Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/191

A way of escape from this conclusion suggested itself to Stokes,^[1] and later to Rankine^[2] and Lord Rayleigh.^[3] What if the aether in a crystal, instead of having its elasticity different in different directions, were to have its rigidity invariable and its inertia different in different directions? This would bring the theory of crystal-optics into complete agreement with Fresnel's and Green's theory of reflexion, in which the optical differences between media are attributed to differences of inertia of the aether contained within them. The only difficulty lies in conceiving how aelotropy of inertia can exist; and all three writers overcame this obstacle by pointing out that a solid which is immersed in a fluid may have its effective inertia different in different directions. For instance, a coin immersed in water moves much more readily in its own plane than in the direction at right angles to this.

Suppose then that twice the kinetic energy per unit volume of the aether within a crystal is represented by the expression

$\rho _{1}\left({\frac {\partial e_{x}}{\partial t}}\right)^{2}+\rho _{2}\left({\frac {\partial e_{y}}{\partial t}}\right)^{2}+\rho _{3}\left({\frac {\partial e_{z}}{\partial t}}\right)^{2}$

and that the potential energy per unit volume has the same value as in space void of ordinary matter. The aether is assumed to be incompressible, so that div e is zero: the potential energy per unit volume is therefore

${\tfrac {1}{2}}n\left\{\left({\frac {\partial e_{z}}{\partial y}}+{\frac {\partial e_{y}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{x}}{\partial z}}+{\frac {\partial e_{z}}{\partial x}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial x}}+{\frac {\partial e_{x}}{\partial y}}\right)^{2}-4{\frac {\partial e_{y}}{\partial y}}{\frac {\partial e_{z}}{\partial z}}\right.$         

$\left.-4{\frac {\partial e_{z}}{\partial z}}{\frac {\partial e_{x}}{\partial x}}-4{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{y}}{\partial y}}\right\}$,

where n denotes as usual the rigidity.

1. Stokes, in a letter to Lord Rayleigh, inserted in his Memoir and Scientific Correspondence, ii, p. 99, explains that the idea presented itself to him while he was writing the paper on Fluid Motion which appeared in Trans. Camb. Phil. Soc., viii (1843), p. 105. He suggested the wave-surface to which this theory leads in Brit. Assoc, Rep., 1862, p. 269.
2. Phil. Mag. (4), i (1851), p. 441.
3. Phil. Mag. (4), xli (1871), p. 519.