjectured to exist for any n general functions of any
(n − 1) variables.

   Development of the method of assigning a superior and
   inferior limit to the roots of any algebraical equation.

Art. (a). Since the articles in the preceding part of this section on the method
of discovering limits to the roots of an algebraical equation were written, the
method of which the germ is therein contained has presented                         p. 529
   itself in a much more fully developed form, which I proceed to exhibit: for
greater simplicity I shall suppose ϕx to be of n − 1, and f x to be of n dimensions

                                        535
in x, and that by means of the ordinary process for common measure (except
that as in Sturm’s theorem the signs of all the remainders are changed) ϕx
                                                                        f x has
been thrown under the form of the improper continued fraction
                              1 1 1             1
                                            ··· ,
                             q1 − q2 − q3 −    qn
where q1 , q2 . . . qn are all restricted to signify simple linear functions of x.
  Suppose the series q1 , q2 , q3 . . . qn to be resolved into the distinct sequences
              q1 q2 · · · qi ,   qi+1 qi+2 · · · qi′ ,     qi′ +1 · · · qi′′ , . . . ,   q(i)+1 · · · qn ,
in such a manner that in each sequence, as qi+1 , qi+2 . . . qi′ , the coefficients of x
have all the same sign, but that in any two adjoining sequences the coefficients
of x have opposite signs, so that for instance in qi and qi+1 the coefficients of x
are unlike, as also in qi′ and qi′ +1 ; there will of course be nothing to preclude
any of these sequences becoming reduced to a single term.
   The first theorem is, that the greatest and least roots of the product of the
cumulants [p. 504 above]
                    [q1 q2 · · · qi ] × [qi+1 qi+2 · · · qi′ ] · · · × [q(i)+1 q(i)+2 · · · qn ]
are superior and inferior limits to the roots of f x. To prove this theorem I
begin with premising the two following lemmas, one virtually and the other
expressly contained in the Philosophical Magazine for the months of September
and October of the present year256 [p. 641 below].                            p. 530
  256
       Each of these two lemmata flows readily from the faculty previously adverted to engaged
by every cumulant of being representable under the form of a determinant. As to the second
lemma, it becomes apparent immediately when the cumulant is so represented, by separating
the matrix into two rectangles and expressing the entire determinant according to a well-
known rule for the decomposition of determinants as a function of the determinants belonging
to these two rectangles taken separately. As to the first lemma, by reason of the cumu-
lant [ω1 ω2 . . . ωi−1 ωi ωi+1 ] being so representable, we know that when [ω1 ω2 . . . ωi−1 ωi ] = 0,
[ω1 ω2 . . . ωi−1 ] and [ω1 ω2 . . . ωi−1 ωi ωi+1 ] must have opposite signs. Suppose, now, that the
theorem is true when the number of elements in the type does not exceed i; then the roots
of [ω1 ω2 . . . ωi−1 ], say of ψi−1 , being called h1 , h2 . . . hi−1 , and of [ω1 ω2 . . . ωi−1 ωi ], say of
ψi , being called k1 , k2 . . . ki , these may be arranged in the following order of magnitude
k1 , h1 , k2 , h2 , k3 . . . ki−1 , hi−1 , ki ; and if the roots of [ω1 ω2 . . . ωi−1 ωi ωi+1 ], say of ψi+1 , be called
l1 , l2 . . . li+1 , from the fact of the leading coefficients in ψi−1 and ψi+1 expanded according to
the powers of x having the same sign, it follows that when x = ∞, ψi−1 and ψi+1 have the same
sign, but they have contrary signs when x = k1 ; but ψi−1 does not change its sign between
x = ∞ and x = k1 , hence ψi+1 does change its sign between x = ∞ and x = k1 , and therefore
a root of ψi+1 lies between ∞ and k1 ; in like manner precisely it may be shown that a root
of ψi+1 lies between −∞ and ki ; and since ψi+1 changes its sign between k1 and k2 , between
k2 and k3 . . . and between ki−1 and ki , ψi+1 must likewise change its sign between one and
the other extremity of each of these intervals, and hence the roots l1 , l2 . . . li+1 are intercalated
between ∞, k1 , k2 . . . ki , −∞, or which is the same thing, k1 , k2 . . . ki are respectively interca-
lated between l1 , l2 . . . li+1 ; consequently, if the theorem is true up to i, it is true for i + 1, and
therefore true universally; but is manifestly true when i = 2, for then x = ±∞ makes [ω1 ω2 ],
that is, ω1 ω2 − 1 positive; but ω1 = 0 makes it negative, which proves the theorem contained in
Lemma A.


                                                         536
   Lemma A. The roots of the cumulant [q1 q2 · · · qi ], in which each element is a
linear function of x, and wherein the coefficient of x for each element has the like
sign, are all real, and between every two of such roots is contained a root of the
cumulant [q1 q2 · · · qi−1 ], and ex converso a root of the cumulant [q2 q3 · · · qi ]; and
(as an evident corollary) for all values of p and p′ intermediate between 1 and i
the greatest root of [q1 q2 · · · qi−1 qi ] will be greater, and the least root of the same
will be less, than the greatest and least roots respectively of [qp qp+1 · · · qp′ −1 qp′ ].
   Lemma B. For all values of the elements q1 q2 · · · qn , the cumulant

   [q1 q2 · · · qω qω+1 qω+2 · · · qn ]
          = [q1 q2 · · · qω−1 qω ] × [qω+1 qω+2 · · · qn ] − [q1 q2 · · · qω−1 ] × [qω+2 · · · qn ].

Thus for example the cumulant [abcd], that is

    abcd − ab − cd − ad + 1 = [ab] × [cd] − [a] × [d] = (ab − 1)(cd − 1) − ad,

and [abcde], that is

          abcde − abc − abe − ade − cde + a + c + e = [abc][de] − [ab][e],

that is
                             = (abc − a − c)(de − 1) − (ab − 1)e.

Art. (β). Also suppose that q1 q2 · · · qω qω+1 · · · qn are all linear functions of x,
and that the coefficients of x have all one (say the positive) sign in q1 , q2 · · · qω ,
and all the contrary signs in qω+1 · · · qn , and let L be not less than the greatest
root of [q1 q2 · · · qω ] or of [qω+1 · · · qn ], and also let Λ be not greater that the least
root of each of these same two cumulants; then by Lemma A, L and Λ will
also be respectively greater than the greatest, and less than the least roots of
[q1 q2 · · · qω−1 ] and of [qω+2 · · · qn ]. Now the coefficient of the highest power of x
in both [q1 q2 · · · qω ] and in [q1 q2 · · · qω−1 ] is positive, but as to [qω+1 · · · qn ] and
[qω+2 · · · qn ] is of contrary signs in the two, namely, negative in that one of those
cumulants which contains an odd, and positive in that one of the two which
contains an even number of elements. Hence by virtue of Lemma B, L and any
quantity greater than L substituted for x will make [q1 q2 · · · qn ] to have always
the same sign, and in like manner it may be shown that Λ and any quantity less
than Λ substituted for x will also cause [q1 q2 · · · qn ] to retain always the same
sign. Hence L and Λ are superior and inferior limits to [q1 q2 · · · qn ]; and the same
reasoning would evidently apply if we had supposed the signs of the coefficients
of x in the first partial series of elements to have been negative, and in the other
series of elements to have been positive.
    The greatest and least roots of

                                    [q1 q2 · · · qω ] × [qω+1 · · · qn ]


                                                   537
evidently satisfy the condition to which L and Λ are subject, and may be taken
in place of L and Λ respectively. They will accordingly be superior and inferior
limits to the cumulant
                              [q1 q2 · · · qω qω+1 · · · qn ].
                                                                                                               p. 531
   Again, by virtue of Lemma B it may readily be shown that
                [q1 q2 · · · qω1 , qω1 +1 qω1 +2 · · · qω2 , qω2 +1 · · · qn ]
                    = [q1 q2 · · · qω1 ] × [qω1 +1 qω1 +2 · · · qω2 ] × [qω2 +1 · · · qn ]
                        − [q1 q2 · · · qω1 −1 ] × [qω1 +2 · · · qω2 ] × [qω2 +1 · · · qn ]
                        − [q1 q2 · · · qω1 ] × [qω1 +1 · · · qω2 −1 ] × [qω2 +2 · · · qn ]
                        + [q1 q2 · · · qω1 −1 ] × [qω1 +2 · · · qω2 −1 ] × [qω2 +2 · · · qn ];
and hence if q1 , q2 · · · qn are all linear functions of x in which the coefficients of x
have all the same algebraical sign in any one (taken per se) of the three series
                           q1 q2 · · · qω1 ,      qω1 +1 · · · qω2 ,     qω2 +1 · · · qn ,
but so that this sign changes in passing from one series to another, it is easily
seen, by the same reasoning as in the preceding case, that the two positive and
two negative products on the right-hand side of the equation all give the same
sign to the coefficient of the highest power of x, and consequently that if L and
Λ be superior and inferior limits to
                          [q1 · · · qω1 ],     [qω1 +1 · · · qω2 ],      [qω2 +1 · · · qn ],
and consequently by Lemma A, to
         [q1 q2 · · · qω1 −1 ],    [qω1 +2 · · · qω2 ],       [qω1 +1 · · · qω2 −1 ],    [qω2 +2 · · · qn ],
and to [qω2 +2 · · · qn ], L or Λ substituted for x will cause [q1 q2 · · · qn ] to retain
always the same sign, and will consequently be superior and inferior limits
thereto; and so in general; whence it follows, returning to the theorem to be
demonstrated, that the greatest and least roots of
                     [q1 q2 · · · qi ] × [qi+1 qi+2 · · · qi′ ] × · · · × [q(i)+1 · · · qn ]
will be superior and inferior limits to the cumulant [q1 q2 · · · qn ], that is to Cf x,
and therefore to f x, as was to be proved.
Art. (γ). The second theorem is the following: if q1 , q2 · · · qn be linear functions
of x, say a1 x + b1 , a2 x + b2 · · · an x + bn , in which the coefficients of x257    p. 532
 257
     If ϕx
        fx
           expanded as a continued fraction by means of the common measure process gives
rise to the quotients q1 , q2 . . . qn , and if L1 , L2 . . . Ln−1 , Ln be the leading coefficients of the
successive simplified residues, (Ln being, in fact, the final simplified residue, that is, the resultant
to ϕx, f x), we must have ϕx = C[q2 , q3 · · · qn ], f x = C[q1 , q2 · · · qn ], where (supposing ϕx to be
of n − 1, and f x of n dimensions in x),
                                              1
                                                                               
                                                       L2n L2n−2 L2n−4 · · ·
                                     C=                                             .
                                             Ln       L2n−1 L2n−3 L2n−5 · · ·



                                                          538
   have all the same sign, and if we take the quantities µ1 , µ2 · · · µn−1 , all having
the same sign as a1 , a2 · · · an , but otherwise arbitrary, and make
                         1                            1                       1                1
k 1 = µ1 ,   k2 = µ2 +      ,       k3 = µ3 +            · · · kn−1 = µn−1 +      ,    kn =          ,
                         µ1                           µ2                     µn−2             µn−1
then the greatest of the quantities
                                k1 − b1          k2 − b2     kn − bn
                                        ,                ···         ,
                                  a1               a2          an
say L, is a superior limit, and the least of the quantities
                          −k1 − b1               −k2 − b2     −kn − bn
                                   ,                      ···          ,
                            a1                     a2           an
say Λ, is an inferior limit to the roots of f x.
   L and any value greater than L substituted for x will evidently make

                                q1 − k 1 ,       q2 − k2 · · · qn − kn ,

all of them positive. Hence, when x = or > L, q1 is positive and > µ1 , and
                                  1         1         1   1
                          q2 −       > k2 −    > µ2 +    − ,
                                  q1        µ1        µ 1 µ1
that is, is positive, and > µ2 ,
                               1          1        1   1
                     q3 −        1 > k3 − µ > µ3 + µ − µ ,
                            q2 − q1        2        2   2

that is, is positive, and > µ3 ,

                                    ························

and
                                             1                    1          1
                     qn −               1                    >          −          ,
                            qn−1 − q −···−  1                    µn−1       µn−1
                                    n−2    q             1

that is, is positive; and consequently the cumulant [q1 q2 q3 · · · qn ], which

                                1                              1
                                                                      !
                    = q1 × q2 −                      × q3 −                 × &c.,
                                q1                          q2 − q11

remains of a constant sign when L and any quantity greater than L is substituted
for x. Hence L is a superior limit. In like manner Λ and any quantity less than
Λ will evidently make q1 + k1 , q2 + k2 · · · qn + kn all of them negative, so that,
when x = or < Λ, q1 is negative, and < −µ1 ,
                          1         1
                   q2 −      < k2 −                   is negative, and < −µ2 ,
                          q1        µ1

                                                      539
                              1          1
                      q3 −       < k3 −      is negative, and < −µ3 ,
                              q2        µ2
                                   ························
and
                                    1           1               1     1    1
                           qn −          −              ··· −      <     −
                                  qn−1         qn−2             q1   µn−1 µn−1
is negative.                                                                               p. 533
   So that [q1 , q2 · · · qn ] for all values of x less than Λ will preserve an invariable
sign, and consequently Λ is an inferior limit to f x.
Art. (δ).     It may be remarked that the quantities
                    1                1                             1                      1        1
      µ1 ,   µ2 +      ,     µ3 +       ,      ···       µn−2 +        ,       µn−1 +         ,
                    µ1               µ2                           µn−3                   µn−2     µn−1
may be derived successively from one another, according to the same law, from
whichever end of the series we begin.
  If we take any two consecutive terms as
                                                    1                   1
                                        µi +             ,    µi+1 +       ,
                                                µi−1                    µi
the effect of diminishing µi is to decrease the first of these two terms, and, pro
tanto, to tend to reduce the limit; but on the other hand, µ1i being increased,
there is brought into play an opposite tendency, which operates pro tanto to
increase the value of the limit.
Art. (ϵ). It is of importance to remark, that by a right selection of the system
of quantities µ1 , µ2 · · · µn−1 , which enter into the composition of k1 , k2 · · · kn , L
may be made to coincide with the greatest root of [q1 , q2 · · · qn ]; and so in like
manner by a right selection of another system of these quantities, whereby to
form k1 , k2 · · · kn , Λ may be made to coincide with the least root of the same.
Thus let µ1 , µ2 · · · µn−1 be so chosen, that

                           q1 − k1 = 0,         q2 − k2 = 0 · · · qn − kn = 0,

are all satisfied by the same value of x. Then
                                                1                        1              1
               q1 = µ 1 ,         q2 = µ2 +        ,         q3 = µ3 +      · · · qn =      ,
                                                µ1                       µ2            µn−1
exist simultaneously. Hence
                                        1                        1            1
                      µ 2 = q2 −           ,    µ3 = q3 −           = q3 −         ,
                                        q1                       µ2        q2 − q1   1

                                                                    1
                              µn−1 = qn−1 −                             1       ,
                                                         qn−2 − q      −···−  1
                                                                   n−3       q 1



                                                         540
                                                             1
                                     µn =                         1       ,
                                              qn−1 − q           −···−  1
                                                             n−2       q 1

which is satisfied by making

                                     [qn , qn−1 , qn−2 · · · q1 ] = 0.

It remains then only to show that the greatest root of x in this equation sub-
stituted for x in q1 , q2 · · · qn will make µ1 , µ2 · · · µn−1 all of one sign, and that
the least root of x similarly substituted, will also make them all of one, but a
contrary sign, which may be proved as follows.                                            p. 534
   We have
           µ 1 = q1 ,    µ2 = [q1 , q2 ] ÷ q1 ,           µ3 = [q1 q2 q3 ] ÷ [q1 , q2 ],          &c.
                         µn−1 = [q1 q2 · · · qn−1 ] ÷ [q1 q2 · · · qn−2 ];

and by Lemma B the superior limit to [q1 q2 · · · qn ] will be a superior limit also
to [q1 q2 · · · qn−2 ], and to

                        [q1 q2 ],       [q1 q2 q3 ],    ...,      [q1 q2 · · · qn−1 ].

Consequently this superior limit will make µ1 , µ2 · · · µn−1 have all the same sign
as that of the coefficients of x in q1 , q2 · · · qn . And in like manner, the inferior
limit to [q1 q2 · · · qn ] will cause µ1 , µ2 · · · µn−1 to have all the contrary sign to that
of these coefficients.
   Thus then we see that when the coefficients of x in the partial quotients to ϕx          fx
expressed as an improper continued fraction form a single series of continuations
of signs, by a right choice of the arbitrary constants µ1 , µ2 · · · µn−1 the superior
or inferior limit given by this new method may severally and separately be made
to coincide with the greatest and least real root, or each in turn with the sole
real root of f x, if there be but one.
Art. (ζ).    The general method of enclosing the roots of f x within limits is
founded upon the combination of the two theorems above demonstrated. An
arbitrary function ϕx, one degree in x below f x, being assumed, and by aid of
the auxiliary function ϕx, f x being thrown under the form

                   C[q1 q2 · · · qi , q1′ q2′ · · · qr′ , q1′′ · · · (q)1 (q)2 · · · (q)(i) ],

in which the coefficient of x is supposed to change sign in the passage from qi to
q1′ , from qr′ to q1′′ , &c., a superior limit is found to each of the cumulants

                   [q1 q2 · · · qi ],     [q1′ q2′ · · · qr′ ] · · · [(q)1 (q)2 · · · (q)(i) ],

taken separately, by means of the second theorem, and then by virtue of the first
theorem the greatest of these superior limits is a superior limit to the cumulant

                                    [q1 q2 · · · qi · · · (q)1 · · · (q)(i) ],

                                                       541
and consequently to f x, and so mutatis mutandis the least of the inferior limits
of the same partial cumulants is an inferior limit to the total cumulant
                          [q1 q2 · · · qi · · · (q)1 (q)2 · · · (q)(i) ].

Art. (η). When all the roots of f x are real, if ϕx be so assumed that all its
roots are intercalated between those of f x, the partial quotients to ϕx
                                                                       f x will form
but one single series. In order that ϕx may fulfil this condition, it is necessary
that the coefficients of ϕx shall be subject to certain conditions                   p. 535
    of inequality, not necessary to be investigated here; but no conditions of
equality, that is, no equations between the coefficients of ϕx, are introduced by
this condition; or in other words, the coefficients of ϕx, the auxiliary function,
are independent and arbitrary within limits; and we have shown that in this
case the auxiliary constants µ1 , µ2 · · · µn−1 may be so determined that the limits
may be made to come separately and respectively into contact with the two
extreme roots. When all the roots of f x are not real, the quotients (however ϕx
is chosen) can no longer be made to form a single series. It still however remains
true, that, by a due choice of the auxiliary function followed by a due choice of
the auxiliary constants, this coincidence may be brought about, so long as there
is a single real root in f x.
    It is rather important to demonstrate this universal possibility of effecting a
coincidence of the limits to the roots with the extreme roots themselves, because
it is the most striking feature which distinguishes the method of limitation here
developed from all others previously brought to light.
Art. (θ).       Before entering upon this demonstration I may make the passing
remark, that every method of root-limitation is implicitly a method of root-
approximation.
    For instance, let e be any given quantity between which and +∞ it is known
that a root of f x lies. Then if we write x = e + y1 , and form the equation
y n f (e + y1 ) = 0, and find L a superior limit to y, it is clear that e + L1 will
lie between e and the root of f x say E, next superior to e. Again, making
x = e + L1 + y1′ , and finding a superior limit L′ to y ′ , we shall have e + L1 + L1′
still nearer to E than e + L1 was; and so we may proceed advancing nearer and
nearer, and always from the same side towards E at each step, and finally obtain
E under the form
                                   1    1     1
                               e + + ′ + ′′ + &c.
                                   L L        L
And in like manner calling E1 the root next below e, we may find
                                      1     1    1
                             E1 = e − − ′ − ′′ , &c.
                                      Λ Λ       Λ
Art. (ι).   In establishing the theorem of coincidence above adverted to, the
following notation will be found very advantageous. Let Ω denote a Type of any
number of Elements, as q1 , q2 · · · qi−1 , qi , and let Ω′ denote this        p. 536


                                              542
    same type when the last element, and ′ Ω the same type when the first element
is cut off, and ′ Ω′ the same type when both extremes are cut off, so that the
apocopated type Ω′ will mean q1 , q2 · · · qi−1 , the apocopated type ′ Ω will mean
q2 q3 · · · qi , and the doubly apocopated type ′ Ω′ will mean q2 , q3 · · · qi−1 .
    If now a type Ω be made up of the types Ω1 , Ω2 · · · Ωi put in apposition, and
if we use in general [Ω] to denote the cumulant corresponding to the type Ω,
there will be a very simple law258 connecting [Ω] with

                            [Ω1 ], [Ω2 ], [Ω3 ] · · · [Ω′i−2 ], [Ωi−1 ], [Ωi ],
                               [Ω′1 ], [Ω′2 ], [Ω′3 ] · · · [Ω′i−2 ], [Ω′i−1 ],
                             [′ Ω2 ], [′ Ω3 ] · · · [′ Ωi−2 ], [′ Ωi−1 ], [′ Ωi ],
                                 [′ Ω′2 ], [′ Ω′3 ] · · · [′ Ω′i−2 ], [′ Ω′i−1 ].

This law will be seen to be obviously deducible by successive steps of expansion
from the fundamental theorem given in Lemma B, Art. (α), for the case of
Ω = Ω1 Ω2 , and will be best understood by showing its operation in a few simple
cases.
   Thus let Ω = Ω1 Ω2 .259 Then

                               [Ω] = [Ω1 ] × [Ω2 ] − [Ω′1 ] × [′ Ω2 ].

Let Ω = Ω1 Ω2 Ω3 . Then

        [Ω] = [Ω1 ] × [Ω2 ] × [Ω3 ] − [Ω′1 ] × [′ Ω2 ] × [Ω3 ] − [Ω1 ] × [Ω′2 ] × [′ Ω3 ]
                + [Ω′1 ] × [′ Ω′2 ] × [′ Ω3 ].

Let Ω = Ω1 Ω2 Ω3 Ω4 .260 Then

           [Ω] = [Ω1 ] × [Ω2 ] × [Ω3 ] × [Ω4 ]
                  − [Ω′1 ] × [′ Ω2 ] × [Ω3 ] × [Ω4 ] − [Ω1 ] × [Ω′2 ] × [′ Ω3 ] × [Ω4 ]
                  − [Ω1 ] × [Ω2 ] × [Ω′3 ] × [′ Ω4 ] + [Ω′1 ] × [′ Ω′2 ] × [′ Ω3 ] × [Ω4 ]
                  + [Ω′1 ] × [′ Ω2 ] × [Ω′3 ] × [′ Ω4 ] + [Ω1 ] × [Ω′2 ] × [′ Ω′3 ] × [′ Ω4 ]
                  − [Ω′1 ] × [′ Ω′2 ] × [′ Ω′3 ] × [′ Ω4 ],
                                                                                                      p. 537
 258
      The cumulant corresponding to any portion or fragment of a type may be said to be a
partial cumulant to the entire type, and a type whose elements are constituted out of the
elements of two or more types placed in juxtaposition may be said to be the aggregate of these
types; the law given in the text above may then be said to have for its object the expansion of
the complete cumulant to any type in terms of complete and partial cumulants to the types of
which the given type is the aggregate.
  259
      The sign of equality is employed here to denote the relation between a concrete whole and
the aggregate of its parts.
  260
      The number of distinct factors entering into these products, taken collectively, is evidently
i + 2(i − 1) + (i − 2), that is 4(i − 1).


                                                      543
   and so in general if Ω = Ω1 Ω2 · · · Ωi , [Ω] may be expanded under the form
of the sum of 2i−1 products separable into i alternately positive and negative
groups containing respectively
                                             1
                    1,     (i − 1),            (i − 1)(i − 2),         ...     (i − 1),       1
                                             2
products.
Art. (κ). In every one of the above groups forming a product the accents enter
in pairs and between contiguous factors, it being a condition that if any Ω have
an accent on the right the next Ω must have one on the left, and if it have one
on the left the preceding Ω must have an accent on the right, and the number
of pairs of accents goes on increasing in each group from 0 to i − 1. This rule
serves completely to define the development in question.261
    For greater brevity let [Ωe ], [Ω′e ], [′ Ωe ], [′ Ω′e ] be denoted respectively by
ωe , ωe′ , ′ ωe , ′ ωe′ , then when the type Ωe consists of a single element,

                                   ωe′ = 1,          ′
                                                         ωe = 1,        ′ ′
                                                                         ωe = 0.

