Seite:Carl Gottfried Neumann - Die elektrischen Kräfte 223.jpg

	Carl Gottfried Neumann: Die elektrischen Kräfte

${\displaystyle {\begin{array}{ll}{\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}{\frac {\omega }{r}}d\left(rj_{1}\right)&=d\left({\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}\omega j_{1}\right)-rj_{1}\cdot \delta \left({\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}{\frac {\omega }{r}}\right),\\\\&=d\left({\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}\omega j_{1}\right)-\omega j_{1}\cdot \delta {\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}-j_{1}{\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}\delta \omega +j_{1}{\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}{\frac {\omega \delta r}{r}},\end{array}}}$

so erhält man sofort:

(${\displaystyle \epsilon }$)

${\displaystyle {\mathfrak {X}}\ dt={\mathsf {Dv}}_{1}\left[d\left({\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}\omega j_{1}\right)-\omega j_{1}\cdot \delta {\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}-j_{1}{\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}\delta \omega -{\frac {\partial j_{1}}{\partial {\mathfrak {x}}_{0}}}\omega \delta r\right].}$

Hiefür aber kann, weil einerseits

${\displaystyle \delta {\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}={\frac {\partial \delta r}{\partial {\mathfrak {x}}_{0}}},}$

und andererseits

${\displaystyle {\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}\delta \omega ={\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}{\frac {d\omega }{dr}}\delta r={\frac {\partial \omega }{\partial {\mathfrak {x}}_{0}}}\delta r}$

ist, auch geschrieben werden:

(${\displaystyle \zeta }$)

${\displaystyle {\mathfrak {X}}\ dt={\mathsf {Dv}}_{1}\left[d\left({\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}\omega j_{1}\right)-\omega j_{1}{\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}-j_{1}{\frac {\partial \omega }{\partial {\mathfrak {x}}_{0}}}\delta r-{\frac {\partial j_{1}}{\partial {\mathfrak {x}}_{0}}}\omega \delta r\right],}$

oder (was dasselbe ist):

(${\displaystyle \eta }$)

${\displaystyle {\mathfrak {X}}\ dt={\mathsf {Dv}}_{1}\left[d\left({\frac {\partial r}{\partial {\mathfrak {x}}_{0}}}\omega j_{1}\right)-{\frac {\partial }{\partial {\mathfrak {x}}_{0}}}\left(\omega j_{1}\delta r\right)\right],}$

oder, falls man für ${\displaystyle \omega }$ seinen Werth (${\displaystyle \beta }$) substituirt:

(${\displaystyle \vartheta }$)

${\displaystyle {\mathfrak {X}}\ dt=4A^{2}{\mathsf {Dv}}_{1}\left[-d\left({\frac {\partial \psi }{\partial {\mathfrak {x}}_{0}}}{\frac {d\psi }{dr}}j_{1}\right)+{\frac {\partial }{\partial {\mathfrak {x}}_{0}}}\left({\frac {d\psi }{dr}}j_{1}\cdot \delta \psi \right)\right].}$

Nun ist aber: ${\displaystyle {\frac {d\psi }{dr}}j_{1}=-\partial _{1}\psi }$. Somit folgt:

${\displaystyle {\begin{array}{lllll}(\iota .)&&{\mathfrak {X}}\ dt=4A^{2}{\mathsf {Dv}}_{1}\left[d\left({\frac {\partial \psi }{\partial {\mathfrak {x}}_{0}}}\cdot \partial _{1}\psi \right)-{\frac {\partial }{\partial {\mathfrak {x}}_{0}}}\left(\partial _{1}\psi \cdot \delta \psi \right)\right].&&{\mathsf {Ebenso\ wird}}:\\\\(\varkappa .)&&{\mathfrak {Y}}\ dt=4A^{2}{\mathsf {Dv}}_{1}\left[d\left({\frac {\partial \psi }{\partial {\mathfrak {y}}_{0}}}\cdot \partial _{1}\psi \right)-{\frac {\partial }{\partial {\mathfrak {y}}_{0}}}\left(\partial _{1}\psi \cdot \delta \psi \right)\right],\\\\(\lambda .)&&{\mathfrak {Z}}\ dt=4A^{2}{\mathsf {Dv}}_{1}\left[d\left({\frac {\partial \psi }{\partial {\mathfrak {z}}_{0}}}\cdot \partial _{1}\psi \right)-{\frac {\partial }{\partial {\mathfrak {z}}_{0}}}\left(\partial _{1}\psi \cdot \delta \psi \right)\right].\end{array}}}$

Durch ${\displaystyle (\iota .)}$ ist der Beweis geliefert für die Formel (11.):

Um den Beweis der folgenden Formel (12.) zu erhalten, bedarf es ebenfalls ziemlich mühsamer Betrachtungen. Es waren [vergl. den Anfang des vorhergehenden Abschnitts (pag. 158)] mit

(${\displaystyle \mu _{1}}$.)

${\displaystyle {\begin{array}{l}x_{0}=C^{1}+C^{11}{\mathfrak {x}}_{0}+C^{12}{\mathfrak {y}}_{0}+C^{13}{\mathfrak {z}}_{0},\\y_{0}=C^{2}+C^{21}{\mathfrak {x}}_{0}+C^{22}{\mathfrak {y}}_{0}+C^{23}{\mathfrak {z}}_{0},\\z_{0}=C^{3}+C^{31}{\mathfrak {x}}_{0}+C^{32}{\mathfrak {y}}_{0}+C^{33}{\mathfrak {z}}_{0}\end{array}}}$