1 0 0 D O C . 9 1 R E L A T I V I T Y & E Q U A T I O N S O F M O T I O N The first row refers to the external field, and the second to the internal field. In or- der for the external field to satisfy the equations (10), the components must depend upon in an appropriate manner if the dependence on is taken as given. Here, again, we do not consider the influence of the spatial inhomogeneity of the external field on the motion of the singularity. After having expressed the total field in the neighborhood of the singularity to first approximation with sufficient precision, we can proceed to calculate the quan- tities that occur in (8), making use of (9). The following types of terms will occur: 1) Squared terms with respect to the external field 2) Squared terms with respect to the internal field 3) Product terms of the components of the internal and the external fields. Correspondingly, is decomposed into three sum terms. Since the are linear in the , the latter will also be composed of the sum of three systems. The first of these systems corresponds to the case that no external field is present, the second to the case that no singularity is present. Only the third corresponds to the coexist- ence of internal and external fields. Only this last system need be considered for the solution of the problem of motion. We therefore consider only the “mixed” terms in .[27] Taking this into account, equations (9), together with (12), (13), and (14), give (15) Here, l, s, t refer only to spacelike indices. After these preparatory considerations, we can now draw conclusions from equation (8). x 1 , x 2 , x 3 x 4 Q i Q i L i g i Q i 8Q st 3 2 st ma l x l 3 ------------- 3---------------------- mx s x t a l x l 5 - – = + 2 3 st e l x l e s x t – e t x s – 8Q 14 i----- 3 x 2 h 3 x 3 h 2 – = 8Q 44 3 2 ma l x l 3 ------------- 2 3 ex l . – =