Page:Elektrische und Optische Erscheinungen (Lorentz) 067.jpg

 force acting on it, N the number of molecules in unit volume. For the equations

$$m\frac{d^{2}\mathfrak{q}_{x}}{dt^{2}}=\mathfrak{K}_{x}$$, etc.

it follows, when we take the averages of second kind and multiply them by eN

$$m\frac{\partial^{2}\mathfrak{M}_{x}}{\partial t^{2}}=eN\overline{\overline{\mathfrak{K}_{x}}}$$, etc.

As regards $$\mathfrak{K}$$, it is at first to note, that by our assumption the fixed parts of the molecule are acting upon the ion by a certain force, that is exactly caused by the displacement $$\mathfrak{q}$$. Let the components of this force be linear, homogeneous functions of $$\mathfrak{q}_{x}, \mathfrak{q}_{y}, \mathfrak{q}_{z}$$, or rather, since only this is relevant for the following, let the averages of those components be given by

{{MathForm2|(55)|$$\left.\begin{array}{c} -\left(s_{1.1}\overline{\overline{\mathfrak{q}_{x}}}+s_{1.2}\overline{\overline{\mathfrak{q}_{y}}}+s_{1.3}\overline{\overline{\mathfrak{q}_{z}}}\right),\\ -\left(s_{2.1}\overline{\overline{\mathfrak{q}_{x}}}+s_{2.2}\overline{\overline{\mathfrak{q}_{y}}}+s_{2.3}\overline{\overline{\mathfrak{q}_{z}}}\right),\\ -\left(s_{3.1}\overline{\overline{\mathfrak{q}_{x}}}+s_{3.2}\overline{\overline{\mathfrak{q}_{y}}}+s_{3.3}\overline{\overline{\mathfrak{q}_{z}}}\right),\end{array}\right\}$$}}

in which certain constants are denoted by s.

We also assume for these forces, that they won't be changed by the translation $$\mathfrak{p}$$, at least not as regards magnitudes of first order.

§ 47. In consequence of the electric motions, also the aether exerts an action upon the ions. This can be derived from formula ($$V_b$$), since $$\mathfrak{E}=\mathfrak{F}$$ as we have seen. If it would be allowed, to put for the electric force $$\mathfrak{E}$$ everywhere the average $$\overline{\mathfrak{E}}$$, that has the same magnitude and direction at all points of an ion, then we would have to add into the expressions (55) only the terms

But this matter isn't all that simple. First, the oscillating ion itself causes a value of $$\mathfrak{E}$$, that is not the same in all points of the particle, so that we could find the corresponding part of $$\mathfrak{K}$$ only by an