Page:Elektrische und Optische Erscheinungen (Lorentz) 018.jpg

${\mathfrak {d}}_{x}=-{\frac {1}{4\pi }}{\frac {\partial \omega }{\partial x}}{\text{, etc.}}$

(6)

and derive from (I)

$\triangle \omega =-4\pi \rho .$

(7)

After we determined $\omega$ from that, ${\mathfrak {d}}_{x},\ {\mathfrak {d}}_{y},\ {\mathfrak {d}}_{z}$ can be calculated from (6).

The first part of the force acting on ponderable matter.

§ 9. According to the older electrostatics, whose conclusions are in agreement with experience, we obtain the force components that act on the volume element in the case previously considered, when we at first determine the "potential function" by means of Poisson's equation, and then multiply its derivatives by $-V^{2}\rho \ d\ \tau$^[1]. Since our formula (7) is in agreement with Poisson's equations, the potential function must coincide with $\omega$; therefore we have to assume as values of the force components

$-V^{2}\rho {\frac {\partial \omega }{\partial x}}\ d\ \tau {\text{, etc.}}$

(8)

If the forces, as it is claimed by Maxwell's theory, shall be caused by the state of the aether, then it is probable that it depends on the dielectric displacement in the considered volume element. Indeed, when we consider (6), we can write for (8)

$4\pi V^{2}{\mathfrak {d}}_{x}\rho \ d\ \tau {\text{, etc.}}$

Therefore I will assume, that in all cases in which a dielectric displacement exists in element $d\tau$, the aether exerts a force with the mentioned components on ponderable matter located at this place, i.e. a force^[2]

1. The factor $V^{2}$ must be added, because we use the electromagnetic system of units.
2. Since this force is the only one, which exists in relation to electrostatic phenomena, it can well be called electrostatic force, although in general it also depends on the motion of ions.