Page:Zur Theorie der Strahlung in bewegten Körpern.djvu/19

or

$Q=2\pi i_{0}h{\overset {\pi /2}{\underset {0}{\int }}}\sin \psi _{1}d\psi _{1}\left({\frac {1}{c_{-,1}}}+{\frac {1}{c_{+,1}}}\right)$ $-8\pi i_{0}h\delta w{\overset {\pi /2}{\underset {0}{\int }}}\sin \psi _{1}\cos \psi _{1}d\psi _{1}\left({\frac {1}{c_{-,1}^{2}}}-{\frac {1}{c_{+,1}^{2}}}\right).$

We can determine the first integral by the aid of equation (21), the second one has the value:

${\frac {2\beta _{1}}{c^{2}(1-\beta _{1}^{2})^{2}}}{\overset {\pi /2}{\underset {0}{\int }}}\sin \psi _{1}\cos ^{2}\psi _{1}d\psi _{1}{\sqrt {1-\beta _{1}^{2}\sin ^{2}\psi _{1}}}$ $={\frac {2}{c^{2}(1-\beta _{1}^{2})^{2}}}\left({\frac {1+\beta _{1}^{2}}{8\beta _{1}}}-{\frac {(1-\beta _{1}^{2})^{2}}{16\beta _{1}^{2}}}\log {\frac {1+\beta _{1}}{1-\beta _{1}}}\right),$

thus it becomes

$Q=h\epsilon _{0}\varkappa _{1}-h\epsilon _{0}{\frac {\delta w}{c}}\left({\frac {1+\beta _{1}^{2}}{2\beta _{1}(1-\beta _{1}^{2})^{2}}}-{\frac {1}{4\beta _{1}^{2}}}\log {\frac {1+\beta _{1}}{1-\beta _{1}}}\right),$

where $\varkappa _{1}$ is the value, which emerges from $\varkappa$ when $\beta _{1}$ is replaced by $\beta$ (see equation (21)). One easily convince oneself, that the bracket expression in the last equation has the value $d\varkappa _{1}/d\beta _{1}$; thus it becomes:

$Q=h\epsilon _{0}\left(\varkappa _{1}-\delta \beta {\frac {dx_{1}}{d\beta _{1}}}\right)=h\epsilon _{0}\varkappa =h\epsilon .$

Our earlier assertion is proven by that. At the beginning, when the velocity of our system was $w$, the energy quantity contained in $B$ had the amount $h(\epsilon +\epsilon ')=h\epsilon _{0}(\varkappa +\tau )$; when the velocity is changed, and the equilibrium of radiation in $R$ is reestablished, then this energy quantity has the amount $h\epsilon _{0}(\varkappa _{1}+\tau _{1})$ (where $\tau _{1}$ is formed from $\beta _{1}$ in the same way again, as $\tau$ from $\beta$). As we know, the fraction $h\epsilon _{0}\varkappa _{1}$ of the heat reservoir of bodies $A$ and $B$ stems from that; since they (as we have just proven) absorb the fraction $h\epsilon _{0}\varkappa$ from the initially existing radiation, we see that when the velocity is changed from $w$ to $w_{1}$, the boundary of the cavity gives off the heat

$h\epsilon _{0}(\varkappa _{1}-\varkappa )$