Page:VaricakRel1912.djvu/23

the length $u'''_{1}$ is increased by u. Thus we eventually come to the equation

(62)	$u'''_{1}=2u-u_{1}$

from which we can see, that for the observer resting in O, the ray is reflected under the angle

(63)	$\varphi '''=\Pi \left(2u-u_{1}\right)$

From Fig. 14 the construction of the reflected ray according to formula (62) can be easily seen. For the construction it is advantageous to take the angle ψ supplementary to $\varphi '''$. By which ratio the angles ψ and φ are related, depends on the direction of the motion of the mirror relative to the light source. In the considered case $\psi >\varphi$, because ψ is related to the smaller perpendicular as parallel angle.

Now we can easily pass to Einstein's formulas. By (62) it is

$\operatorname {th} \,u'''_{1}=-{\frac {\operatorname {th} \,u_{1}-\operatorname {th} \,2u}{1-\operatorname {th} \,2u\ \operatorname {th} \,u_{1}}}$

or

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

Instead of the hyperbolic functions of the perpendiculars $u'''_{1}$ and $u_{1}$, we introduce the spherical functions of the corresponding parallel angles, additionally we replace $\operatorname {th} \,u$ by ${\tfrac {v}{c}}$ and by this way we come to the formula of Einstein^[1]

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

However, in the non-euclidean interpretation it will be completely replaced by the considerably simpler formulas (62) or (63).

By (51) and (62) we can construct, for the frequency $\nu '''$ of the reflected light rays, the formula

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

1. Annalen der Physik, 17, 1905, 915