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In this assumption, that gives us the advantage that no discontinuities must be considered, there is no essential restriction. Because the charge distribution over a surface and a discontinuity of $\rho$ can be treated as limiting cases of states to which that assumption are applicable.

In the cases to be considered, $\rho$ is different from zero only in the interior of a very great number of small spaces which are completely mutually separated. Yet we can start with the general case, that an electric density exists in arbitrary great spaces. Since we think of the electric charges as always connected to ponderable matter, then this would correspond to a continuous distribution of matter.

Ponderable matter, which is not charged, has only to be considered by us, when it exerts molecular forces on the ions. Concerning the electric phenomena, it has no influence at all and everything happens, as if the space where it is located would only contain the aether.

Where $\rho$ is different form zero, equation (3) is not applicable anymore. According to a known theorem from Maxwell's theory, we have for any closed surface $\sigma$, when E represents the entire charge in the interior,

$\int {\mathfrak {d}}_{n}\ d\ \sigma =E=\int \rho \ d\ \tau {,}$

or

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

so everywhere it must be

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

(I)

If the ponderable matter is moving, then — since it carries the charge along with it — at a certain point of space there always exists another $\rho$, and soon it is (when we are dealing with mutually separated ions) different from zero here and there. Yet the condition of the aether has constantly to obey equation (I).

§ 6. The change of ${\mathfrak {d}}$, that happens with time at a