aus Wikisource, der freien Quellensammlung
Achter Abschnitt. §. 100.
§. 100.
Wirkung sämmtlicher Theilchen
ε
′
{\displaystyle \varepsilon '\,}
ε
{\displaystyle \varepsilon \,}
Um die Wirkung sämmtlicher elektrischer Theilchen
ε
′
{\displaystyle \varepsilon '\,}
ε
{\displaystyle \varepsilon \,}
(1)
S
=
ε
∑
(
−
ε
′
r
)
=
ε
V
,
{\displaystyle S=\varepsilon \sum \left(-{\frac {\varepsilon '}{r}}\right)=\varepsilon V,}
ε
′
{\displaystyle \varepsilon '\,}
(
x
,
y
,
z
)
{\displaystyle (x,\,y,\,z)}
D
{\displaystyle D\,}
Weber’s Formel ist
(2a )
D
=
ε
∑
ε
′
c
2
1
r
(
d
r
d
t
)
2
;
{\displaystyle D=\varepsilon \sum {\frac {\varepsilon '}{c^{2}}}{\frac {1}{r}}\left({\frac {dr}{dt}}\right)^{2};}
Riemann’s Formel dagegen
(2b )
D
=
ε
∑
ε
′
c
2
1
r
{
(
d
x
d
t
−
d
x
1
d
t
)
2
+
(
d
y
d
t
−
d
y
1
d
t
)
2
+
(
d
z
d
t
−
d
z
1
d
t
)
2
}
.
{\displaystyle D=\varepsilon \sum {\frac {\varepsilon '}{c^{2}}}{\frac {1}{r}}{\Bigg \lbrace }\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right)^{2}+\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right)^{2}+\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right)^{2}{\Bigg \rbrace }.\,}
b ) die Quadrate ausgerechnet, so zerlegt sich D in drei Bestandtheile, nemlich:
D
=
ε
∑
ε
′
c
2
1
r
{
(
d
x
d
t
)
2
+
(
d
y
d
t
)
2
+
(
d
z
d
t
)
2
}
+
ε
∑
ε
′
c
2
1
r
{
(
d
x
1
d
t
)
2
+
(
d
y
1
d
t
)
2
+
(
d
z
1
d
t
)
2
}
−
2
ε
∑
ε
′
c
2
1
r
{
d
x
d
t
d
x
1
d
t
+
d
y
d
t
d
y
1
d
t
+
d
z
d
t
d
z
1
d
t
}
.
{\displaystyle {\begin{aligned}D&=\varepsilon \sum {\frac {\varepsilon '}{c^{2}}}{\frac {1}{r}}{\Bigg \lbrace }\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}{\Bigg \rbrace }\\&+\varepsilon \sum {\frac {\varepsilon '}{c^{2}}}{\frac {1}{r}}{\Bigg \lbrace }\left({\frac {dx_{1}}{dt}}\right)^{2}+\left({\frac {dy_{1}}{dt}}\right)^{2}+\left({\frac {dz_{1}}{dt}}\right)^{2}{\Bigg \rbrace }\\&-2\varepsilon \sum {\frac {\varepsilon '}{c^{2}}}{\frac {1}{r}}{\Bigg \lbrace }{\frac {dx}{dt}}{\frac {dx_{1}}{dt}}+{\frac {dy}{dt}}{\frac {dy_{1}}{dt}}+{\frac {dz}{dt}}{\frac {dz_{1}}{dt}}{\Bigg \rbrace }.\\\end{aligned}}}
ε
{\displaystyle \varepsilon \,}
v
{\displaystyle v\,}
ε
′
{\displaystyle \varepsilon '\,}
v
′
{\displaystyle v'\,}
D
=
ε
c
2
v
2
∑
ε
′
r
+
ε
c
2
∑
ε
′
r
v
′
2
−
2
ε
c
2
∑
ε
′
r
{
d
x
d
t
d
x
1
d
t
+
d
y
d
t
d
y
1
d
t
+
d
z
d
t
d
z
1
d
t
}
.
=
−
2
ε
c
2
v
2
V
+
ε
c
2
∑
ε
′
r
v
′
2
−
2
ε
c
2
d
x
d
t
∑
ε
′
r
d
x
1
d
t
−
2
ε
c
2
d
y
d
t
∑
ε
′
r
d
y
1
d
t
−
2
ε
c
2
d
z
d
t
∑
ε
′
r
d
z
1
d
t
.
{\displaystyle {\begin{aligned}D={\frac {\varepsilon }{c^{2}}}&v^{2}\sum {\frac {\varepsilon '}{r}}+{\frac {\varepsilon }{c^{2}}}\sum {\frac {\varepsilon '}{r}}v'^{2}\\-2&{\frac {\varepsilon }{c^{2}}}\sum {\frac {\varepsilon '}{r}}{\Bigg \lbrace }{\frac {dx}{dt}}{\frac {dx_{1}}{dt}}+{\frac {dy}{dt}}{\frac {dy_{1}}{dt}}+{\frac {dz}{dt}}{\frac {dz_{1}}{dt}}{\Bigg \rbrace }.\\=&-2{\frac {\varepsilon }{c^{2}}}v^{2}V+{\frac {\varepsilon }{c^{2}}}\sum {\frac {\varepsilon '}{r}}v'^{2}\\&-2{\frac {\varepsilon }{c^{2}}}{\frac {dx}{dt}}\sum {\frac {\varepsilon '}{r}}{\frac {dx_{1}}{dt}}\,\\&-2{\frac {\varepsilon }{c^{2}}}{\frac {dy}{dt}}\sum {\frac {\varepsilon '}{r}}{\frac {dy_{1}}{dt}}\,\\&-2{\frac {\varepsilon }{c^{2}}}{\frac {dz}{dt}}\sum {\frac {\varepsilon '}{r}}{\frac {dz_{1}}{dt}}.\,\\\end{aligned}}}
