4 3 4 D O C . 4 4 3 G E N E R A L R E L A T I V I T Y A N D M O T I O N . (19) In the calculation of (17) one must note, furthermore, that only the products of quantities of the inner field with quantities of the outer field can contribute to the result. For the second-order terms containing only the inner field must cancel each other out, for symmetry reasons the second-order terms containing only the outer field contribute nothing to the integral, because the integration surface is so small. As we have now chosen quasi-Euclidean coordinates, it is practical to choose a sphere as the integration surface, so that (17) takes the form . (17a) The calculation results in . Consequently, we get as the equilibrium condition of the singular point . (20)[35] It can be easily demonstrated that for the case of equilibrium in a stationary field the geodesic equation yields the same condition for the approximation consid- ered.[36] We now proceed to the case where the singular point is situated in a nonstation- ary field.[37] Here, too, equation (15a) applies, as well as the corresponding integral. We transform the singular point to rest, so that the x4-axis is again the singularity in four dimensions. We further select the ’s in such a way that they differ from zero only along a part of the sheath M, but vanish completely on M′. Furthermore, let the ’s be continuous in the vicinity of the x4-axis. As the ’s must vanish at the temporal limit of integration, we cannot choose them to be constant anymore. in (15b) therefore does not vanish. The result of this is a peculiar difficulty. Namely, while the second approximation in the as we saw—has no influence on the ’s, if one limits oneself to terms of second order in , this is not the case with the ’s. For instance, in order to obtain the term exactly up to second order, must be known exactly in terms γ11 γ11 γ11 γ11 – 2m§ r ------- = 2m x1 2 x2 2 x3¹ 2 + + –--------------------------------¸- © ¨ · – = = = = γστ 0 = for σ τ) ≠ ( r const.) = ( tσ 1 x1 r ---- - tσ 2 x2 r ---- - tσ 3 x3- r ----· + + ¹ § Sd ³© ∂xσ ∂γ44== –8πm ∂xσ ∂γ44 == 0= [p. 11] ξα ξα ξα B α gμν’s— tσ α tσ ν B α gμνΓτνξ,τμ α Γτν α