Seite:De Induction in rotirenden Kugeln (Hertz) 008.png

5. Es sei ${\displaystyle \psi \,}$ die Strömungsfunktion einer Kugelschaale vom Radius ${\displaystyle R,\,}$ es sei ${\displaystyle {\mathit {\Psi }}=\int {\frac {\psi ds}{r}}\,}$ das Potential einer Masse, welche auf der Kugelschaale mit der Dichtigkeit ${\displaystyle \psi \,}$ verbreitet ist, so ist das Potential der Strömung: ^[1]

${\displaystyle {\mathit {\Omega }}=-{\frac {1}{R}}{\frac {\partial }{\partial \varrho }}({\mathit {\Psi }}\varrho )\,}$

und die Grössen ${\displaystyle U\ V\ W\,}$ sind:

${\displaystyle {\begin{aligned}&U={\frac {y}{R}}{\frac {\partial {\mathit {\Psi }}}{\partial z}}-{\frac {z}{R}}{\frac {\partial {\mathit {\Psi }}}{\partial y}}={\frac {1}{R}}{\frac {\partial }{\partial \omega _{x}}}{\mathit {\Psi }}\\&V={\frac {z}{R}}{\frac {\partial {\mathit {\Psi }}}{\partial x}}-{\frac {x}{R}}{\frac {\partial {\mathit {\Psi }}}{\partial z}}={\frac {1}{R}}{\frac {\partial }{\partial \omega _{y}}}{\mathit {\Psi }}\\&U={\frac {x}{R}}{\frac {\partial {\mathit {\Psi }}}{\partial y}}-{\frac {y}{R}}{\frac {\partial {\mathit {\Psi }}}{\partial x}}={\frac {1}{R}}{\frac {\partial }{\partial \omega _{z}}}{\mathit {\Psi }}.\end{aligned}}\,}$

Ist ${\displaystyle {\mathit {\Psi }}\,}$ eine homogene Funktion ${\displaystyle n\,}$ten Grades in ${\displaystyle x,y,z,}$ so ist

${\displaystyle {\begin{aligned}{\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}}&=-{\frac {n+1}{R}}{\frac {\partial }{\partial z}}{\mathit {\Psi }}\\{\frac {\partial U}{\partial z}}-{\frac {\partial W}{\partial x}}&=-{\frac {n+1}{R}}{\frac {\partial }{\partial y}}{\mathit {\Psi }}\\{\frac {\partial W}{\partial y}}-{\frac {\partial V}{\partial z}}&=-{\frac {n+1}{R}}{\frac {\partial }{\partial x}}{\mathit {\Psi }}.\end{aligned}}\,}$

Immer ist:

${\displaystyle {\begin{aligned}{\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}}&={\frac {\partial {\mathit {\Omega }}}{\partial z}}\\{\frac {\partial U}{\partial z}}-{\frac {\partial W}{\partial x}}&={\frac {\partial {\mathit {\Omega }}}{\partial y}}\\{\frac {\partial W}{\partial y}}-{\frac {\partial V}{\partial z}}&={\frac {\partial {\mathit {\Omega }}}{\partial x}}.\end{aligned}}\,}$

Man findet diese Formeln entwickelt in Maxwell’s Treatise on electricity, Vol. II, pag. 276. Die Vorzeichen sind dort theilweise andere, es liegt dies daran, dass dort nicht unser Coordinatensystem, sondern das symmetrische angewandt ist. Das hier benutzte Coordinatensystem ist dasjenige, auf welches sich die Helmholtz’schen Formeln beziehen.