Seite:Carl Gottfried Neumann - Die elektrischen Kräfte 179.jpg

	Carl Gottfried Neumann: Die elektrischen Kräfte

Sodann folgt aus (11.a) durch Ableitung nach $s_{1}$ und Multiplication mit $i_{1}$:

(11.c)

${\begin{array}{ll}i_{0}i_{1}{\frac {\partial ^{2}\psi }{\partial s_{0}\partial s_{1}}}&={\frac {\partial ^{2}\psi }{\partial {\mathfrak {x}}_{0}\partial {\mathfrak {x}}_{1}}}{\mathfrak {u}}_{0}{\mathfrak {u}}_{1}+{\frac {\partial ^{2}\psi }{\partial {\mathfrak {x}}_{0}\partial {\mathfrak {y}}_{1}}}{\mathfrak {u}}_{0}{\mathfrak {v}}_{1}+{\frac {\partial ^{2}\psi }{\partial {\mathfrak {x}}_{0}\partial {\mathfrak {z}}_{1}}}{\mathfrak {u}}_{0}{\mathfrak {w}}_{1}\\\\&+{\frac {\partial ^{2}\psi }{\partial {\mathfrak {y}}_{0}\partial {\mathfrak {x}}_{1}}}{\mathfrak {v}}_{0}{\mathfrak {u}}_{1}+{\frac {\partial ^{2}\psi }{\partial {\mathfrak {y}}_{0}\partial {\mathfrak {y}}_{1}}}{\mathfrak {v}}_{0}{\mathfrak {v}}_{1}+{\frac {\partial ^{2}\psi }{\partial {\mathfrak {y}}_{0}\partial {\mathfrak {z}}_{1}}}{\mathfrak {v}}_{0}{\mathfrak {w}}_{1}\\\\&+{\frac {\partial ^{2}\psi }{\partial {\mathfrak {z}}_{0}\partial {\mathfrak {x}}_{1}}}{\mathfrak {w}}_{0}{\mathfrak {u}}_{1}+{\frac {\partial ^{2}\psi }{\partial {\mathfrak {z}}_{0}\partial {\mathfrak {y}}_{1}}}{\mathfrak {w}}_{0}{\mathfrak {v}}_{1}+{\frac {\partial ^{2}\psi }{\partial {\mathfrak {z}}_{0}\partial {\mathfrak {z}}_{1}}}{\mathfrak {w}}_{0}{\mathfrak {w}}_{1}.\end{array}}$

Diese Formeln (11.a,b,c)mögen in abgekürzter Weise angedeutet sein durch:

(12.a)

$i_{0}{\frac {\partial \psi }{\partial s_{0}}}={\mathfrak {S}}_{0}\left({\mathfrak {u}}_{0}{\frac {\partial \psi }{\partial {\mathfrak {x}}_{0}}}\right),$

(12.b)

$i_{1}{\frac {\partial \psi }{\partial s_{1}}}={\mathfrak {S}}_{1}\left({\mathfrak {u}}_{1}{\frac {\partial \psi }{\partial {\mathfrak {x}}_{1}}}\right),$

(12.c)

$i_{0}i_{1}{\frac {\partial ^{2}\psi }{\partial s_{0}\partial s_{1}}}={\mathfrak {S}}_{0}{\mathfrak {S}}_{1}\left({\mathfrak {u}}_{0}{\mathfrak {u}}_{1}{\frac {\partial ^{2}\psi }{\partial {\mathfrak {x}}_{0}\partial {\mathfrak {x}}_{1}}}\right),$

wo alsdann unter ${\mathfrak {S}}_{0}$ eine Summation über ${\scriptstyle \left({{\mathfrak {x}}_{0},{\mathfrak {y}}_{0},{\mathfrak {z}}_{0} \atop {\mathfrak {u}}_{0},{\mathfrak {v}}_{0},{\mathfrak {w}}_{0}}\right)}$, andererseits unter ${\mathfrak {S}}_{1}$ eine Summation über ${\scriptstyle \left({{\mathfrak {x}}_{1},{\mathfrak {y}}_{1},{\mathfrak {z}}_{1} \atop {\mathfrak {u}}_{1},{\mathfrak {v}}_{1},{\mathfrak {w}}_{1}}\right)}$ zu verstehen ist.

