Page:Zur Theorie der Strahlung in bewegten Körpern.djvu/18

 $$c/c_{-}\cos\alpha\,$$ or $$c/c_{+}\cos\alpha\,$$. When the velocity has now the value $$w_1$$, then this relation becomes equal to $$c/c_{-,1}\cos\alpha_1\,$$ or $$c/c_{+,1}\cos\alpha_1\,$$. Thus, from the radiation (31) present in $$R$$ at the beginning, the fraction

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

or

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

is now absorbed. If we want to obtain the fraction $$Q$$ of the whole energy contained in $$R$$ at the beginning, which is absorbed by $$A$$ and $$B$$ after a velocity change of $$\delta w$$, then we have to integrate these expressions with respect to $$\psi_1$$ from 0 to $$\pi/2$$. We preliminarily insert the values from (25) for $$i$$ and $$i'$$, and thus we obtain

$Q=\overset{\pi/2}{\underset{0}{\int}}2\pi\sin\psi_{1}d\psi_{1}i_{0}h\left(\frac{c_{-,1}^{3}}{c_{-}^{4}}+\frac{c_{+,1}^{3}}{c_{+}^{4}}\right).$|undefined

Now it is according to (1):

$c_{-,1}^{2}=c^{2}+(w+\delta w)^{2}-2c(w+\delta w)\cos\varphi=c_{-}^{2}-2\delta w(c\ \cos\varphi-w),$

or by use of (5)

$c_{-,1}^{2}=c_{-}^{2}-2\delta wc_{-}\cos\psi.$

Since we can also set $$c_{-,1}\cos\psi_1\,$$ instead of $$c_{-}\cos\psi\,$$ (within the expression multiplied with the infinitely small magnitude $$\delta w$$) it eventually becomes:

$c_{-}=c_{-,1}\left(1+\delta w\frac{\cos\psi_{1}}{c_{-,1}}\right).$|undefined

Quite analogously it is given:

$c_{+}=c_{+,1}\left(1-\delta w\frac{\cos\psi_{1}}{c_{+,1}}\right).$|undefined

If we use these equations, it becomes.

$Q=2\pi i_{0}h\overset{\pi/2}{\underset{0}{\int}}\sin\psi_{1}d\psi_{1}\left(\frac{1-4\delta w\frac{\cos\psi_{1}}{c_{-,1}}}{c_{-,1}}+\frac{1+4\delta w\frac{\cos\psi_{1}}{c_{+,1}}}{c_{+,1}}\right).$|undefined