Page:Elektrische und Optische Erscheinungen (Lorentz) 035.jpg

$Div\ {\mathfrak {d}}=\rho {,}$

(Ib)

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

(IIb)

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

(IIIb)

$Rot\ {\mathfrak {F}}=-{\dot {\mathfrak {H}}}{,}$

(IVb)

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

(Vb)

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

(VIb)

${\mathfrak {E}}={\mathfrak {F}}+[{\mathfrak {v.H}}].$

(VIIb)

§ 21. From equations (I_a)—(V_a) (§ 19) also some formulas can be derived, any of them contains only one of the magnitudes ${\mathfrak {d}}_{x}$, ${\mathfrak {d}}_{y}$, ${\mathfrak {d}}_{z}$, ${\mathfrak {H}}_{x}$, ${\mathfrak {H}}_{y}$, ${\mathfrak {H}}_{z}$.

At first, it follows from (IV_a)

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

If we consider here what has been said in § 4, h, as well as the relations (I_a), (III_a) and (4_a), we arrive at the three formulas

$V^{2}\Delta {\mathfrak {d}}_{x}-\left({\frac {\partial ^{2}{\mathfrak {d}}_{x}}{\partial t^{2}}}\right)_{1}=V^{2}{\frac {\partial \rho }{\partial x}}+\left({\frac {\partial }{\partial t}}\right)_{1}\left\{\rho \left({\mathfrak {p}}_{x}+{\mathfrak {v}}_{x}\right)\right\}{\text{, etc.}}$

(A)

Similarly, we find

$V^{2}\Delta {\mathfrak {H}}_{x}-\left({\frac {\partial ^{2}{\mathfrak {H}}_{x}}{\partial t^{2}}}\right)_{1}=4\pi V^{2}\left[{\frac {\partial }{\partial z}}\left\{\rho \left({\mathfrak {p}}_{y}+{\mathfrak {v}}_{y}\right)\right\}-\right.$

$\left.-{\frac {\partial }{\partial y}}\left\{\rho \left({\mathfrak {p}}_{z}+{\mathfrak {v}}_{z}\right)\right\}\right]{\text{, etc.}}$

(B)

The last members of these six equations are completely known once we know how the ions are moving.

Application to electrostatics.

§ 22. We want to calculate by which forces the ions act on one another, when all of them are at rest with respect to ponderable matter. In this case a state occurs, where at every point ${\mathfrak {d}}$ and ${\mathfrak {H}}$ are independent of time. We have

$\left({\frac {\partial }{\partial t}}\right)_{1}=-\left({\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}+{\mathfrak {p}}_{y}{\frac {\partial }{\partial y}}+{\mathfrak {p}}_{z}{\frac {\partial }{\partial z}}\right){,}$

(19)

and equations (A) and (B) will be reduced, when for brevity's sake the operation

$\Delta -{\frac {1}{V^{2}}}\left({\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}+{\mathfrak {p}}_{y}{\frac {\partial }{\partial y}}+{\mathfrak {p}}_{z}{\frac {\partial }{\partial z}}\right)^{2}$