Page:Elektrische und Optische Erscheinungen (Lorentz) 039.jpg

 observed in relation to bodies at rest on earth, is the magnetic force (21). At first glance, we might expect a corresponding effect on the current elements. We will return to this question in § 26.

Values of $$\mathfrak{d}$$ and $$\mathfrak{H}$$ at a stationary current.
§ 25. On the basis of equations (A) and (B) we again tackle the problem treated in § 11. We consider, as there, the mean values and take into account that for them the simplification (19) is permitted in stationary states; moreover, we assume at first that the conductors do not have a significant charge, so that $$\bar{\rho}=0$$.

It is near at hand to interpret the vector $$\overline{\rho\mathfrak{v}}$$ as being a "current". We think of it as solenoidally distributed and denote it by $$\bar{\mathfrak{S}}$$, where it remains, however, temporarily undecided whether this is also the mean value of the vector occurring in (4a).

We now derive from (A) and (B)

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

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

If we determine thus the three auxiliary magnitudes $$\chi_{x}$$, $$\chi_{y}$$, $$\chi_{z}$$ by means of the equations

$$\Delta'\chi_{x}=\overline{\mathfrak{S}}_{x},\ \Delta'\chi_{y}=\overline{\mathfrak{S}}_{y},\ \Delta'\chi_{z}=\overline{\mathfrak{S}}_{z}{,}$$

so everywhere we have