Aus (12.c) folgt durch Multiplication mit $2{\frac {\partial \psi }{\partial \tau _{0}}}$:

(13.)

$i_{0}i_{1}\cdot 2{\frac {\partial \psi }{\partial \tau _{0}}}{\frac {\partial ^{2}\psi }{\partial s_{0}\partial s_{1}}}={\mathfrak {S}}_{0}{\mathfrak {S}}_{1}\left({\mathfrak {u}}_{0}{\mathfrak {u}}_{1}\cdot 2{\frac {\partial \psi }{\partial \tau _{0}}}{\frac {\partial ^{2}\psi }{\partial {\mathfrak {x}}_{0}\partial {\mathfrak {x}}_{1}}}\right).$

Zufolge (3.a,b,c) sind ${\mathfrak {x}}_{0},{\mathfrak {y}}_{0},{\mathfrak {z}}_{0},\tau _{0},{\mathfrak {x}}_{1},{\mathfrak {y}}_{1},{\mathfrak {z}}_{1},\tau _{1}$ diejenigen acht coordinirten Argumente, von welchen die Entfernung $r$, mithin auch die Function $\psi =\psi (r)$ in letzter Instanz abhängt. Bei mehrfacher Differentiation nach diesen acht Argumenten wird daher das Resultat unabhängig sein von der Reihenfolge. Somit ist identisch:

(14.)

${\begin{array}{ll}2{\frac {\partial \psi }{\partial \tau _{0}}}{\frac {\partial ^{2}\psi }{\partial {\mathfrak {x}}_{0}\partial {\mathfrak {x}}_{1}}}&={\frac {\partial }{\partial {\mathfrak {x}}_{0}}}\left({\frac {\partial \psi }{\partial \tau _{0}}}{\frac {\partial \psi }{\partial {\mathfrak {x}}_{1}}}\right)+{\frac {\partial }{\partial {\mathfrak {x}}_{1}}}\left({\frac {\partial \psi }{\partial \tau _{0}}}{\frac {\partial \psi }{\partial {\mathfrak {x}}_{0}}}\right)-{\frac {\partial }{\partial \tau _{0}}}\left({\frac {\partial \psi }{\partial {\mathfrak {x}}_{0}}}{\frac {\partial \psi }{\partial {\mathfrak {x}}_{1}}}\right),\\\\&={\frac {\partial \lambda }{\partial {\mathfrak {x}}_{0}}}+{\frac {\partial \mu }{\partial {\mathfrak {x}}_{1}}}-{\frac {\partial \nu }{\partial {\mathfrak {\tau }}_{0}}},\end{array}}$

wo $\lambda ,\mu ,\nu$ die Bedeutungen haben:

(15.)

${\begin{array}{l}\lambda ={\frac {\partial \psi }{\partial \tau _{0}}}{\frac {\partial \psi }{\partial {\mathfrak {x}}_{1}}};\\\\\mu ={\frac {\partial \psi }{\partial \tau _{0}}}{\frac {\partial \psi }{\partial {\mathfrak {x}}_{0}}},\\\\\nu ={\frac {\partial \psi }{\partial {\mathfrak {x}}_{0}}}{\frac {\partial \psi }{\partial {\mathfrak {x}}_{1}}}.\end{array}}$

Durch (14.) geht die Formel (13.) über in:

Empfohlene Zitierweise:
Carl Gottfried Neumann: Die elektrischen Kräfte. B. G. Teubner, Leipzig 1873, Seite 161. Digitale Volltext-Ausgabe bei Wikisource, URL: https://de.wikisource.org/w/index.php?title=Seite:Carl_Gottfried_Neumann_-_Die_elektrischen_Kr%C3%A4fte_179.jpg&oldid=- (Version vom 31.7.2018)