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

	Carl Gottfried Neumann: Die elektrischen Kräfte

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

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

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

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

Nun ist offenbar:

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

und folglich:

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

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

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

Durch Addition der beiden letzten Formeln folgt sofort:

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

wo ${\displaystyle \alpha ,\beta ,\gamma }$ die Bedeutungen haben:

(10.)

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

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