Page:Zur Theorie der Strahlung in bewegten Körpern.djvu/17

we change the velocity $w$ of our system by an infinitely small value $\delta w$. (It can of course be presupposed, that these infinitely small changes occur suddenly.)

Let

$w+\delta w=w_{1}=\beta _{1}c=(\beta +\delta \beta )c\,.$

Thus at the beginning, within $R$ there is a total relative radiation (with respect to $w$) of radiation intensity $i$ or $i'$. According to (10), this radiation is corresponding to an absolute radiation of intensity

$i\cdot {\frac {c^{3}\cos \alpha }{c_{-}^{3}}}$ or $i'\cdot {\frac {c^{3}\cos \alpha }{c_{+}^{3}}}$.

And to this radiation, a total relative radiation with respect to $w+\delta w$ is corresponding again, of intensity

$i\cdot {\frac {c^{3}\cos \alpha }{c_{-}^{3}}}\cdot {\frac {c_{-,1}^{3}}{c^{3}\cos \alpha _{1}}}$ or $i'\cdot {\frac {c^{3}\cos \alpha }{c_{+}^{3}}}\cdot {\frac {c_{+,1}^{3}}{c^{3}\cos \alpha _{1}}}$,

where the index 1 supplemented to the quantities $c_{-},c_{+},\alpha$ means, that these quantities are to be formed by $\psi _{1}$ and $\beta _{1}$ instead of $\psi$ and $\beta$.

Thus we can say: At the beginning, the total relative radiation in $R$ with respect to $w_{1}$, is given by the expressions

$2\pi \cos \psi _{1}\sin \psi _{1}d\psi _{1}i{\frac {c_{-,1}^{3}\cos \alpha }{c_{-}^{3}\cos \alpha _{1}}}$

or

$2\pi \cos \psi _{1}\sin \psi _{1}d\psi _{1}i'{\frac {c_{+,1}^{3}\cos \alpha }{c_{+}^{3}\cos \alpha _{1}}}$.

The density of these radiations is obtained by division by $c_{-,1}\cos \psi _{1}\,$ or $c_{+,1}\cos \psi _{1}\,$. Thus the energy amount of these emphasized radiations in $R$ is equal to (density times volume):

(31a)

$2\pi \sin \psi _{1}d\psi _{1}i{\frac {c_{-,1}^{2}\cos \alpha }{c_{-}^{3}\cos \alpha _{1}}}\cdot h$

or

(31b)

$2\pi \sin \psi _{1}d\psi _{1}i'{\frac {c_{+,1}^{2}\cos \alpha }{c_{+}^{3}\cos \alpha _{1}}}\cdot h.$

Now, the relation of the total to the true relative radiation (the latter is actually absorbed) was equal to $i/i_{0}$ or $i'/i_{0}$; thus when the velocity is equal to