| (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];}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef796cfa04c1250fe3d291f528f440b164bb81b)
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und durch Substitution dieses Werthes (16.) ergiebt sich für das zu berechnende Doppelintegral 
(10.) folgende Darstellung:
| (17.)
|
|
wo 
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/426258a9660e57e51bc917d17dd6675d969d00bc)
|
Um zunächst 
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee8f6a68afe341413669844b20eaa029d60cfb1)
|
Multiplicirt man diese Formel mit 
, und integrirt sodann über sämmtliche Volumelemente des Körpers 
, 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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9d6207691ca4f797c2e2397049484baf79f737)
|
wo 
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1573348e51ee242e03420cbd9def2ec40780fc2)
|
Dabei ist unter 
irgend ein Element der Oberfläche von , und unter 
die auf errichtete innere Normale zu verstehen. Aus (20.) folgt, wenn man für 
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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03477c146bc5095bdac57ad414a38d4c39339262)
|
Hieraus folgt weiter durch Ausführung der Summation 
und mit Rücksicht auf (12.b):