It should be observed that the equations ωe = 0, ωe′ = 0 cannot exist simultane-
ously, for if Ωe represent q1 q2 · · · qi ,

                      ωe = qi ωe′ − ωe′′ ,                ωe′ = qi−1 ωe′′ − ωe′′′ ,    &c.,

so that if ωe = 0 and ωe′ = 0, we have ωe′′ = 0, ωe′′′ = 0, &c., and thus, finally,
−1 = 0, which is absurd.
   Now, if we suppose Ω1 , Ω2 · · · Ωe to be types every element in each of which is
a linear function of x, the coefficients of x in these elements being positive in
 261
     When each partial type Ω consists of a single element, every doubly accented Ω will vanish,
and every singly accented Ω will become unity; hence we may derive the rule for the expansion
of the cumulant [a1 a2 a3 · · · ai ] in terms of a1 , a2 . . . ai , which will accordingly consist of
                                      1                                 1
           a1 a2 a3 · · · ai − Σ           (a1 a2 · · · ai ) ± Σ                  (a1 a2 · · · ai ) ∓ &c.,
                                   ae ae+1                       ae ae+1 af af +1
the indices e and f, e + 1 and f , &c. being understood to be all distinct integers (which agrees
with the known rule for the expression of the denominator of a continued fraction in terms of
the quotients). The number of terms in this expansion, in consequence of the vanishing of the
quantities affected with a double accent, reduces from 2i−1 down to the ith term in the series
commencing with 1, 2, 3, &c. defined by the equation ui+2 = ui+1 + ui , that is
                                       √ i+1              √ i+1
                             1     1+ 5           1    1− 5
                                                    
                            √                  −√                    ;
                              5       2            5      2

the number, therefore, of products in which double accents occur in the general expansion of
[ω1 ω2 · · · ωi ] is
                                        √ i+1              √ i+1
                               1    1+ 5           1    1− 5
                                                     
                       2 i−1
                             −√                +√                    .
                                5      2            5      2



                                                          544
Ω1 , negative in Ω2 , and so on alternately, and Ω is the aggregate of Ω1 , Ω2 · · · Ωe ,
it may easily be made out that each term in the development of ω in terms of
ω1 , ω1′ , ′ ω1 , ω2′ , ′ ω2 , ω2′ , ′ ω2′ , &c. will have the same sign when we give to x a value
which is a superior limit, or an inferior limit to                                                             p. 538
   the roots of each of the cumulants ω1 , ω2 · · · ωe , and consequently to those of
the cumulants ω1′ , ω2′ · · · ωe′ ; ′ ω1 , ′ ω2 · · · ′ ωe ; ′ ω1′ , ′ ω2′ · · · ′ ωe′ ; the products affected
with positive signs being all positive or negative in themselves, and those affected
with negative signs being reversely all negative, or all positive.
   Thus, for example, if

                               Ω = Ω1 Ω2 ,          ω = ω1 ω2 − ω1′ ′ ω2 ,

and the sign of the leading coefficient in ′ ω2 will be the contrary of that in ω2 ,
but ω1 and ω1′ have both the same positive sign; so again if Ω = Ω1 Ω2 Ω3 ,

                       ω = ω1 ω2 ω3 − ω1′ ′ ω2 ω3 − ω1 ω2′ ′ ω3 + ω1′ ′ ω2′ ′ ω3 ,

where the leading coefficients in ω2 and ′ ω2 have contrary signs, as have also
those in ω2 and ω2′ , while ω2 and ω2′ have the same sign; and of course the leading
coefficients in ω1 , ω3 , ω1′ , ′ ω3 have all the same sign, they being all positive, and
so in general. But the superior limit to the roots of any integral algebraical
function of x substituted in place of x causes the signs of the resulting values
of the functions to coincide with the signs of the leading coefficients, so that
in the example last above given, L a superior limit to all the factors in the
several products in the equation substituted for x will make ω1 ω2 ω3 , −ω1′ ′ ω2 ω3 ,
−ω1 ω2′ ′ ω3 , ω1′ ′ ω2′ ′ ω3 to have all the same sign. The like will be true of Λ the
inferior limit; for if Ω1 , Ω2 , Ω3 contain respectively n1 , n2 , n3 elements, the values
of the four products last above written, when x = −∞, will be to the values of
the same when x = +∞ in the respective ratios of

(−)n1 +n2 +n3 : 1,        (−)n1 +n2 +n3 −2 : 1,       (−)n1 +n2 +n3 −2 : 1,        (−)n1 +n2 +n3 −4 : 1,

and so in general. Hence we deduce the theorem, that if the total type Ω represent
the aggregate in apposition of the partial orders Ω1 , Ω2 · · · Ωe (the elements being
understood to be linear functions of x, which are subject to the law of alternation
in the signs of the coefficients of x in passing from one partial type to another),
no superior limit to ω1 , ω2 · · · ωe can make ω vanish unless each separate product
in the expansion of ω in terms of ω1 , ω2 · · · ωe and the appurtenant apocopated
cumulants vanish separately.
Art. (λ). From the above theorem we may deduce the following law, namely,
that if the roots of ω1 , ω2 · · · ωe be supposed to be arranged in order of magnitude,
and λ to be that one of them which is nearest to +∞ or to −∞, then if e is even
it is impossible for λ to be a root of ω. Thus suppose e = 2, and consequently
ω = ω1 ω2 − ω1′ ′ ω2 ; if λ be a root of ω1 and one of the two extremes of the roots

                                                    545
of ω1 , ω2 put in order of magnitude, λ cannot be a root of ω2 , for the roots of
′ ω are confined between the roots of ω ; but                                     p. 539
   2                                    2
    if λ make ω and ω1 each vanish, we must have ω1′ /ω2 = 0, hence ω1′ = 0 as
well as ω1 = 0, which is impossible. In like manner if a root of ω2 were the
extreme root, the same impossibility could be in like manner established.
    Again, suppose e = 4, so that
                                      (
                                               ω1′ ω2 ω2′ ω3 ω3′ ω4
                    ω = ω1 ω2 ω3 ω4 1 −              −      −
                                               ω1 ω2 ω2 ω3 ω3 ω4
                                ω1′ ω2′′ ω3′  ω ′ ω2 ω3′ ω4′  ω′ ω′ ω′
                            +                + 1             + 2 3 4
                                ω1 ω2 ω3      ω1 ω2 ω3 ω4 ω2 ω3 ω4
                                            )
                             ω′ ω′ ω′ ω′
                            − 1 2 3 4 .
                             ω1 ω2 ω3 ω4

Let λ continue to denote one or the other extreme of the roots of ω1 ω2 ω3 ω4 . If
λ makes ω = 0 we have
                ω1 ω2 ω3 ω4 = 0, ω1′ ω2 ω3 ω4 = 0, ω1 ω2′ ω3 ω4 = 0,
                ω1 ω2 ω3′ ω4 = 0, ω1 ω2 ω3 ω4′ = 0, ω1′ ω2′ ω3 ω4 = 0,
                ω1′ ω2 ω3′ ω4 = 0, ω1′ ω2 ω3 ω4′ = 0, ω1 ω2′ ω3′ ω4 = 0,
                ω1 ω2′ ω3 ω4′ = 0, ω1 ω2 ω3′ ω4′ = 0, ω1′ ω2′ ω3′ ω4′ = 0.
Now suppose that λ is a root of ω1 , then the equations remaining to be satisfied
are

      ω1′ ω2 ω3 ω4 = 0,   ω1′ ω2′ ω3 ω4 = 0,    ω1′ ω2 ω3′ ω4 = 0,    ω1′ ω2′ ω3′ ω4 = 0.

Since ω1 and ω1′ cannot both be zero together, λ cannot make ω1′ or ω1′′ zero;
and because λ is an extreme to the roots of ω2 , ω3 , ω4 , λ cannot make ω2′ or
ω2′′ or ω3 or ω3′ or ω4 or ω4′ zero, so that in fact when x = λ none of the singly
accented quantities ω can be zero. As regards the doubly accented quantities ω,
the same thing cannot be affirmed, because if any Ω contains only one element
the corresponding value of ω with a double accent vanishes spontaneously. Again,
any of the unaccented quantities ω may vanish, because we may suppose any
of these to have an extreme root λ. Consequently the first, second and fourth
of the equations remaining to be satisfied, might be satisfied on making the
necessary suppositions as to the form of the quantities ω and the values of the
extreme roots; but the third remaining equation ω1′ ω2 ω3′ ω4 = 0, in which only
singly accented quantities ω occur, remains incapable of being satisfied on any
supposition whatever. And the same thing would be true if we suppose λ to be
a root of any other ω instead of ω1 . Hence λ cannot make ω = 0 when e = 4.
    In like manner, if e be any even number 2e, there will be an equation

                          ω1′ ω2 ω3′ ω4 ω5′ ω6 · · · ω2e−1
                                                      ′
                                                           ω2e = 0,

                                            546
to be satisfied by that value (if it exist) of x which, besides being an extreme
(on either side) of the roots of ω1 , ω2 . . . ω2e arranged in order of magnitude, also
makes ω = 0. But as such equation cannot be satisfied, neither extreme root of
the roots of ω1 , ω2 . . . ω2e can be a root of ω, as was to be proved. Consequently,
unless ϕx is so assumed that the number of changes of sign in the coefficients of
x in the quotients resulting from ϕx   f x expanded as an                               p. 540
   improper continued fraction is even (for if the changes from sequence to
sequence are odd the number of sequences themselves is even), the method of
limitation in the text cannot give the means of drawing either limit indefinitely
near to one or the other extreme roots of f x.
   Art. (µ). It now remains to prove the converse, and to show, first, that
when the number of changes is even, that is, the number of sequences odd, this
coincidence can always be effected; and secondly, that it is always possible when
f x has one or more real roots, so to assume ϕx that the number of sequences
shall be odd.
   The first part of the proposition is easily proved. Thus suppose e = 3, so that

                    ω = ω1 ω2 ω3 − ω1′ ω2 ω3 − ω1 ω2′ ω3 + ω1′ ω2′ ω3 .

If we suppose λ, either extreme of the scale formed by writing in order of
magnitude the roots of ω1 , ω2 , ω3 , to be a root common to ω1 and to ω3 , and if
ω2′ = 0, which last equation may be satisfied by supposing the type Ω2 to consist
of a single element, the separate equations

             ω1 ω2 ω3 = 0,   ω1′ ω2 ω3 = 0,     ω1 ω2′ ω3 = 0,   ω1′ ω2′ ω3 = 0

will all be satisfied; and so in general it may be shown without difficulty that if
e = 2ϵ + 1, and if λ be a root common to

                     ω1 = 0,     ω3 = 0,      ω5 = 0 . . . ω2ϵ+1 = 0,

and if ω2 , ω4 . . . ω2ϵ be all simple linear functions of x, so that consequently
ω2′ = 0, ω4′ = 0 . . . ω2ϵ
                         ′ = 0, each separate term in the development of ω will

vanish singly and separately, and consequently λ will be a root of ω: for since
λ makes ω1 = 0, ω3 = 0 . . . ω2ϵ+1 = 0, every product in the developed form ω,
in which ω1 , ω3 . . . ω2ϵ+1 do not each bear at least one accent, will vanish; and
if we consider any product in which ω1 , ω3 . . . ω2ϵ+1 are all accented, if in any
two of these immediately following one after the other as ω2k−1 , ω2k+1 , an accent
falls to the right of the first, and to the left of the second, the intervening term
ω2k will bear a double accent, and will therefore vanish, since ω2k is supposed
to be a linear function of x; but it is impossible when every ω is accented to
prevent two accents of contiguous odd terms in any such product, from falling
to the right of the left, and to the left of the right, term of the two, since the
contrary would imply that all the accents would fall to the right, or all to the

                                           547
left, which, as above remarked, is impossible, on account of the two extreme
terms being only simply accentable, that is, ω1 only to the right, and ω2ϵ+1 only
to the left. Hence, when x substituted for λ makes ω1 , ω3 . . . ω2ϵ+1 all vanish,
and when ω2 , ω4 . . . ω2ϵ are all linear functions of x, x = λ will be a root of ω. p. 541
   Art. (ν). I believe that the remaining part of the proposition may be rigorously
demonstrated, namely that when any of the roots of f x are real, and the number
of odd integers not exceeding the index of the degree of f x is m, and the number
of imaginary pairs of roots in f x is p, ϕx may be so assumed that the quotients to
ϕx
f x expanded under the form of an improper continued fraction, may be made to
take the form Ω1 , Ω2 , Ω3 , Ω4 . . . Ω2i+1 , where Ω2 , Ω4 . . . Ω2i are linear functions of
x, and i is any number assumed at will, not less than p, and of course not greater
than m; and where ω1 , ω3 . . . ω2i+1 will have in common a root λ, which may be
made at will the greatest or the least root of ω1 ω2 ω3 . . . ω2i+1 ; the investigation,
however, according to the present light which I possess on the subject, appears
complicated and tedious, and therefore, in order that the press, which is waiting
for the completion of these supplemental articles, may not be kept standing,
must be adjourned to some future occasion. For the present I content myself
with showing the truth of the law for the simple case where f x is a cubic function
of x.
   Firstly. If ϕx
               f x gives rise to a single sequence of quotients Ω, we know, from the
theory of intercalations, that it is necessary that all the roots of f x shall be real,
and in order that when this is the case the quotients may form a single sequence
Ω, it is only necessary so to assume ϕx, that its roots may be intermediate
between those of f x.
   Secondly. If the roots of f x are not all real, or if they are all real, but do not
comprise the roots of ϕx intercalated between them, and if for greater brevity
of ratiocination we stipulate that ϕx shall have its leading coefficients of the
same sign as that of the leading coefficient of f x, the leading coefficients of the
three quotients will either bear the respective signs + + −, or the respective
signs + − −, or the respective signs + − −−; in the first and last of these cases
there would be two sequences, and therefore, by what has been shown above, the
method of limitation of the text could not give a limit coincident with a root.
Let us then look to the remaining case, and inquire whether, and how, ϕx may
be assumed so that f x shall become representable to a constant factor près by
the cumulant [p(x − a), −q(x − β), r(x − a)], where p, q, r are all positive, and a
is a root of f x.
   Let this cumulant be called hf x.
   Nothing in point of generality will be lost if we suppose the leading coefficient
of hf x to be −1. We then have

 hf x = [p(x − a), −q(x − β), r(x − a)] = −pqr(x − a)2 (x − b) − (p + r)(x − a)
                                                                                       p. 542



                                         548
   and writing
                             hf x
                                  = x2 + Bx + C
                            x−a
and making x = a, we find from the above identity that

             p + r = a2 + Ba + C,      that is,     p = a2 + Ba + C − r,

and
                              pqr(x − β) = x + a + B,
hence
                      β + a + B = 0,     that is,    β = −B − a,
and
                                                    1        1
             pqr = 1,    and therefore     qr =       = 2             .
                                                    p  a + Ba + C − r
Hence if ϕx be so assumed that the quotients to ϕx
                                                f x are p(x−a), −q(x−β), r(x−a),
we have
      hϕx = [−q(x − β), r(x − a)] = −qr(x + B + a)(x − a) − 1
                                             1
          = −qr(x2 + Bx − a2 − aB) − 1 = − {x2 + Bx − a2 − aB + p}.
                                             p
Hence ϕ(x) is of the form

      m{x2 + Bx − a2 − aB + (a2 + aB + C − r)} = m(x2 + Bx + C − r).

If we call the three roots of f x, a, b, c respectively, we have
                               1                         1
                 q=                        =                         ;
                      r(a2 + Ba + C − r)       r{(a − b)(a − c) − r}
and since q and r are both to be positive, we see that a must be taken the greatest
or least of the three roots if they are all real, so that a2 + Ba + C may be positive,
which it will of course necessarily be if b and c are imaginary; we must also have
a2 + Ba + C − r positive, so that the form of ϕx is m{(x2 − a2 ) + B(x − a) − t},
t being necessarily positive, but otherwise arbitrary, a form containing two
arbitrary constants, one of which is subject to satisfy a certain condition of
inequality; whereas when f x is of such a form as to admit, and ϕx is supposed to
be so assumed as to cause it to come to pass that the quotients to ϕx f x form a single
sequence, then the three coefficients in ϕx remain exempt from all conditions of
equality but are subject to two conditions of inequality. And so in general when
the degree of f x is x and the number of sequences 2i + 1, it is to be inferred
that the n coefficients of ϕx will be subject to satisfy n − i − 1 conditions of
inequality and i conditions of equality.
   Art. (ξ). The theory of the determination of the minimum interval between
either limit determinable by this method and the nearest root, or between the

                                         549
two limits so determinable when ϕx is so assumed that ϕx   f x gives rise to a defined
even number of sequences (which will include the                                       p. 543
   theory of the case where all the roots of f x are imaginary), must be deferred
to an opportunity more favourable for leisurely contemplation. As regards the
application of the theory to the very interesting case of all the roots being
imaginary, the principal point remaining to be cleared up is the determination
of the least value that can be assigned to the greatest, and the greatest value
that can be assigned to the least root of the algebraical product X1 X2 X3 . . . X2n ,
where X1 , X2 . . . X2n are all of them real linear functions of x, subject to the
condition that the cumulant [X1 , X2 , X3 . . . X2n ] shall (to a numerical factor
près) be equal to a given function of the degree 2n in x incapable of changing its
sign, which condition implies, as a necessary consequence, that the coefficients of
x in each of the terms X1 , X2 . . . X2n must be affected with the same algebraical
sign.
   Art. (o). It should be observed that in the application of the above method,
the division of the series of quotients into distinct sequences governed by the
signs of the coefficients of x is introduced for the purpose of drawing the limits
closer to the roots, but is not necessary for the mere object of assigning limits.
   Thus, for instance, if there be two sequences so that

                                 [q1 q2 . . . qi ,    qi+1 qi+2 . . . qi+i′ ],

                                     1                                   1                                  1
                                           2                                2                              2
      q12 = µ21 ,   q22 =       µ2 +             ,    q32 =         µ3 +              . . . qi2 =                    ,
                                     µ1                                  µ2                             µi−1
and
                                                           1                                1
                                                               2                                2
                 2
                qi+1 = ν12 ,         2
                                    qi+2 = ν2 +                             2
                                                                     . . . qi+i ′ =                     ,
                                                           ν1                             νi′ −1
the greatest and least roots of x deduced from these equations will be superior
and inferior limits respectively to the roots of f x; from which it is clear that if
leaving all the other equations unaltered, except those which contain respectively
qi2 and qi+1
         2 , we write in place of these


                                           1                                   1
                                                2                                      2
                      qi2 =         ρ+                ,          2
                                                                qi+1 =           + ν1
                                         µi−1                                  ρ

the roots of the system of i + i′ equations thus modified will à fortiori be limits
to the roots of f x, but then the quantities
                    1                1         1         1       1        1
       µ 1 , µ2 +      . . . µi−1 +      , ρ+      , ν1 + , ν2 +    ...        ,
                    µ1              µi−2      µi−1       ρ       ν1     νi′ −1

form the same single series as would correspond to the two sequences

                                       q1 q2 . . . qi qi+1 . . . qi+i′ ,

                                                          550
                                                                                                                                    p. 544
   treated as a single sequence, and the same is obviously the case for any number
of sequences.262
   Art. (π). If we consider a single sequence as q1 , q2 . . . qn , and write

                      q1 = a1 (x − c1 ),           q2 = a2 (x − c2 ) . . . qn = an (x − cn ),

where a1 , a2 . . . an are supposed to have all the same sign, and write
                                                                            1                                        1
                                                                                2                                       2
   a21 (x − c1 )2 = µ21 ,         a22 (x − c2 )2 = µ2 +                                . . . a2n (x − cn )2 =                   ,
                                                                            µ1                                      µn−1
it seems not unlikely that the interval between the greatest and least of the roots
of the above equations will be a minimum when the interval between any pair is
the same for each pair, that is, when

                               µ1   µ2 + µ 1   µ3 + µ2
                                                    1
                                                                 µ
                                                                            1                       1
                                  =          =         = · · · = n−1 .
                               a1      a2         a3              an
If we assume these equations, and write µ1 = a1 ξ, the equation for determining
ξ will be
                          [a1 ξ, a2 ξ, a3 ξ . . . an ξ] = 0.
If n = 2 this equation becomes a1 a2 ξ 2 − 1 = 0.
   If n = 3, rejecting the factor ξ, it becomes

                                        a1 a2 a3 ξ 2 − (a1 + a3 ) = 0.

If n = 4 it becomes

                            a1 a2 a3 a4 ξ 4 − (a1 a2 + a3 a4 + a1 a4 )ξ 2 + 1 = 0.
 262
       It follows from this, that if q1 , q2 . . . qn be all linear functions of x, and if
                  Q = (q12 − µ21 ){q22 − (µ2 + µ11 )2 }{q32 − (µ3 + µ12 )2 } · · · (qn2 − µ21 ),
                                                                                                         n−1

no root of Q can lie between the extreme roots of the function K, used to denote the cumulant
                                            p           p           p                  p
                                        [       q12 , −     q22 ,    q32 , . . . , ±       qn2 ],
the square roots being understood to be taken so as to make the sign of the coefficients of x all
of them positive; and from a preceding article we know that either extreme root of Q can be
made to coincide with a corresponding extreme root of K. Hence we have an à priori solution
of the following question, namely, “To determine the (n − 1) positive quantities µ1 , µ2 . . . µn−1 ,
so as to make the greatest root of Q a minimum and its least root a maximum;” for the greatest
root of K will be the minimum greatest root of Q, and the least root of K the maximum
least root of Q. Calling these respectively l and λ, the two systems of values of µ1 , µ2 . . . µn−1
required will be obtained by substituting respectively l and λ for x in the equations
                p                 p             1                       p         1                p        1
         µ1 =       q12 ,   µ2 = −    q22 −        ,      µ3 = +          q32 −      . . . µn−1 = ± qn−1
                                                                                                     2
                                                                                                         −      .
                                                µ1                                µ2                       µn−2



                                                                    551
If n = 5, rejecting the factor ξ, it becomes

 a1 a2 a3 a4 a5 ξ 4 − (a1 a2 a3 + a1 a2 a5 + a1 a4 a5 + a3 a4 a5 )ξ 2 + (a1 + a3 + a5 ) = 0,
                                                                                               p. 545
    and so in general, the equation in ξ 2 being always of a degree measured by
the integer nearest to and not exceeding n2 ; and it is easy to be seen that, for all
values of n, the second coefficient divided by the first will be an inferior limit to
ξ 2 (of course actually coinciding with it for the cases of n = 2 and n = 3). Hence
we have the following valuable practical rule for finding a superior and inferior
limit to the cumulant

                       [a1 (x − c1 ), a2 (x − c2 ) . . . an (x − cn )],

where a1 , a2 . . . an have the same sign, namely if C be the greatest, and K be
the least of the quantities c1 , c2 . . . cn , C + ∆ will be a superior, and K − ∆ an
inferior limit, ∆ being taken equal to the positive value of
                       s
                            1     1     1              1
                                +     +      + ··· +         ;
                           a1 a2 a2 a3 a3 a4         an−1 an

and it may be noticed that C and K are the quantities which would them-
selves be the superior and inferior limits to the given cumulant if the series
of terms a1 , a2 . . . an , instead of presenting only a sequence of continuations or
permanencies, presented only a sequence of changes or variations of sign.

                                      Section V.

  On the Theory of Intercalations as applicable to two functions of the same
  degree, and on the formal properties of the Bezoutiant with reference to the
                             method of Invariants.

