Page:Zur Theorie der Strahlung in bewegten Körpern.djvu/13

Thus we can set:

(26a)

${\begin{cases}p_{1}={\frac {i_{0}\left(\beta \sin ^{2}\psi +\cos \psi {\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right)}{c{\sqrt {1-\beta ^{2}\sin ^{2}\psi \left(-\beta \cos \psi +{\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right)}}}}\\\\\qquad ={\frac {i_{0}\left(\cos \psi +\beta {\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right)}{c(1-\beta ^{2}){\sqrt {1-\beta ^{2}\sin ^{2}\psi }}}},\end{cases}}$

(26b)

${\begin{cases}p_{2}={\frac {i_{0}\left(-\beta \sin ^{2}\psi +\cos \psi {\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right)}{c{\sqrt {1-\beta ^{2}\sin ^{2}\psi \left(\beta \cos \psi +{\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right)}}}}\\\\\qquad ={\frac {i_{0}\left(\cos \psi -\beta {\sqrt {1-\beta ^{2}\sin ^{2}\psi }}\right)}{c(1-\beta ^{2}){\sqrt {1-\beta ^{2}\sin ^{2}\psi }}}}.\end{cases}}$

These expressions can now (as already mentioned in the introduction) also be derived from another hypothesis; namely from the assumption, that a moving body emanates in unit time the same amount of waves as at rest, and that also the amplitude of these waves is not changed by the motion of the light source. Only the wavelength experiences a change in accordance with Doppler's principle. If one now assumes in accordance with the old theory of light, that the energy of a wave train is caeteris paribus inversely proportional to the square of the wavelength per unit length, i.e., that one energy quantity (which is inversely proportional to the first power of its length) is connected to one wave, then (since the number of emanated waves is the same in both cases) the radiation of the moving body is related to that of the resting one, by the inverse proportion of the wavelengths. Since this proportion is equal to $(1\mp \beta \cos \varphi )$ according to Doppler's principle, then we obtain:

$i={\frac {i_{0}}{1-\beta \cos \varphi }}={\frac {i_{0}c}{\cos \alpha \cdot c_{-}}}$

or

$i'={\frac {i_{0}}{1+\beta \cos \varphi }}={\frac {i_{0}c}{\cos \alpha \cdot c_{+}}},$

where we have used (7). These expressions are now indeed identical with equations (25). If we insert them into equations (17), we of course can also derive equations (24) and (26).

The thought process stated here, was first applied by Larmor^[1] to the problem of reflection, and then by

1. J. Larmor, Rep. brit. Asa. 1900.