Page:VaricakRel1912.djvu/13

 From that we can see the group property of the displacements along the distance line.

If u is the projection $$NN'$$ of the arc $$MM'$$ in the X-axis, then

thus

By multiplication of the first equations by $$\operatorname{ch}\, Y'=\operatorname{ch}\, Y$$ we have

According to Fig. 2 we have in addition

$$\operatorname{ch}\, OM'=\operatorname{ch}\, X'\ \operatorname{ch}\, Y',$$

or

$$\operatorname{ch}\, r'=\operatorname{ch}\,(X-u)\operatorname{ch}\, Y,$$

that is

Until now we have applied an coordinates; if we want to pass over to coordinates, then we have to consider the transformation formulas (22). By their aid we can bring equations (32) and (33) into the form

If we substitute herein according to formula (6)

$$\operatorname{ch}\, u=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}},\ \operatorname{sh}\, u=\frac{\frac{v}{c}}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}$$

and l = ct, then the transformation in its ordinary form (26) is immediately given. However, we always want to use them in the form (34). Indeed, we can see that the space-time transformation caused by a uniform motion of velocity u, will be completely characterized by the translation of point M representing an elementary event. The inverse transformation is