Page:Zur Theorie der Strahlung in bewegten Körpern.djvu/11

in which they are in $R$, is equal to $h/c{-}\cos \psi$ or $h/c{+}\cos \psi$; they thus contribute the summand

$h\cdot 2\pi \sin \psi \ d\psi \left({\frac {i}{c_{-}}}+{\frac {i'}{c_{+}}}\right)$

to the energy content of $R$. When we integrate this expression with respect to $\psi$ from 0 to $\pi /2$, furthermore dividing by the volume $h$ of space $R$, then we obtain the density $\varrho$ of the total radiation in $R$. If we insert for $i$ and $i'$ their values from (17a) and (17b), then it becomes:

$\varrho =2\pi {\overset {\pi /2}{\underset {0}{\int }}}\sin \psi \ d\psi \left({\frac {i_{0}+wp_{1}}{c_{-}}}+{\frac {i_{0}-wp_{2}}{c_{+}}}\right).$

If we now put

(18)	$\varrho =\epsilon +\epsilon ',$

where

$\epsilon =2\pi i_{0}{\overset {\pi /2}{\underset {0}{\int }}}\sin \psi \ d\psi \left({\frac {1}{c_{-}}}+{\frac {1}{c_{+}}}\right)$

and

(19)	$\epsilon '=2\pi w{\overset {\pi /2}{\underset {0}{\int }}}\sin \psi \ d\psi \left({\frac {p_{1}}{c_{-}}}-{\frac {p_{2}}{c_{+}}}\right)$

If we insert the values from (3a) and (3b) for $c_{-}$ and $c_{+}$ in the first of these expression, it becomes

$\epsilon ={\frac {4\pi i_{0}}{c(1-\beta ^{2})}}{\overset {\pi /2}{\underset {0}{\int }}}\sin \psi \ d\psi {\sqrt {1-\beta ^{2}\sin ^{2}\psi }}.$

We now set the density of energy in a resting cavity

(20)	${\frac {4\pi i_{0}}{c}}=\epsilon _{0}\left(={\frac {4e}{c}}\right)$

($e$ is the "emission capacity") and

$\varkappa ={\frac {1}{1-\beta ^{2}}}{\overset {\pi /2}{\underset {0}{\int }}}\sin \psi \ d\psi {\sqrt {1-\beta ^{2}\sin ^{2}\psi }}$

or after execution of the integration

(21)	$\varkappa ={\frac {1}{2(1-\beta ^{2})}}+{\frac {1}{4\beta }}\log {\frac {1+\beta }{1-\beta }}.$