WILSON AND LEWIS. — RELATIVITY. 487 



values [p] and [pv] of the density and the current density which were 

 in the element (/5' at a time previous by the length of time required 

 for li_<:;ht to pass from dB' to the point in question. From the four 

 dimensional point of view this means that we project the element dS>' 

 parallel to the time-axis upon the hypercone, and take as [p] and [pv] 

 the projections on time and space of the vector q at this point of the 

 hypercone. We then form the integrals 



J^'^dB' and f'J^dB', (133) 



where r is the distance from dB' to the point at which the potential 

 is wanted. 



Let us now consider the element dm of our potential. The vector 

 dQ (corresponding to dS of the figure), being cut out of the hyper- 

 cone, is a singular 3-vector, and its complement dQ* is therefore a 

 singular 1-vector. Hence c/S' is numerically the projection of d^* 

 upon Ici, and it is readily seen that 



Substituting in (132), 



dS* ^ 1 

 d<B' li 



, I'Q W ,^, I'poW W j^, 



dm = 7^ j^ d^ = —— T, d<B . 



li K li li 



But I'W = — ii by (SO) and /4 is equal to l^, that is, to the r in (133). 

 Hence 



m 



= M+^'^^UB'. (134) 



J r 



If we designate the vector and scalar potentials as a and (f> respec- 

 tively, then 



m = a + </)k4. (135) 



We mav show as before ^^ that 



O-m=0 or va+?,t=0. (136) 



at 



We have seen (§ 44) that O 'O P = 0, or C>- p = 0, and consequently 

 <y^Tn. = in the case of a point electron for all points not upon the 



53 A single differentiation under the sign of integration is permissible if Po 

 remains finite; but a second differentiation is not permissible, as is well 

 known in the theory of the potential. 



