Page:Lorentz Simplified1899.djvu/5

 electrostatic phenomena. In these we have $$\mathfrak{v}=0$$ and $$\mathfrak{F}'$$ independent of the time. Hence, by (IIc) and (IIIc)

$$\mathfrak{H}'=0$$,

and by (IVc) and (Ic)

These equations show that $$\mathfrak{F}'$$ depends on a potential &omega;, so that

$$\mathfrak{F}'_{x}=-\frac{\partial\omega}{\partial x'},\ \mathfrak{F}'_{y}=-\frac{\partial\omega}{\partial y'},\ \mathfrak{F}'_{z}=-\frac{\partial\omega}{\partial z'}$$

and

Let S be the system of ions with the translation $$\mathfrak{p}_{x}$$, to which the above formulae are applied. We can conceive a second system S0 with no translation and consequently no motion at all; we shall suppose that S is changed into S0 by a dilatation in which the dimensions parallel to OX are changed in ratio of 1 to k, the dimensions perpendicular to OX remaining what they were. Moreover we shall attribute equal charges to corresponding volume-elements in S and S0; if then $$\varrho_{0}$$ be the density in a point P of S, the density in the corresponding point P0 of S0 will be

$$\varrho_{0}=\frac{1}{k}\varrho$$.

If x, y, z are the coordinates of P, the quantities x', y', z', determined by (1), may be considered as the coordinates of P0.

In the system S0, the electric force, which we shall call $$\mathfrak{E}_{0}$$ may evidently be derived from a potential &omega;0, by means of the equations

$$\mathfrak{E}_{0x}=-\frac{\partial\omega_{0}}{\partial x'},\ \mathfrak{E}_{0y}=-\frac{\partial\omega_{0}}{\partial y'},\ \mathfrak{E}_{0z}=-\frac{\partial\omega_{0}}{\partial z'}$$,