aus Wikisource, der freien Quellensammlung
War
V
2
−
V
1
{\displaystyle V_{2}-V_{1}}
(
V
2
−
V
1
)
0
′
=
(
V
2
−
V
1
)
1
−
(
v
c
)
2
.
{\displaystyle \left(V_{2}-V_{1}\right)_{0}^{'}=\left(V_{2}-V_{1}\right){\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}.}
Wir erhalten nun durch eine ähnliche Rechnung wie die, welche zur Gleichung (46) führte:
(65)
Q
0
′
=
16
3
π
c
K
(
0
)
1
−
(
v
c
)
2
.
{\displaystyle Q'_{0}={\frac {16}{3}}{\frac {\pi }{c}}K(0){\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}.}
Wir gehen jetzt zur Berechnung von
Q
v
′
{\displaystyle Q'_{v}}
p
v
′
{\displaystyle p'_{v}}
U
v
′
{\displaystyle U'_{v}}
v
{\displaystyle v}
K
(
0
)
{\displaystyle K(0)}
K
(
ϑ
,
v
)
=
K
(
0
)
(
1
−
(
v
c
)
2
)
8
3
(
1
−
v
c
cos
ϑ
)
4
,
{\displaystyle K(\vartheta ,v)=K(0){\frac {\left(1-\left({\frac {v}{c}}\right)^{2}\right)^{\frac {8}{3}}}{\left(1-{\frac {v}{c}}\cos \vartheta \right)^{4}}},}
und in (62) außerdem an die Stelle von
(
cos
ϑ
′
+
v
c
)
2
1
−
(
v
c
)
2
{\displaystyle {\frac {\left(\cos \vartheta '+{\frac {v}{c}}\right)^{2}}{1-\left({\frac {v}{c}}\right)^{2}}}}
cos
2
ϑ
′
{\displaystyle \cos ^{2}\vartheta '}
d
p
v
′
=
2
c
K
(
0
)
(
1
−
(
v
c
)
2
)
8
3
(
1
−
v
c
cos
ϑ
)
4
d
Ω
(
1
−
v
c
cos
ϑ
)
2
1
−
(
v
c
)
2
cos
2
ϑ
′
{\displaystyle dp'_{v}={\frac {2}{c}}K(0){\frac {\left(1-\left({\frac {v}{c}}\right)^{2}\right)^{\frac {8}{3}}}{\left(1-{\frac {v}{c}}\cos \vartheta \right)^{4}}}d\Omega {\frac {\left(1-{\frac {v}{c}}\cos \vartheta \right)^{2}}{1-\left({\frac {v}{c}}\right)^{2}}}\cos ^{2}\vartheta '}
oder nach (52):
d
p
v
′
=
2
c
K
(
0
)
(
1
−
(
v
c
)
2
)
5
3
(
1
−
v
c
cos
ϑ
)
4
d
Ω
(
cos
ϑ
−
v
c
)
2
{\displaystyle dp'_{v}={\frac {2}{c}}K(0){\frac {\left(1-\left({\frac {v}{c}}\right)^{2}\right)^{\frac {5}{3}}}{\left(1-{\frac {v}{c}}\cos \vartheta \right)^{4}}}d\Omega \left(\cos \vartheta -{\frac {v}{c}}\right)^{2}}
und:
p
v
′
=
∫
ϑ
=
0
ϑ
=
arccos
v
c
∫
φ
=
0
2
π
2
c
K
(
0
)
(
cos
2
ϑ
−
v
c
)
2
(
1
−
v
c
cos
ϑ
)
4
d
Ω
(
1
−
(
v
c
)
2
)
5
3
,
{\displaystyle p'_{v}={\overset {\vartheta =\arccos {\frac {v}{c}}}{\underset {\vartheta =0}{\int }}}{\overset {2\pi }{\underset {\varphi =0}{\int }}}{\frac {2}{c}}K(0){\frac {\left(\cos ^{2}\vartheta -{\frac {v}{c}}\right)^{2}}{\left(1-{\frac {v}{c}}\cos \vartheta \right)^{4}}}d\Omega \left(1-\left({\frac {v}{c}}\right)^{2}\right)^{\frac {5}{3}},}
p
v
′
=
4
3
π
c
(
1
−
(
v
c
)
2
)
2
3
K
(
0
)
.
{\displaystyle p'_{v}={\frac {4}{3}}{\frac {\pi }{c}}\left(1-\left({\frac {v}{c}}\right)^{2}\right)^{\frac {2}{3}}K(0).}
