Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/235

plane of polarization rotates in unit length of path is a numerical multiple of

${\frac {\mu }{\tau ^{r+s-1}c_{1}^{r+1}}}$,

where τ denotes the period of the light. Now it was shown by Verdet^[1] that the magnetic rotation is approximately proportional to the inverse square of the wave-length; and hence we must have

$r+s=3$;

so that the only equations capable of correctly representing Faraday's effect are either

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

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

The former pair arise, as will appear later, in Maxwell's theory of rotatory polarization: the latter pair, which were suggested in 1868 by Boussinesq.^[2] follow from that physical theory of the phenomenon which is generally accepted at the present time.^[3]

Airy's work on the magnetic rotation of light was limited in the same way as MacCullagh's work on the rotatory power of quartz; it furnished only an analytical representation of the effect, without attempting to justify the equations. The earliest endeavour to provide a physical theory seems to have been made in 1858, in the inaugural dissertations of Carl Neumann,

1. Comptes Rendus, lvi (1863), p. 630.
2. Journal de Math., xi (1868), p. 430.
3. Y and Z being interpreted as components of electric force.