Schwere, Elektricität und Magnetismus/§. 103.

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 Endlich sollen für das elektrische Theilchen ${\displaystyle \varepsilon \,}$ die Bewegungsgleichungen auch aus Weber’s Formel hergeleitet werden:

(1)	${\displaystyle S=\varepsilon V,}$

(2)

${\displaystyle {\begin{aligned}D={\frac {\varepsilon }{c^{2}}}\sum {\frac {\varepsilon '}{r^{3}}}{\Bigg \lbrace }(x-x_{1})\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right)&+(y-y_{1})\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right)\\&+(z-z_{1})\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right){\Bigg \rbrace }^{2}.\,\\\end{aligned}}}$

Hier findet sich

${\displaystyle {\frac {\partial D}{\partial x'}}=a{\frac {dx}{dt}}+b{\frac {dy}{dt}}+c{\frac {dz}{dt}}+k,}$

wobei ${\displaystyle a,\,b,\,c,\,k}$ Functionen von ${\displaystyle x,\,y,\,z}$ sind, welche der partiellen Differentialgleichung genügen

${\displaystyle \Delta _{2}\,\Delta _{2}F=0.}$

Durch Differentiation nach ${\displaystyle t\,}$ erhalten wir

${\displaystyle {\frac {d\left({\frac {\partial D}{\partial x'}}\right)}{dt}}=a{\frac {d^{2}x}{dt^{2}}}+b{\frac {d^{2}y}{dt^{2}}}+c{\frac {d^{2}z}{dt^{2}}}+g,}$

und hier ist die Function ${\displaystyle g\,}$ nur von den Coordinaten ${\displaystyle x,\,y,\,z}$ und den Geschwindigkeiten ${\displaystyle {\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\,}$ abhängig. Demnach lauten die Bewegungsgleichungen:

(3)

${\displaystyle {\begin{aligned}(m-a){\frac {d^{2}x}{dt^{2}}}-b{\frac {d^{2}y}{dt^{2}}}-c{\frac {d^{2}z}{dt^{2}}}&=g-{\frac {\partial D}{\partial x}}+{\frac {\partial S}{\partial x}},\\\,\\-a_{1}{\frac {d^{2}x}{dt^{2}}}+(m-b_{1}){\frac {d^{2}y}{dt^{2}}}-c_{1}{\frac {d^{2}z}{dt^{2}}}&=g_{1}-{\frac {\partial D}{\partial y}}+{\frac {\partial S}{\partial y}},\\\,\\-a_{2}{\frac {d^{2}x}{dt^{2}}}-b_{2}{\frac {d^{2}y}{dt^{2}}}+(m-c_{2}){\frac {d^{2}z}{dt^{2}}}&=g_{2}-{\frac {\partial D}{\partial z}}+{\frac {\partial S}{\partial z}}.\\\end{aligned}}}$

Hier müsste also zunächst eliminirt werden.