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 3 This is related to the following circumstance: Assuming that we have calculated the second approximation according to (18) and (21), it will be singular on the axis, just as . It is thus not uniquely determined by (18) and (21). For in this second approximation, another can be added, which satisfies the equa- tions . [32] Such solutions exist, since we cannot exclude the possibility that they also become singular in the axis. The same holds analogously for the higher approximations. In such a calculational method, it is thus found that the higher approximations are not uniquely determined by the lower ones, which appears to be absurd. This difficulty can be avoided as follows: Assuming that the and the were[33] regular everywhere, then one could require regularity of the higher approx- imations as well. In particular, the could be calculated from (21) as retarded po- tentials. In general, every next higher approximations could be expressed as retard- ed potentials, and they would thus become regular. In that case, there would be no reason to doubt the convergence of the series expansion (5).1 We now calculate the second approximation, by starting with functions and , which are regular everywhere, instead of and . These should have the following properties: 1. Outside a spherical surface with a very small radius r 0 around , they are identical to and 2. they make a continuous transition across that spherical surface to and , and they are regular within the sphere. We suppose that this replacement has been carried out at all the singular points of the system considered. We require of the [35] the following properties: 1. Validity of the equations (21a) throughout the whole space 1 In the choice of the retarded potentials for solving the wave equation, there is, however, a certain arbitrariness with respect to the sign of the time variable. This was already pointed out by Ritz. The Maxwell theory can also not do without this arbitrariness.[34] x 4 g i i and S i x i 0 = x2 2 i 0 = [p. 243] x 4 g i i i g * i * i g i i x 1 x 2 = = x 3 0 = g i i g i i i *i 2S * i =