D O C . 9 1 R E L AT I V I T Y & E Q U AT I O N S O F M O T I O N 1 0 5 Furthermore, we make only a negligible error if we refer the retardation to the cen- ter point of the sphere instead of to the volume element of the integration, so that we obtain: . Applying Gauss’s theorem,1 integrating over the volume of the sphere, and taking into account that on the surface of the sphere, is equal to , this becomes: , (24) where the are functions only of the one argument . We thus finally obtain for the neighborhood of the singular point considered . (25) [35] Equation (18) thus then gives . (26) According to (24), (20), and (15), this means that . (26a) This formula states that, considered from the singular point, the sum of the ponder- omotive forces of gravitation and of the electromagnetic field must vanish. It is clear that (26a) is equivalent to the well-known equations of motion for an electron, which in generally covariant notation are given by . (26b) [36] We have thus shown that the previously hypothetical, presumed law of motion results from the field equations, if one considers a pointlike singularity of static character. But it is still not proven that such singularities, according to the field equations, can perform all the motions that would satisfy the condition (26b). It would indeed be thinkable that in carrying out the corresponding considerations for higher approximations, one would find additional, more limiting constraints. This investigation does not contribute to the understanding of the empirical quantum facts.[37] However, its result remains important: that the law of motion for singularities and the field equations are not independent of one another. 1 The integration over makes no contribution, according to (15) and (20). 1 r x S* i dV S* i S i 1 r S i x r ----- -dS |A i | r ------- - – = A i t r – x i |A i | 2r -------- - – = A i x r ----- -dS Si – 0 = = A i a i m e i + 0 = = [p. 245] A i m g i 2 ds2 d x i ds dx ds dx + i ds dx + 0 = = 4 x S*i4