   Art. 56. If f x and ϕx be any two given functions of x of the same degree m,
we may form a system of m Bezoutics to f and ϕ (as shown in the first section),
the coefficients of the powers of xm−1 , xm−2 . . . x0 in which will compose a square
matrix of m lines of m terms each, which will be symmetrical in respect to the
diagonal which passes through the first coefficient of the first Bezoutic and the last
coefficient of the last Bezoutic; and we may construct a quadratic homogeneous
function of m new variables, such that its determinative matrix shall coincide
with the Bezoutic square so formed. This quadratic form may be considered in
the light of a generating function. All its coefficients will be formed of quantities
obtained by taking any two coefficients in one of the given functions, and two
corresponding coefficients in the other given function, multiplying them in cross
order, and taking the difference; each coefficient of the generating function in
question will consist of one or more such differences, and will thus be of two

                                            552
dimensions altogether, being linear in respect to the coefficients of f , and also
linear in respect to the coefficients of ϕ. This generating function I term the              p. 546
   Bezoutiant, and it may be denoted by the symbol B(f, ϕ): the determinant of
B is of course the resultant to f, ϕ, and the matrix to B is the Bezoutic square to
f, ϕ. Now we have seen that the decrease in the number of continuations of sign
in the series 1, B1 (x), B2 (x) . . . Bm (x) (where B1 (x), B2 (x) . . . Bm (x) are the m
Bezoutics to f, ϕ), as x changes from a to b, measures the number of roots of f x
retained in the effective scale of intercalations taken between the limits a and b.
If we take the entire scale between +∞ and −∞ the total number of effective
intercalations will be the same, whether reckoned by the number of roots of f or
of ϕ remaining; for these two numbers can never differ except by a unit, since
no two of either can ever come together; but the number of each remaining in
the effective scale will be m − 2i and m − 2i′ respectively, i being the number of
pairs of imaginary roots and pairs of unseparated real roots of f , and i′ being
the similar number for ϕ; so that we must have i = i′ .
   Now obviously this number becomes measured by the number of continuations
of sign in the signaletic series 1, (B1 ), (B2 ) . . . (Bm ), where in general (Bi ) denotes
the principal coefficient in Bi (x).
   But (B1 ), (B2 ) . . . (Bm ) are the successive ascending coaxal minor determinants
about the axis of symmetry to the Bezoutic square; and accordingly the number
of continuations just spoken of, measures the number of positive terms in the
Bezoutiant when linearly transformed, so as to contain only positive and negative
squares, or in other words, measures the inertia of the Bezoutiant, the constant
integer which adheres to it under all its real linear transformations.
   Art. 57. This inertia is the same number as, in the case of a homogeneous
quadratic function of three variables used to express a conic referred to trilinear
coordinates, serves to determine whether such conic belongs to the impossible
class or to the possible class of conics, being 3 or 0 in the former case, and 1
or 2 in the latter; or, as in the case of a homogeneous quadratic function of
four variables used to denote a surface referred to quadriplanar or tetrahedral
coordinates, serves to determine whether such surface belongs to the impossible
class or to the class consisting of the ellipsoid and the hyperboloid of two sheets
(which are descriptively indistinguishable), or to the hyperboloid of one sheet,
being 0 or 4 in the first case, 1 or 3 in the second, and 2 in the third. The most
symmetrical (but least expeditious) method of finding the inertia of any quadratic
form is that which corresponds to the method of orthogonal transformations,
and is, in fact, the usual method employed in geometrical treatises on lines and
surfaces of the second degree. If we apply this method to the Bezoutiant B
considered as a homogeneous quadratic function of the m arbitrarily named
variables u1 , u2 , u3 . . . um in order to measure its inertia, that is to say, the
number of effective                                                                          p. 547
   interpositions between the two systems of roots, we must construct the deter-


                                            553
minant
                       d2 B          d2 B          d2 B               d2 B
                            +λ                               ···
                       du21        du1 du2        du1 du3           du1 dum
                         d2 B      d2 B            d2 B               d2 B
                                        +λ                   ···
            D(λ) =     du2 du1     du22           du2 du3           du2 dum .
                          ···         ···           ···                ···
                         d2 B        d2 B          d2 B             d2 B
                                                             ···         +λ
                       dum du1     dum du2        dum du3           du2m

All the roots of D(λ) = 0, as is well known, are real; the inertia of B, being
measured by the number of positive roots of D(−λ), will be equal to the number
of continuations of sign in D(λ) expressed as a function of λ of the mth degree.
   If in f x and ϕx we reverse the order of the coefficients, and f x and ϕx so
transformed become f1 x and ϕ1 x, it is obvious that the roots of f1 and ϕ1 being
the reciprocals of the roots of f and ϕ respectively, the number of effective
intercalations to f1 and ϕ1 must be the same as for f and ϕ. Accordingly we
find that the form of the Bezoutiant to f and ϕ is the same as that of the
Bezoutiant of f1 and ϕ1 , the sole difference (one only of names) being that
B(u1 , u2 . . . um−1 , um ) for the one becomes B(um , um−1 . . . u2 , u1 ) for the other.
The equation D(λ), which determines the inertia of B, remains precisely the
same, as it ought to do, for either of the two systems f and ϕ or f1 and ϕ1 .
   Art. 58. The theory in the preceding articles of this section may be made to
embrace the case involved in Sturm’s theorem; for if

                     f x = a0 xm + a1 xm−1 + · · · + am−1 x + am ,

                 f ′ x = ma0 xm−1 + (m − 1)a1 xm−2 + · · · + am−1 ,
and
              f1 x = mf x − f ′ x = a1 xm−1 + 2a2 xm−2 + · · · + mam ,
the Bezoutian secondaries, or which is the same thing, the simplified Sturmian
residues to f x and f ′ x, will evidently be the same as those to f1 x and f ′ x.
Accordingly, if we form the signaletic series

                          f1 x,   f ′ x,   B1 ,   B2 . . . Bm−1 ,

where B1 , B2 . . . Bm−1 are the Bezoutian secondaries to f1 x and f ′ x, the number
of variations of sign between consecutive terms in this series, when                     p. 548
    x is made +∞, will measure the number of pairs of imaginary roots in f x;
and f x and f ′ x forming always a continuation, and the highest coefficient of f ′ x
being supposed positive, we see that the terms of the rhizoristic series will be
1, (B1 ), (B2 ) . . . (Bm−1 ), consisting of positive unity and the successive ascending
coaxal determinants of the Bezoutian matrix to f ′ x and f1 x. Hence then the


                                            554
form of the Bezoutiant to f ′ x and f1 x will serve to determine the number of pairs
of imaginary, and consequently also the number of real roots to f x. It should
be remarked that the form of the Bezoutiant to f ′ x and f1 x, considered as a
quadratic function of u1 , u2 . . . um−1 and of the coefficients in f x, will remain
unaltered when for f x we write f1 x, for this will change the signs throughout of
f ′ x and f1 x; and consequently the coefficients in the Bezoutiant, which contains
in every term one coefficient from f ′ x, and one from f1 x, will remain unaltered
in sign.
     Art. 59. It appears then from the preceding article, that for every function
of x of the degree m, there exists a homogeneous quadratic function of (m − 1)
variables, the inertia of which augmented by unity will represent the number
of real roots in the given function. Now this inertia itself may be measured
by the number of positive roots of a certain equation in λ formed from the
quadratic function (in fact the well-known equation for the secular inequalities of
the planets), all whose roots will be real. Hence then we are led to the following
remarkable statement. “An algebraical equation of any degree being given, an
equation whose degree is one unit lower may be formed, all the roots of which
shall be real, and of which the number of positive roots shall be one less than the
total number of real roots of the given equation.”
     Let us suppose f x written in its most general form, the first and last as
well as all the intermediate coefficients being anything whatever; by reversing
the order of the coefficients f ′ x will become f1 x and f1 x will become f ′ x; the
Bezoutiant to f1 x and f ′ x (which we may term the Bezoutoid to f x) will remain
unaltered except in sign, and the equation of the (m − 1)th degree in λ formed
from the Bezoutoid remain unchanged; consequently the equation in λ enables
us to substitute, for the purpose of calculating the total number of real roots in
f x, in lieu of Sturm’s auxiliary functions to f x, another set of functions which
remain unaltered when the order of the coefficients is completely
                                                                    reversed, that
is in effect, when we consider the number of real roots of f x1 in lieu of those of
f (x). And of course more generally the equation of the mth degree in λ formed
from the Bezoutiant to any two functions f x and ϕx of the mth degree each
in x, supplies a set of functions for determining the total number of effective
intercalations between the roots of f x and ϕx, which do not alter when we
consider in lieu
                of thesethe
                                                                                    p. 549
     roots of f x and ϕ x . This substitution of functions symmetrically formed
                 1         1

in respect to the two ends of an equation for the purpose of assigning the total
number of real roots in lieu of the unsymmetrical ones furnished by the ordinary
method of M. Sturm, had been long felt by me to be a desideratum, and as an
object the accomplishment of which was indispensable to the ulterior development
of the theory, and it is certain that I did not in anticipation exaggerate the
importance of the result to be attained.
     Art. 60. It may happen that the Bezoutiant to f and ϕ (each of the mth


                                        555
degree) may become a quadratic function of less than m independent variables,
or the Bezoutoid to f (a function in x of the mth degree) of less than (m − 1)
independent variables. This will take place whenever f and ϕ have roots in
common, or whenever f has equal roots. The number of independent relations of
equality between the roots of f and ϕ, and the amount of multiplicity, however
distributed, among the roots of f , will be indicated by the number of orders thus
disappearing out of the general form of the Bezoutiant and Bezoutoid in the
respective cases.263 In what particular mode the form of each would be affected
according to the manner of the distribution of the equalities and the multiplicity
requires a specific discussion, which I must reserve for some future occasion.
   Art. 61. I shall devote the remainder of this memoir to a consideration of
the properties and affinities of Bezoutiants or Bezoutoids, regarded from the
point of view of the Calculus of Invariants. For this purpose it will be more
convenient hereafter to convert all the functions which we are concerned with
into homogeneous forms, and I shall accordingly for the future use f and ϕ to
denote functions each of x and y, which I shall write under the form
                                     1
         f = a0 xm + ma1 xm−1 y + m(m − 1)a2 xm−2 y 2 + · · · + am y m ,
                                     2
                                     1
          ϕ = b0 xm + mb1 xm−1 y + m(m − 1)b2 xm−2 y 2 + · · · + bm y m .
                                     2
In what follows a knowledge of the general principles of the Method of Invariants
is presupposed, but a perusal of my two papers on the Calculus of Forms264
in the Cambridge and Dublin Mathematical Journal, February and May, 1852,
will furnish nearly all the information that is strictly necessary for the present
purpose. The first point to be established is, that B, the                         p. 550
   Bezoutiant of f x and ϕx, is a Covariant to the system f, ϕ; the variables in
B being in compound relation of cogredience with the combinations of powers of
x and y,
                        xm−1 , xm−2 y, xm−3 y 2 . . . y m−1 .
That is to say, I propose to show that if f, g, h, k be any four quantities, taken
for greater simplicity subject to the relation f k − gh = 1, and if on substituting
f x + gy for x and hx + ky for y, f (x, y) becomes
                         1
     A0 xm + mA1 xm−1 y + m(m − 1)A2 xm−2 y 2 + Am y m ,                         say G(x, y),
                         2
and ϕ(x, y) becomes
                           1
       B0 xm + mB1 xm−1 y + m(m − 1)B2 xm−2 y 2 + Bm y m ,                       say T (x, y),
                           2
 263
      I have elsewhere defined how this word order, as here employed, is to be understood. If F ,
a homogeneous function of x1 , x2 . . . xn , can be expressed as a function of u1 , u2 . . . un−i (all
linear functions of x1 , x2 . . . xn ), F is said to be a function of n − i orders, or to have lost i of
the orders belonging to the complete form.
  264
      See pp. 284, 328, 411 above.


                                                 556
and if B ′ (u′1 , u′2 . . . u′m ) be the Bezoutiant to G and T , B(u1 , u2 . . . um ) being
that to f and ϕ, then, on making u1 , u2 . . . um , the same linear functions of
u′1 , u′2 . . . u′m as
(f x+gy)m−1 ,     (f x+gy)m−2 (hx+ky) . . . (f x+gy)(hx+ky)m−2 ,           (hx+ky)m−1 ,
are respectively of
                         xm−1 ,    xm−2 y . . . xy m−2 ,   y m−1 ,
B will become identical with B ′ . I was led to suspect the high probability of
the truth of this proposition concerning the invariance of the Bezoutiant from
the following considerations: Firstly, that for the particular case where f and
ϕ are the differential derivatives in respect to x and y respectively of the same
function F (x, y), the Bezoutiant of f and ϕ, which then becomes the Bezoutoid
of F , determines the number of real factors in F , which obviously remains the
same for all linear transformations of F . Secondly, that taking f and ϕ in their
most general form, the invariant to their Bezoutiant, that is the determinant
of their Bezoutiant, is an invariant of f and ϕ, being in fact the resultant of
these two functions; now as every concomitant (an invariantive form of the most
general kind) to a concomitant is itself a concomitant to the primitive, so it
appeared to me, and is I believe true (although awaiting strict proof), that any
form satisfying certain necessary and tolerably obvious conditions of homogeneity
and isobarism, a concomitant to which is also a concomitant to a given form,
will be itself a concomitant to such form; this principle, if admitted, would be of
course at once conclusive as to the Bezoutiant being an invariantive concomitant
to the functions from which it is derived.
   Art. 61*. Since the publication of the two papers above referred to on the
Calculus of Forms, I have made the important observation that every species of
concomitant, however complex, to a given system of functions, may be treated
as a simple invariant of a system including the given system                           p. 551
   together with an appropriate superadded system of absolute functions; thus an
ordinary covariant involving only one system of variables, as u, v, w . . . cogredient
with x, y, z . . . the variables of a system S, is in fact an invariant of the system
S combined with the system uy − vx, vz − wy, wx − uz, &c., u, v, w . . . being
treated as constants; so again a simple contravariant of S is an invariant of S
combined with the form ux + vy + wz + &c.; so again, to meet the case before us,
a covariant to the binary system f and ϕ expressed as a function of u1 , u2 . . . um ,
where u1 , u2 . . . um are cogredient with xm−1 , xm−2 y . . . y m−1 , may be regarded
as an invariant of the ternary system f, ϕ, Ω, where
                                1
    Ω = u1 y m−1 − mu2 y m−2 x + m(m − 1)u3 y m−3 x2 + · · · + (−)m um xm−1 ,
                                2
(u1 , u2 . . . um being here to be treated as constants); and accordingly the differ-
ential equations which serve to define in the most general and absolute manner

                                           557
such covariant of f, ϕ, or invariant to f, ϕ, Ω, say I, will take the form
       d         d              d         d              d         d
                                                                         

  a0      + b0       + 2  a 1      + b1       + 3   a2      + b2       + · · · 
                                                                                
      da        db             da        db             da        db
                                                                               
         1        1               2        2               3        3

                                                                               
                                                                                

                                                                               
                                                                                
                   d             d
                                                                             
     + m am−1           + bm−1                                                      I = 0,


                 dam            dbm                                          
                                                                              
                                                                              
           d         d         d                         d

                                                                           
                                                                              
     − u1     + 2u2     + 3u3     + · · · + (m − 1)um−1

                                                                             
                                                                              
                                                                             
          du2       du3       du4                       dum

        d          d                d             d
                                                                                   

  am       + bm         + 2 am−1       + bm−1                                        
                                                                                      



     dam−1      dbm−1            dam−2        dbm−2                                  
                                                                                      
                                                                                      
                                                                                      
               d             d                  d      d

                                                                                 
                                                                                      
    + 3 am−2      + bm−2        + · · · + m a1        + b1               I = 0.


           dam−3        dbm−3                 da0         db0         
                                                                       
                                                                      
           d             d                d                        d 

                                                                     
    − um       + 2um−1       + 3um−2          + · · · + (m − 1)u2

                                                                      
                                                                       
                                                                      
         dum−1         dum−2            dum−3                     du1
These equations may be proved to be satisfied when I is taken = B, the
Bezoutiant to f, ϕ, and thus B may be proved to be a covariant to f, ϕ, but the
demonstration is long and tedious. An admirable suggestion, well worthy of its
keen-witted author, for which I am indebted to Mr Cayley, will enable us to
prove the invariantive character of B by a much more expeditious method.
   Art. 62. For greater simplicity begin with considering functions of a single
variable x; and in order to fix the ideas, suppose m to be taken 5, and write

                       f x = ax5 + bx4 + cx3 + dx2 + ex + l,

                      ϕx = αx5 + βx4 + γx3 + δx2 + εx + λ,
and let
                                       f x ϕx′ − f x′ ϕx
                                 ϑ=                      ;
                                            x − x′
this is of course an integral function of x and x′ ,                        p. 552
   since the numerator vanishes when x = x′ ; and we have by performing the
actual operations,

ϑ = (aβ − bα)x4 x′4 + (aγ + cα)x3 x′3 (x + x′ ) + (aδ − dα)x2 x′2 (x2 + xx′ + x′2 )
     + (aε − eα)xx′ (x3 + x2 x′ + xx′2 + x′3 ) + (aλ − lα)(x4 + x3 x′ + x2 x′2 + xx′3 + x′4 )
     + (bγ − cβ)x3 x′3 + (bδ − dβ)x2 x′2 (x + x′ ) + (bε − eβ)xx′ (x2 + xx′ + x′2 )
     + (bλ − lβ)(x3 + x2 x′ + xx′2 + x′3 ) + (cδ − dγ)x2 x′2 + (cε − eγ)xx′ (x + x′ )
     + (cλ − lγ)(x2 + xx′ + x′2 ) + (dε − eδ)xx′ + (dλ − lδ)(x + x′ ) + (eλ − lε);




                                           558
and if we arrange ϑ under the form

            A4,4 x4 x′4 + A4,3 x4 x′3 + A4,2 x4 x′2 + A4,1 x4 x′ + A4,0 x4
            + A3,4 x3 x′4 + A3,3 x3 x′3 + A3,2 x3 x′2 + A3,1 x3 x′ + A3,0 x3
            + A2,4 x2 x′4 + A2,3 x2 x′3 + A2,2 x2 x′2 + A2,1 x2 x′ + A2,0 x2
            + A1,4 xx′4 + A1,3 xx′3 + A1,2 xx′2 + A1,1 xx′ + A1,0 x
            + A0,4 x′4 + A0,3 x′3 + A0,2 x′2 + A0,1 x′ + A0,0 ,

it will readily be perceived that the matrix formed by the twenty-five coefficients,
namely
                           A4,4 A4,3 A4,2 A4,1 A4,0
                           A3,4 A3,3 A3,2 A3,1 A3,0
                           A2,4 A2,3 A2,2 A2,1 A2,0
                           A1,4 A1,3 A1,2 A1,1 A1,0
                           A0,4 A0,3 A0,2 A0,1 A0,0
will be symmetrical about its dexter diagonal (that one, namely, which passes
through A4,4 and A0,0 ), and will be identical with the Bezoutian square corre-
sponding to the system f, ϕ; in fact, using the notation previously employed in
the first section, it becomes

             (0, 1) ( (0, 2)        ( (0, 3)       ( (0, 4)       (0, 5)
                       (0, 3)          (0, 4)          (0, 5)
             (0, 2)                                               (1, 5)
                     +(1, 2)         +(1, 3)           +(1, 4)
                                    
                                     (0, 5)
                      (                            (
                          (0, 4)                       (1, 5)
                                    
             (0, 3)                   +(1, 4)                     (2, 5)     (α)
                          +(1, 3)   +(2, 3)
                                                      +(2, 3)
                      (             (              (
                           (0, 5)      (1, 5)           (2, 5)
             (0, 4)                                               (3, 5)
                          +(1, 4)    +(2, 3)           +(3, 4)
             (0, 5)       (1, 5)      (2, 5)           (3, 5)     (4, 5),
                                                                                       p. 553
   (r, s) being used in general to denote the difference between the cross products
of the coefficients of xr and xs in f and ϕ. Restoring now to m its general value,
and taking f and ϕ homogeneous functions of x and y, and making
                             f (x, y)ϕ(x′ , y ′ ) − f (x′ , y ′ )ϕ(x, y)
                      ϑ=                                                 ,
                                           xy ′ − x′ y
we see without difficulty that

                           ϑ = ΣAr,s {xr y m−1−r x′s y ′m−1−s },

where Ar,s is the term in the rth line and sth column of the Bezoutiant matrix
to f and ϕ. This is the identification, the idea of which, as before observed, is
due to Mr Cayley.

                                             559
   Art. 63. If, now, we consider the system of functions

                     f (x, y) = a0 xm + ma1 xm−1 y + · · · + am y m ,

                      ϕ(x, y) = b0 xm + mb1 xm−1 y + · · · + bm y m ,
          Ω(x, y) = um y m−1 − (m − 1)um−1 y m−2 x + · · · + (−)m u1 xm−1 ,
evidently f (x, y)ϕ(x′ , y ′ ) − f (x′ , y ′ )ϕ(x, y) is a covariant to f and ϕ, and therefore
(which is a mere truism) with the entire system f, ϕ, Ω. So also is xy ′ − x′ y,
and therefore ϑ, the quotient of these two, is a covariant to the system. Hence,
therefore, by virtue of a general theorem given in my Calculus of Forms,
                                               d      d
                                                             
                                            Ω    ′
                                                   ,− ′ ϑ
                                              dy     dx
is a covariant to the system; and, again, therefore,
                                        d     d    d     d
                                                                   
                               Ω           ,−   Ω     ,−   ϑ
                                       dy ′ dx′   dy ′ dx′

is a covariant thereto. Now ϑ is of (m − 1) dimensions in x, y and also of the
same in x′ , y ′ . Consequently this latter form will contain only the quantities
u1 , u2 . . . um−1 , and the coefficients of f and ϕ, so that the powers of x, y; x′ , y ′
will not appear in it.
    Now
                                0 X
                                  0
                       ϑ=                    Ar,s {xr y m−1−r x′s y ′m−1−s },
                                X

                               m−1 m−1
                                                  m−1                              m−2                         m−1
                 d     d                     d                                 d            d              d
                                                                                                      
(−)m−1 Ω            ,−         = um                      +(m−1)um−1                             +· · ·+u1                  ,
                dy ′ dx′                    dx′                               dx′          dy ′           dy ′
                                             m−1                               m−2                          m−1
       d      d                         d                                  d            d              d
                                                                                                  
(−) Ω
    m
         ′
           ,− ′            = um                     +(m−1)um−1                              +· · ·+u1                  ,
      dy     dx                        dx′                                dx′          dy ′           dy ′
                                                                                                              p. 554
   therefore
                               1               d      d    d    d
                                                                                    
                                            Ω      ,− ′ Ω    ,−   ϑ
                 {1 · 2 · 3 · · · (m − 1)}2   dy ′   dx   dy dx
                           0               0 X0
                      =    (Ar,r ur ) + 2       (Ar,s ur us ),
                           X              X
                                  2

                        m−1               m−1 1

r and s being excluded in the latter sum from being made equal; but this latter
expression is the Bezoutiant to f, ϕ. Hence the Bezoutiant of f, ϕ is an invariant
to f, ϕ, Ω, that is a covariant to the system f, ϕ, as was to be proved. The
mode of obtaining the covariant ϑ, used in this and the preceding article, is very



                                                    560
remarkable. I believe that the true suggestive view of the process for finding it,
is to consider
                        f (x, y)ϕ(x′ , y ′ ) − f (x′ , y ′ )ϕ(x, y)
as a concomitant capable of being expressed under the form of a function of ϑ
and ω, ω standing for the universal covariant xy ′ − x′ y; ϑ is then to be considered,
not properly as a quotient, but rather as an invariant of the form ϑω, a function
of ω of the first degree, where ϑ is treated as constant.
   Art. 64. B is not an ordinary covariant of f and ϕ, it belongs to that special
and most important family of invariants to a system to which I have given the
name of Combinants,265 namely Invariants, which, besides the ordinary character
of invariance when linear substitutions are impressed upon the variables, possess
the same character of invariance when linear substitutions are impressed upon
the functions themselves containing the variables; combinants being, as it were,
invariants to a system of functions in their corporate combined capacity quâ
system. That the Bezoutiant possesses this property is evident; for if instead of
f and ϕ we write kf + iϕ and k ′ f + i′ ϕ, any such quantity as ar bs − as br (ar , br
being coefficients in f , and as , bs the corresponding ones in ϕ) becomes

                 (kar + ibr )(k ′ as + i′ bs ) − (kas + ibs )(k ′ ar + i′ br ),

that is
                                 (ki′ − k ′ i)(ar bs − as br ),
so that B, the Bezoutiant, becomes increased in the ratio of (ki′ − k ′ i)m , that is
remains always unaltered in point of form and absolutely immutable, provided
that ki′ − k ′ i be taken, as we may always suppose to be the case, equal to 1.
    We derive immediately from this observation, the somewhat remarkable
geometrical proposition, that the intersections with the axis of x made by any
two curves of the family of curves u = λf (x) + µϕ(x), (f and ϕ being functions
of x of the same degree) give rise to a constant number of effective intercalations,
whatever values be given to λ or µ for the two curves so selected.                     p. 555
    Art. 65. B(u1 , u2 . . . um ) being a covariant of the system f and ϕ, and
u1 , u2 . . . um cogredient with xm−1 , xm−2 y . . . y m−1 , it follows from a general
principle in the theory of invariants, that on making u1 , u2 . . . um respectively
equal to the quantities with which they are cogredient, B will become an ordinary
covariant to f and ϕ. By this transformation B becomes a function of x and y of
the degree 2(m − 1) in x and y conjointly, and linear in respect to the coefficients
of f , and also in respect to those of ϕ. The only covariant capable of answering
this description is what I am in the habit of calling the Jacobian (after the name
of the late but ever-illustrious Jacobi), a term capable of application to any
 265
    For some remarks on the Classification of Combinants, see Cambridge and Dublin Mathe-
matical Journal, November, 1853 [p. 411 above].



                                             561
number of homogeneous functions of as many variables. In the case before us,
where we have two functions of two variables, the Jacobian

                                     df    dϕ
                                     dx    dx          df dϕ df dϕ
                      J(f, ϕ) =      df    dϕ      =         −       .
                                                       dx dy   dy dx
                                     dy    dy

We have then the interesting proposition,266 that the Bezoutiant to two functions,
when the variables in the former are replaced by the combinations of the variables
in the latter, with which they are cogredient, becomes the Jacobian.267 So in
the case of a single function F of the degree m, the Bezoutoid, that is the
Bezoutiant to dF  dx , dy , on making the (m − 1) variables which it contains identical
                       dF

with xm−2 , xm−3 y . . . y m−2 respectively, becomes identical with the Jacobian to
 dx , dy , that is the Hessian of F , namely
dF dF


                                      d2 F      d2 F
                                       dx2      dxdy      .
                                      d2 F      d2 F
                                      dxdy       dy 2
As an example of this property of the Bezoutiant, suppose

        f = ax3 + bx2 y + cxy 2 + dy 3 ,           ϕ = αx3 + βx2 y + γxy 2 + δy 3 .

The Bezoutiant matrix becomes
                           aβ − bα aγ − cα aδ − dα
                                    aδ − dα
                           aγ − cα  +  bγ − cβ
                                            
                                     bγ − cβ
                           aδ − dα   bγ − cβ   cδ − dγ.
                                                                                                p. 556
   The Bezoutiant accordingly will be the quadratic function

(aβ−bα)u21 +{(aδ−dα)+(bγ−cβ)}u22 +(cδ−dγ)u23 +2(aγ−cα)u1 u2 +2(aδ−dα)u3 u1 +2(bγ−cβ

which on making
                         u1 = x2 ,        u2 = xy,            u3 = y 2 ,
becomes
                   Lx4 + M x3 y + N x2 y 2 + P xy 3 + Qy 4 ,               (β)
 266
     I have subsequently found that this proposition is contained under another mode of
statement, at the end of Section 2 of the memoir of Jacobi, “De Eliminatione,” above referred
to.
 267
     For a strict proof of this proposition see Supplement to Third Section of this memoir.


                                             562
where L, M, N, P, Q respectively will be the sum of the terms lying in the
successive bands drawn parallel to the sinister diagonal of the Bezoutiant matrix,
that is
                                  L = aβ − bα,
                                      M = 2(aγ − cα),
                                 N = 3(aδ − dα) + (bγ − cβ),
                                       P = 2(bγ − cβ),
                                         Q = cδ − dγ.
The biquadratic function in x and y, (β), above written, will be found on
computation to be identical in point of form with the Jacobian to f, ϕ, namely

(3ax2 + 2bxy + cy 2 )(βx2 + 2γxy + 3δy 2 ) − (3αx2 + 2βxy + γy 2 )(bx2 + 2cxy + dy 2 ),

this latter being in fact

                        3Lx4 + 3M x3 y + 3N x2 y 2 + 3P xy 3 + 3Qy 4 .

