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

(30)	$X'=X-u,\ Y'=Y,$

thus

(31)	${\begin{array}{l}\operatorname {sh} \,X'=\operatorname {sh} \,X\ \operatorname {ch} \,u-\operatorname {ch} \,X\ \operatorname {sh} \,u,\\\operatorname {sh} \,Y'=\operatorname {sh} \,Y.\end{array}}$

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

(32)	${\begin{array}{rl}\operatorname {sh} \,X'\ \operatorname {ch} \,Y'=&\operatorname {sh} \,X\ \operatorname {ch} \,Y\ \operatorname {ch} \,u-\operatorname {ch} \,X\ \operatorname {ch} \,Y\ \operatorname {sh} \,u\\\operatorname {sh} \,Y'=&\operatorname {sh} \,Y.\end{array}}$

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

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

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

(34)	${\begin{array}{l}x'=x\ \operatorname {ch} \,u-l\ \operatorname {sh} \,u\\y'=y\\l'=l\ \operatorname {ch} \,u-x\ \operatorname {sh} \,u\end{array}}$

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 Lorentz-Einstein 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

(35)	${\begin{array}{l}x=x'\ \operatorname {ch} \,u+l'\ \operatorname {sh} \,u\\y=y',\\l=x'\ \operatorname {sh} \,u+l'\ \operatorname {ch} \,u\end{array}}$