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

$h\cdot2\pi\sin\psi\ d\psi\left(\frac{i}{c_{-}}+\frac{i'}{c_{+}}\right)$|undefined

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:

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

If we now put

where

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

and

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

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

We now set the density of energy in a resting cavity

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

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

or after execution of the integration