is valid . One only has to graphically illustrate the components as it was explained above, and to use Lobachevski an trigonometry for the calculation of the resultant.
v
1
{\displaystyle v_{1}}
v
2
{\displaystyle v_{2}}
α with each other. The lengths
U
1
,
U
2
{\displaystyle U_{1},U_{2}}
u
1
,
u
2
{\displaystyle u_{1},u_{2}}
(4)
v
1
c
=
th
u
1
,
v
2
c
=
th
u
2
,
{\displaystyle {\frac {v_{1}}{c}}=\operatorname {th} \,\ u_{1},\ {\frac {v_{2}}{c}}=\operatorname {th} \,u_{2},}
Then lay off the line
O
A
=
U
1
{\displaystyle OA=U_{1}}
O into the direction of
v
1
{\displaystyle v_{1}}
A
B
=
U
2
{\displaystyle AB=U_{2}}
α . The resultant corresponds to the line
O
B
=
U
{\displaystyle OB=U}
Lobachevski an triangle OAB the relation is given
(5)
ch
u
=
ch
u
1
ch
u
2
+
sh
u
1
sh
u
2
cos
α
{\displaystyle \operatorname {ch} \,u=\operatorname {ch} \,u_{1}\ \operatorname {ch} \,u_{2}+\operatorname {sh} \,u_{1}\operatorname {sh} \,u_{2}\cos \ \alpha }
If we denote herein
(6)
ch
u
=
1
1
−
(
v
c
)
2
,
sh
u
=
v
c
1
−
(
v
c
)
2
,
{\displaystyle \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}}}},}
then we obtain
1
−
(
v
c
)
2
=
1
−
(
v
1
c
)
2
1
−
(
v
2
c
)
2
1
+
v
1
v
2
cos
α
c
2
{\displaystyle {\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}={\frac {{\sqrt {1-\left({\frac {v_{1}}{c}}\right)^{2}}}{\sqrt {1-\left({\frac {v_{2}}{c}}\right)^{2}}}}{1+{\frac {v_{1}v_{2}\cos \alpha }{c^{2}}}}}}
and after some transformations
(
v
c
)
2
=
(
v
1
c
)
2
+
(
v
2
c
)
2
−
(
v
1
v
2
c
2
)
2
−
1
(
1
+
v
1
v
2
cos
α
c
2
)
2
+
1
=
(
v
1
c
)
2
+
(
v
2
c
)
2
−
(
v
1
v
2
c
2
)
2
⋅
(
1
−
cos
2
α
)
+
v
1
v
2
cos
α
c
2
(
1
+
v
1
v
2
cos
α
c
2
)
2
{\displaystyle {\begin{array}{c}\left({\frac {v}{c}}\right)^{2}={\frac {\left({\frac {v_{1}}{c}}\right)^{2}+\left({\frac {v_{2}}{c}}\right)^{2}-\left({\frac {v_{1}v_{2}}{c^{2}}}\right)^{2}-1}{\left(1+{\frac {v_{1}v_{2}\cos \alpha }{c^{2}}}\right)^{2}}}+1\\\\={\frac {\left({\frac {v_{1}}{c}}\right)^{2}+\left({\frac {v_{2}}{c}}\right)^{2}-\left({\frac {v_{1}v_{2}}{c^{2}}}\right)^{2}\cdot \left(1-\cos ^{2}\alpha \right)+{\frac {v_{1}v_{2}\cos \alpha }{c^{2}}}}{\left(1+{\frac {v_{1}v_{2}\cos \alpha }{c^{2}}}\right)^{2}}}\end{array}}}
Eventually, from this it follows
(7)
v
=
v
1
2
+
v
2
2
+
2
v
1
v
2
cos
α
−
(
v
1
v
2
sin
α
c
)
2
1
+
v
1
v
2
cos
α
c
2
{\displaystyle v={\frac {\sqrt {v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos \alpha -\left({\frac {v_{1}v_{2}\sin \alpha }{c}}\right)^{2}}}{1+{\frac {v_{1}v_{2}\cos \alpha }{c^{2}}}}}}
and that is Einstein 's addition law of velocities. In non-euclidean vector notation we can write it in the form
(8)
u
=
u
1
+
u
2
{\displaystyle {\mathfrak {u}}={\mathfrak {u}}_{1}+{\mathfrak {u}}_{2}}
