Page:VaricakRel1912.djvu/23

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

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



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 &psi; supplementary to $$\varphi'''$$. By which ratio the angles &psi; and &phi; are related, depends on the direction of the motion of the mirror relative to the light source. In the considered case $$\psi>\varphi$$, because &psi; is related to the smaller perpendicular as parallel angle.

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

or

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

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