Page:Elektrische und Optische Erscheinungen (Lorentz) 036.jpg

is indicated by $\Delta '$, to

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

(A')

and

$\Delta '{\mathfrak {H}}_{x}=4\pi \left({\mathfrak {p}}_{y}{\frac {\partial \rho }{\partial z}}-{\mathfrak {p}}_{z}{\frac {\partial \rho }{\partial y}}\right){\text{, etc.}}$

(B')

To fulfill these conditions, we determine a function $\omega$ by

$\Delta '\omega =\rho$

and put

${\mathfrak {d}}_{x}={\frac {\partial \omega }{\partial x}}-{\frac {{\mathfrak {p}}_{x}}{V^{2}}}\left({\mathfrak {p}}_{x}{\frac {\partial \omega }{\partial x}}+{\mathfrak {p}}_{y}{\frac {\partial \omega }{\partial y}}+{\mathfrak {p}}_{z}{\frac {\partial \omega }{\partial z}}\right){\text{, etc.,}}$

(20)

${\mathfrak {H}}_{x}=4\pi \left({\mathfrak {p}}_{y}{\frac {\partial \omega }{\partial z}}-{\mathfrak {p}}_{z}{\frac {\partial \omega }{\partial y}}\right){\text{, etc.,}}$

(21)

i.e., values that really satisfy the fundamental equations (I_a) - (IV_a).

From (V_a) it also follows

${\mathfrak {E}}_{x}=4\pi \left(V^{2}-{\mathfrak {p}}^{2}\right){\frac {\partial \omega }{\partial x}}{\text{, etc.,}}$

(22)

so that the sought forces are found.

Without prejudice to generality, we may assume that the translation happens in the direction of the x-axis. It is then ${\mathfrak {p}}_{y}={\mathfrak {p}}_{z}=0$, and the formula for the determination of $\omega$ will be transformed into

$\left(1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}\right){\frac {\partial ^{2}\omega }{\partial x^{2}}}+{\frac {\partial ^{2}\omega }{\partial y^{2}}}+{\frac {\partial ^{2}\omega }{\partial z^{2}}}=\rho .$

(23)

§ 23. To clearly define the meaning of the above formulas, we will compare the considered system $S_{1}$ with a second one $S_{2}$. The latter should not be moved, and it arises from $S_{1}$ by increasing all the dimensions that have the direction of the x-axis (therefore the relevant dimensions of the ions as well), in the ratio $\textstyle {\sqrt {V^{2}-{\mathfrak {p}}^{2}}}$ to $V$, or: between the coordinates x, y, z of a point of $S_{1}$ and the coordinates $x',y',z'$ of the same corresponding point of $S_{2}$, we let remain the relations