Page:Elektrische und Optische Erscheinungen (Lorentz) 037.jpg

$x=x'{\sqrt {1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}}},\ y=y',\ z=z'$

(24)

In addition, the mutually corresponding volume elements, and therefore also the ions, shall have the same charges in $S_{1}$ and $S_{2}$.

If we apply to all magnitudes, which are related to the second system, a prime so they can be distinguished, then

$\rho '=\rho {\sqrt {1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}}}{,}$

and

${\frac {\partial ^{2}\omega '}{\partial x'^{2}}}+{\frac {\partial ^{2}\omega '}{\partial y'^{2}}}+{\frac {\partial ^{2}\omega '}{\partial z'^{2}}}=\rho '=\rho {\sqrt {1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}}}.$

Then the equation (23) can be written in the form

${\frac {\partial ^{2}\omega }{\partial x'^{2}}}+{\frac {\partial ^{2}\omega }{\partial y'^{2}}}+{\frac {\partial ^{2}\omega }{\partial z'^{2}}}=\rho$

then

$\omega ={\frac {\omega '}{\sqrt {1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}}}}{,}$

and since in the second system

${\mathfrak {E'}}_{x}=4\pi V^{2}{\frac {\partial \omega '}{\partial x'}}{\text{, etc.}}$

thus

${\mathfrak {E}}_{x}={\mathfrak {E'}}_{x},\ {\mathfrak {E}}_{y}={\sqrt {1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}}}{\mathfrak {E'}}_{y},\ {\mathfrak {E}}_{z}={\sqrt {1-{\frac {{\mathfrak {p}}^{2}}{V^{2}}}}}{\mathfrak {E'}}_{z}.$

The same relations, as they exist between the components of ${\mathfrak {E}}$ and ${\mathfrak {E}}'$, also exist, since the charges in $S_{1}$ and $S_{2}$ are equal, between the force components acting on an ion.

If in the second system at certain places ${\mathfrak {E}}'=0$, then ${\mathfrak {E}}$ vanishes at the corresponding points of the first system.

§ 24. Several implications of this theorem are obvious. From ordinary electrostatics, we know for example that an excess of positive (or negative) ions can be distributed over a