Page:Lorentz Simplified1899.djvu/5

( 431 )

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

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

and by (IV_c) and (I_c)

${\frac {\partial {\mathfrak {F}}'_{z}}{\partial y'}}-{\frac {\partial {\mathfrak {F}}'_{y}}{\partial z'}}=0,\ {\frac {\partial {\mathfrak {F}}'_{x}}{\partial z'}}-{\frac {\partial {\mathfrak {F}}'_{z}}{\partial x'}}=0,\ {\frac {\partial {\mathfrak {F}}'_{y}}{\partial x'}}-{\frac {\partial {\mathfrak {F}}'_{x}}{\partial y'}}=0$, $Div'\ {\mathfrak {F}}'={\frac {4\pi }{k}}V^{2}\varrho$.

These equations show that ${\mathfrak {F}}'$ depends on a potential ω, 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

${\frac {\partial ^{2}\omega }{\partial x'^{2}}}+{\frac {\partial ^{2}\omega }{\partial y'^{2}}}+{\frac {\partial ^{2}\omega }{\partial z'^{2}}}=-{\frac {4\pi }{k}}V^{2}\varrho$.

(2)

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 ω₀, by means of the equations

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