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

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 $i'\cdot\frac{c^{3}\cos\alpha}{c_{+}^{3}}\cdot\frac{c_{+,1}^{3}}{c^{3}\cos\alpha_{1}}$,|undefined

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

$2\pi\cos\psi_{1}\sin\psi_{1}d\psi_{1}i\frac{c_{-,1}^{3}\cos\alpha}{c_{-}^{3}\cos\alpha_{1}}$|undefined

or

$2\pi\cos\psi_{1}\sin\psi_{1}d\psi_{1}i'\frac{c_{+,1}^{3}\cos\alpha}{c_{+}^{3}\cos\alpha_{1}}$.|undefined

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):

or

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