Page:VaricakRel1912.djvu/18

 we want to write this in the form

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

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

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

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 an 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 &alpha;, or the perpendicular a (of which &alpha; is the parallel angle) instead of any angle &alpha;.