# Page:LorentzGravitation1916.djvu/18

and ${\displaystyle {\overline {X_{1}}}{\overline {d\sigma }}}$ in the first of the integrals (16) annul each other. It will be clear now that the whole integral vanishes and that similar considerations may be applied to the other three.

So we have proved that under the special assumptions made the left hand side of (10) will vanish in the special case that the directions of the coordinates are perpendicular to each other. This conclusion likewise holds for an other set of coordinates if only the assumption made at the beginning of this § is fulfilled. This is obvious, as we can pass from mutually perpendicular coordinates ${\displaystyle x_{1},\dots x_{4}}$ to arbitrarily chosen other ones ${\displaystyle x'_{1},\dots x'_{4}}$ which fulfil this latter condition by linear transformation formulae with constant coefficients. The ${\displaystyle x}$- and the ${\displaystyle x'}$-components of the vector

${\displaystyle \left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]}$

are then connected by homogeneous linear formulae with coefficients which have the same value at all points of the surface ${\displaystyle \sigma }$. Hence if, as has been shown above, the four ${\displaystyle x}$-components of the vector

${\displaystyle \int \left\{\left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]\right\}d\sigma }$

vanish, the four ${\displaystyle x'}$-components are now seen to do so likewise.[1]

§ 20. The above considerations were intended to prepare a corollary which will be of use in the treatment of the integral on the left hand side of (10), if we now leave the special assumptions made above and suppose the quantities ${\displaystyle g_{ab}}$ to be functions of the coordinates while also the rotations ${\displaystyle \mathrm {R} _{e}}$ and ${\displaystyle \mathrm {R} _{h}}$ may change from point to point.

This corollary may be formulated as follows: If all dimensions of the limiting surface ${\displaystyle \sigma }$ are infinitely small of the first order, the integral

${\displaystyle \int \left\{\left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]\right\}_{x}d\sigma }$

will be of the fourth order.

In order to make this clear let us suppose that in the calculation of the integral we confine ourselves to quantities of the third order. The surface ${\displaystyle \sigma }$ being already of that order we may then omit all infinitesimal values in the quantities by which ${\displaystyle d\sigma }$ is multiplied;

1. In the above considerations difficulties might arise if the vector ${\displaystyle \mathrm {N} }$ lay on the asymptotic cone of the indicatrix, our definition of a vector of the value 1 would then fail (comp. note 2, p. 1345). With a view to this we can choose the form of the extension ${\displaystyle \Omega }$ (§ 13) in such a way that this case does not occur, a restriction leading to a boundary with sharp edges.