From Wikisource
Jump to navigation Jump to search
This page has been proofread, but needs to be validated.

direction of one of the coordinates e. g. of over the distance . We had then to keep in mind that for the two sides the values of , which have opposite signs, are a little different; and it was precisely this difference that was of importance. In the calculation of the integral


however it may be neglected. Hence, when we express the components in terms of the quantities , we may give to these latter the values which they have at the point .

Let us consider two sides situated at the ends of the edges and whose magnitude we may therefore express in -units if are the numbers which are left of 1, 2, 3, 4 when the number is omitted. For the part contributed to (38) by the side we found in § 26

We now find for the part of (39) due to the two sides

where the first integral relates to and the second to . It is clear that but one value of , viz. has to be considered. As everywhere in and everywhere in it is further evident that the above expression becomes

This is one part contributed to the expression (36). A second part, the origin of which will be immediately understood, is found by interchanging and . With a view to (37) and because of

we have for each term of (36) another by which it is cancelled. This is what had to be proved.

§ 31. Now that we have shown that equation (32) holds for each element we may conclude by the considerations of § 21 that this is equally true for any arbitrarily chosen magnitude and shape of the extension . In particular the equation may be applied to an element and by considerations exactly similar to