Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/473

value as found from the first term, namely, eE/(κ² – mn²). The equation thus becomes

$\mathbf {r} ={\frac {e\mathbf {E} }{\kappa ^{2}-mn^{2}}}+{\frac {e^{2}}{(\kappa ^{2}-mn^{2})^{2}}}[\mathbf {{\dot {E}}.H} ]$.

If P denote^[1] the electric moment per unit volume, we have

P = er × the number of such systems in unit volume of the medium;

so P must be of the form

${\frac {\epsilon -1}{4\pi c^{2}}}\mathbf {E} +\sigma [\mathbf {{\dot {E}}.H} ]$,

where ε evidently represents the dielectric constant of the medium, and σ is the coefficient which measures the magnetic rotatory power. In the magneto-optic term we may replace H by K, the external magnetic force, since this is large compared with the magnetic force of the luminous vibrations. Thus if D denote the electric induction, we have

$\mathbf {D} =\epsilon \mathbf {E} /4\pi c^{2}+[\mathbf {{\dot {E}}.K} ]$.

Combining this with the usual electromagnetic equations,

${\begin{matrix}{\text{curl }}\mathbf {H} &=&4\pi \mathbf {\dot {D}} ,\\{\text{curl }}\mathbf {E} &=&-\mathbf {\dot {H}} ,\end{matrix}}$

we have

$-{\text{curl curl }}\mathbf {E} =\epsilon \mathbf {\ddot {E}} /c^{2}+4\pi \sigma [\mathbf {{\overset {...}{E}}.K} ]$.

When a plane wave of light is propagated through the medium in the direction of the lines of magnetic force, and the axis of x is taken parallel to this direction, the equation gives

${\begin{cases}{\frac {\partial ^{2}E_{y}}{\partial x^{2}}}&=&{\frac {\epsilon }{c^{2}}}{\frac {\partial ^{2}E_{y}}{\partial t^{2}}}+4\pi \sigma K{\frac {\partial ^{3}E_{z}}{\partial t^{3}}}\\{\frac {\partial ^{2}E_{z}}{\partial x^{2}}}&=&{\frac {\epsilon }{c^{2}}}{\frac {\partial ^{2}E_{z}}{\partial t^{2}}}-4\pi \sigma K{\frac {\partial ^{3}E_{y}}{\partial t^{3}}}\end{cases}}$

and these equations, as we have seen,^[2] are competent to explain the rotation of the plane of polarization.

1. Cf. p. 428.
2. Cf. p 215.