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

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

(55)

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

$e{\overline {\mathfrak {E}}}_{x},\ e{\overline {\mathfrak {E}}}_{y},\ e{\overline {\mathfrak {E}}}_{z}$

(56)

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