Extrait of the Supplement to the Memoir on double refraction

English translation of A. Fresnel, "Extrait du Supplément au Mémoire sur la double réfraction" (read 13 Jan. 1822?), printed in Oeuvres complètes d'Augustin Fresnel, vol. 2 (1868), pp. 335–42,  with the corresponding extract from the "Table Analytique" in Oeuvres complètes..., vol. 3 (1870), at pp. 645–7.

Translator's abstract:

The hypothesis entertained in the First Memoir—that the wave-normal speeds in a biaxial birefringent crystal are given by the maximum and minimum radii of the diametral section of an ellipsoid in a plane parallel to the wavefront—is an approximation valid for weakly birefringent crystals. It cannot be true in general, because it does not give the correct result for calcite—a strongly birefringent (albeit uniaxial) crystal. But the author has accounted for all these cases by supposing the existence of three perpendicular directions (axes of elasticity) in which a displacement produces a restoring force parallel thereto. Then, for a general orientation of the wavefront, the permitted polarizations (those which can propagate unchanged) are those for which the restoring force is coplanar with the displacement and the wave-normal. It turns out that the associated directions of displacement are those in which the component of the restoring force parallel to the displacement is a maximum or a minimum (per unit displacement). Hence the wave-normal speed is indeed given by the maximum and minimum radii of the diametral section of a certain surface in a plane parallel to the wavefront. The equation of this surface, which the author calls the surface of elasticity, turns out to be

    v² = a² cos² X + b² cos² Y + c² cos² Z ,

where v is the wave-normal speed,  X,Y, Z are the angles between the wave-normal velocity and the coordinate axes, and the constants a,b,c  are the semi-axes of the surface. This surface is indeed well approximated by an ellipsoid when a,b,c  are not too different. Like an ellipsoid, it has the property that there are generally two directions in which a plane through the center cuts the surface in a circle, and these directions merge when two of the constants a,b,c  are equal—explaining why there are at most two optical axes, and sometimes only one.

If the object point is sufficiently distant from the crystal that the waves can be taken as plane, the refractions can be worked out by knowing the wave-normal speed as a function of direction. If the object point is closer, however, it becomes necessary to know the shape of the wave surface (secondary wavefront) within the crystal. This surface is tangential to all the plane wavefronts, with all orientations, that travel from the origin in unit time. For a uniaxial crystal, the wave surface reduces to an ellipsoid of revolution, in agreement with Huygens' theory.

That the wave-normal speed is proportional to the square root of the elasticity in play can be shown by analogy with waves on a stretched string.

The author concludes by drawing attention to the extreme economy of assumptions by which he accounts for the laws of polarization and double refraction of both biaxial and uniaxial crystals.