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which can be represented for the unit of charge by

${\mathfrak {E}}_{1}=4\pi V^{2}{\mathfrak {d}}.$

§ 10. Let two stationary ions with charges e and $e'$ be given, whose dimensions are small in relation to distance r. To find the force that acts on the first one, we have to decompose it into space elements, to apply on any of them the previous theorem, and then to integrate. Thereby ${\mathfrak {d}}$ may be considered as composed of the dielectric displacements, that stem from the first and the second particle. We easily find that the first part of ${\mathfrak {d}}$ doesn't contribute anything to the total force. The second part has (within the first ion) everywhere the direction of r and the magnitude $e'/4\pi r^{2}$; so e will be repulsed by $e'$ by the force

$V^{2}{\frac {ee'}{r^{2}}}.$

As this is in agreement with Coulomb's laws, it is clear that the theory of ions, as regards the ordinary problems, leads back to the older way of treatment.

Electric currents in ponderable conductors.

§ 11. In a ponderable conductor, in which a current flows through, and in which innumerable ions are in motion according to our view, ${\mathfrak {d}}$, ${\mathfrak {S}}$ and ${\mathfrak {H}}$ are changing in an irregular way from point to point. Yet from equations (II) and (III) it follows

${\begin{array}{l}Div\ {\bar {\mathfrak {H}}}=0{,}\\Rot\ {\bar {\mathfrak {H}}}=4\pi {\bar {\mathfrak {S}}}{;}\\\end{array}}$

since ${\bar {\mathfrak {H}}}$ coincides with ${\mathfrak {H}}$ in measurable distance from the conductor, and the action into the outside will only be determined by the average current ${\bar {\mathfrak {S}}}$. It is this current, with which the ordinary theory (which neglects molecular processes) is dealing.