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	Carl Gottfried Neumann: Die elektrischen Kräfte

(16.)

${\displaystyle i_{0}i_{1}2{\frac {\partial \psi }{\partial \tau _{0}}}{\frac {\partial ^{2}\psi }{\partial s_{0}\partial s_{1}}}={\mathfrak {S}}_{0}{\mathfrak {S}}_{1}\left[{\mathfrak {u}}_{0}{\mathfrak {u}}_{1}\left({\frac {\partial \lambda }{\partial {\mathfrak {x}}_{0}}}+{\frac {\partial \mu }{\partial {\mathfrak {x}}_{1}}}\right)\right]-{\frac {\partial }{\partial \tau _{0}}}\left[{\mathfrak {S}}_{0}{\mathfrak {S}}_{1}\left({\mathfrak {u}}_{0}{\mathfrak {u}}_{1}\nu \right)\right];}$

und durch Substitution dieses Werthes (16.) ergiebt sich für das zu berechnende Doppelintegral ${\displaystyle {\mathsf {K}}}$ (10.) folgende Darstellung:

(17.)

${\displaystyle {\mathsf {K}}=\Lambda +{\mathsf {M}}-{\frac {\partial {\mathsf {N}}}{\partial \tau _{0}}},}$

wo ${\displaystyle \Lambda ,{\mathsf {M,N}}}$ die Werthe haben:

(18.)

${\displaystyle {\begin{array}{l}\Lambda =\Sigma \Sigma \ {\mathsf {Dv}}_{0}{\mathsf {Dv}}_{1}\left[{\mathfrak {S}}_{0}{\mathfrak {S}}_{1}\left({\mathfrak {u}}_{0}{\mathfrak {u}}_{1}{\frac {\partial \lambda }{\partial {\mathfrak {x}}_{0}}}\right)\right],\\\\{\mathsf {M}}=\Sigma \Sigma \ {\mathsf {Dv}}_{0}{\mathsf {Dv}}_{1}\left[{\mathfrak {S}}_{0}{\mathfrak {S}}_{1}\left({\mathfrak {u}}_{0}{\mathfrak {u}}_{1}{\frac {\partial \mu }{\partial {\mathfrak {x}}_{1}}}\right)\right],\\\\{\mathsf {N}}=\Sigma \Sigma \ {\mathsf {Dv}}_{0}{\mathsf {Dv}}_{1}\left[{\mathfrak {S}}_{0}{\mathfrak {S}}_{1}\left({\mathfrak {u}}_{0}{\mathfrak {u}}_{1}\nu \right)\right].\end{array}}}$

Um zunächst ${\displaystyle \Lambda }$ näher zu bestimmen, sei bemerkt, dass

(19.)

${\displaystyle {\begin{array}{ll}{\mathfrak {S}}_{0}\left({\mathfrak {u}}_{0}{\mathfrak {u}}_{1}{\frac {\partial \lambda }{\partial {\mathfrak {x}}_{0}}}\right)&={\mathfrak {u}}_{1}\left({\mathfrak {u}}_{0}{\frac {\partial \lambda }{\partial {\mathfrak {x}}_{0}}}+{\mathfrak {v}}_{0}{\frac {\partial \lambda }{\partial {\mathfrak {y}}_{0}}}+{\mathfrak {w}}_{0}{\frac {\partial \lambda }{\partial {\mathfrak {z}}_{0}}}\right),\\\\&={\mathfrak {u}}_{1}\left[\left({\frac {\partial \left(\lambda {\mathfrak {u}}_{0}\right)}{\partial {\mathfrak {x}}_{0}}}+\cdots \right)-\lambda \left({\frac {\partial {\mathfrak {u}}_{0}}{\partial {\mathfrak {x}}_{0}}}+\cdots \right)\right].\end{array}}}$

Multiplicirt man diese Formel mit ${\displaystyle {\mathsf {Dv}}_{0}}$, und integrirt sodann über sämmtliche Volumelemente des Körpers ${\displaystyle A}$, so ergiebt sich in bekannter Weise:

(20.)

${\displaystyle \Sigma \ {\mathsf {Dv}}_{0}\left[{\mathfrak {S}}_{0}\left({\mathfrak {u}}_{0}{\mathfrak {u}}_{1}{\frac {\partial \lambda }{\partial {\mathfrak {x}}_{0}}}\right)\right]=\Sigma \ {\mathsf {D}}o_{0}\left(\lambda F_{0}{\mathfrak {u}}_{1}\right)+\Sigma \ {\mathsf {Dv}}_{0}\left(\lambda E_{0}{\mathfrak {u}}_{1}\right),}$

wo ${\displaystyle E_{0},F_{0}}$ die Bedeutungen haben:

(21.)

${\displaystyle {\begin{array}{l}E_{0}=-\left[{\frac {\partial {\mathfrak {u}}_{0}}{\partial {\mathfrak {x}}_{0}}}+{\frac {\partial {\mathfrak {v}}_{0}}{\partial {\mathfrak {y}}_{0}}}+{\frac {\partial {\mathfrak {w}}_{0}}{\partial {\mathfrak {z}}_{0}}}\right],\\\\F_{0}=-\left[{\mathfrak {u}}_{0}\cos \left(N_{0},{\mathfrak {x}}_{0}\right)+{\mathfrak {v}}_{0}\cos \left(N_{0},{\mathfrak {y}}_{0}\right)+{\mathfrak {w}}_{0}\cos \left(N_{0},{\mathfrak {z}}_{0}\right)\right].\end{array}}}$

Dabei ist unter ${\displaystyle {\mathsf {D}}o_{0}}$ irgend ein Element der Oberfläche von ${\displaystyle A}$, und unter ${\displaystyle N_{0}}$ die auf ${\displaystyle {\mathsf {D}}o_{0}}$ errichtete innere Normale zu verstehen. Aus (20.) folgt, wenn man für ${\displaystyle \lambda }$ seine eigentliche Bedeutung (15.) substituirt:

(22.)

${\displaystyle \Sigma \ {\mathsf {Dv}}_{0}\left[{\mathfrak {S}}_{0}\left({\mathfrak {u}}_{0}{\mathfrak {u}}_{1}{\frac {\partial \lambda }{\partial {\mathfrak {x}}_{0}}}\right)\right]=\Sigma \ {\mathsf {D}}o_{0}\left(F_{0}{\frac {\partial \psi }{\partial \tau _{0}}}{\frac {\partial \psi }{\partial {\mathfrak {x}}_{1}}}{\mathfrak {u}}_{1}\right)+\Sigma \ {\mathsf {Dv}}_{0}\left(E_{0}{\frac {\partial \psi }{\partial \tau _{0}}}{\frac {\partial \psi }{\partial {\mathfrak {x}}_{1}}}{\mathfrak {u}}_{1}\right).}$

Hieraus folgt weiter durch Ausführung der Summation ${\displaystyle {\mathfrak {S}}_{1}}$ und mit Rücksicht auf (12.b):