Seite:De Induction in rotirenden Kugeln (Hertz) 035.png

ferner:

${\displaystyle {\begin{aligned}{\frac {\partial V}{\partial x}}-{\frac {\partial U}{\partial y}}&=-m\varrho ^{m-2}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right)-\varrho ^{m}(n+1){\frac {\partial \chi _{n}}{\partial z}}\\{\frac {\partial W}{\partial y}}-{\frac {\partial V}{\partial z}}&=-m\varrho ^{m-2}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial x}}-nx\chi _{n}\right)-\varrho ^{m}(n+1){\frac {\partial \chi _{n}}{\partial x}}\\{\frac {\partial U}{\partial z}}-{\frac {\partial W}{\partial x}}&=-m\varrho ^{m-2}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial y}}-ny\chi _{n}\right)-\varrho ^{m}(n+1){\frac {\partial \chi _{n}}{\partial y}}.\end{aligned}}\,}$

Daraus folgt:

${\displaystyle O=-n(n+1)\varrho ^{m}\chi _{n}.\,}$

Also werden die Bedingungen für ${\displaystyle \varphi \,}$:

${\displaystyle {\begin{aligned}{\mathit {\Delta }}\varphi =&-\omega m(m+2n+3)\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right)\varrho ^{m-2}\\&-2\omega (n+1)\varrho ^{m}{\frac {\partial \chi _{n}}{\partial z}};\end{aligned}}\,}$

an der Grenze:

${\displaystyle {\frac {\partial \varphi }{\partial \varrho }}=-(m+n+1){\frac {\omega }{\varrho }}\varrho ^{m}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right).\,}$

Diesen Bedingungen aber genügt ${\displaystyle \varphi ,\,}$ denn es ist

erstens:

${\displaystyle {\begin{aligned}{\mathit {\Delta }}\varphi =&-\omega \left[{\mathit {\Delta }}\left(\varrho ^{m+2}{\frac {\partial \chi _{n}}{\partial z}}\right)-2n{\frac {\partial }{\partial z}}(\varrho ^{m}\chi ^{n})-nz{\mathit {\Delta }}(\varrho ^{m}\chi ^{n})\right]\\=&-\omega \left\lbrace \varrho ^{n}{\frac {\partial \chi _{n}}{\partial z}}[(m+2)(m+2n+1)-2n]\right.\\&\left.-nz\varrho ^{m-2}\chi _{n}[m(m+2n+1)+2m]\right\rbrace \\=&-\omega \left\lbrace m(m+2n+3)\varrho ^{m-2}\left(\varrho ^{2}{\frac {\partial \chi _{n}}{\partial z}}-nz\chi _{n}\right)\right.\\&\left.+2(n+1)\varrho ^{n}{\frac {\partial \chi _{n}}{\partial z}}\right\rbrace ,\end{aligned}}\,}$

wonach die erste Bedingung erfüllt ist;

zweitens ist ${\displaystyle \varphi \,}$ das Produkt aus einer Funktion der Winkel und ${\displaystyle \varrho ^{m+n+1},\,}$ also

${\displaystyle {\frac {\partial \varphi }{\partial \varrho }}={\frac {m+n+1}{\varrho }}\varphi ,\,}$