Page:Elektrische und Optische Erscheinungen (Lorentz) 111.jpg

 under continuing omission of magnitudes of second order,

If we want, on that basis, to calculate the energy which flows more out- than inwards between the times $$t_{0}-T$$ and $$t_0$$, and consequently, by remarking the latter, integrate with respect to time. As regards the two latter terms, we can also think of a surface, that progresses with velocity $$\mathfrak{p}$$.

§ 81. To also arrange the integration of the first term in such a way, that we have to deal with such a movable surface, we at first set for the increase of the integral $$V^{2}\int[\mathfrak{d}'.\mathfrak{H}']_{n}d\sigma$$ at certain t, when we displace the surface $$\sigma$$ in the direction of $$\mathfrak{p}$$ about an infinitely small distance $$\epsilon$$, the sign

$$\varkappa\ \epsilon$$,

where $$\varkappa$$ is of course a very special function of t. Furthermore, we think of a surface $$\sigma_0$$, which falls into $$\sigma$$ at time $$t_0$$, yet which is rigidly connected with earth. Then, at time t the "distance" of $$\sigma$$ and $$\sigma_0$$ has the value $$\mathfrak{p}(t_{0}-t)$$, which is to be considered as infinitely small, and our integral for the fixed surface $$\sigma$$ amounts

$$\mathfrak{p}\varkappa(t_{0}-t)$$,

more than for $$\sigma_0$$. The time integral, about which we speak eventually, is thus about

greater than the time integral taken for $$\sigma_0$$, and, since the latter vanishes by (106), we have only to deal with the value (108).

By the way, in $$\varkappa$$ we don't have to consider the magnitudes containing $$\mathfrak{p}$$, and thus we may understand, since with this omission