The remark is not without some interest, that in fact the Bezoutiant, which is
capable (as has been shown already) of being mechanically constructed, gives the
best and readiest means of calculating the Jacobian; for in summing the sinister
bands transverse to the axis of symmetry the only numerical operation to be
performed is that of addition of positive integers, whereas the direct method
involves the necessity of numerical subtractions as well as additions, inasmuch
as the same terms will be repeated with different signs. Thus if

                      f = ax5 + bx4 y + cx3 y 2 + dx2 y 3 + exy 4 + ly 5 ,

                     ϕ = αx5 + βx4 y + γx3 y 2 + δx2 y 3 + εxy 4 + λy 5 ,
using (r, s) in the ordinary sense that has been considered throughout, we obtain
by taking the sum of the sinister bands in (α)268 for the value of B when we
write x4 , x3 y, x2 y 2 , xy 3 , y 4 in place of u1 , u2 , u3 , u4 , u5 ,

         (0, 1)x8 + 2(0, 2)x7 y + {3(0, 3) + (1, 2)}x6 y 2 + {4(0, 4) + 2(1, 3)}x5 y 3
            + {5(0, 5) + 3(1, 4) + (2, 3)}x4 y 4 + {4(1, 5) + 2(2, 4)}x3 y 5
            + {3(2, 5) + (3, 4)}x2 y 6 + 2(3, 5)xy 7 + (4, 5)y 8 .
                                                                                          p. 557
   The direct process requires the calculation of

(5ax4 + 4bx3 y + 3cx2 y 2 + 2dxy 3 + ey 4 )(βx4 + 2γx3 y + 3δx2 y 2 + 4εxy 3 + 5λy 4 )
   − (5αx4 + 4βx3 y + 3γx2 y 2 + 2δxy 3 + εy 4 )(bx4 + 2cx3 y + 3dx2 y 2 + 4exy 3 + 5ly 4 ),
 268
       Vide Art. 62 [p. 552 above].


                                              563
each coefficient of which will contain the numerical factor 5; so that to reduce
the Jacobian to its simplest form each coefficient will necessitate the employment
of additions, subtractions, and a division, instead of additions merely, as when
the Bezoutic square is employed. For instance, to find the coefficient of x4 y 4
from the above expression (α) we have to calculate
                   1
                     {25(0, 5) + 16(1, 4) + 9(2, 3) + 4(3, 2) + (4, 1)},
                   5
that is
                    1
                      {25(0, 5) + (16 − 1)(1, 4) + (9 − 4)(2, 3)},
                    5
which is 5(0, 5) + 3(1, 4) + (2, 3), agreeing with what has been found above for
the value of such coefficient, by a simple process of counting. The same remark
will, of course, also apply to the computation of the Hessian of F by means of
its Bezoutoid.
   Art. 66. This relation between the Bezoutiant and the Jacobian led me to
inquire whether, as would at first sight appear probable, the Bezoutiant were
the only lineo-linear quadratic function of m variables covariantive to f and ϕ
(the word lineo-linear being used to denote the form of coefficients, such as those
in the Bezoutiant, linear in respect of the coefficients in f and the coefficients
of ϕ). If so, then there would have existed a method of performing the inverse
process of recovering the Bezoutiant from the Jacobian, almost as simple as
that of deriving the Jacobian from the Bezoutiant. On investigating the matter,
however, I found that such is by no means the case,269 but that there exists a
whole family of independent                                                         p. 558
 269
    This might have been concluded immediately from the following observation. Let J, the
Jacobian of f and ϕ, be expressed under the form
                                          1
   A0 x2m−2 + (2m − 2)A1 x2m−3 y +          (2m − 2)(2m − 3)A2 x2m−4 y 2 + · · · + A2m−2 y 2m−2 ,
                                          2
then we know [p. 282 above] from the Calculus of Forms, that, D being taken to represent the
persymmetrical Determinant
                              A0          A1     A2     ...   Am−1
                              A1          A2     A3     ...    Am
                              A2          A3     A4     ...   Am+1     ,
                              ···         ···    ···           ···
                             Am−1         Am    Am+1    ...   A2m−2
D = 0 is the condition to be satisfied in order that J may be representable under the form of
the sum of powers of (m − 1) linear functions of x and y, and D itself is an invariant to J, and
consequently an invariant and (as is obvious from its form) a combinative invariant to f and ϕ.
Moreover, which is more immediately to the point, we know that the quadratic form
                                                          (m − 1)(m − 2)
                                                                             
 A0 u21 + 2A1 {u1 (m − 1)u2 } + A2       (m − 1)u22 + 2u1                u3       + &c. + A2m−2 u2m ,
                                                                 2
will be an invariant to f, ϕ and Ω (this last quantity Ω being defined as in p. [551]), and a
combinative covariant to f and ϕ in the same sense precisely as the Bezoutiant is a covariant


                                                 564
   lineo-linear quadratic covariants of m variables to every two homogeneous
functions of x and y of the mth degree. I have, moreover, I believe, succeeded in
determining the number of such lineo-linear quadratic forms for any value of m,
of which all the rest, in whatever manner obtained, may be expressed as linear
functions, the coefficients of the linear relations moreover being abstract numbers;
in other words, I have succeeded in forming the fundamental or constituent scale
of lineo-linear quadratic forms of m variables covariantive to f and ϕ; a result of
too great interest, as exhibiting the affinities of the Bezoutiant to its cognate
forms, to be altogether passed over in silence. Supposing the number of linearly
independent forms of the kind to be ν, then speaking à priori any of the forms
taken at random might seem to be equally eligible to form one of the ν included
in the fundamental scale, combined with any (ν − 1) others independent inter se,
and of which the selected one is also independent. In fact, however, this is not
so; for it will always be more satisfactory to contemplate the fundamental scale
of forms as generated successively or simultaneously by a uniform process; and
in the case before us, the process which I have hit upon, and which I believe is
the simplest that can be employed for generating the fundamental scale, will be
found not to include directly the Bezoutiant among the number. There will thus
arise two subjects of inquiry; firstly, the mode of forming the fundamental scale,
and proving its fundamental character; secondly, determining the numerical
relations between the Bezoutiant and the scale; subjects which I reserve for some
future occasion, and may probably be considered as exhausting the matter of

to the same, and like the Bezoutiant is lineo-linear in respect of the coefficients of f and ϕ. If
we operate with the symbol E, where E represents
                        d            d                     d               d
                 v12       + 2v1 v2     + (v22 + 2v1 v3 )     + &c. + vm
                                                                       2
                                                                                ,
                       dA0          dA1                   dA2            dA2m−2
upon K any invariant of f and ϕ, we shall obtain EK, a quadratic function of v1 , v2 . . . vm ,
which by the rules of the Calculus of Forms we know will be a contravariant to f and ϕ, and
the matrix corresponding to which must evidently be persymmetrical. It is an interesting
subject of inquiry, which I reserve for some future occasion, to determine the Co-bezoutiant,
the Discriminant of which must be employed for K, so that when this discriminant is operated
upon by E, the matrix corresponding to EK may become identical (term for term) with the
matrix which is the inverse to the Bezoutiant matrix, which inverse, as Jacobi has so simply and
beautifully demonstrated, possesses this persymmetrical character. Vide the “De Eliminatione,”
Section 5. The investigation of the arithmetical connexion between the Q of this note and
the fundamental Co-bezoutiants must be also similarly reserved. I believe it to be generally
true, and have verified the fact for the case of two cubic functions, that EQ gives a quadratic
form such that the corresponding matrix is the inverse to the matrix of Q. The calculations
necessary for extending the verification of this remarkable proposition for functions of x, y
exceeding the third degree (notwithstanding that they are much abbreviated by the application
of the rules of the calculus) still remain excessively laborious. The abbreviation alluded to
consists in confining the verification in question to the comparison of either one of the two
unreiterated terms at opposite corners of the matrix to EQ with the corresponding term in the
inverse matrix of Q; if these coincide, it is easy to prove that every other pair of corresponding
terms in the two matrices must also coincide respectively with one another.



                                                565
this Section.                                                                         p. 559
   which connect that very important form, perhaps of all its kind the most
important, with the forms comprised in the fundamental or constituent scale.
These questions I propose to consider more fully at a future period. For the
present I shall content myself with giving a method of forming the constituent
scale (without, however, seeking the proof of all the forms extra to such assumed
scale being linear functions of those comprised within it), and with determining
the numerical relations between the forms in this scale and the Bezoutiant for
a limited number of values of m. All the forms which we are seeking, besides
being lineo-linear quadratics, must also be combinative invariants to f and ϕ,
remaining (as forms) unaltered for any linear substitutions impressed either upon
the variables or upon the functions containing the variables.
   Art. 67. I must here premise that if there be any two forms of the same
degree (and that degree odd) in x and y, a combinant may be formed from them,
which will be linear in respect to each set of coefficients270 . Thus calling the two
functions
                                    1
       a0 x2n+1 + (2n + 1)a1 x2n y + (2n + 1)2n a2 x2n−1 y 2 + · · · + a2n+1 y 2n+1 ,
                                    2
                                     1
       α0 x2n+1 + (2n + 1)α1 x2n y + (2n + 1)2n α2 x2n−1 y 2 + · · · + α2n+1 y 2n+1 ,
                                     2
the lineo-linear combinant in question will be

                            1                    (2n + 1)(2n)(2n − 1)
T = a0 α2n+1 −(2n+1)a1 α2n + (2n+1)2n a2 α2n−1 +                      a3 α2n−2 &c.− &c.
                            2                           1·2·3
which, using our customary notation, will be of the form

                                (2n + 1)2n                    (2n + 1)(2n)(2n − 1) · · · (n + 2
(0, 2n+1)−(2n+1)(1, 2n)+                   (2, 2n−1)±&c.+(−)n
                                   1·2                                  1 · 2 · 3···n
As a corollary to this proposition (which, as well as the proposition itself, will be
needed for the purposes of the ensuing determination), taking any function of
an even degree in x, y, F (x, y), there will exist a combinant to dFdx and dy , by
                                                                              dF

virtue of what has been stated above, which will be                                   p. 560
   Mr Cayley’s well-known quadrinvariant to F ; namely, if

                        F = a0 x2n + 2na1 x2n−1 y + · · · + a2n y 2n ,
 270
     I may add here incidentally (although not wanted for our present purposes) that as a
combinant in which each set of coefficients enters linearly can always be formed to a system
of functions two in number of as many variables and of any odd degree, so reciprocally can
a combinant in which each set of coefficients enters linearly be always formed to a system of
functions each of the degree 2, of which and of the variables contained in them, the number is
any odd integer [cf. p. 606 below].




                                             566
this will be
                        2n(2n − 1)                   1     2n(2n − 1) · · · (n + 1) 2
a0 a2n − 2na1 a2n−1 +              a2 a2n−2 + · · · + (−)n                         an .
                            2                        2           1 · 2···n
The proposition itself is easily proved; first, the expression T being expressed
entirely in terms of quantities of the form (r, s) remains unaltered for linear
substitutions impressed upon the forms f and ϕ; it remains then only to show
that T satisfies the differential equations to T treated as a mere invariant, namely
           d       d       d                                        d
                                                                                  
       a0
      
            + 2a1   + 3a2   + · · · + (2n + 1)a2n                                 
                                                                                   
          da1        da2           da3                            da2n+1
                                                                                   
             d       d       d                                             d           T = 0,
       + α0   + 2α1   + 3α2   + · · · + (2n + 1)α2n
      
                                                                                  
                                                                                   
                                                                                   
               dα1         dα2           dα3                            dα2n+1
and
                d                   d                    d
                                                                              
         a2n+1
        
                   + 2a2n        + · · · + (2n + 1)a1          
                                                                
               da2n        da2n−1                       da0
                                                                
                    d            d                           d   T = 0.
         + α2n+1      + 2α2n          + · · · + (2n + 1)α1
        
                                                               
                                                                
                  dα2n        dα2n−1                        dα0
From the hemihedral symmetry of T , which only changes its sign when the order
of the coefficients in f and ϕ is simultaneously reversed, it is obvious that one
of these equations cannot be satisfied without the other being so too. Looking
then exclusively at the first of them, we see that this is satisfied by virtue of the
equations
                            d                   d
                                                    
                        a0     + (2n + 1)α2n           T = 0,
                           da1               dα2n+1
                                 d              d
                                                        
                             2a1    + 2nα2n−1        T = 0,
                                da2           dα2n
                             ··························· ,
                                            d     d
                                                            
                         (2n + 1)a2n        + α0                 T = 0.
                                     da2n+1      dα1
Hence then the differential equations to T being satisfied proves that it is an
invariant, and, as above observed, its form shows upon its face that it is a
combinant.
   Precisely in the same way it may be demonstrated, that to two functions each
of the same even degree 2m as
                                        2m(2m − 1)
         a0 x2m + 2ma1 x2m−1 y +                   a2 x2m−2 y 2 + · · · + a2m y 2m ,
                                            2
                                                                                                p. 561
  and
                                        2m(2m − 1)
        α0 x2m + 2mα1 x2m−1 y +                    α2 x2m−2 y 2 + · · · + α2m y 2m ,
                                            2

                                                567
there will be a quantity
                               2m(2m − 1)
G = a0 α2m − 2ma1 α2m−1 +                 a2 α2m−2 ± &c. − 2mα1 a2m−1 + α0 a2m ,
                                   2
which, although not a combinant, will satisfy the differential equations necessary
to prove it to be an ordinary invariant to the two given functions.
   Art. 68. Now let us consider the three forms, f, ϕ and the subsidiary form Ω,
where
                     f = a0 xm + ma1 xm−1 y + · · · + am y m ,
                       ϕ = b0 xm + mb1 xm−1 y + · · · + bm y m ,
            Ω = u1 y m−1 − (m − 1)u2 y m−2 x + &c. + (−)m−1 um xm−1 ,
where u1 , u2 . . . um are to be treated as constants. Make

                                 1
                                                                   2i+1
                                                   d     d
                                                  
              E2i+1 f =                         ξ    +η                    f,
                        m(m − 1) . . . (m − 2i)   dx    dy

                                  1
                                                                   2i+1
                                                    d     d
                                                  
               ′
              E2i+1 ϕ=                           ξ    +η                   ϕ,
                         m(m − 1) . . . (m − 2i)   dx    dy
i being any integer such that 2i + 1 does not exceed m, and now consider
            ′
E2i+1 f, E2i+1  ϕ as two functions of the degree 2i + 1 in ξ, η (x and y being
regarded as constants); and by virtue of the formula in the last article, form
Ti , the lineo-linear combinant of E2i+1 f and E2i+1 ϕ; Ti will then be lineo-linear
in respect to the coefficients in f and ϕ, and of the degree 2{m − (2i + 1)} in
respect to x and y. Again, let

                                   1
                                                                    2i
                                                     d     d
                                                      
              E2i Ω =                             ξ    +η                  Ω.
                      m(m − 1) . . . (m − 2i + 1)   dx    dy
E2i Ω treated as a function of ξ and η of the degree 2i will furnish a quadrinvariant
Qi of the degree 2(m − 1 − 2i) in respect of x and y, and quadratic in respect
of the system u1 , u2 . . . um . We have thus two forms, Ti and Qi , each of the
same even degree 2{m − (2i + 1)} in respect of x, y. Forming between these
the lineo-linear invariant Gi , Gi will be a function lineo-linear in respect of
the coefficients of f and ϕ, and quadratic in respect of the system u1 , u2 . . . um .
Moreover, Gi will (by the general principle of successive concomitance) be an
invariant in respect to the system f, ϕ, Ω, and combinative in respect to f and ϕ.
Thus then Gi for all admissible values of i will belong to the family of forms to
which the Bezoutiant is to be referred.
   It requires to be noticed that when i is taken zero, so that Ti and Gi are of
the degree 2(m − 1), E2i for this case must be taken equal to Ω2 , which               p. 562
   evidently fulfils the required conditions of being of the degree 2(m−1) in (x, y),
and quadratic in respect of the coefficients of Ω. If, now, m be even, we may take

                                         568
for 2i + 1 successively all the odd numbers from 1 to (m − 1) inclusively, and
there will be 12 m forms Gi ; when m is odd we may take for 2i + 1 successively all
the odd numbers from 1 to m, and the number of forms of Gi will be 12 (m + 1). It
should be observed, that when m is odd and 2i + 1 = m, Ti will become identical
with the lineo-linear combinant to f and ϕ, and Qi with the quadrinvariant to
Ω; and no power of x or y will enter into either, so that Gm will become simply
Tm × Qm . I am now able to enunciate the proposition, that G0 , G1 . . . G m2 −1 ,
when m is even, and G0 , G1 . . . G m−1 , when m is odd, form the constituent scale
                                      2
of forms, of which the Bezoutiant and all other lineo-linear quadratic functions of
m variables, which are combinants of the system f, ϕ, will be numerically-linear
functions. I propose to term the members of this scale Co-bezoutiants.
   As regards the present memoir, I shall content myself with exhibiting a
partial verification of this law as regards the connection of the Bezoutiant with
the G scale of Co-bezoutiants, and a complete determination of the numerical
multipliers which express this connection for the cases comprised between m = 2
and m = 6 taken inclusively. It is impossible to predict for what ulterior purposes
in the development of the Calculus of Invariants these numbers may or may
not be required, and it seems to me desirable that a commencement of a table
containing them should be made and placed on record. The remaining pages of
this memoir will accordingly be devoted to the ascertainment of them.
   The theory of the Bezoutoid being included within that of the Bezoutiant,
need not hereafter call for any special attention; I may merely notice that the
Bezoutoid to a function of the degree m is a numerico-linear function of 12 (m − 3)
of the G’s if m be odd, and 21 (m − 4) of the G’s if m be even.
   It will be more convenient hereafter to denote the G’s as G1 , G3 , G5 respectively,
in lieu of G0 , G1 , G2 , &c., and to continue at the same time to give to the T ’s
and Q’s the same subscripts as the corresponding G’s.
   Art. 69. Firstly. Suppose m = 2,

     f = ax2 + 2bxy + cy 2 ,      ϕ = αx2 + 2βxy + γy 2 ,          Ω = u1 y − u2 x.
                                                                                          p. 563
   Then
                          E1 f = (ax + by)ξ + (bx + cy)η,
                         E1 ϕ = (αx + βy)ξ + (βx + γy)η,
                 T1 = (ax + by)(βx + γy) − (bx + cy)(αx + βy)
                    = (aβ − bα)x2 + (aγ − cα)xy + (bγ − cβ)y 2 ,
                        Q1 = Ω2 = u21 y 2 − 2u1 u2 xy + u22 x2 ,
and therefore

                G1 = (aβ − bα)u21 + (aγ − cα)u1 u2 + (bγ − cβ)u22 .


                                          569
Let us now form in the usual manner the Bezoutiant to f, ϕ; this is the quadratic
function which corresponds to the matrix
                                                              )
                             (2aβ − 2bα), (aγ − cα)
                                                                  ,
                              (aγ − cα), (2bγ − 2cβ)

that is
    1
      B = (aβ − bα)u21 + (aγ − cα)u1 u2 + (bγ − cβ)u22 = G1              or   B = 2G1 .
    2
Secondly. Suppose m = 3.

                            f = ax3 + 3bx2 y + 3cxy 2 + dy 3 ,

                           ϕ = αx3 + 3βx2 y + 3γxy 2 + δy 3 ,
                               Ω = u1 y 2 − 2u2 yx + u3 x2 .
We have then

                E1 f = (ax2 + 2bxy + cy 2 )ξ + (bx2 + 2cxy + dy 2 )η,

               E1 ϕ = (αx2 + 2βxy + γy 2 )ξ + (βx2 + 2γxy + δy 2 )η,
      T1 = (ax2 + 2bxy + cy 2 )(βx2 + 2γxy + δy 2 )
             − (bx2 + 2cxy + dy 2 )(αx2 + 2βxy + γy 2 )
          = (aβ − bα)x4 + 2(aγ − cα)x3 y + {3(bγ − cβ) + (aδ − dα)}x2 y 2
             + 2(bδ − dβ)xy 3 + (cδ − dγ)y 4 ,
     Q1 = Ω2 = u21 y 4 − 4u1 u2 y 3 x + (4u22 + 2u1 u3 )y 2 x2 − 4u2 u3 yx3 + u23 x4 .
Supplying for facility of computation the reciprocals of the binomial coefficients
to the index 4, namely
                                  1    1      1
                            1, − ,       , − , 1,
                                  4    6      4
we obtain
                                                       2
                                                                                 
    G1 = (aβ − bα)u21 + 2(aγ − cα)u1 u2 + 2(bγ − cβ) + (aδ − dα) u22
                                                       3
                         1
         + {(bγ − cβ) + (aδ − dα)}u1 u3 + 2(bδ − dβ)u2 u3 + (cδ − dγ)u23 .
                         3
It will here and henceforth be more useful to employ [r, s] to denote, not the
difference of the cross products of the (r + 1)th and (s + 1)th entire            p. 564
   coefficients divided each by its appropriate binomial coefficient. We may then
write
     G1 = [0, 1]u21 + 2[0, 2]u1 u2 + ([1, 2] + 31 [0, 3])u1 u3 + (2[1, 2] + 23 [0, 3])u22
           + 2[1, 3]u2 u3 + [2, 3]u23 .

                                            570
Again,

G3 = {(aδ−dα)−3(bγ−cβ)}(u1 u3 −u22 ) = ([0, 3]−3[1, 2])u1 u3 −([0, 3]−3[1, 2])u22 .

Hence
    1
G1 − G3 = [0, 1]u21 +2[0, 2]u1 u2 +2[1, 2]u1 u3 +([0, 3]+[1, 2])u22 +2[1, 3]u2 u3 +[2, 3]u23 .
    3
But, again, the Bezoutiant of f, ϕ corresponds to the matrix

                            3[0, 1],      3[0, 2],     [0, 3],
                            3[0, 2], [0, 3] + 9[1, 2], 3[1, 3],
                             [0, 3],      3[1, 3],     [2, 3].

Hence summing the sinister bands to form the coefficients, we have

B = 3[0, 1]u21 +6[0, 2]u1 u2 +(3[0, 3]+9[1, 2])u22 +6[1, 3]u2 u3 +[2, 3]u23 = 3G1 −G3 .

Thirdly. Suppose m = 4,

                      f = ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 ,

                     ϕ = αx4 + 4βx3 y + 6γx2 y 2 + 4δxy 3 + ϵy 4 ,
                        Ω = u1 y 3 − 3u2 y 2 x + 3u3 yx2 − u4 x3 .
Then

        E3 f = (ax + by)ξ 3 + 3(bx + cy)ξ 2 η + 3(cx + dy)ξη 2 + (dx + ey)η 3 ,

therefore
          T3 = {(ax + by)(δx + ϵy) − (αx + βy)(dx + ey)}
               − 3{(bx + cy)(γx + δy) − (βx + γy)(cx + dy)}
             = ([0, 3] − 3[1, 2])x2 + ([0, 4] − 2[1, 3])xy + ([1, 4] − 3[2, 3])y 2

and
             Q3 = (u1 y − u2 x)(u3 y − u4 x) − (u2 y − u3 x)2
                 = (u1 u3 − u22 )y 2 − (u1 u4 − u2 u3 )xy + (u2 u4 − u23 )x2 .
Hence supplying the binomial reciprocals
                                            1
                                      1,   − ,     1,
                                            2
we have
                                   1
G3 = ([0, 3]−3[1, 2])(u1 u3 −u22 )+ ([0, 4]−2[1, 3])(u1 u4 −u2 u3 )+([1, 4]−3[2, 3])(u2 u4 −u23 ).
                                   2
                                                                                            p. 565


                                           571
  Again,

      T1 = (ax3 + 3bx2 y + 3cxy 2 + dy 3 )(βx3 + 3γx2 y + 3δxy 2 + ϵy 3 )
            − (αx3 + 3βx2 y + 3γxy 2 + δy 3 )(bx3 + 3cx2 y + 3dxy 2 + ey 3 )
          = [0, 1]x6 + 3[0, 2]x5 y + (3[0, 3] + 6[1, 2])x4 y 2 + ([0, 4] + 8[1, 3])x3 y 3
            + (3[1, 4] + 6[2, 3])x2 y 4 + 3[2, 4]xy 5 + [3, 4]y 6 ,

and
  Q1 = Ω2 = u21 y 6 − 6u1 u2 y 5 x + (9u22 + 6u1 u3 )y 4 x2 − (2u1 u4 + 18u2 u3 )x3 y 3
                 + (9u23 + 6u2 u4 )y 2 x4 − 6u3 u4 yx5 + u24 x6 .

Hence, supplying the reciprocal binomial coefficients,
                              1        1             1      1       1
                        1,   − ,         ,       −      ,      ,   − ,    1,
                              6       15             20     15      6
we find
                                                     1         2
                                                                    
          G1 = [0, 1]u21 + 3[0, 2]u1 u2 +              [0, 3] + [1, 2] (9u22 + 6u1 u3 )
                                                     5         5
                    1            8
                                            
                +      [0, 4] + [1, 3] (u1 u4 + 9u2 u3 )
                    20           20
                    1          2
                                    
                +     [1, 4] + [2, 3] (9u23 + 6u2 u4 ) + 3[2, 4]u3 u4 + [3, 4]u24 .
                    5          5
Now the Bezoutic square, taking account of the binomial factors in f and ϕ, may
be written under the form
                         4[0, 1],  6[0, 2],   4[0, 3],    [0, 4],
                                   4[0, 3]     [0, 4]
                         6[0, 2],           ,          , 4[1, 4],
                                  +24[1, 2] +16[1, 3]
                                   [0, 4]     4[1, 4]
                         4[0, 3],           ,          , 6[2, 4],
                                  +16[1, 3] +24[2, 3]
                         [0, 4],   4[1, 4],   6[2, 4],    [3, 4].

