This expression contains the correct number of constants, namely, four: three of them represent the optical constants of a biaxal crystal, and one (namely, μ) represents the square of the velocity of propagation of longitudinal waves. It is found that the two sheets of the wave-surface which correspond to the two distortional waves form a Fresnel's wave-surface, the third sheet, which corresponds to the longitudinal wave, being an ellipsoid. The directions of polarization and the wave-velocities of the distortional waves are identical with those assigned by Fresnel, provided it is assumed that the direction of vibration of the aether-particles is parallel to the plane of polarization; but this last assumption is of course inconsistent with Green's theory of reflexion and refraction.
In his Second Theory, Green, like Cauchy, used the condition that for the waves whose fronts are parallel to the coordinate planes, the wave-velocity depends only on the plane of polarization, and not on the direction of propagation. He thus obtained the equations already found by Cauchy—
G - f = H - g = I - h.
The wave-surface in this case also is Fresnel's, provided it is assumed that the vibrations of the aether are executed at right angles to the plane of polarization.
The principle which underlies the Second Theories of Green and Cauchy is that the aether in a crystal resembles an elastic solid which is unequally pressed or pulled in different directions by the unmoved ponderable matter. This idea appealed strongly to W. Thomson (Kelvin), who long afterwards developed it further,[1] arriving at the following interesting result:—Let an incompressible solid, isotropic when unstrained, be such that its potential energy per unit volume is
,
where q denotes its modulus of rigidity when unstrained, and
- ↑ Proc. R. S. Edin. xv (1887), p. 21: Phil. Mag. xxv (1888) p. 116: Baltimore Lectures (ed. 1904), pp. 228-259.
