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Zweiter Abschnitt. §. 31.
−
(
4
π
)
2
ρ
=
(
∂
(
4
π
V
′
)
∂
r
′
)
a
+
0
−
(
∂
(
4
π
V
′
)
∂
r
′
)
a
−
0
.
{\displaystyle -(4\;\pi )^{2}\,\rho =\left({\frac {\partial (4\pi V')}{\partial r'}}\right)_{a+0}-\left({\frac {\partial (4\pi V')}{\partial r'}}\right)_{a-0}\,.}
(
∂
(
4
π
V
′
)
∂
r
′
)
a
+
0
=
+
a
2
∫
0
2
π
∂
φ
∫
0
π
(
∂
2
U
∂
r
∂
r
′
)
r
=
a
,
r
′
=
a
f
(
θ
,
φ
)
sin
θ
d
θ
{\displaystyle \left({\frac {\partial (4\pi V')}{\partial r'}}\right)_{a+0}=+\,a^{2}\int \limits _{0}^{2\pi }\partial \varphi \int \limits _{0}^{\pi }\left({\frac {\partial ^{2}U}{\partial r\,\partial r'}}\right)_{r=a,\,r'=a}\,f(\theta ,\,\varphi )\sin \theta \,d\theta }
(
∂
(
4
π
V
′
)
∂
r
′
)
a
−
0
=
−
a
2
∫
0
2
π
∂
φ
∫
0
π
(
∂
2
U
∂
r
∂
r
′
)
r
=
a
,
r
′
=
a
f
(
θ
,
φ
)
sin
θ
d
θ
.
{\displaystyle \left({\frac {\partial (4\pi V')}{\partial r'}}\right)_{a-0}=-\,a^{2}\int \limits _{0}^{2\pi }\partial \varphi \int \limits _{0}^{\pi }\left({\frac {\partial ^{2}U}{\partial r\,\partial r'}}\right)_{r=a,\,r'=a}\,f(\theta ,\,\varphi )\sin \theta \,d\theta .}
(5)
−
(
4
π
)
2
ρ
=
2
a
2
∫
0
2
π
∂
φ
∫
0
π
(
∂
2
U
∂
r
∂
r
′
)
r
=
a
,
r
′
=
a
f
(
θ
,
φ
)
sin
θ
d
θ
.
{\displaystyle -(4\;\pi )^{2}\,\rho =2a^{2}\int \limits _{0}^{2\pi }\partial \varphi \int \limits _{0}^{\pi }\left({\frac {\partial ^{2}U}{\partial r\,\partial r'}}\right)_{r=a,\,r'=a}\,f(\theta ,\,\varphi )\sin \theta \,d\theta .}
Der bei
∂
2
U
∂
r
∂
r
′
{\displaystyle {\frac {\partial ^{2}U}{\partial r\,\partial r'}}}
r
=
a
{\displaystyle r=a\!}
r
′
=
a
{\displaystyle r'=a\!}
Es bleibt noch übrig, in (4) und (5) die Function
U
{\displaystyle U\!}
V
′
{\displaystyle V'\!}
θ
′
=
0
,
φ
′
{\displaystyle \theta '=0,\;\varphi '}
U
{\displaystyle U\!}
(
∂
U
∂
r
)
r
=
a
=
−
a
−
r
′
cos
θ
(
a
2
+
r
′
2
−
2
a
r
′
cos
θ
)
3
2
+
a
r
′
(
a
−
a
2
r
′
cos
θ
)
(
a
2
+
(
a
2
r
′
)
2
−
2
a
a
2
r
′
cos
θ
)
3
2
=
(
r
′
a
)
2
(
a
−
a
2
r
′
cos
θ
)
−
(
a
−
r
′
cos
θ
)
(
a
2
+
r
′
2
−
2
a
r
′
cos
θ
)
3
2
=
r
′
2
−
a
2
a
(
a
2
+
r
′
2
−
2
a
r
′
cos
θ
)
3
2
.
{\displaystyle {\begin{aligned}\left({\frac {\partial U}{\partial r}}\right)_{r=a}=&-{\frac {a-r'\cos \theta }{(a^{2}+r'^{2}-2\,a\,r'\cos \theta )^{\frac {3}{2}}}}\\&+{\frac {{\frac {a}{r'}}\left(a-{\frac {a^{2}}{r'}}\,\cos \theta \right)}{\left(a^{2}+\left({\frac {a^{2}}{r'}}\right)^{2}-2\,a\,{\frac {a^{2}}{r'}}\,\cos \theta \right)^{\frac {3}{2}}}}\\=&{\frac {\left({\frac {r'}{a}}\right)^{2}\left(a-\,{\frac {a^{2}}{r'}}\,\cos \theta \right)-(a-r'\cos \theta )}{(a^{2}+r'^{2}-2\,a\,r'\cos \theta )^{\frac {3}{2}}}}\\=&{\frac {r'^{2}-a^{2}}{a(a^{2}+r'^{2}-2\,a\,r'\cos \theta )^{\frac {3}{2}}}}.\,\\\end{aligned}}}