Hence the Bezoutiant B becomes
          B = 4[0, 1]u21 + 12[0, 2]u1 u2 + (4[0, 3] + 24[1, 2])u22 + 2[0, 4]u1 u4
                + (2[0, 4] + 32[1, 3])u2 u3 + 8[1, 4]u2 u4 + ([1, 4] + 24[2, 3])u23
                + 12[2, 4]u3 u4 + [3, 4]u24 .

And we ought to have B = cG1 + eG3 , to satisfy which equation we must
manifestly have c = 4; to find e, compare the coefficients of u22 ; this gives
                                      36         72
               4[0, 3] + 24[1, 2] =      [0, 3] + [1, 2] + e(3[1, 2] − [0, 3]);
                                      5          5

                                                 572
accordingly we ought to be able to satisfy the two equations
                                 36                72
                                    − e = 4,          + 3e = 24,
                                 5                  5
each of which accordingly we find is satisfied by the equality e = 16
                                                                   5 . Substituting
in the equation for B above written, we thus obtain
                                                     16
                                       B = 4G1 +        G3 ,
                                                      5
which will be found to be identically true.                                      p. 566
   Art. 70. We may now see our way to a more concise mode of obtaining the
numerical coefficients, by which they may in fact be computed and verified with
comparatively little labour, connecting the Bezoutiant with the Co-bezoutiant
forms of the constituent scale. It will not fail to have been remarked, that
throughout the preceding determinations I have presumed the truth of the
formula, which admits of an immediate verification, that for all values of m and
ω we have the identical equation

                                              1
                   ω 
         d     d
                                                                                                    
    ξ      +η             c0 xm + mc1 xm−1 y + m(m − 1)c2 xm−2 y 2 + · · · + mcm−1 xy m−1 + cm y m
        dx    dy                              2
                                             1
                                                                                         
= m(m−1) · · · (m−ω+1) L0 ξ ω + ωL1 ξ ω−1 η + ω(ω − 1)L2 ξ ω−2 η 2 + · · · + Lω η ω ,
                                             2
where
                                            m−ω−1
L0 = c0 xm−ω + (m − ω)c1 xm−ω−1 y + (m − ω)        c2 xm−ω−2 y 2 + · · · + cm−ω y m ,
                                               2
                                            m−ω−1
L1 = c1 xm−ω + (m − ω)c2 xm−ω−1 y + (m − ω)        c3 xm−ω−2 y 2 + · · · + cm−ω+1 y m ,
                                               2
   ···························
                                              m−ω−1
Lω = cω xm−ω + (m − ω)cω+1 xm−ω−1 y + (m − ω)          cω+2 xm−ω−2 y 2 + · · · + cm y m .
                                                 2
Let us now proceed to determine by an abridged method the linear relations
corresponding to the cases of m = 5, m = 6, and first for m = 5. Let

                   f = ax5 + 5bx4 y + 10cx3 y 2 + 10dx2 y 3 + 5exy 4 + hy 5 ,

                   ϕ = αx5 + 5βx4 y + 10γx3 y 2 + 10δx2 y 3 + 5ϵxy 4 + ηy 5 ,
                      Ω = u1 y 4 − 4u2 y 3 x + 6u3 y 2 x2 − 4u4 yx3 + u5 x4 .
In forming G5 , G3 , G1 let us confine our attention to the terms u21 , u1 u3 , u1 u4 .
A comparison of the coefficients of these with those in the Bezoutiant (B) will
be sufficient for assigning the three numerical quantities which connect B with


                                               573
G1 , G3 , G5 . I omit u1 u2 , because G1 is the only one of the G’s for any value of
m which contains u21 or u1 u2 , and in G1 the terms containing u21 and u1 u2 are

                              [0, 1]u21 + (m − 1)[0, 2]u1 u2 ,

and the corresponding part of the Bezoutiant is

                           m[0, 1]u21 + m(m − 1)[0, 2]u1 u2 ;

so that if we write
                           B = c1 G1 + c3 G3 + c5 G5 + &c.,
the two terms u21 and u1 u2 will only enable us to form one equation with the c’s,
namely, c1 = m. Again, instead of considering the entire coefficients              p. 567
    of u1 u3 and u1 u4 , it will be sufficient to take a single argument of either
of these coefficients (in the forms to be compared), as for instance [0, 3] and
[1, 3]. Then c1 being known, c3 , c5 will be determined; but for the purposes of
verification I shall furthermore compute the whole of the coefficient of u1 u5 .
    Accordingly, calculating the G system in reverse order, we have

               G5 = {[0, 5] − 5[1, 4] + 10[2, 3]}{u1 u5 − 4u2 u4 + 3u23 }
                   = {[0, 5] − 5[1, 4] + 10[2, 3]}u1 u5 + · · · ,

E3 f = (ax2 +2bxy+cy 2 )ξ 3 +3(bx2 +2cxy+dy 2 )ξ 2 η+3(cx2 +2dxy+ey 2 )ξη 2 +(dx2 +2exy+f y 2
                                       E3 ϕ = &c. &c.;
therefore
T3 = {(ax2 + 2bxy + cy 2 )(δx2 + 2ϵxy + ηy 2 ) − (αx2 + 2βxy + γy 2 )(dx2 + 2exy + hy 2 )}
     − 3{(bx2 + 2cxy + dy 2 )(γx2 + 2δxy + ϵy 2 ) − (βx2 + 2γxy + δy 2 )(cx2 + 2dxy + ey 2 )}
   = ([0, 3] − 3[1, 2])x4 + (2[0, 4] + · · · )x3 y + {[0, 5] + [1, 4] − 8[2, 3]}x2 y 2 + &c.

The number −8 results from the calculation 1 − 3(4 − 1) = −8. Again,

E2 Ω = (u1 y 2 −2u2 yx+u3 x2 )ξ 2 −2(u2 y 2 −2u3 yx+u4 x2 )ξη+(u3 y 2 −2u4 yx+u5 x2 )η 2 ,

therefore
                Q3 = (u1 y 2 − 2u2 yx + u3 x2 )(u3 y 2 − 2u4 yx + u5 x2 )
                      − (u2 y 2 − 2u3 yx + u4 x2 )2
                    = u1 u3 y 4 − 2u1 u4 y 3 x + u1 u5 y 2 x2 + &c.,
all the terms and parts of terms unexpressed being free of u1 , and therefore not
necessary for our purpose. Hence supplying the reciprocal factors
                                         1     1
                                  1,    − ,      ,    ··· ,
                                         4     6

                                            574
we have
                                               1
    G3 = [0, 3]u1 u3 + ([0, 4] + · · · )u1 u4 + {[0, 5] + [1, 4] + [2, 3]}u1 u5 + &c.
                                               6
Again, expressing E1 f and E1 ϕ in the usual way,

T1 = (ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 )(βx4 + 4γx3 y + 6δx2 y 2 + 4ϵxy 3 + ηy 4 )
      − (αx4 + 4βx3 y + 6γx2 y 2 + 4δxy 3 + ϵy 4 )(bx4 + 4cx3 y + 6dx2 y 2 + 4exy 3 + hy 4 )
   = [0, 1]x8 + 4[0, 2]x7 y + (6[0, 3] + · · · )x6 y 2 + (4[0, 4] + · · · )x5 y 3
      + ([0, 5] + 15[1, 4] + 20[2, 3])x4 y 4 + &c.

(where it may be observed that the numbers 15 and 20 in the coefficient of x4 y 4
arise from the quantities 42 − 1, 62 − 42 ).                                      p. 568
   Again,

  Q1 = Ω2 = u21 x8 + 8u1 u2 x7 y + 12u1 u3 x6 y 2 − 8u1 u4 x5 y 3 + 2u1 u5 x4 y 4 + &c.

Hence supplying the multipliers
                                  1      1            1      1
                           1,    − ,        ,    −       ,      ,   &c.
                                  8      28           56     70
we have
                                 18              4             1
G1 = [0, 1]u21 +4[0, 2]u1 u2 +      [0, 3]u1 u3 + [0, 4]u1 u4 + {[0, 5]+15[1, 4]+20[2, 3]}u1 u5 .
                                 7               7             35
Again, the Bezoutiant

B = 5[0, 1]u21 + 2 · 10[0, 2]u1 u2 + 2 · 10[0, 3]u1 u3 + 2 · 5[0, 4]u1 u4 + 2[0, 5]u1 u5 + &c.

Accordingly, if we write B = c1 G1 + c3 G3 + c5 G5 , we have, as above remarked,
c1 = 5; and to determine c3 , c5 , we have, by comparing the coefficients of
u1 u3 , u1 u4 in B, G1 , G3 , G5 ,
                                   90                        20
                            20 =      + c3 ,          10 =      + c3 .
                                    7                        7
These two equations, then, as it turns out, are not independent, but are satisfied
simultaneously by
                                           50
                                      c3 = .
                                            7
Finally, equating the coefficients of the several arguments in u1 u5 , we have
                          1    50 1
               0=5×          +   × + c5           from the argument [0, 5],
                          35   7  6
                         15 50 1
              0=5×          +   × + 5c5               from the argument [1, 4],
                         35   7  6

                                                575
                     20 50 1
              0=5×      +    × + 10c5              from the argument [2, 3].
                     35    7    6
The first of which equations gives
                                            1 25   14  2
                               c5 = 2 −      −   =    = ;
                                            7 21   21  3
the second gives
                                            7 25  2
                                    c5 =     +   = ,
                                            3 21  3
and the third gives
                                 20 2      2
                                    + = .
                                    c5 =
                                 21 3      3
We have thus abundantly verified the accuracy of the calculation, and there
results the relation
                                    50      2
                        B = 5G1 + G3 + G5 .
                                     7      3
Lastly, let m = 6,

         f = ax6 + 6bx5 y + 15cx4 y 2 + 20dx3 y 3 + 15ex2 y 4 + 6hxy 5 + ly 6 ,

         ϕ = αx6 + 6βx5 y + 15γx4 y 2 + 20δx3 y 3 + 15ϵx2 y 4 + 6ηxy 5 + λy 6 ,
           Ω = u1 y 5 − 5u2 y 4 x + 10u3 y 3 x2 − 10u4 y 2 x3 + 5u5 yx4 − u6 x5 .
                                                                                                  p. 569
   I shall here confine myself to the determination of a single argument in each
of the terms u21 , u1 u2 , u1 u3 , u1 u4 , u1 u5 , u1 u6 ; this will be ample for the purpose
of verification, as the equation to be assigned is of the form

                                B = c1 G1 + c3 G3 + c5 G5 .

The arguments which I select as the most simple, will be those expressed by the
symbols (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0, 6) respectively; then we have

T5 = (ax+by)(ηx+λy)∓&c.−(hx+ly)(αx+βy) = ([0, 5]+· · · )x2 +([0, 6]+· · · )xy+(· · · )y 2 ,

Q5 = (u1 y − u2 x)(u5 y − u6 x) ∓ &c. = (u1 u5 + · · · )y 2 − (u1 u6 + · · · )yx + (· · · )x2 .
Hence supplying the binomial reciprocals
                                              1
                                       1,    − ,     1,
                                              2
                                              1
                 G5 = ([0, 5] + · · · )u1 u5 + ([0, 6] + · · · )u1 u6 + &c.
                                              2
Again,

T3 = (ax3 +· · · )(δx3 +3ϵx2 y+3ηxy 2 +λy 3 )∓&c.−(dx3 +3ex2 y+3hxy 2 +ly 3 )(αx3 +· · · )

                                             576
= ([0, 3] + · · · )x6 + (3[0, 4] + · · · )x5 y + (3[0, 5] + · · · )x4 y 2 + ([0, 6] + · · · )x3 y 3 + &c.
Q3 = (u1 y 3 + &c.)(u3 y 3 + 3u4 y 2 x + 3u5 yx2 − u6 x3 ) ∓ &c.
    = (u1 u3 + · · · )y 6 − (3u1 u4 + · · · )y 5 x + (3u1 u5 + · · · )y 4 x2 − (u1 u6 + · · · )y 3 x3 + &c.,
and the reciprocal binomial multipliers will be
                                         1        1           1
                                 1,     − ,          ,    −      ,   &c.
                                         6        15          20
Hence
                         3             3             1
       G3 = [0, 3]u1 u3 + [0, 4]u1 u4 + [0, 5]u1 u5 + [0, 6]u1 u6 + &c. &c.
                         2             5             20
Finally,

   T1 = (ax5 + &c.)(βx5 + 5γx4 y + 10δx3 y 2 + 10ϵx2 y 3 + 5ηxy 4 + λy 5 ) − &c.

      = ([0, 1] + · · · )x10 + 5([0, 2] + · · · )x9 y + (10[0, 3] + · · · )x8 y 2
         + (10[0, 4] + · · · )x7 y 3 + (5[0, 5] + · · · )x6 y 4 + ([0, 6] + · · · )x5 y 5 + &c.
Q1 = Ω2 = u21 y 10 +(10u1 u2 +· · · )y 9 x+(20u1 u3 +· · · )y 8 x2 +(20u1 u4 +· · · )y 7 x3 +(10u1 u5 +· · ·
                                                                                                            p. 570
   and supplying the numerical series
                              1       1            1         1            1
                    1,    −      ,       ,    −       ,         ,    −       ,   &c.,
                              10      45          120       210          252
we have
                                     40              5             5               1
G1 = [0, 1]u21 +5[0, 2]u1 u2 +          [0, 3]u1 u3 + [0, 4]u1 u4 + [0, 5]u1 u5 +     [0, 6]u1 u6 +&c.
                                     9               3             21             126
Again, the Bezoutiant

= 6[0, 1]u21 +30[0, 2]u1 u2 +40[0, 3]u1 u3 +30[0, 4]u1 u4 +12[0, 5]u1 u5 +2[0, 6]u1 u6 +&c. &c. = B

Hence making
                                     B = c1 G1 + c3 G3 + c5 G5 ,
from u21 and u1 u2 we obtain respectively

                                                  c1 = 6,

                                              5c1 = 30;
hence from u1 u3 and u1 u4 we obtain respectively
                    240                       30 3                40
                        + c3 = 40,               + c3 = 30 or c3 = ;
                     9                         3  2                3

                                                   577
hence from u1 u5 and u1 u6 we obtain respectively
               5    40 3                                                  10  18
          6×      +   × + c5 = 12,           that is      c5 = 12 − 8 −      = ,
               21   3  5                                                   7  7
              1    40   1  1                               1         2  1  9
        6×       +    ×   + c5 = 2,             that is      c5 = 2 − −   = ;
             126    3   20 2                               2         3 21  7
hence
                                                18
                                         c5 =      ,
                                                7
and the equation sought for is
                                             40     18
                               B = 6G1 +        G3 + G5 .
                                             3      7
  Art. 71. The following table exhibits the relations between the Bezoutiant
and the correspondent system of Co-bezoutiants for all values of m between 1
and 6 under a synoptical form.

                          m = 1,     B = G1 ,
                          m = 2,     B = 2G1 ,
                          m = 3,     B = 3G1 − G3 ,
                          m = 4,     B = 4G1 + 16
                                               5 G3 ,
                          m = 5,     B = 5G1 + 7 G3 + 32 G5 ,
                                               50

                          m = 6,     B = 6G1 + 40
                                               3 G3 + 7 G 5 .
                                                       18

                                                                                                 p. 571
  These series could if wanted be easily extended, and the calculation of the
coefficients reduced to a mere mechanical procedure.
  If we suppose m to be 2i or 2i − 1, we have the equation

                         B = c1 G1 + c3 G3 + · · · + c2i−1 G2i−1 ,

and it appears from the foregoing instances that the comparison of the coefficients,
either of u21 , or of u1 u2 on the two sides of the equation, will serve to give c1
(m being known), c3 may be found by a comparison of the coefficients either of
u1 u3 , or of u1 u4 , and so on for c5 . . . c2i−1 ; all the coefficients in the equation
for B above given, thus admitting of being found separately and successively
and in two modes, so that there is a check at each step upon the correctness
of the computations: the only exception to this last remark is (when m is
odd) for the last coefficient of which the above condensed method affords only
a single determination. I need hardly add the remark, that in substituting
xm−1 , xm−2 y, . . . , xy m−2 , y m−1 in place of u1 , u2 . . . um , respectively, all the G’s
become (to a numerical factor près) identical with one another and with the
Jacobian to the system (f, ϕ).


                                            578
   Art. 72. The foregoing theory took its origin (as will have been readily
imagined) in meditations growing out of the celebrated theorem of M. Sturm.
There appear to be several directions in which a development or extension of
the subject matter of that theorem may be sought for. Thus a theory may be
constructed relative to a single function of one or more variables, viewed in
all cases as representing a geometrical locus. In the limiting case, when this
locus becomes a system of points in a right line, we have the theorem of Sturm;
generally the theory will be that of contours. Or, again, a theory may be formed
in which the number of functions is always kept equal to that of the variables.
We have then a theory of discrete points corresponding to roots, the number of
real ones of which comprised within given limits it is the object of such theory to
determine. M. Hermite, in a memoir recently presented to the French Institute,
appears to have made a valuable addition to the Sturmian theory extended in this
direction, to which the beautiful researches of M. Cauchy and the joint labours
of MM. Liouville and Sturm, with reference to the disposition of the imaginary
roots of equations appear to have led the way. Finally, the number of variables
may be supposed to be arbitrarily increased, but made always inferior by a unit
to the number of the functions in which they are contained, or which comes
to the same thing, we may construct the theory of a system of homogeneous
functions equal in number to the variables in them, which in its simplest case
becomes the theory of Intercalations which has been here partially considered,
and which (as has been shown) embraces (not as a particular case, but as an
implied consequence and easily extricated result) the theorem of M. Sturm.          p. 572


                General and Concluding Supplement.

Art. (r).     The expressions given in Art. (n) [p. 507 above] for the partial
quotients of the continued fraction represented by ϕx   f x , are restricted to the
supposition of all these partial quotients (except the first) being linear in x;
when the first partial quotient is linear the formula (B) of that article continues
applicable on replacing (Di hθ ) by 1. I was forcibly struck by the peculiarity of
these formulae not ceasing to be true in consequence of the first partial quotient
being supposed non-linear; and reflecting upon this, I was soon led to perceive
that all the partial quotients might be supposed to be arbitrary integral functions
of x, and the formulae would still continue to apply to any such of them as might
happen to be linear, although, as it were, imbedded among a group of other
non-linear partial quotients. From this it was but an easy step to perceive that
the formulae (A) and (B) must admit of extension to the representation of partial
quotients of any form, and that the dimorphism of the representation of the
linear partial quotients could only be a consequence of the equation in integers
u + v = 1 having two solutions u = 0, v = 1 and u = 1, v = 0. I now proceed
to enunciate the very remarkable general theorem (or as it may perhaps not
inappropriately be termed Algebraical Porism), by virtue of which any partial

                                        579
quotient of a given degree in x belonging to an infinite continued fraction, all
of whose partial quotients are algebraical functions of x, may be expressed to
a constant factor près, by means of the numerator and denominator (or if we
please either one of these) of the convergent immediately antecedent to and of
the numerator and denominator of any convergent not antecedent to the partial
quotient which is to be determined.
Art. (s).    Theorem. Let Q1 , Q2 . . . Qi , Qi+1 . . . Qn , &c. each of an arbitrary
degree in x, be the n first partial quotients of an algebraical continued fraction;
let Qi+1 be the partial quotient to be determined and of the given degree ωi+1 ;
let
                                     1              ϕi (x)
                                                =          ,
                           Q1 −        1
                                         1          fi (x)
                                        Q2 −
                                                 Q3 −···− 1
                                                         Qi

and
                                            1                               Φ(x)
                                                                        =         ;
                       Q1 − Q −                    1
                                                     1                      F (x)
                                 2                      1
                                      Q3 −···−
                                                 Qi − 1 −···− 1
                                                     Qi+1    Qn

let u and v be any couple of integers of the ωi+1 + 1 couples which satisfy the
equation v + u = ωi+1 ; then, as usual, denoting the products of the differences
of each of one set of terms from each of another set, by writing the former under
the latter, and calling η1 , η2 . . . ηµ the µ roots of Φ(x), and h1 , h2 . . . hm p. 573
   the m roots of F (x), (Φ and F being supposed respectively of µ and m
dimensions in x), and forming the disjunctive equations

                                θ1 , θ2 , θ3 . . . θν = 1, 2, 3 . . . µ,

                               t1 , t2 , t3 . . . tm = 1, 2, 3 . . . m,
we have the following equation, wherein ϕ and f are written for ϕi and fi ,
                   (
Qi+1 = Ku,v × Σ {(ϕηθ1 ϕηθ2 . . . ϕηθv )2 × (f ht1 f ht2 . . . f htu )2 }
                          "                                     #       "                                  #
                               ηθ1      ηθ2         . . . ηθv                ht1       ht2    . . . ht u
                                                                    ×
                              htu+1    htu+2        . . . htm               ηθv+1     ηθv+2   . . . ηθµ
                       ×"                                       #       "                                  #
                               ηθ1      ηθ2         . . . ηθv                ht1       ht2    . . . htu
                                                                    ×
                              ηθv+1    ηθv+2        . . . ηθµ               htu+1     htu+2   . . . htm
                                                                                                               )
                       × {(x − ηθ1 )(x − ηθ2 ) . . . (x − ηθv )}{(x − ht1 )(x − ht2 ) . . . (x − htu )} ,

and moreover the different values of Ku,v depending upon the different modes of
breaking up ωi+1 into two parts u and v are all (to a numerical factor près) equal
to one another. Thus then the theorem pointed at in Art. (p) is discovered, and

                                                    580
the way laid open (by an unexpected channel) for a complete discussion of the
theory of the singular cases which may occur in the expansion of any rational
algebraical fraction under the form of a continued fraction.
Art. (t). In the above expression, if we suppose ωi+1 = 1, we have u = 1 and
v = 0, or u = 0 and v = 1, and remembering that
         "                            #                              "                           #
              h                                                            η
                                          = Φh          and                                           = F η,
             η1 , η2 . . . ηµ                                             h1 , h2 . . . hm
   "                              #                                  "                                #
       ht1                                     ′                          ηθ1
                                      = F ht1           and                                               = Φ′ ηθ1 ,
       ht2 , ht3      . . . htm                                           ηθ2 , ηθ3       . . . ηθµ
Qi+1 becomes by virtue of the general formula representable under either of the
equivalent forms
                           F ηθ                                                              Φht
                                                                                                          
       K0,1 Σ       (ϕηθ )2 ′ (x − ηθ )                 and          K1,0 Σ          (f ht )2 ′ (x − ht )        ,
                           Φ ηθ                                                              F ht
K0,1 and K1,0 being either equal, or differing only in the sign, agreeably to the
formulae (A) and (B) [p. 508 above].
Art. (u).       It may be worth while to notice, that, although (of course) these
formulae and the general formula of Art. (s), when supposed converted into
functions of x and of the coefficients of F and of Φ by the reduction, integration
and summation of the symmetrical functions of the roots which enter into them
remain universally valid, and subject to no cases of exception,                      p. 574
   yet antecedently to these processes being performed the formulae as they
stand may become illusory when any relations of equality exist between the roots
of Φ inter se, or between the roots of F inter se. Thus in the case before us, if Φ
have equal roots the formula commencing with K0,1 is illusory, and if F have
equal roots the other of the two formulae becomes illusory.
   Let us take the second of these and suppose that F (x) has k1 roots c1 , k2
roots c2 . . . kp roots cp , we may pass to the actual case from any case where the
roots are infinitesimally near to the actual roots of F (x), and all infinitesimally
different from one another. Moreover the choice of the infinitesimal variations
being arbitrary, let the k1 roots c1 be replaced by a group of roots
                       c1 + δ,       c1 + δρ1 ,            c1 + δρ21 . . . c1 + δρk11 −1 ,
where ρ1 is a prime root of the equation ρk11 = 0, and δ is an infinitesimal
quantity, and suppose each of the other groups to be varied in an analogous
manner. Then it may easily be shown from this that the second of the formulae
in question will become
                                                   k−1
                               p
                               X               d
                                              dct          {(f ct )2 (Φct )(x − ct )}
                         K10         kt                             k                      ,
                                                                d
                               t=1
                                                               dct        F ct

                                                           581
and similarly, the twin formula becomes
                                                κ−1
                            π
                            X               d
                                           dγθ          {(ϕγθ )2 (F ′ γθ )(x − γθ )}
                      K01         κθ                              κ                           .271
                            θ=1
                                                              d
                                                             dγθ        Φγθ

 Corresponding modifications will admit of being made by aid of a like method
in the general formula of Art. (s) upon a similar supposition as to equalities
springing up between the roots of f x per se and of ϕ(x) per se, or between the
roots of f x and ϕx inter se.                                                   p. 575

Art. (v).        If in Art. (s) we take i = 0, the formula for Qi+1 will become
                      "                                            #        "                                      #
                           ηθ1         ηθ2         . . . ηθv                     ht1         ht2       . . . htu
                                                                        ×
                          htu+1       htu+2        . . . htm                    ηθv+1       ηθv+2      . . . ηθµ
       Q1 = Ku,v "                                                 #       "                                       #
                           ηθ1         ηθ2         . . . ηθv                     ht1         ht2    . . . htu
                                                                       ×
                          ηθv+1       ηθv+2        . . . ηθµ                    htu+1       htu+2   . . . htm

                    ×{(x − ηθ1 ) · · · (x − ηθv )}{(x − ht1 ) · · · (x − htu )},
u and v being any two integers whose sum is ω1 , which is identical (as it ought
to be) with the expression virtually contained in the formulae of Section II. for
the syzygetic multiplier of Φ(x) in the syzygetic equation connecting F x and Φx
with their first residue when Φx is supposed to be ω1 dimensions in x lower than
F x identical, videlicet, in other words, with the integer part of the algebraical
fraction
                                       F (x)
                                             .
                                       Φ(x)

Art. (w).        When Φ(x) = F ′ (x),

                                        Φ(h1 ), Φ(h2 ) . . . Φ(hωi+1 )
                                  "                                                     #
                                        h1               h2             . . . hωi+1
                                      h1+ωi+1        h2+ωi+1            . . . hm
 271
    For in general if ρ is a prime root of the equation ρω = 1, and if f x have ω roots all equal
to c and ψx is any other function of x and if δ is an infinitesimal quantity, then rejecting all
powers of δ higher than the (ω − 1)th degree,

                    ψ(c + δ)      ψ(c + ρδ)      ψ(c + ρ2 δ)              ψ(c + ρω−1 δ)
                                +              +                + · · · +
                    f ′ (c + δ)   f ′ (c + ρδ)   f ′ (c + ρ2 δ)           f ′ (c + ρω−1 δ)
               1
       =   d ω
                        {ψ(c + δ) + ρψ(c + ρδ) + ρ2 ψ(c + ρ2 δ) + · · · + ρω−1 ψ(c + ρω−1 δ)}
           dc
               f c δ ω−1
                                       d ω−1                                d ω−1
                                                                               
                                              ψc ωδ ω−1                            ψc
                                =      dc
                                          d ω
                                                                  =ω       dc
                                                                              d ω
                                                                                     .
                                          dc
                                              f c δ ω−1                      dc
                                                                                  fc



                                                          582
becomes identical with
                                1
                            (−) 2 ωi+1 (ωi+1 −1) ωi+1 ζ(h1 , h2 . . . hωi+1 ),

and we may consequently (using an extreme term in the forms in the polymorphic
scale of forms representing Qi+1 ), write
                1
Qi+1 = (−) 2 ωi+1 (ωi+1 −1) ωi+1 K0,ωi+1 Σ{ζ(h1 , h2 . . . hωi+1 )(fi h1 )2 (fi h2 )2 · · ·
                                                      × (fi hωi+1 )2 (x − h1 )(x − h2 ) . . . (x − hωi+1 )}.

