Seite:De Induction in rotirenden Kugeln (Hertz) 065.png

${\displaystyle {\begin{aligned}&U\ &{\text{durch }}\ &{\frac {\partial M}{\partial z}}-{\frac {\partial N}{\partial y}}\\&V&&{\frac {\partial N}{\partial x}}-{\frac {\partial L}{\partial z}}\\&W&&{\frac {\partial L}{\partial y}}-{\frac {\partial M}{\partial x}}.\end{aligned}}\,}$

Danach ist weiter zu ersetzen:

${\displaystyle 1)\,}$

${\displaystyle {\begin{aligned}&{\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}}\ &{\text{durch }}\ &{\mathit {\Delta }}N+{\frac {\partial \chi }{\partial z}}\\&{\frac {\partial U}{\partial z}}-{\frac {\partial W}{\partial x}}&&{\mathit {\Delta }}M+{\frac {\partial \chi }{\partial y}}\\&{\frac {\partial W}{\partial y}}-{\frac {\partial V}{\partial z}}&&{\mathit {\Delta }}L+{\frac {\partial \chi }{\partial x}},\end{aligned}}\,}$

da

${\displaystyle 2)\,}$

${\displaystyle {\frac {\partial L}{\partial x}}+{\frac {\partial M}{\partial y}}+{\frac {\partial N}{\partial z}}=-\chi \,}$ ist.

Nun ist aber

${\displaystyle 3)\,}$

${\displaystyle {\begin{aligned}&{\mathit {\Delta }}L&=-4\pi \lambda \\&{\mathit {\Delta }}M&=-4\pi \mu \\&{\mathit {\Delta }}N&=-4\pi \nu \end{aligned}}\,}$

${\displaystyle 4)\,}$

${\displaystyle {\begin{aligned}{\frac {\lambda }{\theta }}&={\frac {\partial V}{\partial z}}-{\frac {\partial W}{\partial y}}-{\frac {\partial \chi }{\partial x}}\\{\frac {\mu }{\theta }}&={\frac {\partial W}{\partial x}}-{\frac {\partial U}{\partial z}}-{\frac {\partial \chi }{\partial y}}\\{\frac {\nu }{\theta }}&={\frac {\partial U}{\partial y}}-{\frac {\partial V}{\partial x}}-{\frac {\partial \chi }{\partial z}}.\end{aligned}}\,}$

Indem man hiernach ${\displaystyle L,\ M,\ N\,}$ aus den Ausdrücken 1) eliminirt, und die von den Strömen direct ausgeübten Kräfte ^[1] ${\displaystyle {\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}}\,}$ etc. addirt, erhält man für die Ausdrücke, welche jetzt an Stelle der ${\displaystyle {\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}}\,}$ etc. zu treten haben die folgenden: