Page:VaricakRel1912.djvu/18

we want to write this in the form

(41)	$F=\sin \nu \left({\frac {l-x\ \cos \varphi -y\ \cos \psi }{c}}\right)$

By means of transformation (35) this goes over into

${\begin{array}{ll}F'&=\sin \nu \left({\frac {x'\operatorname {sh} \,u+l'\operatorname {ch} \,u-\left(x'\operatorname {ch} \,u+l'\operatorname {sh} \,u\right)\cos \varphi -y'\cos \psi }{c}}\right)\\\\&=\sin \nu \left({\frac {l'(\operatorname {ch} \,u-\cos \varphi \ \operatorname {sh} \,u)-x'(\operatorname {ch} \,u\cos \varphi -\operatorname {sh} \,u)-y'\cos \psi }{c}}\right)\end{array}}$

or

(42)	$F'=\sin \nu (\operatorname {ch} \,u-\cos \varphi \ \operatorname {sh} \,u\left({\frac {l'-x'\cos \varphi '-y'\cos \psi '}{c}}\right)$

We see, that F and $F'$ can be represented in the same form, we only have to put

(43)	$\nu '=\nu \left(\operatorname {ch} \,u-\cos \varphi \ \operatorname {sh} \,u\right)$

(44)	$\cos \varphi '={\frac {\operatorname {ch} \,u\ \cos \varphi -\operatorname {sh} \,u}{\operatorname {ch} \,u-\cos \varphi \ \operatorname {sh} \,u}}={\frac {\cos \varphi -\operatorname {th} \,u}{1-\cos \varphi \ \operatorname {th} \,u}}$

(45)	$\cos \psi '={\frac {\cos \psi }{\operatorname {ch} \,u-\cos \varphi \ \operatorname {sh} \,u}}$

If we consider formulas (1) and (6), then we immediately obtain Einstein's formulas

(46)	$\nu '=\nu \cdot {\frac {1-{\frac {v}{c}}\cos \varphi }{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}$

(47)	$\cos \varphi '={\frac {\cos \varphi -{\frac {v}{c}}}{1-{\frac {v}{c}}\cos \varphi }}$

(48)	$\cos \psi '={\frac {\cos \psi {\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}{1-{\frac {v}{c}}\cos \varphi }}$

To interpret this relation in a geometrical way, we want to resort to formulas (43)-(45), which will be altered by us so that only distances occur in them. It is a great advantage of Lobachevskian geometry, that we can express lengths and angles by magnitudes of the same kind, by either the introduction of the related parallel angle $\alpha =\Pi (a)$ instead of length α, or the perpendicular a (of which α is the parallel angle) instead of any angle α.^[1]

1. More about that in F. Engel, Nikolaj Iwanowitsch Lobatschefskij. Zwei geometrische Abhandlungen, 1898, 244