Page:Elektrische und Optische Erscheinungen (Lorentz) 034.jpg

$Rot\ {\mathfrak {H}}=4\pi {\mathfrak {S}}{,}$

(IIIa)

$-4\pi V^{2}Rot\ {\mathfrak {d}}=\left({\frac {\partial {\mathfrak {H}}}{\partial t}}\right)_{1}{,}$

(IVa)

${\mathfrak {E}}=4\pi V^{2}{\mathfrak {d}}+[{\mathfrak {p}}.{\mathfrak {H}}]+[{\mathfrak {v.H}}].$

(Va)

§ 20. For some purposes, a different form of some equations is more appropriate.

The first of the three (IV) summarized relations is namely

$-4\pi V^{2}\left({\frac {\partial {\mathfrak {d}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {d}}_{y}}{\partial z}}\right)={\frac {\partial {\mathfrak {H}}_{x}}{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial {\mathfrak {H}}_{x}}{\partial x}}-{\mathfrak {p}}_{y}{\frac {\partial {\mathfrak {H}}_{x}}{\partial y}}-{\mathfrak {p}}_{z}{\frac {\partial {\mathfrak {H}}_{x}}{\partial z}}{,}$

where, by equation (II_a), we can write for the last three members

$\left({\mathfrak {p}}_{x}{\frac {\partial {\mathfrak {H}}_{y}}{\partial y}}-{\mathfrak {p}}_{y}{\frac {\partial {\mathfrak {H}}_{x}}{\partial y}}\right)-\left({\mathfrak {p}}_{z}{\frac {\partial {\mathfrak {H}}_{x}}{\partial z}}-{\mathfrak {p}}_{x}{\frac {\partial {\mathfrak {H}}_{z}}{\partial z}}\right){,}$

which is nothing else than the first component of

$Rot\ [{\mathfrak {p.H}}].$

Accordingly, we obtain instead of (IV_a)

$Rot\ \left\{4\pi V^{2}{\mathfrak {d}}+[{\mathfrak {p.H}}]\right\}=-{\dot {\mathfrak {H}}}.$

Furthermore, the current ${\mathfrak {S}}$ can be entirely eliminated. The first of equations (III_a) becomes, when we consider (4_a) and (I_a),

${\frac {\partial {\mathfrak {H}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {H}}_{y}}{\partial z}}=4\pi \rho \left({\mathfrak {p}}_{x}+{\mathfrak {v}}_{x}\right)+4\pi \left({\frac {\partial {\mathfrak {d}}_{x}}{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial {\mathfrak {d}}_{x}}{\partial x}}-{\mathfrak {p}}_{y}{\frac {\partial {\mathfrak {d}}_{x}}{\partial y}}-\right.$

$\left.-{\mathfrak {p}}_{z}{\frac {\partial {\mathfrak {d}}_{x}}{\partial z}}\right)=4\pi \rho {\mathfrak {v}}_{x}+4\pi \left\{\left({\mathfrak {p}}_{x}{\frac {\partial {\mathfrak {d}}_{y}}{\partial y}}-{\mathfrak {p}}_{y}{\frac {\partial {\mathfrak {d}}_{x}}{\partial y}}\right)-\left({\mathfrak {p}}_{z}{\frac {\partial {\mathfrak {d}}_{x}}{\partial z}}-\right.\right.$

$\left.\left.-{\mathfrak {p}}_{x}{\frac {\partial {\mathfrak {d}}_{z}}{\partial z}}\right)\right\}+4\pi {\frac {\partial {\mathfrak {d}}_{x}}{\partial t}}.$

By that it follows, if we define a new vector ${\mathfrak {H}}'$ by means of the equation

${\mathfrak {H}}'={\mathfrak {H}}-4\pi [{\mathfrak {p.d}}]{,}$

thus

$Rot\ {\mathfrak {H}}'=4\pi \rho {\mathfrak {v}}+4\pi {\dot {\mathfrak {d}}}.$

If we now introduce the sign ${\mathfrak {F}}$ for the electric force-action on stationary ions, we obtain the following set of formulas