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

under continuing omission of magnitudes of second order,

$V^{2}[{\mathfrak {d}}'.{\mathfrak {H}}']_{n}+4\pi V^{2}\left\{{\mathfrak {p}}_{n}{\mathfrak {d}}^{2}-{\mathfrak {d}}_{n}({\mathfrak {p}}_{x}{\mathfrak {d}}_{x}+{\mathfrak {p}}_{y}{\mathfrak {d}}_{y}+{\mathfrak {p}}_{z}{\mathfrak {d}}_{z})\right\}+$ $+{\frac {1}{4\pi }}\left\{{\mathfrak {p}}_{n}{\mathfrak {H}}^{2}-{\mathfrak {H}}_{n}({\mathfrak {p}}_{x}{\mathfrak {H}}_{x}+{\mathfrak {p}}_{y}{\mathfrak {H}}_{y}+{\mathfrak {p}}_{z}{\mathfrak {H}}_{z})\right\}$

(107)

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

${\mathfrak {p}}\int _{t_{0}-T}^{t_{0}}\varkappa (t_{0}-t)dt$

(108)

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