Page:Elektrische und Optische Erscheinungen (Lorentz) 060.jpg

between the surfaces $\Sigma _{1}$, and $\Sigma _{2}$, (Fig. 2). If we speak in the latter case about a "limiting surface", then we shall mean by that, for example, a surface $\Sigma$ halfway between $\Sigma _{1}$ and $\Sigma _{2}$.

We will always calculate by averages, and namely not only by those defined in §4 l, but sometimes also by others, that come into consideration when the relevant magnitude only exist in a single point Q, for example in one point of the various molecules, or if we have reason to consider only the values of a function in such points. Such a average of second kind we distinguish from the averages of first kind by a double horizontal prime, and besides we follow a similar calculation rule as during the last calculation. Namely we understand under the value of ${\overline {\overline {\phi }}}$ in a point P the arithmetic average of the values of $\varphi$ in the points Q, so far as they are present within the sphere I around P (as mentioned in §4, l).

By the assumption made about the radius R (§4), all "rapid" variations are vanished from the averages; however, concerning the velocity of the remaining variations, we have to distinguish between the interior of the body and the border. If we are positioning in Figures 1 and 2 the surfaces $\sigma _{1}$ and $\sigma _{2}$ in such a way, so that in the first figure they are both distant from $\Sigma$ by R, while in the second this distance exists, first, between $\Sigma _{1}$ and $\sigma _{1}$, and second, between $\Sigma _{2}$ and $\sigma _{2}$, then for the calculation of ${\overline {\overline {\phi }}}$ or ${\overline {\phi }}$ in the points, that are outside of the layer ($\sigma _{1}$, $\sigma _{2}$), only the values of $\varphi$ come into play. While the averages, although in a completely steady way, can be considerably different from $\sigma _{1}$ to $\sigma _{2}$, we want to assume, that the variations from point to point are much slower in the interior of the body. This will be indeed satisfied in the problems to be considered, when only the wavelength $\lambda$ is many times greater than the distance a of $\sigma _{1}$ and $\sigma _{2}$

We even want to assume, that between $\lambda$ and a,