FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 239 



where in the last term but one the gradient operation only affects the E functions. 

 Now 



div (E+E ) = o, 



and thus the resultant force may be taken as distributed throughout the volume with 

 intensity at each point equal to 



(IV) B + i [** E + E ]-grad ( 



in agreement with the general result derived above. The local terms in I" and E may 

 again be presumed to balance out with other forces of a type not at present under 

 i-eview. 



13. The two new potentials A L> and <f>.,, introduced in the analysis of the last 

 paragraph, are the general forms of the potentials analogous to the ordinary scalar 

 and vector potentials of this theory, and they satisfy similar equations. We have, 

 in fact, 



Curl A 2 = E + 4 



where C is the total current density of electric flux including the effective representa- 

 tion of the magnetism. Thus 



Curl Curl A, = Curl E + 4ir fCurl C dt 



= -]^ + 47r['CurlC c2/. 

 c at J 



Thus 



iv rr>* 1 d*A.: I idfa f' -, ,,-, , 



grad div A,- V~A 2 = a j-^ grad -^ + 477 Curl C dt 



C Cf/t C _ Cti' J 



whilst since div B = 0, we have also 



VU a +- div A a =0. 

 c dt 



We may now adopt one of a number of alternatives. The simplest one is got by 



taking ^ = 0, when we also have 



div A 2 = 

 with, therefore, 



c dr 

 The last equation really involves the first, for 



so that div A,, must be zero as it has no singularities. 

 VOL. ccxx. A. 2 r, 



