Page:A Dynamical Theory of the Electromagnetic Field.pdf/39

Rh a quantity essentially positive; so that, when the primary electrification is in one direction, the secondary discharge is always in the same direction as the primary discharge.

 PART VI. — ELECTROMAGNETIC THEORY OF LIGHT.

(91) At the commencement of this paper we made use of the optical hypothesis of an elastic medium through which the vibrations of light are propagated, in order to show that we have warrantable grounds for seeking, in the same medium, the cause of other phenomena as well as those of light. We then examined electromagnetic phenomena, seeking for their explanation in the properties of the field which surrounds the electrified or magnetic bodies. In this way we arrived at certain equations expressing certain properties of the electromagnetic field. We now proceed to investigate whether these properties of that which constitutes the electromagnetic field, deduced from electromagnetic phenomena alone, are sufficient to explain the propagation of light through the same substance.

(92) Let us suppose that a plane wave whose direction cosines are $$l, m, n$$ is propagated through the field with a velocity $$V$$. Then all the electromagnetic functions will be functions of

$w=lx+my+nz-Vt$

The equations of Magnetic Force (B), p. 482, will become

$\begin{array}{l} \mu\alpha=m\frac{dH}{dw}-n\frac{dG}{dw},\\ \\\mu\beta=n\frac{dF}{dw}-l\frac{dH}{dw},\\ \\\mu\gamma=l\frac{dG}{dw}-m\frac{dF}{dw}.\end{array}$

If we multiply these equations respectively by $$l, m, n$$, and add, we find

which shows that the direction of the magnetization must be in the plane of the wave.

(93) If we combine the equations of Magnetic Force (B) with those of Electric Currents (C), and put for brevity

{{MathForm2|(64)|$$\left.\begin{array}{l} 4\pi\mu p'=\frac{dJ}{dx}-\nabla^{2}F,\\ \\4\pi\mu q'=\frac{dJ}{dy}-\nabla^{2}G,\\ \\4\pi\mu r'=\frac{dJ}{dz}-\nabla^{2}H,\end{array}\right\} $$}}