Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/450

for the specific inductive capacity in terms of the number and circumstances of the electrons.^[1]

Returning now to the case in which the dielectric is supposed to be in motion, the equation for the polarization may be written

$4\pi c^{2}\mathbf {P} =(\mu ^{2}-1)\mathbf {E} ^{\prime };\qquad \qquad (2)$

from this equation, Fresnel's formula for the velocity of light in a moving dielectric may be deduced. For, let the axis of z be taken parallel to the direction of motion of the dielectric, which is supposed to be also the direction of propagation of the light; and, considering a plane-polarized wave, take the axis of x parallel to the electric vector, so that the magnetic vector must be parallel to the axis of y. Then equation (III) above becomes

$-\partial H_{y}/\partial z=4\pi \mathbf {\dot {D}} _{x}+4\pi w\partial P_{x}/\partial z$;

equation (IV) becomes (assuming B equal to II, as is always the case in optics),

$-\partial E_{x}/\partial z={\dot {H}}_{y}$.

The equation which defines the electric induction gives

$D_{x}=(1/4\pi c^{2})E_{x}+P_{x}$;

and equations (1) and (2) give

$4\pi c^{2}P_{x}=(\mu ^{2}-1)(E_{x}-wH_{y})$.

Eliminating D., Px, and Ily, we have

$c^{2}{\frac {\partial ^{2}E_{x}}{\partial z_{2}}}={\frac {\partial ^{2}E_{x}}{\partial t^{2}}}+\left(\mu ^{2}-1\right)\left({\frac {\partial }{\partial t}}+w{\frac {\partial }{\partial z}}\right)^{2}E_{x}$;

or, neglecting w/cº,

${\frac {\partial ^{2}E_{x}}{\partial z^{2}}}={\frac {\mu ^{2}}{c^{2}}}{\frac {\partial ^{2}E_{x}}{\partial t^{2}}}+{\frac {}{c^{2}}}{\frac {\partial ^{2}E_{x}}{\partial t\partial z}}$.^[2]

Substituting $E_{x}=e^{n(t-z/V){\sqrt {-1}}}$, so that V denotes the velocity of light in the moving dielectric with respect to the fixed aether we have

$c^{2}=\mu ^{2}V^{2}-2w(\mu ^{2}-1)V$,

1. Cf. p. 211.
2. This equation was first given as a result of the theory of electrons by Lorentz in the last chapter of his memoir of 1892, Arch. Néerl. xxv, p. 525. It was also given by Larmor, Phil. 'Trans., clxxxv (1894), p. 891.