Page:LorentzGravitation1916.djvu/58

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Each value of e occurring twice, i.e. combined with the two values different from e which a can take, we have in addition to (118)

-2\int\frac{\partial\nu}{\partial n}d\sigma

so that (117) becomes

E_{2}=\frac{c}{2\varkappa}\int\frac{\partial\nu}{\partial n}d\sigma

As now outside the sphere

\nu=-\frac{\varkappa}{r}\int\limits _{0}^{a}\varrho\ dr

we have for every closed surface that does not surround the sphere E_{2}=0, but for every surface that does

E_{2}=2\pi c\int\limits _{0}^{a}\varrho\ dr (119)

As to E_{1} we remark that substituting (65) in (41) and taking into consideration (64) we find,

G=\varkappa T,\ Q=\varkappa\sqrt{-g}T (120)

From this we conclude that E_{1} is zero if there is no matter inside the surface \sigma. In order to determine E_{1} in the opposite case, we remember that G is independent of the choice of coordinates. To calculate this quantity we may therefore use the value of T indicated in § 56, which is sufficient to calculate E_{1} as far as the terms of the first order. We have therefore

G=\frac{\varkappa}{r^{2}}\varrho

and if, using further on rectangular coordinates, we take for \sqrt{-g} the normal value c,

Q=\frac{c\varkappa}{r^{2}}\varrho

From this we find by substitution in (114) for the case of the closed surface a surrounding the sphere

E_{1}=-2\pi c\int\limits _{0}^{a}\varrho\ dr

This equation together with (119) shows that in (113) when integrated over the whole space the terms of the first order really cancel each other. In order to calculate those of the second order