Page:Lorentz Simplified1899.djvu/4

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$Div'\ {\mathfrak {F}}'={\frac {4\pi }{k}}V^{2}\varrho -4\pi k{\mathfrak {p}}_{x}\varrho {\mathfrak {v}}_{x}$,

(Ic)

$Div'\ {\mathfrak {H}}'=0$,

(IIc)

$\left.{\begin{array}{l}{\frac {\partial {\mathfrak {H}}'_{z}}{\partial y'}}-{\frac {\partial {\mathfrak {H}}'_{y}}{\partial z'}}=4\pi k^{2}\varrho {\mathfrak {v}}_{x}+{\frac {k^{2}}{V^{2}}}{\frac {\partial {\mathfrak {F}}'_{x}}{\partial t'}}\\\\{\frac {\partial {\mathfrak {H}}'_{x}}{\partial z'}}-{\frac {\partial {\mathfrak {H}}'_{z}}{\partial x'}}=4\pi k\varrho {\mathfrak {v}}_{y}+{\frac {k^{2}}{V^{2}}}{\frac {\partial {\mathfrak {F}}'_{y}}{\partial t'}}\\\\{\frac {\partial {\mathfrak {H}}'_{y}}{\partial x'}}-{\frac {\partial {\mathfrak {H}}'_{x}}{\partial y'}}=4\pi k\varrho {\mathfrak {v}}_{z}+{\frac {k^{2}}{V^{2}}}{\frac {\partial {\mathfrak {F}}'_{z}}{\partial t'}}\end{array}}\right\}$

(IIIc)

$\left.{\begin{array}{l}{\frac {\partial {\mathfrak {F}}'_{z}}{\partial y'}}-{\frac {\partial {\mathfrak {F}}'_{y}}{\partial z'}}=-{\frac {\partial {\mathfrak {H}}'_{x}}{\partial t'}}\\\\{\frac {\partial {\mathfrak {F}}'_{x}}{\partial z'}}-{\frac {\partial {\mathfrak {F}}'_{z}}{\partial x'}}=-{\frac {\partial {\mathfrak {H}}'_{y}}{\partial t'}}\\\\{\frac {\partial {\mathfrak {F}}'_{y}}{\partial x'}}-{\frac {\partial {\mathfrak {F}}'_{x}}{\partial y'}}=-{\frac {\partial {\mathfrak {H}}'_{z}}{\partial t'}}\end{array}}\right\}$

(IVc)

$\left.{\begin{array}{l}{\mathfrak {E}}_{x}={\mathfrak {F}}'_{x}+k{\frac {{\mathfrak {p}}_{x}}{V^{2}}}({\mathfrak {v}}_{y}{\mathfrak {F}}'_{y}+{\mathfrak {v}}_{z}{\mathfrak {F}}'_{z})+({\mathfrak {v}}_{y}{\mathfrak {H}}'_{z}-{\mathfrak {v}}_{z}{\mathfrak {H}}'_{y})\\\\{\mathfrak {E}}_{y}={\frac {1}{k}}{\mathfrak {F}}'_{y}-k{\frac {{\mathfrak {p}}_{x}}{V^{2}}}{\mathfrak {v}}_{x}{\mathfrak {F}}'_{y}+({\frac {1}{k}}{\mathfrak {v}}_{z}{\mathfrak {H}}'_{x}-{\mathfrak {v}}_{x}{\mathfrak {H}}'_{z})\\\\{\mathfrak {E}}_{z}={\frac {1}{k}}{\mathfrak {F}}'_{z}-k{\frac {{\mathfrak {p}}_{x}}{V^{2}}}{\mathfrak {v}}_{x}{\mathfrak {F}}'_{z}+({\mathfrak {v}}_{x}{\mathfrak {H}}'_{y}-{\frac {1}{k}}{\mathfrak {v}}_{y}{\mathfrak {H}}'_{x})\end{array}}\right\}$.

(Vc)

Putting ${\mathfrak {v}}=0$ in the three last equations we see that

${\mathfrak {F}}'_{x},\ {\frac {1}{k}}{\mathfrak {F}}'_{y},\ {\frac {1}{k}}{\mathfrak {F}}'_{z}$

are the components of the electric force that would act on a particle at rest.

§ 5. We shall begin with an application of the equations to