Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/328

Maxwell further assumed that the current thus produced interacts dynamically with the luminous motion in such a manner that the kinetic energy of the medium contains a term proportional to the scalar product of ė and curl ∂e/∂θ. The total kinetic energy of the medium may therefore be written

${\tfrac {1}{2}}\rho \mathbf {\overset {.}{e}} ^{2}+{\tfrac {1}{2}}(\mathbf {{\overset {.}{e}}.} \mathrm {curl} \ \partial \mathbf {e} /\partial \theta )$,

where p denotes the density of the medium, and o denotes a constant which measures the capacity of the medium to rotate the plane of polarization of light in a magnetic field.

The equation of motion may now be derived as in the elastic-solid theories of light: it is

$\rho \mathbf {\overset {..}{e}} +n\nabla ^{2}e-\sigma {\frac {\partial ^{2}}{\partial t\partial \theta }}\mathrm {curl} \ \mathbf {e}$.

When the light is transmitted in the direction of the lines of force, and the axis of x is taken parallel to this direction, the equation reduces to

${\begin{cases}\rho {\frac {\partial ^{2}e_{y}}{\partial t^{2}}}=n{\frac {\partial ^{2}e_{y}}{\partial x^{2}}}+\sigma \mathbf {H} {\frac {\partial ^{3}e_{z}}{\partial t\partial x^{2}}},\\\rho {\frac {\partial ^{2}e_{z}}{\partial t^{2}}}=n{\frac {\partial ^{2}e_{z}}{\partial x^{2}}}-\sigma \mathbf {H} {\frac {\partial ^{3}e_{y}}{\partial t\partial x^{2}}};\end{cases}}$,

and these equations, as we have seen,^[1] furnish an explanation of Faraday's phenomenon.

It may be remarked that the term

${\tfrac {1}{2}}\sigma (\mathbf {{\overset {.}{e}}.} \mathrm {curl} \ \partial \mathbf {e} /\partial \theta )$

in the kinetic energy may by partial integration be transformed into a term

${\tfrac {1}{2}}\sigma (\mathrm {curl} \ \mathbf {{\overset {.}{e}}.} \partial \mathbf {e} /\partial \theta )$,^[2]

together with surface-terms; or, again, into

$-{\tfrac {1}{2}}\sigma (\mathrm {curl} \ \mathbf {e.} \partial \mathbf {\overset {.}{e}} /\partial \theta )$,

together with surface-terms. These different forms all yield

1. Cf. p. 215.
2. This form was suggested by FitzGerald six years later, Phil. Trans., 1880, p. 691: FitzGerald's Scientific Writings, p. 45.