Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/167

reject the first theory of crystal-optics in favour of the second. After 1836 he consistently adhered to the view that the vibrations of the aether are performed at right angles to the plane of polarization. In that year he made another attempt to frame a satisfactory theory of reflexion,^[1] based on the assumption just mentioned, and on the following boundary-conditions:—At the interface between two media curl e is to be continuous, and (taking the axis of x normal to the interface) $\partial e_{x}/\partial x$ is also to be continuous.

Again we find no very satisfactory reasons assigned for the choice of the boundary-conditions, and as the continuity of e itself across the interface is not included amongst the conditions chosen, they are obviously open to criticism; but they lead to Fresnel's sine- and tangent-equations, which correctly express the actual behaviour of light.^[2] Cauchy remarks that in order to justify them it is necessary to abandon the assumption of his earlier theory, that the density of the aether is the same in all material bodies.

It may be remarked that neither in this nor in Cauchy's earlier theory of reflexion is any trouble caused by the appearance of longitudinal waves when a transverse wave is reflected, for the simple reason that he assumes the boundary-conditions to be only four in number; and these can all be satisfied without the necessity for introducing any but transverse vibrations.

These features bring out the weakness of Cauchy's method of attacking the problem. His object was to derive the properties of light from a theory of the vibrations of elastic solids. At the outset he had already in his possession the differential equations of motion of the solid, which were to be his starting-point, and the equations of Fresnel, which were to be his goal. It only

1. Comptes Rendus, ii. (1836), p. 341: "Mémoire sur la dispersion de la lumière" (Nouveaux exercices de Math., 1836), p. 203.
2. These boundary-conditions of Cauchy's are, as a matter of fact, satisfied by the electric force in the electro-magnetic theory of light. The continuity of curl e is equivalent to the continuity of the magnetic vector across the interface, and the continuity of $\partial e_{x}/\partial x$ leads to the same equation as the continuity of the component of electric force in the direction of the intersection of the interface with the plane of incidence.

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