Seite:De Induction in rotirenden Kugeln (Hertz) 013.png

entsprechend für ${\displaystyle V\,}$ und ${\displaystyle W;\,}$ ausserdem die gewöhnlichen Stetigkeitsbedingungen. Allen diesen Bedingungen ist genügt, wenn ^[1] man setzt:

${\displaystyle {\begin{aligned}U_{i}&={\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}\left\lbrace -{\frac {1}{n+1}}{\frac {\partial }{\partial x}}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right)+x{\frac {\partial \chi _{n}}{\partial z}}\right\rbrace \\V_{i}&={\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}\left\lbrace -{\frac {1}{n+1}}{\frac {\partial }{\partial y}}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right)+y{\frac {\partial \chi _{n}}{\partial z}}\right\rbrace \\W_{i}&={\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}\left\lbrace -{\frac {1}{n+1}}{\frac {\partial }{\partial z}}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right)+z{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right\rbrace \end{aligned}}\,}$

${\displaystyle {\begin{aligned}U_{a}(\varrho )&=\left({\frac {R}{\varrho }}\right)^{2n+1}U_{i}\left({\frac {1}{\varrho }}\right)\\V_{a}(\varrho )&=\left({\frac {R}{\varrho }}\right)^{2n+1}V_{i}\left({\frac {1}{\varrho }}\right)\\W_{a}(\varrho )&=\left({\frac {R}{\varrho }}\right)^{2n+1}W_{i}\left({\frac {1}{\varrho }}\right).\end{aligned}}\,}$

Aus diesen ${\displaystyle U\ V\ W\,}$ wollen wir die magnetisirenden Kräfte im Innern, nämlich

${\displaystyle {\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}},\,}$ etc.

berechnen, und dieselben

${\displaystyle =-{\frac {n+1}{R}}{\frac {\partial {\mathit {\Psi }}}{\partial z}},\,}$ etc.

setzen, wir erhalten so die Funktion ${\displaystyle {\mathit {\Psi }}\,}$ (§ 1,5). Wir finden nämlich:

${\displaystyle {\begin{aligned}-{\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}{\frac {\partial }{\partial z}}\left(x{\frac {\partial \chi _{n}}{\partial y}}-y{\frac {\partial \chi _{n}}{\partial z}}\right)&=-{\frac {n+1}{R}}{\frac {\partial }{\partial z}}{\mathit {\Psi _{i}}}\\-{\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}{\frac {\partial }{\partial x}}\left(x{\frac {\partial \chi _{n}}{\partial y}}-y{\frac {\partial \chi _{n}}{\partial x}}\right)&=-{\frac {n+1}{R}}{\frac {\partial }{\partial x}}{\mathit {\Psi _{i}}}\\-{\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}{\frac {\partial }{\partial y}}\left(x{\frac {\partial \chi _{n}}{\partial y}}-y{\frac {\partial \chi _{n}}{\partial x}}\right)&=-{\frac {n+1}{R}}{\frac {\partial }{\partial y}}{\mathit {\Psi _{i}}}.\end{aligned}}\,}$

Also ergiebt sich

${\displaystyle {\begin{aligned}{\mathit {\Psi _{i}}}&={\frac {4\pi R^{2}}{(2n+1)(n+1)}}{\frac {\omega }{k}}\left(x{\frac {\partial \chi _{n}}{\partial y}}-y{\frac {\partial \chi _{n}}{\partial x}}\right)\\&={\frac {4\pi R^{2}}{(2n+1)(n+1)}}{\frac {\omega }{k}}\chi _{n}',\end{aligned}}\,}$