Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/195

amounts in passing through a given thickness of the substance: at any stage they may be recompounded into a plane-polarized ray, the azimuth of whose plane of polarization varies with the length of path traversed.

It is readily seen from this that a ray of light incident on a crystal of quartz will in general bifurcate into two refracted rays, each of which will be elliptically polarized, i.e. will be capable of resolution into two plane-polarized components which differ in phase by a definite amount. The directions of these refracted rays may be determined by Huygens' construction, provided the wave-surface is supposed to consist of a sphere and spheroid which do not touch.

The first attempt to frame a theory of naturally active bodies was made by MacCullagh in 1836.^[1] Suppose a plane wave of light to be propagated within a crystal of quartz. Let (x, y, z) denote the coordinates of a vibrating molecule, when the axis of x is taken at right angles to the plane of the wave, and the axis of z at right angles to the axis of the crystal. Using Y and Z to denote the displacements parallel to the axes of y and z respectively at any time t, MacCullagh assumed that the differential equations which determine Y and Z are

${\frac {\partial ^{2}Y}{\partial t^{2}}}=c_{1}^{2}{\frac {\partial ^{2}Y}{\partial x^{2}}}+\mu {\frac {\partial ^{3}Z}{\partial x^{3}}}$

${\frac {\partial ^{2}Z}{\partial t^{2}}}=c_{2}^{2}{\frac {\partial ^{2}Z}{\partial x^{2}}}-\mu {\frac {\partial ^{3}Y}{\partial x^{3}}}$

where μ denotes a constant on which the natural rotatory property of the crystal clepends. In order to avoid complications arising from the ordinary crystalline properties of quartz, we shall suppose that the light is propagated parallel to the optic axis, so that we can take c₁ equal to c₂.

Assuming first that the beam is circularly polarized, let it be represented by

$Y=A\sin {\frac {2\pi }{\tau }}(lx-t)$, $Z=\pm A\cos {\frac {2\pi }{\tau }}(lx-t)$,

1. Trans, Royal Irish Acad., xvii.; MacCullagh's Coll. Works, p. 63.