Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/149

displacement), which urges the molecules of the medium parallel to the wave-front. Hence the velocity of propagation of a wave, measured at right angles to its front, is proportional to the square root of the component, along the direction of displacement, of the elastic force per unit displacement, and the velocity of propagation of such a plane-polarized wave as we have considered is proportional to the radius vector of the surface of elasticity in the direction of displacement.

Moreover, any displacement in the given wave-front can be resolved into two, which are respectively parallel to the two axes of the diametral section of the surface of elasticity by a plane parallel to this wave-front; and it follows from what has been said that each of these component displacements will be propagated as an independent plane-polarized wave, the velocities of propagation being proportional to the axes of the section,^[1] and therefore inversely proportional to the axes of the section of the inverse surface of this with respect to the origin, which is the ellipsoid

${\frac {x^{2}}{\epsilon _{1}}}+{\frac {y^{2}}{\epsilon _{2}}}+{\frac {z^{2}}{\epsilon _{3}}}=1$.

But this is precisely the result to which, as we have seen, Fresnel lad been led by purely geometrical considerations; and thus his geometrical conjecture could now be regarded as substantiated by a study of the dynamics of the medium.

It is easy to determine the wave-surface or locus at any instant—say, t = 1—of a disturbance originated at some previous instant—say,t=0—at some particular point—say, the origin. For this wave-surface will evidently be the envelope of plane waves emitted from the origin at the instant t = 0—that is, it will be the envelope of planes

$lx+my+nz-v=0$,

where the constants l, m, n, v are connected by the identical equation$l^{2}+m^{2}+n^{2}=1$,

1. It is evident from this that the optic axes, or lines of single wave-velocity, along which there is no double refraction, will be perpendicular to the two circular sections of the surface of elasticity.

K