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	Carl Gottfried Neumann: Die elektrischen Kräfte

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

(11.c)

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

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

(12.a)

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

(12.b)

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

(12.c)

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

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

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

(13.)

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

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

(14.)

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

wo ${\displaystyle \lambda ,\mu ,\nu }$ die Bedeutungen haben:

(15.)

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

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