Page:Lorentz Simplified1899.djvu/9

 $$-\frac{\mathfrak{p}_{x}}{V^{2}}\int\varrho\mathfrak{v}_{x}\mathfrak{F}'_{y}d\tau$$,

or, since we may replace $$\mathfrak{F}'_{y}$$ by $$\mathfrak{F}'_{0y}$$ and $$\varrho$$ by $$\varrho_{0}$$,

$$-\frac{\mathfrak{p}_{x}}{V^{2}}\mathfrak{v}_{x}\int\varrho_{0}\mathfrak{F}'_{0y}d\tau$$,

which vanishes on account of (5).

Hence, as far as regards the resultant force, we may put $$\mathfrak{E}=\mathfrak{F}'$$, that is to say, we may take $$\mathfrak{F}'$$ as the electric force, acting not only on ions at rest, but also on moving ions.

The equations will be somewhat simplified, if, instead of $$\mathfrak{F}'$$, we introduce the already mentioned difference $$\mathfrak{F}'-\mathfrak{F}'_{0}$$. In order to do this, we have only twice to write down the equations (Ic)—(IVc), once for the vibrating system and a second time for the same system in a state of rest; and then to subtract the equations of the second system from those of the first. In the resulting equations, I shall, for the sake of brevity, write $$\mathfrak{F}'$$ instead of $$\mathfrak{F}'-\mathfrak{F}'_{0}$$, so that henceforth $$\mathfrak{F}'$$ will denote not the total electric force, but only the part of it that is due to the vibrations. At the same time we shall replace the value of $$\varrho$$, given above, by

$$\varrho_{0}-\mathfrak{a}_{x}\frac{\partial\varrho_{0}}{\partial x'}-\mathfrak{a}_{y}\frac{\partial\varrho_{0}}{\partial y'}-\mathfrak{a}_{z}\frac{\partial\varrho_{0}}{\partial z'}$$.

We may do so, because we have supposed $$\mathfrak{a}_{x},\mathfrak{a}_{y},\mathfrak{a}_{z}$$ to have the same values all over an ion, and because $$\varrho_{0}$$ is independent of the time, so that

$$\frac{\partial\varrho_{0}}{\partial x}=\frac{\partial\varrho_{0}}{\partial x'}$$.

Finally we have