Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/454

Budde and FitzGerald^[1] had advanced in a similar case; a conductor carrying a constant electric current and moving with the earth would exert a force on electric charges at relative rest in its vicinity, were it not that this force induces on the surface of the conductor itself a compensating electrostatic charge, whose action annuls the expected effect.

The most satisfactory method of discussing the influence of the terrestrial motion on electrical phenomena is to transform the fundamental equations of the aether and electrons to axes moving with the earth. Taking the axis of x parallel to the direction of the earth's motion, and denoting the velocity of the earth by w, we write

$x=x_{1}+wt,\qquad y=y_{1},\qquad z=z_{1}$,

so that (x₁, y₁, z₁) denote coordinates referred to axes moving with the earth. Lorentz completed the change of coordinates by introducing in place of the variable t a "local time" t₁ defined by the equation

$t=t_{1}+wx_{1}/c^{2}$.

It is also necessary to introduce, in place of d and h, the electric and magnetic forces relative to the moving axes: these are^[2]

$\mathbf {d} _{1}=\mathbf {d} +[\mathbf {w.h} ]$

$\mathbf {h} _{1}=\mathbf {h} +(1/c^{2})[\mathbf {d.w} ]$;

and in place of the velocity v of an electron referred to the original fixed axes, we must introduce its velocity v₁, relative to the moving axes, which is given by the equation

$\mathbf {v} _{1}=\mathbf {v} -\mathbf {w}$.

The fundamental equations of the aether and electrons, referred to the original axes, are

${\begin{matrix}{\text{div }}\mathbf {d} &=&4\pi c^{2}\rho ,\qquad \qquad &{\text{curl }}\mathbf {d} &=&-\mathbf {\dot {h}} ,\\{\text{div }}\mathbf {h} &=&0,\qquad \qquad &{\text{curl }}\mathbf {h} &=&(1/c^{2})\mathbf {\dot {d}} +4\pi \rho \mathbf {v} ,\end{matrix}}$

$\mathbf {F} =\mathbf {d} +[\mathbf {v.h} ]$,

where F denotes the ponderomotive force on a particle carrying a unit charge.

1. Cf. p. 263.
2. Cf. pp. 365, 366.