Page:VaricakRel1912.djvu/12

thus

$l^{2}-x^{2}-y^{2}=\operatorname {ch} ^{2}Z$

and eventually

(24)	$l^{2}-x^{2}-y^{2}-z^{2}=1$

This relation is valid between the Weierstrass coordinates of every single point. It is known which role this invariant plays in Minkowski's four-dimensional interpretation of relativity theory.

5. The Lorentz-Einstein transformation.

The Galilei-Newton transformation

(25)	$x'=x-vt,\ y'=y,\ z'=z,\ t'=t$

represents the translation along the X-axis in euclidean space. The Lorentz-Einstein transformation

(26)	$x'={\frac {x-vt}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}},\ y'=y,\ z'=z,\ t'={\frac {t-{\frac {v}{c^{2}}}x}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}$

similarly can be interpreted as a translation along the X-axis in Lobachevskian space.

If we remain in the plane then we can say: The Lorentz-Einstein transformation defines a motion along the distance line with the X-axis as its center line.^[1]

This distance line Y = b is the location of the points having a constant distance b from the X-axis. The length of its arc between two points M and $M'$ is (Fig. 6).

(27)	$s=(X-X')\operatorname {ch} \,b.$

The displacement by the distance s along that equidistant line is defined by the equations

(28)	$X'=X-{\frac {s}{\operatorname {ch} \,b}},\ Y'=Y.$

For the passage from $M'$ to $M''$ we have

$X''=X'-{\frac {s'}{\operatorname {ch} \,b}},\ Y''=Y'.$

or

(29)	$X''=X-{\frac {s+s'}{\operatorname {ch} \,b}},\ Y''=Y.$

1. On the transformations of the Lobachevskian plane see my relevant papers in Rad jugoslavenske akademije 165, 50-80, 236-244, 1906, or the short excerpt therefrom in Jahresber. d. Deutsch. Mathematiker-Ver. 17, 80-83, 1908.