Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/475

and propagated parallel to the axis of z, the electric vector being parallel to the axis of x. Thus the equations of motion reduce to

${\begin{cases}{\frac {\partial ^{2}E_{x}}{\partial z^{2}}}&=&{\frac {1}{c^{2}}}{\frac {\partial ^{2}E_{x}}{\partial t^{2}}}+4\pi {\frac {\partial ^{2}P_{x}}{\partial t^{2}}}\\E_{x}&=&\alpha {\frac {\partial ^{2}P_{x}}{\partial t^{2}}}+\beta {\frac {\partial ^{2}P}{\partial t}}+\gamma P_{x}.\end{cases}}$

For E_x, and P_x we may substitute exponential functions of

$n{\sqrt {-1}}(t-z\mu /c)$,

where n denotes the frequency of the light, and μ the quasi-index of refraction of the metal: the equations then give at once

$(\mu ^{2}-1)(-\alpha n^{2}+\beta n{\sqrt {-1}}+\gamma )=4\pi c^{2}$.

Writing $\nu (1-\kappa {\sqrt {-1}})$ for μ, so that ν is inversely proportional to the velocity of light in the medium, and κ denotes the coefficient of absorption, and equating separately the real and imaginary parts of the equation, we obtain

${\begin{cases}\nu ^{2}(1-\kappa ^{2})&=&1+{\frac {4\pi c^{2}(\gamma -\alpha n^{2})}{\beta ^{2}n^{2}+(\gamma -\alpha n^{2})^{2}}};\\\nu ^{2}\kappa &=&{\frac {2\pi c^{2}\beta n}{\beta ^{2}n^{2}+(\gamma -\alpha n^{2})^{2}}}\end{cases}}$

When the wave-length of the light is very large, the inertia represented by the constant α has but little influence, and the equations reduce to those of Maxwell's original theory^[1] of the propagation of light in metals. The formulae were experimentally confirmed for this case by the researches of E. Hagen and H, Rubens^[2] with infra-red light; a relation being thus established between the ohmic conductivity of a metal and its optical properties with respect to light of great wavelength.

When, however, the luminous vibrations are performed more rapidly, the effect of the inertia becomes predominant; and

1. Cf. p. 290.
2. Berlin Sitzungsber., 1903, pp. 269, 410; Ann. d. Phys. xi (1903), p. 873; Phil. Mag. vii (1904), p. 157.