Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/234

In the year following Faraday's discovery, Airy^[1] suggested a way of representing the effect analytically; as might have been expected, this was by modifying the equations which had been already introduced by MacCullagh for the case of naturally active bodies. In MacCullagh's equations

${\begin{cases}{\frac {\partial ^{2}Y}{\partial t^{2}}}=c_{1}^{2}{\frac {\partial ^{2}Y}{\partial x^{2}}}+\mu {\frac {\partial ^{2}Z}{\partial x^{3}}}\\{\frac {\partial ^{2}Z}{\partial t^{2}}}=c_{1}^{2}{\frac {\partial ^{2}Z}{\partial x^{2}}}-\mu {\frac {\partial ^{2}Y}{\partial x^{3}}}\end{cases}}$,

the terms ${\tfrac {\partial ^{3}Z}{\partial x^{3}}}$ and ${\tfrac {\partial ^{3}Y}{\partial x^{3}}}$ change sign with x, so that the rotation of the plane of polarization is always right-handed or always left-handed with respect to the direction of the beam. This is the case in naturally-active bodies; but the rotation due to a magnetic field is in the same absolute direction whichever way the light is travelling, so that the derivations with respect to x must be of even order. Airy proposed the equations

${\begin{cases}{\frac {\partial ^{2}Y}{\partial t^{2}}}=c_{1}^{2}{\frac {\partial ^{2}Y}{\partial x^{2}}}+\mu {\frac {\partial Z}{\partial t}}\\{\frac {\partial ^{2}Z}{\partial t^{2}}}=c_{1}^{2}{\frac {\partial ^{2}Z}{\partial x^{2}}}-\mu {\frac {\partial Y}{\partial t}}\end{cases}}$,

where μ denotes a constant, proportional to the strength of the magnetic field which is used to produce the effect. He remarked, however, that instead of taking $\mu {\tfrac {\partial Z}{\partial t}}$ and $\mu {\tfrac {\partial Y}{\partial t}}$ as the additional terms, it would be possible to take $\mu {\tfrac {\partial ^{3}Z}{\partial t^{3}}}$ and $\mu {\tfrac {\partial ^{3}Y}{\partial t^{3}}}$, or $\mu {\tfrac {\partial ^{3}Z}{\partial x^{2}\partial t}}$ and $\mu {\tfrac {\partial ^{3}Z}{\partial x^{2}\partial t}}$, or any other derivates in which the number of differentiations is odd with respect to t and even with respect to x. It may, in fact, be shown by the method previously applied to MacCullagh's formulae that, if the equations are ${\begin{cases}{\frac {\partial ^{2}Y}{\partial t^{2}}}=c_{1}^{2}{\frac {\partial ^{2}Y}{\partial x^{2}}}+\mu {\frac {\partial ^{r+s}Z}{\partial x^{r}\partial t^{s}}}\\{\frac {\partial ^{2}Z}{\partial t^{2}}}=c_{1}^{2}{\frac {\partial ^{2}Z}{\partial x^{2}}}-\mu {\frac {\partial ^{r+s}Y}{\partial x^{r}\partial t^{s}}}\end{cases}}$, where (r + s) is an odd number, the angle through which the

1. Phil. Max. xxviii (1848) p. 469.