Page:LorentzGravitation1916.djvu/28

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

\int u_{b}\sum(c)\frac{\partial\pi{}_{ba}}{\partial x_{c}}\mathrm{x}_{c}d\sigma (39)

however it may be neglected. Hence, when we express the components u_{b} in terms of the quantities \psi_{ab}, we may give to these latter the values which they have at the point P.

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

\psi{}_{be}dx_{j}dx_{k}dx_{l}

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

\psi{}_{be}\sum(c)\frac{\partial\pi{}_{ba}}{\partial x_{c}}\left[\int\limits _{2}\mathrm{x}_{c}d\sigma-\int\limits _{1}\mathrm{x}_{c}d\sigma\right]

where the first integral relates to \Sigma_{2} and the second to \Sigma_{1}. It is clear that but one value of c, viz. e has to be considered. As everywhere in \Sigma_{1}:\mathrm{x}_{c}=0 and everywhere in \Sigma_{2}:\mathrm{x}_{c}=dx_{e} it is further evident that the above expression becomes

\psi{}_{eb}\frac{\partial\pi{}_{ba}}{\partial x_{c}}dW

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

\psi{}_{eb}=-\psi{}_{be}

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 \left(dx_{1},\dots dx_{4}\right) we may conclude by the considerations of § 21 that this is equally true for any arbitrarily chosen magnitude and shape of the extension \Omega. In particular the equation may be applied to an element \left(dx'_{1},\dots dx'_{4}\right) and by considerations exactly similar to