D O C . 4 4 3 G E N E R A L R E L AT I V I T Y A N D M O T I O N 4 3 5 of second order, because contains a zero order component ( ). Therefore, the ’s themselves have to be known exactly in terms of second order. Thus, we could not be content with solutions to the linearized field equations. This difficulty does, however, seem to be solvable in the following way. One sets . (18a) Now let us again take to be given by (19), and as referring to the outer field and continuous in the vicinity of the singularity. Let be a quantity of sec- ond order that is proportional to the “mass” m and to the outer field. It now seems that with such an approach we can do justice to the gravitational field equations up to second order if we neglect terms that are proportional to and if we neglect terms that are quadratic in the outer field ( ). Here the dependence of the on r is not of the character (as is the case for ), but is rather of the character . From this it follows that the second-order term has no influence on the in- tegral of , which extends over the infinitely narrow sheath M. It is then easy to prove that the integral of over M vanishes given a suitable choice of coordinates. For we can choose the coordinates in such that the ’s vanish on the singular line (x4-axis). (The axes are to be orthogonal to the singular line, and the coordinate unit equal to the measuring unit on all four axes). Let us further assume that, particularly for such an (undistorted) coordinate system, the singularity is centrally symmetric, i.e., that the field is computable from (19). This is, actually, not a necessary hypothesis.[38] But by assuming it we simplify the calculation greatly, and the hypothesis is confirmed in that it leads to the vanishing of the integral of over M for any choice of . The course of the calculation shall be demonstrated with respect of the first term of . In the vicinity of the singularity, the ’s and ’s behave, apart from finite terms, as , the Γ’s as . Only the parts of Γ proportional to can contribute anything to our integral. Furthermore, since the terms proportional to , as well as those quadratic in do not interest us, the above term of can be substituted by gμν δμν gμν gμν δμν – γμν γμν εμν + + + = γμν γμν εμν m2 γμν εμν r 1– γμν r0 εμν B α B α γμν x1, x2, x3 γ B α ξα [p. 12] B α gμνΓτνξ,τμ α gμν gμν r 1– r 2– r 2– m2 γ = B α