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From that formula it follows

$e^{-u}={\sqrt {\frac {1-{\frac {v}{c}}}{1+{\frac {v}{c}}}}}$

and in this way we obtain Einstein's expression

(54)	$\nu '=\nu {\sqrt {\frac {1-{\frac {v}{c}}}{1+{\frac {v}{c}}}}}$

From (53) it additionally follows

$\nu '=\nu \left(1-u+{\frac {u^{2}}{2!}}-{\frac {u^{3}}{3!}}+\dots \right)$

Higher powers of u can be neglected for small values; and if we replace the velocity u by the reduced velocity ${\tfrac {v}{c}}$, then we obtain the expression of Doppler's principle of ordinary mechanics:

(55)	$\nu '=\nu \left(1-{\frac {v}{c}}\right)$

Note, that in the primed reference frame $v'=-v$.

In euclidean geometry the distance lines and the limiting circles are reduced to parallels to the given straight line. Expression (55) can be easily illustrated by intersections of parallel transversals between the legs of an angle, because of relation $\nu :\nu '=c:\left(c-v\right)$.

9. Aberration

In the primed System $S'$ the given light ray encloses the angle $\varphi '$ with the $X'$-axis. If we denote by $u'_{1}$ the distance corresponding to the parallel angle $\varphi '$, then (47) goes over into

$\operatorname {th} \,u'_{1}={\frac {\operatorname {ch} \,u\ \operatorname {th} \,u_{1}-\operatorname {sh} \,u}{\operatorname {ch} \,u-\operatorname {th} \,u_{1}\operatorname {sh} \,u}}=\operatorname {th} \,\left(u_{1}-u\right)$

or

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

Based on that aberration equation we have the following construction of the diverted light ray. A light ray T coming from an infinitely distant light source J, is striking the x-axis at point M under the acute angle φ. If we make $MP=u_{1}$, then $\varphi =\Pi \left(u_{1}\right)$. Then, we lay off the line $MN=u$ toward the increasing abscissa, and from N we apply the Lobachevskian parallel to T. This parallel $T'$ encloses the angle $\varphi '$ with the X-axis, because $\varphi '=\Pi \left(u_{1}-u\right)=\Pi \left(u'_{1}\right)$.