Art. (x). The following observations will serve to complete the theory of the
singular cases in the expansion of an algebraical continued fraction.
   Preserving the notation of Art. (s), let

                             σi = m − (ω1 + ω2 + · · · + ωi−1 + 1).

Then (calling the roots of F x, h1 , h2 . . . hm ) the (i)th simplified residue to FΦxx ,
in accordance with the general formulae for the residues in the second section
(for greater simplicity selecting an extreme term of the polymorphic scale), will
be represented by
             Φh1 , Φh2 , Φh3 . . . Φhσi
   Σ"                                               # (x − h1 )(x − h2 )(x − h3 ) . . . (x − hσi ),
           h1        h2         h3      . . . hσi
         h1+σi      h2+σi     h3+σi     . . . hm

which will be of the form Li xσi − ωi + 1 + &c., all the terms containing higher
powers of x vanishing by the coefficients becoming zero. If in the above expression
we should use σi′ in lieu of σi , where σi′ is σi diminished by any integer inferior
to ωi , we should get other forms of the same residue, but                                p. 576
    these will all be of higher dimensions in the roots or coefficients than the
one just given, and in fact the forms thus obtained corresponding to the values
σi , σi − 1, σi − 2 . . . σi − ωi + 1 substituted for σi′ in succession, would, by aid of
the relations of condition between the coefficients of Φx and F x implied in the
value of ωi , admit of being exhibited as a scale in which each form would be an
exact algebraical product of the form which precedes it, multiplied by a function
of the coefficients, and did space permit thereof it would be perfectly easy to give
the forms of these multiplicators. But I pass on to the representation of what is
more material, namely, the form of the complete residue in the case supposed,
merely observing (as an obiter dictum) that the existence of each singular partial
quotient (meaning thereby a quotient non-linear in x) only affects the form of
the single simplified residue in immediate connexion with itself, and not at all
the form of the other residues antecedent or subsequent to that one.
Art. (y).  Let the ith simplified residue be called Ri and the corresponding
complete residue [Ri ], then applying a method similar to the method given in

                                                     583
Section I., we shall find that
                                                ω +ω            ω +ω
                                             i
                                           Li−2 i+1   i
                                                    Li−4 i+1
                                                             &c.
                           (−)i [Ri ] =         ω +ω            ω +ω                Ri ,
                                             i
                                           Li−1 i+1   i
                                                    Li−3 i+1
                                                             &c.

Li representing the leading coefficient in the ith simplified residue, and the sign
of interrogation (?) denoting some function of ω1 , ω2 . . . ωi (possibly a constant)
remaining to be determined. And reverting to Art. (s), the quantity that would
be called K0,ωi according to the notation employed in the formulae expressing
Qi+1 in that article, will (abstraction being made of the algebraical sign and
using for greater brevity (ι), (ι − 1), &c. to express 1 + ωi , 1 + ωi−1 , &c.) come to
be represented by
                                  (ι−1) (ι−3) (ι−5)
                               Li−1 Li−3 Li−5 &c.
                                         (ι) (ι−2) (ι−4)
                                     Li Li−2 Li−4 &c.
a similar convention being supposed to be made respecting the numerator and
denominator of each convergent as was made respecting them in the particular
case treated of in Art. (f ), page [502].
Art. (z). I will merely add a very few words in generalization of the method of
limiting the roots of f x given in the Supplement to the fourth Section [p. 528
above]. As an inferior limit to f x is identical with a superior limit to f (−x), we
may confine our attention to superior limits alone. Suppose then that
              ϕx                                            1
                 =                                               1                                   .
              fx   Q1 −                                                1
                             Q2 −···−                                      1
                                        Qi −
                                               Q′ −                            1
                                                1 Q′ −···−                          1
                                                    2      Q′µ −···−                    1
                                                                                           1
                                                                           (Q)1 −
                                                                                    (Q)2 −···− 1
                                                                                              (Q)ν

                                                                                                         p. 577
    where the partial quotients Q are each of any arbitrary degree in x, and
have all one algebraical sign in the coefficients of the highest powers of x
from Q1 to Qi , and all the same sign (contrary to the former), in the coef-
ficients of the highest powers of x from Q′i to Q′′i , and so on alternately, then
firstly a superior limit to the superior limits of the cumulants [Q1 , Q2 . . . Qi ],
[Q′1 , Q′2 . . . Q′i ], . . . [(Q)1 , (Q)2 . . . (Q)i ] will be a superior limit to f x, so that it re-
mains only to give a rule for finding a superior limit to a cumulant [Q1 , Q2 , Q3 . . . Qi ],
which, secondly, is to be found by making

             Q1 − M1 = 0, Q2 − M2 = 0, Q3 − M3 = 0 . . . Qi − Mi = 0,

where
                                               1                               1              1
            M 1 = µ1 ,      M2 = µ2 +             ,    M3 = µ3 +                  . . . Mi =      ,
                                               µ1                              µ2            µi−1


                                                      584
µ1 , µ2 . . . µi−1 being any quantities entirely independent and arbitrary except
in regard to their being all of the same sign as the leading coefficients in the
elements Q1 , Q2 . . . Qi .
    We may then find L1 , L2 . . . Li any superior limits to the roots of x in these i
equations respectively; L, the greatest of these, will be a superior limit to the
proposed cumulant [Q1 , Q2 . . . Qi ]; and it may be observed that M1 , M2 . . . Mi
are the general values which satisfy the equation

                                           1
                         M1 −                               = 0,
                                               1
                                M2 −
                                                       1
                                       M3 − · · · −
                                                       Mi
subject to the condition that for all values of e
                                          1
                                               1
                         Me −
                                                   1
                                Me−1 −
                                                            1
                                         Me−2 − · · · −
                                                            M1
shall have a given invariable sign. The first part of the process, as just shown,
consists in separating the type of the total cumulant which represents f x into
partial types, the point for each fracture of the total type being marked by a
change of sign in the elements of the type for the value x = +∞; it is easily seen
therefore from this, that if FΦxx is the generatrix of the cumulant in question, the
number of such fractures (that is, the number one less than the number of partial
cumulants) will be the number of changes of algebraical sign in the signaletic
series, consisting of the leading coefficients in F x and in each of the odd-placed
complete residues respectively, together with the number of changes of sign in
the signaletic series, consisting of the leading coefficients in Φx and in each of
the even-placed complete residues respectively.
   The syzygetic theory of two algebraical functions, and the allied theory of
algebraical continued fractions with their principal applications, may, I think,
now be said to be completely made out, as well for the singular cases as for the
general hypothesis.                                                                  p. 578

Art. (ψ). I will conclude with observing that the theory within developed
gives the means of transforming (explicitly and without the aid of symmetrical
functions) into an algebraical continued fraction, any given sum of algebraical
fractions of the form
                      c1     c2     c3             cn
                          +      +       + ··· +        ,
                    x − h1 x − h2 x − h3         x − hn


                                         585
where each c and h is supposed known. For let the above sum be called FΦxx ; then
if he , ce be used to denote any pair of corresponding terms of the h series and
the c series, we have
                                    Φhe
                                           = ce ,
                                    F ′ he
as is well known and easily proved. Again, if Di x represent the simplified
denominator of the ith convergent to the continued fraction equal to FΦxx which
is to be found, say
                                                      1
                                                                                    ,
                                                               1
                    (A1 x + B1 ) −
                                                                          1
                                            (A2 x + B2 ) − · · · −
                                                                     (An x + Bn )
we have [p. 476 above]


              Φh1 , Φh2 . . . Φhi
  Di x = Σ                                 (x − h1 )(x − h2 ) . . . (x − hi )
             h1 ,     h2 . . . hi
            hi+1 , hi+2 . . . hn
             (i−1)i ζ(h1 , h2 . . . hi )Φh1 , Φh2 . . . Φhi
       = Σ(−) 2                                             (x − h1 )(x − h2 ) . . . (x − hi )
                            F ′ h1 F ′ h2 . . . F ′ hi
               (i−1)i
       = (−)      2     Σ{c1 c2 . . . ci ζ(h1 , h2 . . . hi )(x − h1 )(x − h2 ) . . . (x − hi )}.
Therefore
(Di h1 )2 = {Σ(c2 c3 . . . ci+1 )ζ(h2 , h3 . . . hi+1 )(h1 − h2 )(h1 − h3 ) . . . (h1 − hi+1 )}2
                                        1                      1
          = {Σ(c2 c3 . . . ci+1 )ζ 2 (h1 , h2 . . . hi+1 )ζ 2 (h2 , h3 . . . hi+1 )}2 ;
and the simplified (i + 1)th quotient, that is, the value of Ai+1 x + Bi+1 , when
divested of the allotrious factor, has been proved [cf. p. 508 above] to be equal to
                                                     Φh1
                                        Σ(Di h1 )2          (x − h1 );
                                                     F ′ h1
it is therefore now known as a rational and integral function of x; h1 , h2 . . . hn ; c1 , c2 . . . cn .
The allotrious factor itself is made up of the product of squares of quantities all
of the same form as the leading coefficient in Di x, which, from what has been
shown above, is seen to be equal to
                                  i−1
                             (−) 2 Σ{(c1 c2 . . . ci )ζ(h1 , h2 . . . hi )}.
Hence each term in the continued fraction
                                      1
                                                                                    ,
                                                               1
                    (A1 x + B1 ) −
                                                                          1
                                            (A2 x + B2 ) − · · · −
                                                                     (An x + Bn )

                                                     586
                                                                                     p. 579
  which is to be made equal to
                        c1     c2             cn
                             +      + ··· +        ,
                      x − h1 x − h2         x − hn
is completely assigned in terms of x and the given quantities c and h.
Art. (ω). The number of effective intercalations between the roots of Φx, F x is
easily seen to be equal to the excess of the number of positive real numerators
over the number of negative real numerators in the partial fractions of which
F x is the sum, and hence we see à priori, as an obvious consequence of a simple
Φx

extension of the reasoning in Art. 47 [p. 515 above], that the inertia of the
quadratic function
                                                                    1
                                                          2             
              Σ ce u1 + he u2 + h2e u3 + · · · + hn−1
                                                  e   un                       ,
                                                                  x − he
where ce = FΦh
             ′ h , will represent the value of the index in question. So too we
                e
                e
may see that the formulae given for the residues to f x, f ′ x in Art. 46 continue
to apply to the residues F x, Φx. That is to say, these residues when divided
out by F x will be respectively represented by the successive principal coaxal
determinants to the matrix
                            S0 , S1 ,  S2         ...   Sm−1 ,
                            S1 , S2 ,  S3         ...    Sm ,
                            S2 , S3 ,  S4         ...   Sm+1 ,
                            ···  ···   ···                ···
                          Sm−1 , Sm , Sm+1        . . . S2m−2 ,
where in general
                            c1           c2                   cn
                   Sr =          hr1 +        hr2 + · · · +       hr ,
                          x − h1       x − h2               x − hn n
and using the same matrix as above written with S ′ substituted for S, where in
general
            Sr′ = c1 (x − h1 )hr1 + c2 (x − h2 )hr2 + · · · + cn (x − hn )hrn ,
the successive principal coaxal determinants of the new matrix represent the
successive denominators to the convergents of the continued fraction which
expresses FΦxx .
   The expression for the numerators to the convergents may also, there is no
doubt, be obtained by some simple modification (dependent on introducing the
quantities c1 , c2 . . . cn ) of the formula in Art. 41, p. [492].
   I annex, more with the hope of suggesting than (in all instances) of conveying
a full conception of the force of the definitions, a Glossary, or rather a Repertory
of the principal terms of art employed in the preceding pages, which might
otherwise be apt to occasion some difficulty to persons unfamiliar with the
subject.                                                                             p. 580


                                          587
Glossary of New or Unusual Terms, or of Terms Used in a New or
            Unusual Sense, in the Preceding Memoir.

Allotrious.— The allotrious factor to a residue or quotient in the process of
common measure applied to two algebraical functions is the constant factor of
which such residue or quotient must be divested in order to become an integral
and irreducible function.

Apocopated.— Applied to a type in the Theory of Cumulants, denotes a type
the final or initial element of which has been taken away. If both are taken away,
the type is said to be doubly apocopated.

Bezoutic.— For definition of Primary and Secondary Bezoutics see first Section.
Bezoutiant to two functions, each of degree n, is a homogeneous quadratic
invariantive function of n variables, the form of which serves to assign the index
of the scale of the effective intercalations of the real roots of the two given
functions.

Bezoutoid.— The Bezoutiant to two homogeneous functions obtained by
differentiation from one homogeneous function of two variables. The Bezoutoid
to a given function of m dimensions in the variables is accordingly a quadratic
function of (m − 1) variables, the form of which is sufficient for determining the
number of real roots in the given function.

Characteristic.— The employment of this word has been avoided in the
preceding memoir; but as it contains an idea of capital importance in analysis,
and especially in all inquiries of the kind here treated of, I subjoin the definition
of its meaning. The characteristic of a simple condition of any kind is the
rational integral function (in its lowest terms) whose evanescence necessarily
and universally implies and is implied by the satisfaction of such condition. A
simple condition has always a single characteristic, abstraction being made of
the algebraical sign, which remains indeterminate. In like manner, a multiple
condition, or a system of conditions, will have for its characteristic a plexus
of rational integral functions, whose evanescence necessarily and universally
implies and is implied by the satisfaction of such multiple condition or system of
conditions. The number of functions in the characteristic plexus will however in
general greatly exceed the index of the multiplicity of the conditions, and need
not always be a unique system. There are however exceptions to this: thus the
duplex condition, that a biquadratic function of x shall contain a cubic factor,
or that a curve of the third degree shall have a cusp, will each be definitely
characterized by a plexus of two functions, and no more.


                                        588
   The spirit of the higher analysis resides, and is to be sought for, in the logic
of characteristics.

Co-bezoutiant.— Any homogeneous quadratic function similar in form and
in its property of invariance to the Bezoutiant.                       p. 581


Cogredient and Contragredient.— A system of variables is cogredient to
another system when it is subject to undergo simultaneously therewith linear
substitutions of a like kind, and contragredient when it is subject to undergo
linear substitutions simultaneously therewith but of a contrary kind.

Combinant.— A function of the quantities appearing in a given set of functions
which remains unaltered as well for linear substitutions impressed upon the
variables as for linear combinations of the functions themselves.

Concomitant.— Nomen generalissimum for a form invariantively connected
with a given form or system of forms.

Conjunctive.— A syzygetic function of a given set of functions. Any function
which universally, and subject to no cases of exception, vanishes when a certain
number of other functions all vanish together must be a conjunctive (that is
a syzygetic function), or a root of a conjunctive of such functions. But if its
vanishing is subject to cases of exception, then all that can be predicated of it is
that it is syzygetically related to such functions, but it may, and usually does
happen, that it will be syzygetically related to them in more than one way.

Contravariant.— A function which stands in the same relation to the prim-
itive function from which it is derived as any of its linear transforms to an
inversely derived transform of its primitive.

Covariant.— A function which stands in the same relation to the primitive
function from which it is derived as any of its linear transforms to a similarly
derived transform of its primitive.

Cumulant.— The denominator of the simple algebraical fraction which ex-
presses the value of an improper continued fraction. See Type, infra.

Determinant.— This word is used throughout in the single sense, after which
it denotes the alternate or hemihedral function the vanishing of which is the
condition of the possibility of the coexistence of a system of a certain number of
homogeneous linear equations of as many variables.

                                        589
Dialytic.— If there be a system of functions containing in each term different
combinations of the powers of the variables in number equal to the number of
the functions, a resultant may be formed from these functions by, as it were,
dissolving the relations which connect together the different combinations of
the powers of the variables, and treating them as simple independent quantities
linearly involved in the functions. The resultant so formed is called the Dialytic
Resultant of the functions supposed; and any method by which the elimination
between two or more equations can be made to depend on the formation of such
a resultant is called a dialytic method of elimination. In such method accordingly
the process of elimination between equations of a higher degree than the first is
always reduced to a question of elimination between equations which are of the
first degree only.

Discriminant.— The resultant of the n differential coefficients of a homoge-
neous function of n variables. See Resultant, infra.                         p. 582


Disjunctive.— A disjunctive equation is a relation between two sets of quanti-
ties such that each one of either set is equal according to some unspecified order
of connexion with some one of the other set.

Effective scale of intercalations.— The series of the real roots of two
functions of x written in order of magnitude after repeated processes of removing
pairs of roots belonging to either the same function (when not separated by
roots of the other function): the roots of the two functions follow each other
alternately.

Effluent.— From every homogeneous function of any number i of variables of
the degree mm′ , where m, m′ are any two integers, may be formed (as shown
in the Calculus of Forms, Section II.) a covariantive function of the degree m
and of µ variables, where µ is the number of permutations that can be obtained
by dividing m′ into i parts (zeros admissible), in which all the coefficients are
numerical multiples of the given coefficients; covariants so formed may be termed
effluents of their primitive. An example of this occurs in the footnote to Section
V., [p. 557], where the quantity there called Q is a quadratic effluent of the
Jacobian.

Element.—      A simple component of the type to a cumulant. See Cumulant,
supra.

Emanant.— The result of operating any number of times (suppose i times)
upon a given homogeneous function of any number of variables x, y, z . . . t with


                                       590
the operative symbol
                                   d       d      d               d
                                                                  
                             x′      + y′    + z′    + · · · + t′    ,
                                  dx      dy      dz              dt
is called the ith emanant of the function operated upon. Every emanant is a
covariant to its primitive, the new variables x′ , y ′ , z ′ . . . t′ being cogredient with the
variables x, y, z . . . t with which they are respectively associated. E2i+θ f, E2i+θ ϕ,
page [561], are emanants of f and ϕ. The process of emanation is one of incessant
occurrence in the theory of invariants. When the order of the emanant is the
same as the degree of the function (supposed to be rational and integral) from
which the emanant proceeds, the form of the original function is reproduced in
the final emanant, the names only of the variables being changed.

Endoscopic, Exoscopic.— When the coefficients of the functions concerned
in any investigation are regarded as integral indecomposable monads, the method
is called exoscopic, and endoscopic when the coefficients are treated with reference
to their internal constitution as composed of roots or other elements.
    In addition to the examples in the footnote to Section I., these words have a
marked and most important application in the theory of Invariants, especially of
two variables.

Form.— Any function may be regarded as an opus operandum; the matter
operated upon being the variables, and the substance of the operations being
the form, which resides in the function as the soul in the body. A form is always
common to an infinity of functions, but for greater brevity may be and frequently
is called by the name of some specified function in which it is contained.        p. 583


Fundamental.— The fundamental scale of a system of Invariants or Concomi-
tants is a set of the same, whereof every other is a Rational Integral Function.

Hessian or Hessean.— Named after Dr Otto Hesse, of Königsberg (the
worthy pupil of his illustrious master, Jacobi, but who, to the scandal of the
mathematical world, remains still without a Chair in the University which he
adorns with his presence and his name), is the Jacobian to the differential
coefficients of a homogeneous function of any number of variables. It is to a
Jacobian what a Bezoutoid is to a Bezoutiant, or a Discriminant to a Resultant.

Hyperdeterminants.— See Memoir of Mr Cayley, Cambridge and Dublin
Mathematical Journal, May 1845, and Crelle’s Journal of about the same date.




                                                591
Improper.— Continued fraction is a continued fraction differing only from an
ordinary one in the circumstance of negative signs being substituted for positive
signs to connect the terms.

Inertia.— The unchangeable number of integers in the excess of positive over
negative signs which adheres to a quadratic form expressed as the sum of positive
and negative squares, notwithstanding any real linear transformations impressed
upon such form.

Intercalations.— The theory of intercalations is the theory of the relative
distribution of the real roots, or point-roots, of two or more equations, but in
this theory the number of roots mutually interposed is to be taken only with
reference to the number 2 as a modulus.

Invariance.— The property (under prescribed or implied conditions) of re-
maining invariable.

Invariant.— A function of the coefficients of one or more forms which remains
unaltered when these undergo suitable linear transformations.

Inverse.— The inverse to a given square matrix is formed by selecting in its
turn each component of the given matrix, substituting unity in its place, making
all the other components in the same line and column therewith zero, and finally
writing the value of the determinant corresponding to the matrix thus modified
in lieu of the selected component. If the determinant to the matrix be equal to
unity, its second inverse, that is the inverse to its inverse, will be identical, term
for term, with the original matrix.

Jacobian.— The Jacobian to n homogeneous functions of n variables is the
determinant represented by the symmetrical collocation in a square of the n
differential coefficients of each of the n functions.

Kenotheme.— A finite system of discrete points defined by one or more
homogeneous equations in number one less than the number of variables contained
therein.

Limiting Series.— One set of quantities whose extreme values are exterior
to the extreme values of a second set is set to limit the latter.

Matrix.—      A square or rectangular arrangement of terms in lines and columns. p. 584


                                         592
Minor Determinant.— Any determinant represented by a square group of
terms arbitrarily chosen out of a matrix is a minor determinant thereto. The
simple terms of the matrix are the last minors, and of course if the matrix is a
square, it will itself in its totality represent a single complete determinant.

Monotheme.— A line, or finite system of lines, defined by one or more
homogeneous equations two less in number than the number of the variables
contained therein.

Order.— The orders of a homogeneous function are the linear functions of
the variables the least in number by aid of which the function admits of being
expressed.

Persymmetrical.— A symmetrical matrix, in which all the terms in the
diagonal bands transverse to the axis of symmetry are identical, is said to be
persymmetrical. Example. An addition table.

Quadrinvariant.— An invariant of which the terms are quadratic functions
of the coefficients of the primitive.

Relation (simple and compound).—             Vide Substitution, infra.

Resultant.— The resultant of n homogeneous general functions of n variables
is that function of their coefficients which, equated to zero, expresses in the
simplest terms the condition of the possibility of their coexistence.

Rhizoristic.— A rhizoristic series is a series of disconnected functions which
serve to fix the number of real roots of a given function lying between any
assigned limits.

Signaletic.— A signaletic or Semaphoretic series is a sequence of disjunctive
terms, considered solely with reference to the algebraical signs of plus and minus
which they respectively carry.

Singular.— A proper algebraical function of a given degree, n, in one variable
in its most general form, will, in respect to that variable, be of the nth degree
in the denominator and the (n − 1)th degree in the numerator, and will admit
of being represented by a continued algebraical fraction of n terms, all of them
linear.
   But for particular values of, or relations among, the coefficients entering into
the given fraction this mode of representation fails, and the continued fraction,

                                       593
instead of consisting of linear terms n in number, will consist of terms, some
of them at least, non-linear, and fewer than n in number. These then are the
singular cases (or cases of singularity) in the theory of the development of an
algebraical fraction under the continued fraction form; and it will be seen that
according to this definition the case of the development of any proper algebraical
fraction in which the degree of the numerator is more than one unit below that
of the denominator, belongs (strictly speaking) to the class of singular cases;
and this view of the case supposed is perfectly correct and conformable to the
analogies of the subject.                                                          p. 585


Substitution (linear, similar or contrary).— A linear substitution is said
to be impressed upon a system of variables when each variable is replaced by
a linear conjunctive of all the variables. The matrix formed by the coefficients
of substitution arranged in regular order is called the Matrix of Substitution,
and is of course a square. When two substitutions (impressed on two systems
of variables) have the same matrix, they are said to be similar, and contrary
when their matrices are contrary, that is mutually inverse to each other. When
two systems of variables are supposed to be subject to the condition that their
substitutions are always similar or always contrary, they are said to be related
or in simple relation, the relation being of cogredience in the one case and of
contragredience in the other.
   When a linear substitution is impressed upon a system of independent variables,
a corresponding linear substitution is necessarily impressed at the same time
upon every complete system of homogeneous combinations (that is, products
and powers and products of powers) of these variables, the matrix to which
latter substitution will consist of terms which will be functions (depending upon
the degree of the homogeneous combinations) of the terms of the matrix to the
primitive substitution. This matrix may be termed a compound matrix, having
the primitive matrix for its base.
   If, now, two systems of independent variables are subject to be synchronously
impressed with substitutions, the matrices to which (not being both of them
simple matrices) have for their bases matrices which are either similar or contrary,
these two systems will be said to be in compound relation of cogredience in the
one case, and of contragredience in the other.

