Page:VaricakRel1912.djvu/24

From this it follows

${\frac {\nu '''}{\nu }}=\operatorname {ch} \,2u-\operatorname {th} \,u_{1}\operatorname {sh} \,2u=\operatorname {ch} ^{2}u+\operatorname {sh} ^{2}u-2\operatorname {sh} \,u\ \operatorname {ch} \,u\ \operatorname {th} \,u_{1}$

or

(67)	${\frac {\nu '''}{\nu }}=\operatorname {ch} ^{2}u\left(1-2\operatorname {th} \,u\ \operatorname {th} \,u_{1}+\operatorname {th} ^{2}u\right)$

By introducing the parallel angle we obtain Einstein's expression

(68)	${\frac {\nu '''}{\nu }}={\frac {1-2{\frac {v}{c}}\cos \varphi +\left({\frac {v}{c}}\right)^{2}}{1-\left({\frac {v}{c}}\right)^{2}}}={\frac {A'''}{A}}$

because the amplitudes were transformed in the same way as the frequencies. Fig. 15 is the geometrical image of formula (66).

The relations of the amplitudes and the frequencies of the incident and reflected light can be illustrated by the relation of the arc of two distance lines between common perpendiculars. The parameter of these distance lines are $OA=u_{1},\ OA'''=u_{1}-2u$. We can easily see that we obtain $\nu ''(A''')$ by mirroring $\nu (A)$ on $\nu '(A')$. In the same way the angle $\varphi '''$ can be defined by mirroring the incident ray on the aberrated ray.

For a normally incident light ray we have $\varphi =0$, also $u_{1}=\infty$, the formula (66) goes over to

(69)	$\nu '''=\nu e^{-2u}$

The relation of the frequencies and the amplitudes can in this case be illustrated as the relation of two coaxial limiting arcs. The corresponding figure would be the same as in Fig. 11, we only have to take $AA_{1}=2u$.

The formula (51) for Doppler's principle and the aberration equation (56) have the same structure as the formula (66) for the frequency, the amplitude, and the formula (62) for the reflection angle of light reflected by a moving mirror. From this it follows, that the same light ray appears to be of the same constitution to an observer moving with the doubled velocity 2u, as it would appear to a stationary observer after the reflection by a mirror moving with velocity v. In both cases the motion must have the same direction.