Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/391

will be remembered^[1] that in the clastic-solid theory of Boussinesq, the rotation of the plane of polarization of saccharine solutions had been represented by substituting the equation

$\mathbf {e} ^{\prime }=\mathbf {Ae} +\mathbf {B} {\text{curl }}\mathbf {e}$

in place of the usual equation

$\mathbf {e} ^{\prime }=\mathbf {Ae}$.

Goldhammer now proposed to represent rotatory power in the electromagnetic theory by substituting the equation

$\mathbf {E} =(4\pi c^{2}/\epsilon )\mathbf {D} +k{\text{ curl }}\mathbf {D}$,

in place of the customary equation

$\mathbf {E} =(4\pi c^{2}/\epsilon )\mathbf {D}$:

the constant k being a measure of the natural rotatory power of the substance concerned. The remaining equations are as usual,

${\text{curl }}\mathbf {H} =4\pi \mathbf {\dot {D}} ,\qquad -{\text{curl }}\mathbf {E} =\mathbf {H}$.

Eliminating H and E, we have

$\mathbf {\ddot {D}} =(c^{2}/\epsilon )\nabla ^{2}\mathbf {D} +(k/4\pi )\nabla ^{2}{\text{curl }}\mathbf {D}$.

For a plane wave which is propagated parallel to the axis of x, this equation reduces to

${\begin{cases}{\frac {\partial ^{2}D_{y}}{\partial t^{2}}}={\frac {c^{2}}{\epsilon }}{\frac {\partial ^{2}D_{y}}{\partial x^{2}}}-{\frac {k}{4\pi }}{\frac {\partial ^{3}D_{z}}{\partial x^{3}}}\\{\frac {\partial ^{2}D_{z}}{\partial t^{2}}}={\frac {c^{2}}{\epsilon }}{\frac {\partial ^{2}D_{z}}{\partial x^{2}}}-{\frac {k}{4\pi }}{\frac {\partial ^{3}D_{y}}{\partial x^{3}}}\end{cases}}$

and, as MacCullagh had shown in 1836,^[2] these equations are competent to represent the rotation of the plane of polarization.

In the closing years of the nineteenth century, the general theory of aether and electricity assumed a new form. But before discussing the memoirs in which the new conception was unfolded, we shall consider the progress which had been made since the middle of the century in the study of conduction in liquid and gaseous media.

1. Cf. p. 186.
2. Cf. p. 175.

2 B 2