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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 (4_a).

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 $\chi _{x}$, $\chi _{y}$, $\chi _{z}$^[1] 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

${\overline {{\mathfrak {d}}_{x}}}=-{\frac {1}{V^{2}}}\left({\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}+{\mathfrak {p}}_{y}{\frac {\partial }{\partial y}}+{\mathfrak {p}}_{z}{\frac {\partial }{\partial z}}\right)\chi _{x}{\text{, etc.,}}$

(25)

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

(26)

1. These magnitudes are only different by a constant factor from the components of the vector potential, when ${\mathfrak {p}}=0$.