Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/171

elastic constants already introduced; by substituting this value of φ in the general variational equation

$\iiint \rho \left\{{\frac {\partial ^{2}e_{x}}{\partial t^{2}}}\partial e_{x}+{\frac {\partial ^{2}e_{y}}{\partial t^{2}}}\partial e_{x}y+{\frac {\partial ^{2}e_{z}}{\partial t^{2}}}\partial e_{z}\right\}\ dx\ dy\ dz=-\iiint \partial \phi \ dx\ dy\ dz$

(where ρ denotes the density), the equation of motion may be deduced.

But this method does more than merely furnish the equation of motion

$\rho {\ddot {\mathbf {e} }}=-\left(k+{\frac {4}{3}}n\right)\ \mathrm {grad\ div} \ \mathbf {e} -n\ \mathrm {curl\ curl} \ \mathbf {e}$;

or,

$\rho {\ddot {\mathbf {e} }}=-\left(k+{\frac {1}{3}}n\right)\mathrm {grad\ div} \ \mathbf {e} +n\nabla ^{2}\mathbf {e}$,

which had already been obtained by Cauchy; for it also yields the boundary-conditions which must be satisfied at the interface between two elastic media in contact; these are, as might be guessed by physical intuition, that the three components of the displacement^[1] and the three components of stress across the interface are to be equal in the two media. If the axis of x be taken normal to the interface, the latter three quantities are

$\left(k-{\frac {2}{3}}n\right)\mathrm {div} \ \mathbf {e} +2n{\frac {\partial e_{x}}{\partial _{x}}}$, $n\left({\frac {\partial e_{z}}{\partial x}}+{\frac {\partial e_{x}}{\partial z}}\right)$, and $n\left({\frac {\partial e_{x}}{\partial y}}+{\frac {\partial e_{y}}{\partial x}}\right)$.

The correct boundary-conditions being thus obtained, it was a simple matter to discuss the reflexion and refraction of an incident wave by the procedure of Fresnel and Cauchy. The result found by Green was that if the vibration of the aethereal molecules is executed at right angles to the plane of incidence, the intensity of the reflected light obeys Fresnel's sine-law, provided the rigidity n is assumed to be the same for all media, but the inertia ρ to vary from one medium to another. Since the sine-law is known to be true for light polarized in the plane of incidence, Green's conclusion confirmed the hypotheses of

1. These first three conditions are of course not dynamical but geometrical.