Page:VaricakRel1912.djvu/7

thus in the same way as in classical theory, whose formula

${\mathfrak {v}}={\mathfrak {v}}_{1}+{\mathfrak {v}}_{2}$

is only valid at first approximation, because one can put ${\mathfrak {v}}={\mathfrak {u}}$ only for small velocities. If $v_{1}$ and $v_{2}$ are small compared to the speed of light, then one can neglect the last term in the numerator and denominator of expression (7), and one gets the ordinary formula

(9)	$v={\sqrt {v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos \alpha }}$

If one defines parameter c of our Lobachevskian space as infinitely small, then it goes over into euclidean space, and formula (7) is exactly reduced to (9). If velocities $v_{1}$ and $v_{2}$ enclose the angle α = 0, i.e. they lie in the same direction, then according to (5)

(10)	$u=u_{1}+u_{2}$

The resulting reduced velocity v follows from the formula

$\operatorname {th} \,u={\frac {\operatorname {th} \,u_{1}+\operatorname {th} \,u_{2}}{1+\operatorname {th} \,u_{1}\operatorname {th} \,u_{2}}}$

or

(11)	$v={\frac {v_{1}+v_{2}}{1+{\frac {v_{1}v_{2}}{c^{2}}}}}$

Although in the arithmetical sense the resultant is smaller as the sum of the components, it will be represented as in ordinary mechanics by a length equal to the sum of the lengths representing the components. If we compose two equal velocities $U_{1}$ in the same direction, then the resultant will be represented by the length $2U_{1}$.

On substitution (1) that paved me the way to the non-euclidean interpretation of relativity theory, I want to remark that Minkowski once put^[1]

(12)	$-i\ \operatorname {tg} \,i\ \psi ={\frac {e^{\psi }-e^{-\psi }}{e^{\psi }+e^{-\psi }}}=q$

i.e., the expression of the velocity relation as tangens hyperbolicus, but he didn't pay further attention to the middle term of this relation. Also Herglotz had spoken out the conviction, that non-euclidean geometry con be applied in a useful way

1. H. Minkowski, Zwei Abhandlungen über die Grundgleichungen der Elektrodynamik, 1910, p. 10, formula 2.