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Anziehung eines homogenen elliptischen Cylinders.
(18)
Z
=
−
2
ρ
x
z
γ
1
−
β
2
∫
σ
∞
s
+
β
2
(
1
+
s
β
2
)
(
1
+
s
γ
2
)
⋅
∂
t
∂
s
d
s
t
t
−
x
2
+
Ψ
(
y
,
z
)
.
{\displaystyle Z=-{\frac {2\,\rho \,x\,z}{\gamma ^{1}-\beta ^{2}}}\int \limits _{\sigma }^{\infty }{\frac {s+\beta ^{2}}{\sqrt {\left(1+{\frac {s}{\beta ^{2}}}\right)\left(1+{\frac {s}{\gamma ^{2}}}\right)}}}\cdot {\frac {{\frac {\partial t}{\partial s}}ds}{t{\sqrt {t-x^{2}}}}}+\Psi (y,\,z).}
(19)
Z
=
−
2
ε
β
γ
π
ρ
z
γ
1
−
β
2
+
Ψ
(
y
,
z
)
−
4
ε
ρ
z
1
−
γ
2
β
2
∫
σ
∞
arcsin
1
−
x
2
t
⋅
d
1
+
s
γ
2
1
+
s
β
2
,
{\displaystyle {\begin{aligned}Z=&-{\frac {2\,\varepsilon \,\beta \,\gamma \,\pi \,\rho \,z}{\gamma ^{1}-\beta ^{2}}}+\Psi (y,\,z)\\&-{\frac {4\,\varepsilon \,\rho \,z}{1-{\frac {\gamma ^{2}}{\beta ^{2}}}}}\int \limits _{\sigma }^{\infty }\arcsin {\sqrt {1-{\frac {x^{2}}{t}}}}\cdot d{\sqrt {\frac {1+{\frac {s}{\gamma ^{2}}}}{1+{\frac {s}{\beta ^{2}}}}}},\\\end{aligned}}}
ε
=
+
1
{\displaystyle \varepsilon =+1\,}
x
>
0
{\displaystyle x>0\,}
ε
=
−
1
{\displaystyle \varepsilon =-1\,}
x
<
0
{\displaystyle x<0\,}
z
{\displaystyle z\,}
x
{\displaystyle x\,}
Die Function
Ψ
(
y
,
z
)
{\displaystyle \Psi (y,\,z)\,}
Φ
(
y
,
z
)
{\displaystyle \Phi (y,\,z)\,}
(20)
Ψ
(
y
,
z
)
=
2
ε
π
ρ
z
γ
2
−
β
2
⋅
σ
′
+
β
2
(
1
+
σ
′
β
2
)
(
1
+
σ
′
γ
2
)
{\displaystyle \Psi (y,\,z)=2\,\varepsilon \,\pi \,\rho {\frac {z}{\gamma ^{2}-\beta ^{2}}}\cdot {\frac {\sigma '+\beta ^{2}}{\sqrt {\left(1+{\frac {\sigma '}{\beta ^{2}}}\right)\left(1+{\frac {\sigma '}{\gamma ^{2}}}\right)}}}}
Nun bleibt von den Gleichungen (8) noch die zweite zu beweisen.
Aus der Gleichung (15) leiten wir her
∂
Y
∂
z
=
∂
Φ
∂
z
−
2
ρ
x
y
z
β
2
γ
2
∫
σ
∞
s
d
s
t
t
−
x
2
⋅
1
(
1
+
s
β
2
)
3
(
1
+
s
γ
2
)
3
,
{\displaystyle {\frac {\partial Y}{\partial z}}={\frac {\partial \Phi }{\partial z}}-{\frac {2\,\rho \,x\,y\,z}{\beta ^{2}\,\gamma ^{2}}}\int \limits _{\sigma }^{\infty }{\frac {s\,ds}{t{\sqrt {t-x^{2}}}}}\cdot {\frac {1}{\sqrt {\left(1+{\frac {s}{\beta ^{2}}}\right)^{3}\left(1+{\frac {s}{\gamma ^{2}}}\right)^{3}}}},}
∂
Z
∂
y
=
∂
Ψ
∂
y
−
2
ρ
x
y
z
β
2
γ
2
∫
σ
∞
s
d
s
t
t
−
x
2
⋅
1
(
1
+
s
β
2
)
3
(
1
+
s
γ
2
)
3
.
{\displaystyle {\frac {\partial Z}{\partial y}}={\frac {\partial \Psi }{\partial y}}-{\frac {2\,\rho \,x\,y\,z}{\beta ^{2}\,\gamma ^{2}}}\int \limits _{\sigma }^{\infty }{\frac {s\,ds}{t{\sqrt {t-x^{2}}}}}\cdot {\frac {1}{\sqrt {\left(1+{\frac {s}{\beta ^{2}}}\right)^{3}\left(1+{\frac {s}{\gamma ^{2}}}\right)^{3}}}}.}
