Page:VaricakRel1912.djvu/9

In hyperbolic geometry the sum of two angles in any triangle is less than two right angles. In triangle OAB it is thus

$\alpha _{1}+\alpha _{2}<{\frac {\pi }{2}},$

If we lay off the length $u_{2}$ in the direction OC perpendicular to OA and apply under a right angle the line $u_{1}$, then we come to point D different from B. In the older mechanics those points coincide. Thus if we compose these velocities in reverse order, we obtain a resultant of same magnitude but different direction. The direction difference

(15)	$\delta =\sphericalangle BOD={\frac {\pi }{2}}-\left(\alpha _{1}+\alpha _{2}\right)$

can easily be represented as a function of the components.

If we introduce into the formula

(16)	$\operatorname {cotg} \ \delta ={\frac {\operatorname {tg} \,\alpha _{1}+\operatorname {tg} \,\alpha _{2}}{1-\operatorname {tg} \,\alpha _{1}\operatorname {tg} \,\alpha _{2}}}$

the values taken from the Lobachevskian triangle OAB

$\operatorname {tg} \,\alpha _{1}={\frac {\operatorname {th} \,u_{1}}{\operatorname {sh} \,u_{2}}},\ \operatorname {tg} \,\alpha _{2}={\frac {\operatorname {th} \,u_{2}}{\operatorname {sh} \,u_{1}}}$

then we obtain

(17(	$\operatorname {cotg} \,\delta ={\frac {\operatorname {th} \,u_{1}\operatorname {sh} \,u_{1}+\operatorname {th} \,u_{2}\operatorname {sh} \,u_{2}}{\operatorname {sh} \,u_{1}\operatorname {sh} \,u_{2}-\operatorname {th} \,u_{1}\operatorname {th} \,u_{2}}}$

By (1) and (6) this goes over into

(18)	$\operatorname {cotg} \,\delta ={\frac {v_{1}^{2}{\sqrt {1-\left({\frac {v_{2}}{c}}\right)^{2}}}+v_{2}^{2}{\sqrt {1-\left({\frac {v_{1}}{c}}\right)^{2}}}}{v_{1}v_{2}\left(1-{\sqrt {1-\left({\frac {v_{1}}{c}}\right)^{2}}}{\sqrt {1-\left({\frac {v_{2}}{c}}\right)^{2}}}\right)}}$

We also want to express this in a different way. The direction difference of the resultant is equal to the defect of triangle OAB. The content of a Lobachevskian triangle is equal to its defect. According to the known formula^[1] for the defect we can put

(19)	$\operatorname {th} \,{\frac {\delta }{2}}={\frac {e^{u_{1}}-1}{e^{u_{1}}+1}}\cdot {\frac {e^{u_{2}}-1}{e^{u_{2}}+1}}$

or

(20)	$\operatorname {th} \,{\frac {\delta }{2}}=\operatorname {th} \,{\frac {u_{1}}{2}}\operatorname {th} \,{\frac {u_{2}}{2}}$

1. H. Liebmann, Nichteuklidische Geometrie, p. 149,