Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/204

This theory, in one modification or another, held its ground until forty years later it was overthrown by the facts of anomalous dispersion.

The distinction between aether and ponderable matter was more definitely drawn in memoirs which were published independently in 1841–2 by F. E. Neumann^[1] and Matthew O'Brien.^[2] These authors supposed the ponderable particles to remain sensibly at rest while the aether surges round them, and is acted on by them with forces which are proportional to its displacement. Thus^[3] the equation of motion of the aether becomes

$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-(k+{\tfrac {4}{3}}n){\text{grad div }}\mathbf {e} -n{\text{ curl curl }}\mathbf {e} -C\mathbf {e}$,

where C denotes a constant on which the phenomena of dispersion depend. For polarized plane waves propagated parallel to the axis of x, this equation becomes

$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=n{\frac {\partial ^{2}\mathbf {e} }{\partial x^{2}}}-C\mathbf {e}$;

and substituting

$\mathbf {e} =e^{{\frac {2\pi {\sqrt {-1}}}{\tau }}(t-{\frac {x}{V}})}$,

where τ denotes the period and V the velocity of the light, we have

${\frac {n}{V^{2}}}=\rho -{\frac {C}{r\pi ^{2}}}r^{2}$,

an equation which expresses the dependence of the velocity on the period.

The attempt to represent the properties of the aether by those of an elastic solid lost some of its interest after the rise of the electromagnetic theory of light. But in 1867,

1. Berlin Abhandlungen aus dem Jahre 1841, Zweiter Teil, p. 1: Berlin, 1843.
2. Trans. Camb. Phil. Soc. vii (1842), p. 397.
3. O'Brien, loc. cit., §§ 15, 28.