Page:LorentzGravitation1916.djvu/28

 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

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

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

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

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

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