Page:Lorentz Simplified1899.djvu/7

( 433 )

§ 6. We shall now shew how our general equations (I_c) — (V_c) may be applied to optical phenomena. For this purpose we consider a system of ponderable bodies, the ions in which are capable of vibrating about determinate positions of equilibrium. If the system be traversed by waves of light, there will be oscillations of the ions, accompanied by electric vibrations in the aether. For convenience of treatment we shall suppose that, in the absence of lightwaves, there is no motion at all; this amounts to ignoring all molecular motion.

Our first step will be to omit all terms of the second order. Thus, we shall put k=1, and the electric force acting on ions at rest will become ${\mathfrak {F}}'$ itself.

We shall further introduce certain restrictions, by means of which we get rid of the last term in (I_c) and of the terms containing ${\mathfrak {v}}_{x},{\mathfrak {v}}_{y},{\mathfrak {v}}_{z}$ in (V_c).

The first of these restrictions relates to the magnitude of the displacements ${\mathfrak {a}}$ from the positions of equilibrium. We shall suppose them to be exceedingly small, even relatively to the dimensions of the ions and we shall on this ground neglect all quantities which are of the second order with respect to ${\mathfrak {a}}$.

It is easily seen that, in consequence of the displacements, the electric density in a fixed point will no longer have its original value $\varrho _{0}$ but will have become

$\varrho =\varrho _{0}-{\frac {\partial }{\partial x}}(\varrho _{0}{\mathfrak {a}}_{x})-{\frac {\partial }{\partial y}}(\varrho _{0}{\mathfrak {a}}_{y})-{\frac {\partial }{\partial z}}(\varrho _{0}{\mathfrak {a}}_{z})$.

Here, the last terns, which evidently must be taken into account, have the order of magnitude ${\frac {c\ \varrho _{0}}{a}}$, if c denotes the amplitude of the vibrations; consequently, the first term of the right-hand member of (Ic) will contain quantities of the order

${\frac {V^{2}c\ \varrho _{0}}{a}}$.

(3)

On the other hand, if T is the time of vibration, the last term in (I_c) will be of the order

${\frac {{\mathfrak {p}}_{x}\varrho _{0}c}{T}}$,

(4)