Page:Elektrische und Optische Erscheinungen (Lorentz) 012.jpg

often use the three individual equations. If they have the same form, so that they transform into one another by cyclic permutation of the letters, then we can restrict ourselves to only writing down the first equation, and to sketch the two others by "etc.".

l. We will often have to consider bodies with molecular structure. Then functions arise, whose value quickly changes in the individual molecules and in the interspaces, and namely in a highly irregular way, as the molecules themselves are not always structured and oriented regularly. In those cases it is recommended, to calculate with averages, which we define as follows:

We describe around center-point P a sphere of area I, and calculate for it, when $\phi$ is the magnitude to be considered, the integral $\int \phi \ d\ \tau$. Then we call

${\frac {1}{I}}\int \phi \ d\ \tau {,}$

(2)

for which we want to write ${\bar {\phi }}$, the "average of $\phi$ at point P".

If we give to the sphere, where ever P may lie, always the same magnitude, then ${\bar {\phi }}$ can obviously only depend on t and the coordinates x, y, z of point P. It is clear that also ${\bar {\phi }}$ will show "rapid" changes from point to point, as long the sphere encloses only a few molecules, yet by a continuing increase the changes will step back more and more. We think for once and for all time a certain R as chosen, which is just as great that — with respect to the degree of exactitude that can be reached by the observations — we can neglect the rapid changes in ${\bar {\phi }}$. Then only the slow changes from point to point remain, that are accessible to our senses, and in all real cases they proceed so slow, that they hardly appear in spaces which are considerably greater as the sphere I. In these cases, ${\bar {\phi }}$