Page:LorentzGravitation1916.djvu/28

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

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

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

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

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

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

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

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

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

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

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

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