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

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

(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