Syrrhizoristic.— A syrrhizoristic series is a series of disconnected functions
which serve to determine the effective intercalations of the real roots of two
functions lying between any assigned limits.

Syzygetic.— A syzygetic function or conjunctive of a number of given rational
integral functions is the sum of these affected respectively with arbitrary func-
tional multipliers, which are termed the syzygetic multipliers. When a syzygetic

                                        594
function of a given set of functions can be made to vanish, they are said to be
syzygetically related.

Transform.—       Equivalent to the French noun substantive “transformée.”

Type.— The type of a cumulant is the series of the simple elements (or quo-
tients), arranged in a fixed order, of which the cumulant is composed.

Umbral.— The umbral notation is a notation according to which simple
quantities are denoted by syllables, instead of by single letters (the composition
of these syllables being governed by the mode in which the quantities which they
express are obtained); and the single letters of such syllables are termed umbral
quantities or umbrae.

Weight.— In this memoir (throughout the earlier sections) the weight of any
quantity composed of the product of the coefficients of any given function or p. 586
   functions of x is used to denote the number of roots of x appertaining to the
given function or functions which must be employed to express such quantity.
More generally, when dealing with a system of homogeneous functions, the weight
of a quantity may be defined with respect to any selected variable therein as the
sum of the weights in respect to such variable of the several coefficients of which
the quantity is composed (the weight of each several coefficient meaning the
index of the power of the selected variable in that term of the given function or
functions which is affected with such coefficient). These two definitions of weight
may be perfectly well reconciled with each other by understanding the weight
of a quantity formed from the coefficients of a function or system of functions
of x to mean the weight, in respect to unity, of such quantity when the given
functions are treated as homogeneous functions of x and 1.

Zeta.— The symbol ζ (preceding a row of bracketed terms) is used to denote
the product of the squared differences of the terms which it affects.

[ ].— A bracket of this form, when enclosing a superior and an inferior row
of terms m and n in number respectively, indicates the mn products of the
differences obtained by subtracting each term in the second row from each term
in the first row; when enclosing an arrangement of terms in a single line, it is
used to denote the cumulant of which such an arrangement is the type.




                                       595
                                              58.
On the Conditions Necessary and Sufficient to be Satisfied in
 order that a Function of any Number of Variables may be
 Linearly Equivalent to a Function of any Less Number of
                        Variables
                  [Philosophical Magazine, V. (1853), pp. 119–126]
                                                                                                      p. 587
   In the Cambridge and Dublin Mathematical Journal for November 1850272 ,
I defined an order as signifying any linear function of a given set of variables,
and spoke of a general function of n variables as losing r orders when the
relation between its coefficients is such that it is capable of being expressed as a
function of (n − r) orders only. It will be highly convenient to preserve the same
nomenclature for the purposes of the present investigation.
   Dr Otto Hesse, in a long memoir in Crelle’s Journal, the contents of which
have been described to me273 but which I have not yet been able to procure, has
given a rule for determining the analytical conditions for the loss of one order.
I propose to give a more simple and comprehensive scheme of conditions than
Professor Hesse appears to have discovered, applicable not to this case only, but
to that of the loss of any number whatever of orders, and shall moreover show in
what relation the substituted orders stand to the given variables.
   Dr Hesse’s rule had been previously stated by me in the 4th section of my
Calculus of Forms (Cambridge and Dublin Mathematical Journal, May 1852274 )
as applicable to the case of a general function of the 3rd degree                      p. 588
   of three variables becoming the representative of three right lines diverging
from the same point, which is the case of a cubic function of three variables
becoming a function of two linear functions of these variables, that is to say, losing
one order: this, perhaps, might have been noticed in the Professor’s memoir. I
gave also another rule for the same case; but the true fundamental scheme of
conditions about to be set forth will be seen to embrace as mere corollaries all
such and such-like rules, which in fact supply more or less arbitrary combinations
of the conditions, rather than the naked conditions themselves in their simple
form and absolute totality.
 272
      p. 171 above.
 273
      A distinguished mathematical friend in Paris communicated to me with great admiration
Professor Hesse’s result overnight. I ventured to affirm that, to one conversant with the calculus
of forms, the problem could offer no manner of difficulty. An hour’s quiet reflection in bed the
following morning, or morning after, sufficed to disclose to me the true principle of the solution.
[Cf. Noether, Math. Annal. L. (1898) p. 138. Ed.]
  274
     Vide Vol. VII. p. 187 [p. 335 above]. “When U represents a pencil of three rays meeting in
a point, dSda
              = 0, dS
                    db
                       = 0, &c., and also therefore T = 0” (S and T being the two Aronholdian
invariants of U , and a, b, c, &c. the coefficients of U ); “also in place of this system may be
substituted the system obtained by taking all the coefficients of the Hessian zero.”


                                               596
   I shall call the function to be dealt with U , and shall consider U to be a
homogeneous275 rational function of m dimensions in respect of x1 , x2 . . . xn , and
shall inquire what are the conditions which must obtain when U is capable of
being expressed as a function of only (n − r) orders, say l1 , l2 . . . ln−r , each of
which is of course a homogeneous linear function of the given n variables.
   Let the term derivative of U be understood to mean any result obtained
by differentiating U any number of times with respect to one or more of the
variables x1 , x2 . . . xn . The first derivatives will be of (m − 1) dimensions, the
second derivatives of (m − 2) dimensions, and so on; and finally, the (m − 1)th
derivatives will be homogeneous linear functions of x1 , x2 . . . xn . Suppose U to
be expressible as a function of l1 , l2 . . . ln−r . It is immediately obvious that the
derivatives from the 1st to the (m − 1)th inclusive will be all expressible as
homogeneous functions of l1 , l2 . . . ln−r , and vanish when these vanish. But this
statement is in substance pleonastic; for by means of Euler’s well-known law,
any derivative of U , say K, may be expressed (to a numerical factor près) under
the form of
                                 dK       dK                dK
                              x1     + x2      + · · · + xn     ,
                                 dx1      dx2               dxn
and consequently, whenever the linear derivatives of U vanish, all the upper
derivatives of U , including U itself, must vanish at the same time. The number
of these linear derivatives, say ν, will be the number of terms in a homogeneous
function of n variables of (m − 1) dimensions, that is to say,
                                 n(n − 1) . . . (n − m + 2)
                                                            .
                                     1 · 2 . . . (m − 1)
Again, if all the ν linear derivatives vanish when the (n − r) equations l1 = 0, l2 =
0 . . . ln−r = 0 are satisfied, r being greater than zero, this can only happen by
virtue of these ν derivatives being linear functions of (n − r)                       p. 589
    of them. Now, conversely, I shall prove, that if it be true that all the linear
derivatives of U are linear functions (n − r) of them, then U may be expressed as
a function of these (n − r); and this rule, as will be immediately made apparent,
will give the necessary and sufficient conditions for the loss of r orders in the
most simple and complete form by which they admit of being expressed. For
the proof of the rule, only one additional remark has to be made in addition to
that already made, of the vanishing of the linear derivatives necessarily implying
the simultaneous evanescence of all the other derivatives; this additional remark
being, that if the derivatives of any class, linear or otherwise, quà one set of
variables, become all zero, the derivatives of the same class, quà any other set
of variables linear functions of the first set and the same in number, will also
become zero, for they are evidently expressible as linear functions of the first set.
 275
    It is a common error to regard homogeneity of expression as merely a means for satisfying
the desire for symmetry; the ground of its application and utility in analysis lies, in fact, much
deeper; it is essentially a method and a power.


                                              597
   Now let d1 , d2 . . . dn−r be any (n − r) linear derivatives of U , of which all
the other of the ν derivatives of this class are linear functions, so that they
vanish when these (n − r) vanish, and let U be expressed as a function of
(d1 , d2 . . . dn−r ; x1 , x2 . . . xr ). Then we may write

                U = ϕm,0 + ϕm−1,1 + ϕm−2,2 + · · · + ϕ1,m−1 + ϕ0,m ,

where in general ϕm−e,e denotes a function homogeneous and of m − e dimensions
in respect to d1 , d2 . . . dn−r , and homogeneous and of e dimensions in respect
to x1 , x2 . . . xr . Now the linear derivatives of U all vanish when d1 = 0, d2 =
0 . . . dn−r = 0 for all values of x1 , x2 . . . xr . Hence U = 0 on the same supposition,
and hence ϕ0,m is similarly zero. Also the first derivatives of U , quà d1 , d2 . . . dn−r ,
must vanish on the same supposition. Hence ϕ1,m−1 is identically zero; and so by
taking the 2nd, 3rd . . . up to the (m − 1)th or linear derivatives of U in respect
to d1 , d2 . . . dn−r , we find successively ϕ2,m−2 , ϕ3,m−3 . . . ϕm−1,1 each identically
zero, and consequently

                            U = ϕm,0 = ϕ(d1 , d2 . . . dn−r ),

as was to be proved. To express the fact of the ν derivatives being linear functions
of (n − r) of them, form a rectangular matrix with the coefficients of the ν linear
derivatives. This matrix will be n terms in breadth and ν terms in depth. Let
r = 1: it is a direct consequence of the rule which has been established, that
every full determinant consisting of a square n terms by n terms that can be
formed out of this rectangular matrix must be zero: again, let r = 2; all the first
minors, that is to say, all the determinants composed of squares (n − 1) terms
by (n − 1) terms, must be zero, and so in general a loss of r orders will require
that the (r − 1)th minors shall all vanish; if r = n, the (n − 1)th minors, that is
the simple terms of the matrix which are all coefficients of U , must vanish, or
in other words, when the function is of zero order all the coefficients vanish (an
obvious truism).                                                                     p. 590
   Thus, then, we see that the true rule for the loss of one order in a polynomial
of any degree is precisely the same as the well-known rule for the loss of one
order in a quadratic function; the speciality in the latter case consisting merely
in the fact that ν being equal to n, the rectangular matrix becomes a square,
and there is only one full determinant. Moreover, for any other value of r the
above rule coincides with that given by me some time back in the Philosophical
Magazine for the case of quadratic functions.
   Professor Hesse’s rule for finding conditions applicable to the loss of one
order is, as I have already stated, a consequence of the more simple scheme of




                                           598
conditions above given. It consists in forming the determinant
                          d2 U       d2 U            d2 U
                             2                ···
                          dx1      dx1 dx2          dx1 dxn
                          d2 U       d2 U            d2 U
                                              · · ·
                        dx2 dx1      dx22           dx2 dxn ,
                           ···        ···     ···     ···
                          d2 U       d2 U            d2 U
                                              ···
                        dxn dx1 dxn dx2              dx2n
and equating the coefficients of this determinant fully developed separately to
zero276 . The attachment of the Professor to this particular form of covariant
(I use the language of the calculus of forms) is readily intelligible, seeing the
admirable application which he has made of it to the canonization of the cubic
function of three variables, but it is really foreign to the nature of the present
question; the coefficients of this covariant may easily be shown to be merely
the full determinants of the n × ν rectangular matrix above described, or linear
functions of these said determinants with numerical coefficients. Hence the
ground of its applicability.
   Returning to the rule of the matrix, if we suppose the number of variables to
be two, and call the coefficients of U
                                       1
                         a0 , na1 ,       n(n − 1)a2 . . . an ,
                                       2
 276
     A form capable of being so derived I have elsewhere termed (in compliment to M. Hesse)
the Hessian of the function to which it appertains. This is the trivial name which is much
needed on account of the frequent occurrence of the form, and has been adopted by Mr Salmon
in his admirable treatise on the higher plane curves. In systematic nomenclature it would
be termed the discriminant of the quadratic emanant, or more briefly, the quadremanative
discriminant. I have discovered quite recently that the long sought for symmetrical, and by far
the most easy practical process for discovering the number of the real roots of an equation, is
contained in, and may be deduced immediately from, a certain transformation of its Hessian!
There are frequent cases occurring in the calculus of forms of interchange between the degree of
a function and the number of variables which it contains. Thus, to select a striking example
(although one where the interchange is not exact), the theory of the real and imaginary roots
or factors of a homogeneous function of two variables and of the nth degree may be shown
to be immediately dependent upon the determination of the specific nature of a concomitant
homogeneous function of the 2nd degree and of (n − 1) variables. For instance, if any ordinary
algebraical equation of the 5th degree be given, a homogeneous quadratic function of four
variables may be constructed, representing, consequently, a surface of the 2nd degree [the
coefficients of which (as indeed is true whatever be the degree of the equation) will be quadratic
functions of the coefficients of the given equation]; and such that, according as the surface so
represented belongs to the class of (1), impossible surfaces; (2), the ellipsoid or hyperboloid of
two sheets; (3), the hyperboloid of one sheet; the given equation will have 5, 3, or only 1 real
root! Moreover, an equality between two of the roots of the equation will be denoted by the
loss of one order in the associated quadratic function; and so many orders altogether will be
lost as there are independent equalities existing between the roots. An entirely new light is thus
thrown on M. Sturm’s theorem; and the number of real and imaginary roots in an equation
is for the first time made to depend upon the signs of functions symmetrically constructed in
respect to the two ends of the equation, which has long been felt as a desideratum.


                                              599
our rectangle becomes
                                       a0 ,  a1
                                       a1 ,  a2
                                       a2 ,  a3
                                       ···   ···
                                      an−1 , an
                                                                                           p. 591
  and the conditions become
                       a0 a2 − a21 = 0,       a0 a3 − a1 a2 = 0,
                         a1 a3 − a22 = 0,         ············ ,
                          ············ ,          ············ ,
             an−2 an − a2n−1 = 0,         an−3 an − an−2 an−1 = 0,    &c.,

all of which equations are obviously true (when the function loses an order, that
is to say, becomes a perfect power) and are satisfied (special cases excepted)
when any (n − 1) independent equations out of the entire number obtain; so
that the number of conditions implied in the property to be represented is in
exact conformity with the number of independent equations derived from the
matrix, that is equations which, when satisfied, will in general cause all the rest
to be satisfied. This conformity manifests itself also in the case of a quadratic
function of n variables. But except in these two limiting (and, in an occult
sense, reciprocal) cases of a function of two variables of the nth degree, or of the
degree 2 and n variables, this conformity in measure as the degree or number of
variables rises, although it must substantially continue to exist, becomes, and in
an accelerated degree, less and less apparent.
   Thus, take the simple case of a cubic function of three variables, and let us
confine ourselves to the consideration of the conditions which must be satisfied
when it loses a single order. Let U be written out at length,

ax3 + by 3 + cz 3 + 3hyz 2 + 3izx2 + 3jxy 2 + 3h′ y 2 z + 3i′ z 2 x + 3j ′ x2 y + 6mxyz.
                                                                                           p. 592
  The matrix formed out of the coefficients of the linear derivatives becomes
                                    a, j ′ , i
                                    j, b, h′
                                    i′ , h, c
                                               .
                                    m, h′ , h
                                    i, m, i′
                                    j ′ , j, m

Now by the homaloidal law, if the terms in this rectangle were all unlike, the
number of full determinants (3 terms by 3 terms) whose evanescence (except
for special values) determines the evanescence of all the rest, should be (6 −
3 + 1)(3 − 3 + 1), that is 4; but in the actual case, since the evanescence of

                                            600
all the full determinants is a necessary consequence of the function becoming
a cubic function of two orders (that is, breaking up into the product of three
linear functions of x, y, z), and as this decomposability, as is well known, implies
only the existence of three affirmative conditions, the four full determinants

            a j′ i             a j′ i              a j′ i             a j′ i
            j b h′             j b h′              j b h′             j b h′        277

            i′ h c             m h′ h              i m i′             j′ j m

Thus, if we take the three full determinants that can be formed out of the matrix

                                             a, a′ ,
                                             b, β,
                                             c, γ,

that is
                           aβ − bα,         bγ − cβ,        ca − aγ,
these are in syzygy, for we can form the equation

                       c(aβ − bα) + a(bγ − cβ) + b(ca − aγ) = 0.

This, however, is not the only equation of the kind that can be formed, for

                       γ(aβ − bα) + a(bγ − cβ) + β(ca − aγ) = 0

is also identically true. We see in this case that the evanescence of any two of
the three functions                                                                  p. 593
   which in the general case would be entirely independent, in this case cease to
be so; and the vanishing of three of them must draw along with it by necessary
implication (except for special values) the evanescence of the 4th, for thus only
can the necessary conformity between the number of affirmative conditions and
the number of unimplicated equations come to take effect. The clear and direct
putting in evidence of this peculiar species of implication demands and deserves
to be minutely considered; and as it must in part borrow its explanation from
the very little yet known of syzygetic relations, so it must also throw new light on
 277
    That is to say, a syzygetic relation must connect these four determinants. I may as well
here repeat, that when the vanishing of a set of i rational integral functions necessarily, and
without cases of exception, implies the vanishing of another rational integral function, then this
function is termed a syzygetic function of the others; and some power of it must be expressible
under the form of a sum of i binary products of rational integral functions, one factor of each
of which products must be one of the i given functions. When the vanishing of all but one of a
set of functions in general necessarily implies the vanishing of that one, but subject to cases of
exception for specific values of the variables, then it can only be affirmed that the functions of
the set are in syzygy; that is to say, that the sum of the products of each of them respectively
by some rational integral function will be zero: the equation expressing this relation is termed
a syzygetic equation.


                                              601
that great and important, but as yet unformed and scarcely more than nascent
theory.
    In conclusion, it is apparent from the demonstration above given, that when
U , a function of n variables, becomes expressible as a function of (n − r) orders,
these orders may be taken respectively any independent linear functions of the
linear derivatives of U , which remark completes the theory of functions subject to
the loss of one or more orders. It is obvious (and I am indebted to my esteemed
friend Mr Cayley for the remark), that the conditions furnished as above by the
(m − 1)th, that is linear derivatives, are identical with and may be more elegantly
replaced by those involved in the assertion of the existence of linear relations
between the 1st or (m − 1)th degreed derivatives, and we have then this very
simple rule; if ϕ, a function of x1 , x2 . . . xn , is expressible as a function of n − r
linear functions of x1 , x2 . . . xn , it is necessary and sufficient that r independent
linear relations shall exist between
                                 dϕ      dϕ      dϕ
                                     ,       ...     .
                                 dx1     dx2     dxn
aβ − bα; bγ − cβ; ca − aγ will in general imply the third, subject, however, to
special cases of exception. Thus, if the 1st and 2nd vanish, the 3rd must vanish
unless b and β both vanish; if the 2nd and 3rd vanish, the 1st must vanish unless
c and γ both vanish; if the 3rd and 1st vanish, the second will vanish unless
a and α both vanish. It will thus be seen that a peculiar species of astricted
syzygy obtains between the three proposed functions, which enables us to affirm
that in general, and except under extra special conditions, all three must vanish
simultaneously. If two out of the three vanish, and the 3rd does not vanish, it is
not merely (as might at the first blush of the theory of syzygy be conjectured)
because some one other function vanishes in its place, but necessarily because
a plurality of entirely independent functions (two simple letters as it happens
here) each separately vanish. Thus we see how all but one of a set of functions
X1 , X2 . . . Xn may in general, and yet not universally, necessarily vanish when
all the rest vanish: to say that one syzygetic equation such as

                          X1 X1′ + X2 X2′ + · · · + Xn Xn′ = 0

obtains, is not enough to explain the circumstances of the case; the fact is,
that several distinct systems of values of X1′ , X2′ . . . Xn′ will be found capable of
satisfying the equation, so that each of the functions X1 , X2 . . . Xn will have a
system of syzygetic factors attached to it, and these unrelated, in the wide sense
that, if we take Xr , Xs , Xt , any two of the syzygetic factors attached to Xn , they
will not be in syzygy with X1 , X2 . . . Xn−1 ; so that when these (n − 1) functions
vanish, the vanishing of Xn and Xn′ represents two distinct and completely
independent conditions. Thus, in fine, the mutual implication of functions will in
general denote the possibility of forming a series of syzygetic equations between
them,—a remark, this, of no minor importance.                                           p. 594


                                          602
   This rule itself also, it is evident, is capable of an independent and immedi-
ate demonstration by means of integrating the partial differential equation or
equations by which it admits of being expressed. The above theory may readily
be extended to functions of several systems of variables. Thus, for instance, the
determinant
                                      a, b, c
                                     a′ , b′ , c′
                                     a′′ , b′′ , c′′
vanishing will be indicative of the function
                           
                            a xu + b xv + c xw
                           
                               + a′ yu + b′ yv + c′ yw,
                            + a′′ zu + b′′ zv + c′′ zw
                           


being linearly equivalent to a function of the form
                              (
                                  Ax′ u′ + Bx′ v ′ ,
                                  + Cy ′ u′ + Dy ′ v ′ ,

that is losing an order in respect of each of the two systems x, y, z; u, v, w; and
so in general.




                                         603
                                          59.
 On Mr Cayley’s Impromptu Demonstration of the Rule for
   Determining at Sight the Degree of any Symmetrical
Function of the Roots of an Equation Expressed in Terms of
                     the Coefficients
                [Philosophical Magazine, V. (1853), pp. 199–202]
                                                                                          p. 595
   For a considerable time past, among the few cultivators of the higher algebra,
a proposition relative to the theory of the symmetrical functions of the roots
of an equation has been in private circulation, which, to say nothing of the
important applications of which it has been found susceptible to the calculus
of forms, merits (by reason of its extreme simplicity), although, strange to say,
it has, I believe, not yet obtained, a place in elementary treatises on algebra.
The proposition alluded to I have reason to think first came to be observed in
connexion with my well-known formulae for Sturm’s auxiliary functions in terms
of the roots given in this Magazine. The theorem is briefly as follows. If a, b, c,
&c. be the roots of an equation
                          xn + p1 xn−1 + p2 xn−2 + &c. = 0,
any symmetric function such as Σaα bβ cγ . . ., where α, β, γ . . . are positive integers
arranged according to the order of their magnitudes in a descending (or, to
speak more strictly, non-ascending) order, when expressed as a function of the
coefficients, will be made up of terms of the form pθ11 pθ22 pθ33 . . . pθkk , such that
θ1 + θ2 + θ3 + . . . + θk will be equal to α for some terms, but will for no term
exceed α; α being, as above described, that one of the indices α, β, γ . . . which is
not less than any of the others.
   I had prepared, and indeed despatched, a somewhat elaborate proof of this
theorem for the Cambridge and Dublin Mathematical Journal; but on proceeding
to explain my method to Mr Cayley, elicited from that sagacious analyst the
following excellent impromptu, which I think too valuable to be lost; and as it is
now a twelvemonth or two since our conversation on the subject took place, and
the author has not cared to put it on record, I feel                                      p. 596
   myself under an obligation so to do, the more so as it entirely supersedes
the comparatively inelegant demonstration of my own which I had previously
intended to publish.
   The method rests essentially on the following well-known theorem given by
Euler relative to the partition of numbers; to wit, that the number of ways of
breaking up a number n into parts is the same, whether we impose the condition
that the number of parts in any partitionment shall not exceed m, or that
the magnitude of any one of the parts shall not exceed m. Of this rule more
hereafter—for the present to its application to the matter in hand.

