We shall need to distinguish between the vector operators 



V =: X 3— + J -^— + k ^5— and V = x ^rp + j -i^ 1- k ^r— 



P dx -^ dy dz d^ -^ 9n 3^ 



The Poincare transformation is as follows: 



VpX 



'f^dx- r 



^"^ dx - I ^ dS 



■PQ 



dx - 



PQ 



1 '^PQ 



dS 



4tt 



P in D 



+ { 27T > V , I P on S 

 ) P in E 



(53) 



where n is the unit vector in the direction of the outward normal to S. 

 First suppose that P is in E. Then we have 



if.-'^^ l[-{iy&'] 



dx 



SM-[['-'^'^i 



V X I B^ dS + 

 ^ ' ^PQ 



- vV .V -— Idx 

 PQ PQ. 



•4 PQ -^n PQ 



(54) 



21 



