Page:Elektrische und Optische Erscheinungen (Lorentz) 068.jpg

integration over the space in which the ion is located. Second, even if we could neglect this, for the calculation of ${\overline {\overline {\mathfrak {K}}}}$ the average ${\overline {\overline {\mathfrak {E}}}}$ is of relevance, not the average ${\overline {\mathfrak {E}}}$, and it is not allowed, to mutually interchange both. Of course nothing would be in the way, in so far the motions of ions that cause the electric force ${\mathfrak {E}}$, take place in the distance P from the considered point that is much greater as the distance of the molecules, but ${\mathfrak {E}}$ is partly caused by molecules that are located more nearly — we want to say, by the oscillations within the sphere I drawn around P — and an inequality ${\overline {\mathfrak {E}}}$ and ${\overline {\overline {\mathfrak {E}}}}$ is very well possible for a irregular distribution of the thus produced states in the aether.

When we now, in agreement with these remarks and to obtain ${\overline {\overline {\mathfrak {K}}}}$, add to the expressions (55) not only the values (56) but also certain supplementary terms

${\mathfrak {k}}_{x},\ {\mathfrak {k}}_{y},\ {\mathfrak {k}}_{z}$

and thus put

$m{\frac {\partial ^{2}{\mathfrak {M}}_{x}}{\partial t^{2}}}=-eN\left(s_{1.1}{\overline {\overline {{\mathfrak {q}}_{x}}}}+s_{1.2}{\overline {\overline {{\mathfrak {q}}_{y}}}}+s_{1.3}{\overline {\overline {{\mathfrak {q}}_{z}}}}\right)+e^{2}N{\overline {{\mathfrak {E}}_{x}}}+eN{\mathfrak {k}}_{x}$,etc.

(57)

then we can maintain for the magnitudes ${\mathfrak {k}}$, that they only depend on processes within sphere I. Additionally it is given, that also the supplementary terms only exist during the displacement of the ions from their equilibrium positions and — since ${\mathfrak {q}}$ can be considered as infinitely small — they must be linear, homogeneous functions of the magnitudes ${\mathfrak {q,{\dot {q}}}}$, etc., or rather of their averages. In consequence of equations (48), also ${\mathfrak {k}}$ are homogeneous, linear functions of the values of ${\mathfrak {M}}_{x},{\mathfrak {M}}_{y},{\mathfrak {M}}_{z},{\dot {\mathfrak {M}}}_{x}$, etc. in the various points of the spherical space I. Eventually we still have to consider, that all these values can be expressed by application of Taylor's theorem by the values, which will be assumed by ${\mathfrak {M}}_{x},{\mathfrak {M}}_{y},{\mathfrak {M}}_{z},{\dot {\mathfrak {M}}}_{x}$, etc., and the derivatives with respect to x, y, z in the considered point P, the center of the sphere. All these values thus are linearly included into the expressions for ${\mathfrak {k}}_{x},{\mathfrak {k}}_{y},{\mathfrak {k}}_{z}$.

To which extend these latter ones must contain the translation velocity ${\mathfrak {p}}$,