aus Wikisource, der freien Quellensammlung
64
U
durch
∂
M
∂
z
−
∂
N
∂
y
V
∂
N
∂
x
−
∂
L
∂
z
W
∂
L
∂
y
−
∂
M
∂
x
.
{\displaystyle {\begin{aligned}&U\ &{\text{durch }}\ &{\frac {\partial M}{\partial z}}-{\frac {\partial N}{\partial y}}\\&V&&{\frac {\partial N}{\partial x}}-{\frac {\partial L}{\partial z}}\\&W&&{\frac {\partial L}{\partial y}}-{\frac {\partial M}{\partial x}}.\end{aligned}}\,}
Danach ist weiter zu ersetzen:
1
)
{\displaystyle 1)\,}
∂
V
∂
x
−
∂
U
∂
y
durch
Δ
N
+
∂
χ
∂
z
∂
U
∂
z
−
∂
W
∂
x
Δ
M
+
∂
χ
∂
y
∂
W
∂
y
−
∂
V
∂
z
Δ
L
+
∂
χ
∂
x
,
{\displaystyle {\begin{aligned}&{\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}}\ &{\text{durch }}\ &{\mathit {\Delta }}N+{\frac {\partial \chi }{\partial z}}\\&{\frac {\partial U}{\partial z}}-{\frac {\partial W}{\partial x}}&&{\mathit {\Delta }}M+{\frac {\partial \chi }{\partial y}}\\&{\frac {\partial W}{\partial y}}-{\frac {\partial V}{\partial z}}&&{\mathit {\Delta }}L+{\frac {\partial \chi }{\partial x}},\end{aligned}}\,}
da
2
)
{\displaystyle 2)\,}
∂
L
∂
x
+
∂
M
∂
y
+
∂
N
∂
z
=
−
χ
{\displaystyle {\frac {\partial L}{\partial x}}+{\frac {\partial M}{\partial y}}+{\frac {\partial N}{\partial z}}=-\chi \,}
ist.
Nun ist aber
3
)
{\displaystyle 3)\,}
Δ
L
=
−
4
π
λ
Δ
M
=
−
4
π
μ
Δ
N
=
−
4
π
ν
{\displaystyle {\begin{aligned}&{\mathit {\Delta }}L&=-4\pi \lambda \\&{\mathit {\Delta }}M&=-4\pi \mu \\&{\mathit {\Delta }}N&=-4\pi \nu \end{aligned}}\,}
4
)
{\displaystyle 4)\,}
λ
θ
=
∂
V
∂
z
−
∂
W
∂
y
−
∂
χ
∂
x
μ
θ
=
∂
W
∂
x
−
∂
U
∂
z
−
∂
χ
∂
y
ν
θ
=
∂
U
∂
y
−
∂
V
∂
x
−
∂
χ
∂
z
.
{\displaystyle {\begin{aligned}{\frac {\lambda }{\theta }}&={\frac {\partial V}{\partial z}}-{\frac {\partial W}{\partial y}}-{\frac {\partial \chi }{\partial x}}\\{\frac {\mu }{\theta }}&={\frac {\partial W}{\partial x}}-{\frac {\partial U}{\partial z}}-{\frac {\partial \chi }{\partial y}}\\{\frac {\nu }{\theta }}&={\frac {\partial U}{\partial y}}-{\frac {\partial V}{\partial x}}-{\frac {\partial \chi }{\partial z}}.\end{aligned}}\,}
Indem man hiernach
L
,
M
,
N
{\displaystyle L,\ M,\ N\,}
aus den Ausdrücken 1) eliminirt, und die von den Strömen direct ausgeübten Kräfte [1]
∂
V
∂
x
−
∂
U
∂
y
{\displaystyle {\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}}\,}
etc. addirt, erhält man für die Ausdrücke, welche jetzt an Stelle der
∂
V
∂
x
−
∂
U
∂
y
{\displaystyle {\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}}\,}
etc. zu treten haben die folgenden:
↑ Die Differentialgleichungen. WS: Die Randnotiz wurde als Fußnote übertragen.