Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/312

We may all modern theories of dispersion. The earliest of these was devised by Maxwell, who, in the Cambridge Mathematical Tripos for 1869,^[1] published the results of the following investigation:—

A model of a dispersive medium may be constituted by embedding systems which represent the atoms of ponderable matter in a medium which represents the aether. picture each atom^[2] as composed of a single massive particle supported symmetrically by springs from the interior face of a massless spherical shell: if the shell be fixed, the particle will be capable of executing vibrations about the centre of the sphere, the effect of the springs being equivalent to a force on the particle proportional to its distance from the centre. The atoms thus constituted may be supposed to occupy small spherical cavities in the aether, the outer shell of each atom being in contact with the aether at all points and partaking of its motion. An immense number of atoms is supposed to exist in each unit volume of the dispersive medium, so that the medium as a whole is fine-grained.

Suppose that the potential energy of strain of free aether per unit volume is

${\tfrac {1}{2}}\mathbf {E} \left({\frac {\partial \eta }{\partial x}}\right)^{2}$,

where η denotes the displacement and E an elastic constant; so that the equation of wave-propagation in free aether is

$\rho {\frac {\partial ^{2}\eta }{\partial t^{2}}}=E{\frac {\partial ^{2}\eta }{\partial x^{2}}}$,

where ρ denotes the aethereal density.

Then if σ denote the mass of the atomic particles in unit volume, (η + ζ) the total displacement of an atomic particle at the place x at tine t, and σp²ζ the attractive force, it is evident that for the compound medium the kinetic energy per unit volume is

${\tfrac {1}{2}}\rho {\frac {\partial \eta }{\partial t}}^{2}+{\tfrac {1}{2}}\sigma \left({\frac {\partial \eta }{\partial t}}+{\frac {\partial \zeta }{\partial t}}\right)^{2}$,

1. Cambridge: Calendar, 1869; republished by Lord Rayleigh, Phil. Mag. xlviii (1899), p. 151.
2. This illustration is due to W. Thomson.