Page:Elektrische und Optische Erscheinungen (Lorentz) 112.jpg

$V^{2}\int [{\mathfrak {d}}'.{\mathfrak {H}}']_{n}d\sigma$,

is the energy flux, under

$\varkappa \ \epsilon$

the difference calculated for the unit time, and under

$\varkappa \ \epsilon \ dt$

the difference calculated for the element dt, of the energy fluxes through two fixed surfaces that are mutually distant by the length ε.

Now, let $Q\epsilon$ be the energy, which at time t is more surrounded by our surface $\sigma$ in its fixed location, as when this surface would be displaced by $\epsilon$ in the direction of ${\mathfrak {p}}$; then we immediately see, that

$\varkappa \epsilon dt={\frac {dQ}{dt}}\epsilon dt$, $\varkappa ={\frac {dQ}{dt}}$

By that, and furthermore by partial integration, (108) is transformed into

${\mathfrak {p}}\int _{t_{0}-T}^{t_{0}}{\frac {dQ}{dt}}(t_{0}-t)dt=-{\mathfrak {p}}TQ_{t=t_{0}-T}+{\mathfrak {p}}\int _{t_{0}-T}^{t_{0}}Q\ dt$,

or

$-{\mathfrak {p}}TQ_{t=t_{0}}+{\mathfrak {p}}\int _{t_{0}-T}^{t_{0}}Q\ dt$,

since, except magnitudes of order ${\mathfrak {p}}$, Q has again the original value after the expiration of time T.

§ 82. Until now, we only spoke about the first member in (107). If we denote the two other members by A, then we have

$-{\mathfrak {p}}TQ_{t=t_{0}}+{\mathfrak {p}}\int _{t_{0}-T}^{t_{0}}Q\ dt+\int _{t_{0}-T}^{t_{0}}dt\int A\ d\sigma$

the complete value of the energy, that travelled outwards through $\sigma$. If we add the increase of the energy in the interior of $\sigma$, and the work of the forces, by which the