Page:LorentzGravitation1916.djvu/19

we may therefore neglect the infinitesimal changes of the quantities ${\displaystyle g_{ab}}$ over the extension considered, and also those of ${\displaystyle \mathrm {R} _{e}}$ and ${\displaystyle \mathrm {R} _{h}}$. By this we just come to the case considered in § 19. Thus it is evident, that as regards quantities of the third order the first part of (10) is 0. From this it follows that in reality it is at least of the fourth order.

§ 21. Let us now return to the general case that the extension ${\displaystyle \Omega }$ to which equation (10) refers, has finite dimensions. If by a surface ${\displaystyle {\bar {\sigma }}}$ this extension is divided into two extensions ${\displaystyle \Omega _{1}}$ and ${\displaystyle \Omega _{2}}$, the quantities on the two sides in (10) each consist of two parts referring to these extensions. For the right hand side this is immediately clear and as to the quantity on the left hand side, it follows from the consideration that the contributions of a to the integrals over the boundaries of ${\displaystyle \Omega _{1}}$ and ${\displaystyle \Omega _{2}}$ are equal with opposite signs. In the two cases namely we must take for ${\displaystyle \mathrm {N} }$ equal but opposite vectors.

Also, if the extension ${\displaystyle \Omega }$ is divided into an arbitrary number of parts, each term in (10) will be the sum of a number of integrals, each relating to one of these parts.

By surfaces with the equations ${\displaystyle x_{1}=\mathrm {const.} ,\dots x_{4}=\mathrm {const} .}$ we can divide the extension ${\displaystyle \Omega }$ into elements which we shall denote by ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$. As a rule there will be left near the surface ${\displaystyle \sigma }$ certain infinitely small extensions of a different form. From the preceding § it is evident that, in the calculation of the integrals, these latter extensions may be neglected and that only the extensions ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$ have to be considered. From this we can conclude that equation (10) is valid for any finite extension, as soon at it holds for each of the elements ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$.

§ 22. We shall now show what equation (10) becomes for one element ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$. Besides the infinitesimal quantities ${\displaystyle x_{1},\dots x_{4}}$, occurring in the equation

${\displaystyle F=\sum (ab)g_{ab}x_{a}x_{b}=\epsilon ^{2}}$

of the indicatrix we introduce four other quantities ${\displaystyle \xi _{1},\dots \xi _{4}}$, which we define by

 ${\displaystyle \xi _{a}={\frac {1}{2}}{\frac {\partial F}{\partial x_{a}}}}$ (18)

or

 ${\displaystyle \left.{\begin{array}{c}\xi _{1}=g_{11}x_{1}+g_{12}x_{2}+\dots +g_{14}x_{4}\\\cdots \cdots \cdots \cdots \cdots \cdots \\\cdots \cdots \cdots \cdots \cdots \cdots \\\xi _{4}=g_{41}x_{1}+g_{42}x_{2}+\dots +g_{44}x_{4}\end{array}}\right\}}$ (19)

with the equalities ${\displaystyle g_{ba}=g_{ab}}$.