Schwere, Elektricität und Magnetismus:286

 

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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Fertig! Dieser Text wurde zweimal anhand der Quelle Korrektur gelesen. Die Schreibweise folgt dem Originaltext.

Sechster Abschnitt. §. 78.

${\begin{aligned}&{\frac {\partial }{\partial y}}\,\left\{{\frac {\partial (U_{1}-u_{1})}{\partial y}}-{\frac {\partial (U_{2}-u_{2})}{\partial x}}\,\right\}\\=&{\frac {\partial }{\partial z}}\,\left\{{\frac {\partial (U_{3}-u_{3})}{\partial x}}-{\frac {\partial (U_{1}-u_{1})}{\partial z}}\,\right\},\\\,\\&{\frac {\partial }{\partial z}}\,\left\{{\frac {\partial (U_{2}-u_{2})}{\partial z}}-{\frac {\partial (U_{3}-u_{3})}{\partial y}}\,\right\}\\=&{\frac {\partial }{\partial x}}\,\left\{{\frac {\partial (U_{1}-u_{1})}{\partial y}}-{\frac {\partial (U_{2}-u_{2})}{\partial x}}\,\right\},\\\,\\&{\frac {\partial }{\partial x}}\,\left\{{\frac {\partial (U_{3}-u_{3})}{\partial x}}-{\frac {\partial (U_{1}-u_{1})}{\partial z}}\,\right\}\\=&{\frac {\partial }{\partial y}}\,\left\{{\frac {\partial (U_{2}-u_{2})}{\partial z}}-{\frac {\partial (U_{3}-u_{3})}{\partial y}}\,\right\}.\\\end{aligned}}$

 Diese partiellen Differentialgleichungen sind erfüllt, wenn man setzt:

${\frac {\partial (U_{1}-u_{1})}{\partial y}}={\frac {\partial (U_{2}-u_{2})}{\partial x}},\,$ ${\frac {\partial (U_{2}-u_{2})}{\partial z}}={\frac {\partial (U_{3}-u_{3})}{\partial y}},\,$ ${\frac {\partial (U_{3}-u_{3})}{\partial x}}={\frac {\partial (U_{1}-u_{1})}{\partial z}}.\,$

Darin spricht sich aber aus, dass die Differenzen $U_{1}-u_{1},\,U_{2}-u_{2},\,U_{3}-u_{3}$ die resp. nach $x,\,y,\,z$ genommenen Derivirten einer und derselben Function $F(x,\,y,\,z)$ sind.

 Um nun die Functionen $u_{1},\,u_{2},\,u_{3}$ völlig zu bestimmen, darf man noch eine Gleichung hinzufügen. Wir wählen die Gleichung:

(3)	${\frac {\partial u_{1}}{\partial x}}+{\frac {\partial u_{2}}{\partial y}}+{\frac {\partial u_{3}}{\partial z}}=0.\,$

 Durch sie gehen die Gleichungen (2) in folgende über:

(4)

${\frac {\partial ^{2}u_{1}}{\partial x^{2}}}+{\frac {\partial ^{2}u_{1}}{\partial y^{2}}}+{\frac {\partial ^{2}u_{1}}{\partial z^{2}}}=4\pi i_{1},$ ${\frac {\partial ^{2}u_{2}}{\partial x^{2}}}+{\frac {\partial ^{2}u_{2}}{\partial y^{2}}}+{\frac {\partial ^{2}u_{2}}{\partial z^{2}}}=4\pi i_{2},$ ${\frac {\partial ^{2}u_{3}}{\partial x^{2}}}+{\frac {\partial ^{2}u_{3}}{\partial y^{2}}}+{\frac {\partial ^{2}u_{3}}{\partial z^{2}}}=4\pi i_{3}.$