                                          604
   Since a, b, c . . . are the roots of xn + p1 xn−1 + . . ., we have
                             p1 = a + b + c + . . . ,
                             p2 = ab + ac + bc + . . . ,
                             p3 = abc + abd + acd + . . . ,
                                ············
                                ············ .
Let α + β + γ + . . . = n, none of the quantities α, β, γ . . . being greater than m,
but α, β, γ . . . being otherwise arbitrary and capable of becoming equal to any
extent inter se. Also let λ + µ + ν + . . . = n, the number of quantities λ, µ, ν,
&c. being never greater than m, but the quantities themselves being otherwise
arbitrary, and being capable of becoming equal to any extent inter se. By Euler’s
rule the number of systems α, β, γ . . . is the same as of the systems λ, µ, ν . . ., say
P for each. For any system λ, µ, ν . . ., we shall have pλ pµ pν . . ., by virtue of the
equations above written, expressible as the sum of terms of the form Σaα bβ cγ . . .;
it may easily be made ostensible, that all the combinations of α, β, γ . . . subject
to the above prescribed conditions must come into evidence by giving λ, µ, ν . . .
all the variations of which they admit; but this is also immediately obvious
indirectly from the consideration, that were it otherwise, linear relations would
subsist between the different values of pλ pµ pν . . ., which is obviously absurd.
Hence, then, we shall be able to express the P quantities of the form pλ pµ . . .
by means of linear functions of the P quantities Σaα bβ cγ . . .; and conversely, by
solving the linear equations thus arising, the P quantities Σaα bβ cγ . . . may be
expressed in terms of the quantities pλ pµ . . .; consequently Σaα bβ cγ . . ., where m
is greater or not less than any of the quantities β, γ . . ., will be expressible by
means of combinations pλ pµ . . ., where the number of coefficients pλ pµ . . . (any
number of which may become identical) is for some of the combinations as great
as, but for none of the combinations greater than m, as was to be proved. It
will of course be seen that, for the purposes of the demonstration above given, it
would have been sufficient                                                                p. 597
   to have been able to assume that the number of partitions, when the greatest
part is not allowed to exceed m, is not greater than the number of partitions when
the number of parts in any one partitionment does not exceed m. The equality
of these two numbers would then evince itself in the course of the demonstration
as a consequence of this assumption.
   A word now as to Euler’s beautiful law upon which the above demonstration
is based.
   A corollary from it, obtained by subtracting the equation which it gives when
the limiting number is taken (m − 1) from the equation which it gives when the
limiting number is m, will be the following proposition. The number of modes
of partitioning n into m parts is equal to the number of modes of partitioning
n into parts, one of which is always m, and the others m or less than m. This

                                          605
proposition was mentioned to me by Mr N. M. Ferrers278 , whose demonstration
of it (probably not different from that of Euler’s for the other proposition, of
which it may be viewed as a corollary) is so simple and instructive, that I am
sure every logician will be delighted to meet with it here or elsewhere. It affords
a most admirable example of that rather uncommon kind of reasoning whereby
two abstract integers are proved to be equal indirectly, by showing that neither
can be greater than the other.
   If there be a group of A’s and a group of B’s, and every A can be shown to
produce a B, and every B can be shown to produce an A, no matter whether
the A producing a B is the same as, or different from, the A produced by that
B, it is obvious that the number of A’s cannot exceed that of the B’s, nor of the
B’s that of the A’s, and the two numbers will therefore be equal.
   Take any such grouping as 3, 3, 2, 1, say A. This may be written as

                                       1, 1, 1
                                       1, 1, 1
                                       1, 1,
                                       1,

and by reading off the columns as lines, may be transformed into the group

                                     1, 1, 1, 1
                                     1, 1, 1
                                     1, 1

that is 4, 3, 2, say B.                                                            p. 598
   In A the number of parts is 4. In B the greatest part is 4; the others might
be (although they happen not in this particular instance to be) 4, but cannot
be greater than 4. And so every A in which the number of parts is 4 will give
rise to a B in which 4 is one of the parts, and every other part is 4 or less, and
evidently (although, as above remarked, this is immaterial to the demonstration)
every such B gives reciprocally the same A from which it is itself derived; hence
the number of A’s and B’s is equal. This is the theorem which, for the sake of
distinction, I have called the Corollary to Euler’s. Euler’s own is proved by the
same diagram; for if we define A as a grouping where the number of parts does
not exceed 4, we get a definition of B as a grouping where the greatest part does
not exceed 4, and so in general. We see that this theorem may be varied also by
affirming that the number of ways in which n may be broken up, so that there
shall never be less than m parts, is the same as the number of ways in which it
may be broken up into parts, the greatest of which in any one way is not less
than m. So, again, a similar diagram makes it apparent, that if we break up each
of i numbers into parts so that the sum of the greatest parts shall not exceed
 278
     I learn from Mr Ferrers that this theorem was brought under his cognizance through a
Cambridge examination paper set by Mr Adams of Neptune notability.


                                          606
(or be less than) m, the number of ways in which this can be done will be the
same as the number of ways in which these i numbers can be simultaneously
partitioned so that the total number of parts in any simultaneous partitionment
shall never exceed (or never be less than) m; and doubtless an extensive range of
analogous general theorems relative to the partitioning of numbers may be struck
out by aid of the same diagram, by no means easily demonstrable unless this
simple mode of conversion happen to be thought of, but in that event becoming
intuitively apparent. This mode of conversion is precisely that (only applied to a
more general state of things) whereby, in elementary arithmetic, it is established
that m times n is the same as n times m. A consideration of the process by
which the mind satisfies itself of the universality of this law, has been always
sufficient to convince me of the absurdity of ascribing to an inductive process
the capacity of the human mind for forming general ideas concerning necessary
relations.




                                       607
                                               60.
A Proof that all the Invariants279 to a Cubic Ternary Form
 are Rational Functions of Aronhold’s Invariants and of a
      Cognate Theorem for Biquadratic Binary Forms
            [Philosophical Magazine, V. (1853), pp. 299–303, 367–372]
                                                                                                      p. 599
   Although contrary to the order of exposition indicated in the title to this
paper, I shall, as the simpler case, begin with establishing the theorem for a
biquadratic form, say F in x, y. Let

                       F = ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 ,
                        s = ae − 4bd + 3c2 ,
                        t = ace − ad2 − c3 − b2 e + 2bcd,

s and t are the two well-known invariants of F . I propose to prove that there
can exist no other invariants to F except such as are explicit rational functions
of s and t.
   Let F , by means of the substitution of f x + gy for x, and f ′ x + g ′ y for y,
be made to take the form f1 = x4 + y 4 + 6mx2 y 2 . Then by the characteristic
property of invariants, if I(a, b, c, d, e) be any invariant to F of the degree q, we
must have
                   I(1, 0, m, 0, 1) = (f g ′ − f ′ g)2q I(a, b, c, d, e);
and it will be sufficient to prove that I(1, 0, m, 0, 1), or say more simply I(m),
can only have the two radically distinct forms corresponding to s and t, that is

                          (s) = 1 − 3m2        and    (t) = m − m3 ,

any other admissible form of I being a rational explicit function of these two. p. 600
   It may be shown280 that the parameter m in f1 will have six different√ values
and no more. In the first place, if we write ix for x in f1 (i meaning −1), it
is obvious that m becomes −m. Again, let x + iy and x − iy be substituted in
place of x and y respectively; then calling (f ) the value assumed by f1 , when
 279
      A Constant in analysis is any quantity which in its own nature, or by the explicit conditions
to which it is subjected, is incapable of change. An Invariant is an expression apparently liable
to change, but which, owing to certain compensations in the modifying tendencies impressed
upon it, remains as a whole unaltered. The former may be compared to a fixed point or system
in mechanics; the latter to a point or system free to move, but kept at rest under the combined
operation of contending forces.
  280
      See Addendum [p. 607 below].




                                               608
this substitution is made,

                    (f ) = (x + iy)4 + (x − iy)4 + 6m(x2 + y 2 )2
                       = (2 + 6m)(x4 + y 4 ) + (−12 + 12m)x2 y 2
                                                −1 + m 2 2
                                                          
                       = (2 + 6m) x4 + y 4 + 6          x y .
                                                 1 + 3m
Hence if we write
                            1               i
                                   x+             y         for x,
                       (2 + 6m)1/4    (2 + 6m)1/4

and
                             1               i
                                    x−             y        for y,
                        (2 + 6m)1/4    (2 + 6m)1/4
and call what f1 becomes after these substitutions f2 ,

                             f2 = x4 + y 4 + 6γ(m)x2 y 2 ,
              −1 + m
γ(m) denoting         .
              1 + 3m
  In like manner, by writing in f2
                           1                 i
                                   x+                y          for x,
                    {2 + 6γ(m)}1/4    {2 + 6γ(m)}1/4

and
                           1                 i
                                   x−                y          for y,
                    {2 + 6γ(m)}1/4    {2 + 6γ(m)}1/4
we obtain
                             f3 = x4 + y 4 + 6γ 2 (m)x2 y 2 ,
where
                              −1 + m
                           −1 +
                γ 2 (m) =     1 + 3m = −2 − 2m = −1 − m ;
                              −1 + m   −2 + 6m   −1 + 3m
                          1+3
                              1 + 3m
γ(m) is a periodic function of m of the third order, for we find

                                        −(1 + 3m) − (−1 + m)
              γ 3 (m) = γ 2 {γ(m)} =                          = m.
                                        −(1 + 3m) + 3(−1 + m)

It will of course be observed, also, that

                  γ 2 (m) = −γ(−m) and           γ(m) = −γ 2 (−m).
                                                                         p. 601




                                           609
  Hence

    (−γ)(−γ)(m) = −γ 3 (−m) = m,                (−γ 2 )(−γ 2 )(m) = −γ 3 (−m) = m.

So that, in fact, the six values of the parameter are

                              m,  γ(m),   γ 2 (m),
                              −m, −γ(m), −γ 2 (m),

forming two cycles, having the remarkable property that the terms in the same
cycle are periodic functions of the third order of one another, and each term in
one cycle is a periodic function of the second order of every term in the other
cycle.
   The modulus of substitution for passing from f1 to f2 , that is the square of
the determinant
                                 1              i
                           (2 + 6m) 1/4   (2 + 6m)1/4
                                 1             −i       ,
                           (2 + 6m)1/4 (2 + 6m)1/4
is
                             (−2i)2             −2
                                     , or           .
                             2 + 6m          1 + 3m
So that if I(m) be the value of any
                                   invariant
                                            of the degree q, corresponding to
                                    m−1
the form f1 , and consequently I              the same for f2 , we must have
                                   1 + 3m
                                       1 + 3m                m−1
                                               q                   
                        I(m) =                       I              .
                                         −2                  1 + 3m
In like manner, by means of f3 it may be shown that we must have the further
equation
                                1 − 3m q     m+1
                                                 
                      I(m) =              I           .
                                  −2         1 − 3m
These equations are easily verified for the values of (s) and (t).
  Thus
                            (1 + 3m)2                  m−1
                                                (                          2 )
            (s) = 1 + 3m2 =                        1+3
                                4                      3m + 1
                  (1 − 3m)2          m+1
                               (                        2 )
                =                1+3                             ,
                      4              1 − 3m

                             (1 + 3m)3                 m−1                 m−1
                                                   (                              3 )
             (t) = m − m = −
                          3
                                                              −
                                 8                     3m + 1             3m + 1
                   (1 − 3m)3           m+1               m+1
                                   (                                3 )
                =−                            −                             ;
                       8               1 − 3m            1 − 3m

                                             610
                                                                                        p. 602
   and it is moreover obvious, that the values of (s) and (t) might have been
found à priori by means of these functional equations.
   The essential point of inference for my present purpose from the equations
above, which are of the form
                                     m−1                      m+1
                                                                
                 I(m) = H × I                     =K ×I              ,
                                     3m + 1                   1 − 3m

is this, that if I(m) contain any power of m, say mi , it must also contain (m − 1)i
and (m + 1)i ; in a word, (m3 − m)i , which, by the way, it may be noticed, is (t)i .
Now, if possible, let there be any invariant Iq (m) of the qth degree in m which
is not a rational function of (s) and (t). If we make

                                     2x + 3y = q,

as many integer solutions as exist of this equation (in which zero values of x and
y are admissible), so many functions of the form (s)x (t)y may be formed of the
degree q in m, and all of them of course invariantive functions.
    As regards the general nature of any invariantive function in m, since the
change of x into −x in x4 + y 4 + 6mx2 y 2 introduces no change into the invariant
if q be even, but changes the sign if q be odd, it follows that Iq (m) is of the form
ϕ(m2 ) when q is even, and of the form mϕ(m2 ) when q is odd.
    Let µ be the number of solutions of the equation in integers above written.
Then, by linearly combining all the different values of (s)x (t)y with Iq (m), it is
obvious that we may form a new invariant, say Iq′ , in which the µ first occurring
powers of m will be wanting, that is in which the indices 0, 2, 4 . . . (2µ − 2) will
be wanting when q is even, and 1, 3, 5 . . . (2µ − 1) when q is odd. Hence in the
former case the new invariant will contain m2µ , and in the latter case m2µ+1 ; and
therefore, by virtue of what has been shown already, Iq′ will contain (m3 − m)2µ
in the one case and (m3 − m)2µ+1 in the other.
    Firstly, let q = 6i, or 6i+2, or 6i+4; then µ = i+1; and therefore (m3 −m)2i+2 ,
which is of the degree 6i + 6 in m, is contained as a factor in I which is of the
degree q only, a quantity less than 6i + 6, which is absurd.
    Again, secondly, let q = 6i + 1, then µ = i; and (m3 − m)2µ+1 is of the degree
6i + 3 in m, and is contained as a factor in I, which is of the degree 6i + 1, which
is again absurd.
    Finally, if q = 6i + 5, or 6i + 3, µ = i + 1; and the factor (m3 − m)2µ+1 is of
the degree 6i + 9, that is, in each case, greater than q, which is absurd, and thus
the theorem is completely demonstrated.
    It may for a moment be objected, that we have been dealing only with a
particular form x4 + 6mx2 y 2 + y 4 , instead of the general form

                      ax4 + 4bx3 y + 6cx2 y 2 + 4dxy 3 + ey 4 ;

                                         611
                                                                                                    p. 603
   but the latter is always reducible to the former by means of a definite linear
substitution; and if we call the modulus of the substitution, that is the square of
the determinant formed by the coefficients of substitution, M , to every general
invariant Iq of the qth degree, to the latter corresponds a partial form (Iq ) of
invariant to the former, such that
                                               1
                                        Iq =      (Iq );
                                               Mq
and consequently, since every (I) is a rational function of (s) and (t), so must
every I be the same of s and t; unless, indeed, it were possible to have
                                                1
                                        Iq =        (Iq′ ),
                                               M q′
q ′ being different from and greater than q: but if this were the case, since
       1
Iq = q (Iq ), a power of M the modulus would necessarily be an invariant; but
      M
in passing from x4 + y 4 + 6mx2 y 2 to x4 + y 4 + 6γ(m)x2 y 2 , 1 + 3m becomes the
modulus, which we know is not an invariant. Hence the proposition is completely
established for the case of the biquadratic function (x, y)4281 .
    Now let us proceed to Aronhold’s famous S and T , the invariants to the
general cubic function (x, y, z)3 , forms equally dear to the analyst and geometer.
(Vide Mr Salmon’s Higher Plane Curves passim.)
    The method will be precisely the same as that applied to s and t282 .
    We commence with the canonical form

                                  x3 + y 3 + z 3 + 6mxyz.

On substituting x + y + z, x + ρy + ρ2 z, x + ρ2 y + ρz for x, y, z, where ρ is the
cube root of unity, the above quantity takes the form

                         (3 + 6m){x3 + y 3 + z 3 + 6β(m)xyz},

where
                                        18 − 18m    1−m
                             β(m) =               =        ,
                                        6(3 + 6m)   1 + 2m
a periodic function in m of the second order only, for
                                         1 + 2m − 1 + m
                            β 2 (m) =                   = m.
                                        1 + 2m + 2 − 2m
                                                                                                    p. 604
 281
     I have made a tacit assumption throughout the foregoing demonstration (which is, however,
capable of an easy proof), namely that if any fractional function of the coefficients of any form
be invariantive, the numerator and denominator must be separately invariants.
 282
     The S is Mr Cayley’s property, the T belongs to Professor Boole, having been by him
imparted, in the infancy of the theory, to Mr Cayley, by whom it was first given to the world,
at least in its character as an Invariant.


                                               612
   But if we write for x in the original form ρx, it becomes
                              x3 + y 3 + z 3 + 6ρmxyz;
and if for x we write ρ2 x, it becomes
                              x3 + y 3 + z 3 + 6ρ2 mxyz.
Hence we can by linear substitutions obtain from x3 + y 3 + z 3 + 6mxyz the three
additional forms
                          x3 + y 3 + z 3 + 6β(m)xyz,
                             x3 + y 3 + z 3 + 6γ(m)xyz,
                             x3 + y 3 + z 3 + 6δ(m)xyz,
where
                         1−m                    1 − ρm     ρ2 − m
               β(m) =           ,   γ(m) = ρ2           =         ,
                         1 + 2m                 1 + 2ρm   1 + 2ρm
                                  1 − ρ2 m      ρ−m
                         δ(m) = ρ           =           .
                                 1 + 2ρ m
                                        2     1 + 2ρ2 m
In all, there will be twelve values of m forming three remarkable compound
cycles,
                        m,     β(m),      γ(m),    δ(m),
                       ρm, ρβ(m), ργ(m), ρδ(m),
                       ρ2 m, ρ2 β(m), ρ2 γ(m), ρ2 δ(m).
It would be beside my present object to seek to develope fully the functional
relations in which the several terms of these cycles stand to one another: the
interesting relations
                         β 2 (m) = γ 2 (m) = δ 2 (m) = m,
                              βγ(m) = γβ(m) = δ(m),
                              γδ(m) = δγ(m) = β(m),
                              δβ(m) = βδ(m) = γ(m),
have been already283 stated by me in another place (Cambridge and Dublin
Mathematical Journal, March 1851284 ).
   The (S) of the canonical form corresponding to the S of the general form is
m−m4 ; and the (T ) corresponding to the T of the general form is 1−20m3 −8m6 .
(See my Calculus of Forms285 , Cambridge and Dublin Mathematical Journal,
February 1852.) It is my object to show that any other invariant (I) to the
canonical form must be a rational function of S and T .
   In the first place, I observe that every invariant to any function of an odd
degree i of any odd number p of variables must be of even dimensions; for if the
degree of the dimensions be q, and D the determinant of the                      p. 605
 283
     p. 192 above.
 284
    Vide Addendum [p. 607 below].
 285
     p. 311 above.


                                         613
   coefficients of substitution, the invariant to the transform becomes the original
                                                   iq
invariant affected with a factor Diq/p , where        must be an even integer, since
                                                   p
otherwise the sign of this multiplier would be equivocal and indeterminable;
hence when i and p are both odd, q must be even. Thus, then, I(m) in the case
before us must be an even-degreed function of m. Moreover, since the change of
x into ρx converts m into ρm, and Iq (m) into ρq Iq (m), for D becomes ρ when
x, y, z become ρx, y, z, Iq (m) must be of the form ϕ(m3 ), m2 ϕ(m3 ), mϕ(m3 ),
according as the index q is of the form 6i, 6i + 2, 6i + 4.
   By precisely the same reasoning as was applied to the preceding case of (s)
and (t), we see that any invariant of m which contains mc must also contain
(1 − m)c , (1 − ρm)c , (1 − ρ2 m)c , that is must contain (m − m4 )c , which in fact
is (S)c . If, now, we consider any invariant of the qth degree in m, I(m), and
suppose it to be other than a rational function of (S) and (T ), and if we take µ
to denote the number of the solutions of

                                      4x + 6y = q,

it will follow that we may form an invariant I ′ (m), which, when q is of the form
12i or 12i + 6, will contain m, and consequently (m − m4 )3µ+2 as a factor; and in
like manner when q is of the form 12i + 2 or 12i + 8, will contain (m − m4 )3µ+2 as
a factor; and when q is of the form 12i + 4 or 12i + 10 will contain (m − m4 )3µ+1
as a factor. Now when
                                   q = 12i,     µ = i + 1,
                               q = 12i + 6,     µ = i + 1;

when
                               q = 12i + 2,     µ = i,
                               q = 12i + 8,     µ = i + 1;
when
                               q = 12i + 10,    µ = i + 1,
                                q = 12i + 4,    µ = i + 1.
Hence the factors dividing Iq in these several cases will be of the respective
degrees

       12i + 12, ; 12i + 12;       12i + 8, 12i + 12;        12i + 16, 12i + 16;

corresponding to q, being of the several values

            12i, 12i + 6;        12i + 2, 12i + 8;      12i + 10, 12i + 4;

which is clearly impossible. This proves the theorem in question (the passage
being made from the canonical to the general form, as in the former part of
this investigation), to wit, that S and T form what I have elsewhere termed

                                          614
a fundamental scale of invariants to the cubic ternary form, entering as the
exclusive ingredients into every other invariant that can be derived from such
form.                                                                                    p. 606
   A word of warning is necessary before I lay down my pen: that there can
be only two algebraically independent invariants to (x, y)4 or (x, y, z)3 , is an
immediate consequence of the canonical form of each having but one parameter;
so in general there can be at most but (n − 2) absolutely independent invariants
of (x, y)n ; but the point established in the preceding investigation goes to show
that there can exist no other invariants than such as are rational functions of s
and t in the one case, and S and T in the other. I shall take some other occasion
to establish a similar conclusion for the forms (x, y)5 and (x, y)6 .
   I have shown that there exist three invariants to the one of the degrees 4,
8, 12, and four to the other of the degrees 2, 4, 6, 10; and I shall demonstrate
that any other invariant to either form must be a rational function of those
above stated. For the cubic form (x, y)3 we know that there is but one invariant,
namely its discriminant. Thus, then, for n = 3, n = 4, n = 5, n = 6 the number of
absolutely independent invariants is n−2, and the number of linearly independent
invariants is no greater. But this result is by no means generally true. It may be
proved by means of a great law of reciprocity286 which I myself originated, but
unfortunately threw aside, and which M. Hermite has since demonstrated, that
there are more than five linearly independent invariants to (x, y)8 , and more
than ten, in fact twelve at least, to (x, y)12 ; that is to say, it is impossible in the
latter case to find ten of which all the rest shall be rational functions, although
an algebraical equation connects any 11. So, again, if we take a system of two
cubic equations, there are only five absolutely independent invariants; but there
are not less than seven linearly independent fundamental invariants,                     p. 607

 286
     The theorem of reciprocity alluded to in the text is the following:—If to any function (x, y)n
there exists an invariant of the order m in the coefficients, then to (x, y)m there exists an
invariant of the order n in the coefficients; or more generally, which is M. Hermite’s addition, if
to any system of functions (x, y)n1 , (x, y)n2 . . . (x, y)ni there exists an invariant of the several
dimensions m1 , m2 . . . mi in the respective sets of coefficients, then conversely to a system
(x, y)m1 , (x, y)m2 . . . (x, y)mi there exists an invariant of the dimensions n1 , n2 . . . ni in the
respective sets of coefficients.
   I had previously shown in this Magazine [p. 279 above], that Mr Cayley’s formulae for finding
the number of biquadratic invariants to any function (x, y)n , given in that remarkable paper of
his on linear transformations [Cayley’s Collected Papers, Vol. I., p. 95], where first dawned upon
the world the clear and full-formed idea of invariants (the most original and important infused
into analysis since the discovery of fluxions), could be expressed by means of the number of
solutions of the equation in integers 2x + 3y = n, the square of the quadratic invariant (which
only exists for even values of n) counting for one in the fundamental biquadratic scale; this
is of course a direct consequence, through the law of reciprocity, of the fundamental scale to
(x, y)4 consisting of a quadratic and a cubic invariant. My discovery of the fundamental scale of
invariants to (x, y)5 and (x, y)6 now enables us, through the same law of reciprocity, to express
the number of distinct Quintic and Sextic invariants to (x, y)n , namely as being the number of
integer solutions of x + 2y + 3z = n4 in the one case, and of x + 2y + 3z + 5t = n2 in the other.



                                                615
   of which any other invariant must be a rational function. In fact, if we take
for our two cubics
                       U = ax3 + 3bx2 y + 3cxy 2 + dy 3 ,
                        V = αx3 + 3βx2 y + 3γxy 2 + δy 3 ,
the five coefficients of the powers of λ, in the discriminant of U + λV , each
of which is of four dimensions in the two sets of coefficients combined, are all
invariants of the system; but there will be besides two more, one of which is a
Combinant of six dimensions, being the resultant of U and V ; the other is a
Combinant of two dimensions only, namely aδ − 3bγ + 3cβ − dα. These seven
together form the fundamental constituent scale.
   The two last-mentioned may be expressed algebraically (by the introduction of
square roots) as functions of the other five, but of course not as rational functions
of the same. My attention was more particularly called to the search of a proof
of the completeness of the Aronholdian system of invariants, by an inquiry as
to the possibility of rigidly demonstrating that there could exist no others not
made up of these, addressed to me in the spring of last year by one of the most
gifted geometers of this or any other country. A morning or two after the inquiry
reached me, in a walk before breakfast by the side of the ornamental water in
St James’s Park (a time and place by no means, according to my experience,
unfavourable to the inspirations of the analytic muse), I had the satisfaction of
falling upon the rather piquant demonstration above given, which essentially
rests upon a principle, requiring no harder exercise of faith than the concession
of the impossibility of a greater being contained in or proceeding out of a less.

                                   Addendum.

On the nature of the three Cycles of four terms each which contain the twelve
 values of the parameter to the canonical form of a cubic function of three
                                  variables.

   The equations given in the text [p. 604 above] show that each term in any one
cycle is a periodic function of the second order of each other term in the same
cycle. Moreover, it may be shown that each term in any one cycle is a periodic
function of the third order of every term in either of the other two cycles; a sort
of relation between the cycles taken per se, and with one another, precisely the
inverse of what obtains (as already shown) for the two cycles of three terms
containing the six values of the parameter to the biquadratic function of two
variables. For as regards that case, it was shown                                   p. 608
   in the first part of this paper that the terms in the same cycle are periodic
functions of the third order of one another, and of the second order of each of
those not in the same cycle with themselves.


                                        616
  If we make
                      1−m                     ρ2 − m                  ρ−m
       m = A,                = B,                    = C,                     = D,
                      1 + 2m                 1 + 2ρm                1 + 2ρ2 m

                ρA = A′ ,      ρB = B ′ ,          ρC = C ′ ,      ρD = D′ ,
           ρ2 A = A′′ ,      ρ2 B = B ′′ ,         ρ2 C = C ′′ ,     ρ2 D = D′′ .
The following table will exhibit all the ternary periods that can be formed
between the terms of the several cycles:—

          (1)    AB ′ D′′ , (4) BA′ C ′′ ,    (7) CA′ D′′ , (10)          DA′ B ′′ ,
          (2)    AC ′ B ′′ , (5) BC ′ D′′ , (8) CB ′ A′′ ,         (11)   DB ′ C ′′ ,
          (3)    AD′ C ′′ , (6) BD′ A′′ , (9)          CD′ B ′′ , (12)    DC ′ A′′ .

For instance, as an example of the meaning of the table, take line (8), namely
CB ′ A′′ . This indicates that A′′ is formed from B ′ and C from A′′ in the same
way as B ′ from C, and of course A′′ from C in the same way as C from B ′ and
B ′ from A′′ , &c. By means of this table it will easily be seen that a term in each
of two cycles being given, the term in the third which forms with the given two
a ternary period may immediately be assigned.
   The remarks which I have to add on the nature of the equations for finding
the parameter m, as well for (x, y)4 as for (x, y, z)3 , will be given hereafter.




                                             617
