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

(39) |

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