Page:Elektrische und Optische Erscheinungen (Lorentz) 062.jpg

 § 41. The rule contained in the formula

$$\overline{\epsilon}=\frac{1}{l}\int\epsilon d\mathfrak{r}$$

can be express somewhat differently. Namely we shall choose for the point P an infinite amount, we want to say k, locations that are uniformly distributed over the sphere I, and take the arithmetic mean of the values of $$\epsilon$$ that are valid for these locations, i.e. we put

Any ion, that has its equilibrium position in the interior of I, will (during its displacement) now pass through some positions that are connected with the element $$d\sigma$$ and thus add some terms to the sum $$\Sigma\epsilon$$. We obtain the whole sum, if we at first add to one another the terms that stem from a certain ion, and than sum over all ions.

Let Q be the equilibrium position of the considered ions, and Q the new position; so $$QQ'=\mathfrak{q}$$. The length and the direction of this line are given, as well as the direction and magnitude of $$d\sigma$$. Whether the particle hits the surface element and provides the part e for the sought sum, only depends on the relative positions of P and Q. Thus we can, instead of giving k positions in the sphere l to P, also let remain the point at its position and lead point Q around a sphere I. As $$QQ'$$ now hits the fixed element $$d\sigma$$, when Q lies in a certain, easily specifiable cylinder of area $$\mathfrak{q}_{n}d\sigma$$, then the number of "relevant" positions is related to the integer k, as the area of that cylinder is related to the area of sphere I. This number is thus

$$\frac{k}{I}\mathfrak{q}_{n}d\sigma$$,

and the sum $$\Sigma\epsilon$$, as far it is caused by the ion Q,

$$\frac{k}{I}e\mathfrak{q}_{n}d\sigma$$.