Seite:Carl Gottfried Neumann - Die elektrischen Kräfte 262.jpg

	Carl Gottfried Neumann: Die elektrischen Kräfte

Multiplicirt man daher auf beiden Seiten mit ${\mathsf {D}}s\ {\mathsf {D}}s_{1}$, und integrirt über alle ${\mathsf {D}}s_{1}$ des geschlossenen Stromes, so ergiebt sich:

$0=\Sigma _{1}\left[{\mathsf {D}}s\ {\mathsf {D}}s_{1}\left(-{\frac {\partial x_{1}}{\partial s_{1}}}{\frac {\partial {\frac {1}{r}}}{\partial s}}-{\frac {3\Theta \Theta _{1}\left(x-x_{1}\right)}{r^{3}}}+{\frac {{\mathsf {E}}\left(x-x_{1}\right)}{r^{3}}}\right)\right].$

Addirt man aber diese Formel, nachdem sie zuvor mit $A^{2}JJ_{1}$ multiplicirt worden ist, zur Formel (7.), so folgt:

(8.)	$X=A^{2}JJ_{1}\cdot \Sigma _{1}\left[{\mathsf {D}}s\ {\mathsf {D}}s_{1}\left(-{\frac {\partial x_{1}}{\partial s_{1}}}{\frac {\partial {\frac {1}{r}}}{\partial s}}-{\frac {{\mathsf {E}}\left(x-x_{1}\right)}{r^{3}}}\right)\right].$

Nun ist offenbar:

${\frac {\partial x_{1}}{\partial s_{1}}}{\frac {\partial {\frac {1}{r}}}{\partial s}}=\left({\frac {\partial {\frac {1}{r}}}{\partial x}}{\frac {\partial x}{\partial s}}+{\frac {\partial {\frac {1}{r}}}{\partial y}}{\frac {\partial y}{\partial s}}+{\frac {\partial {\frac {1}{r}}}{\partial z}}{\frac {\partial z}{\partial s}}\right){\frac {\partial x_{1}}{\partial s_{1}}},$

und folglich:

${\mathsf {D}}s\ {\mathsf {D}}s_{1}{\frac {\partial x_{1}}{\partial s_{1}}}{\frac {\partial {\frac {1}{r}}}{\partial s}}=\left({\frac {\partial {\frac {1}{r}}}{\partial x}}{\mathsf {D}}x+{\frac {\partial {\frac {1}{r}}}{\partial y}}{\mathsf {D}}y+{\frac {\partial {\frac {1}{r}}}{\partial z}}{\mathsf {D}}z\right){\mathsf {D}}x_{1},$

wo ${\mathsf {D}}x,\ {\mathsf {D}}y,\ {\mathsf {D}}z$ und ${\mathsf {D}}x_{1},\ {\mathsf {D}}y_{1},\ {\mathsf {D}}z_{1}$ die rechtwinkligen Projectionen von ${\mathsf {D}}s$ und ${\mathsf {D}}s_{1}$ vorstellen. — Andererseits ist:

${\mathsf {D}}s\ {\mathsf {D}}s_{1}{\frac {{\mathsf {E}}\left(x-x_{1}\right)}{r^{3}}}=-{\frac {\partial {\frac {1}{r}}}{\partial x}}\left({\mathsf {D}}x\ {\mathsf {D}}x_{1}+{\mathsf {D}}y\ {\mathsf {D}}y_{1}+{\mathsf {D}}z\ {\mathsf {D}}z_{1}\right).$

Durch Addition der beiden letzten Formeln folgt sofort:

(9.)	${\mathsf {D}}s\ {\mathsf {D}}s_{1}\left({\frac {\partial x_{1}}{\partial s_{1}}}{\frac {\partial {\frac {1}{r}}}{\partial s}}+{\frac {{\mathsf {E}}\left(x-x_{1}\right)}{r^{3}}}\right)=-\gamma {\mathsf {D}}y+\beta {\mathsf {D}}z,$

wo $\alpha ,\beta ,\gamma$ die Bedeutungen haben:

(10.)

${\begin{array}{l}\alpha ={\frac {\partial {\frac {1}{r}}}{\partial y}}{\mathsf {D}}z_{1}-{\frac {\partial {\frac {1}{r}}}{\partial z}}{\mathsf {D}}y_{1},\\\\\beta ={\frac {\partial {\frac {1}{r}}}{\partial z}}{\mathsf {D}}x_{1}-{\frac {\partial {\frac {1}{r}}}{\partial x}}{\mathsf {D}}z_{1},\\\\\gamma ={\frac {\partial {\frac {1}{r}}}{\partial x}}{\mathsf {D}}y_{1}-{\frac {\partial {\frac {1}{r}}}{\partial y}}{\mathsf {D}}x_{1}.\end{array}}$

Durch Benutzung von (9.) gewinnt die zu berechnende Componente $X$ (8.) folgendes Aussehen:

Empfohlene Zitierweise:
Carl Gottfried Neumann: Die elektrischen Kräfte. B. G. Teubner, Leipzig 1873, Seite 244. Digitale Volltext-Ausgabe bei Wikisource, URL: https://de.wikisource.org/w/index.php?title=Seite:Carl_Gottfried_Neumann_-_Die_elektrischen_Kr%C3%A4fte_262.jpg&oldid=- (Version vom 31.7.2018)