Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/313

and the potential energy per unit volume is

${\tfrac {1}{2}}E(\partial \eta /\partial x)^{2}+{\tfrac {1}{2}}\sigma p^{2}\zeta ^{2}$.

The equations of motion, derived by the process usual in dynamics, are

${\begin{cases}\rho {\frac {\partial ^{2}\eta }{\partial t^{2}}}+\sigma \left({\frac {\partial ^{2}\eta }{\partial t^{2}}}+{\frac {\partial ^{2}\zeta }{\partial t^{2}}}\right)-E{\frac {\partial ^{2}\eta }{\partial x^{2}}}=0,\\\sigma \left({\frac {\partial ^{2}\eta }{\partial t^{2}}}+{\frac {\partial ^{2}\zeta }{\partial t^{2}}}\right)+\sigma p^{2}\zeta =0.\end{cases}}$

Consider the propagation, through the medium thus constituted, of vibrations whose frequency is n, and whose velocity of propagation in the medium is v; so that η and ζ are harmonic functions of n(t - x/v). Substituting these values in the differential equations, we obtain

${\frac {1}{v^{2}}}={\frac {\rho }{E}}+{\frac {\sigma p^{2}}{E(p^{2}-n^{2})}}$.

Now, ρ/E has the value 1/c², where c denotes the velocity of light in free aether; and c/v is the refractive index μ of the medium for vibrations of frequency n. So the equation, which may be written

$\mu ^{2}=1+{\frac {\sigma p^{2}}{\rho (p^{2}-n^{2})}}$,

determines the refractive index of the substance for vibrations of any frequency n. The same formula was independently obtained from similar considerations three years later by W. Sellmeier.^[1]

If the oscillations are very slow, the incident light being in the extreme infra-red part of the spectrum, n is small, and the equation gives approximately μ² = (ρ + σ)/ρ: for such oscillations, each atomic particle and its shell move together as a rigid body, so that the effect is the same as if the aether were simply loaded by the masses of the atomic particles, its rigidity remaining unaltered.

1. Ann. d. Phys. cxlv (1872), pp. 399, 520: cxlvii (1872), pp. 385, 525. Cf. also Helmholtz, Ann. d. Phys. cliv (1875), p. 582.