1 0 4 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 2. Validity of the equations (18a) outside the spherical surface. If we now go to the limit r 0 = 0, then the go over in that we are seeking. We thus obtain directly , (22) where dV is the spacelike volume element for the integration and  its spatial dis- tance from the point considered. For each of the four coordinates , it follows from (22) that . Then , (23) where the vertical bars indicate that in , the argument is to be inserted instead of t. We now have to express the fact that the right-hand side of (23) van- ishes outside the sphere of radius r 0 . Since outside the sphere, becomes iden- tical to , the integrand is zero there, and we need only integrate over the inter- nal volume of the sphere. Since we must also go to the limit r0 = 0, we can replace  by the distance r of the point considered from the singularity.1 The integral thus becomes: . 1 The integration over the other singularities evidently gives no appreciable contribution in the im- mediate neighborhood of .  x   i * 0 = * i  i * i 1 2 ------ – S* i t  –   dV ----------------------- = x   x  S* i t  –   dV -----------------------     S* i x  -------- -t  – dV -------- = [p. 244]  x  * i 1 2 –  S x  ---------i  -dV ---------------- =  x  S* i t  – S* i S i 1 r --  x  S* i |dV | x 1 x 2 x 3 0 = = =