FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 219 



separately in the dynamical variational equation. The coefficients of the latter 

 variations lead to the equations 



-- 

 c at c dt 



CurlA 1 = i -, 

 c dt 



from which it follows that the multipliers <j> and Aj are the ordinary scalar and vector 

 potentials of the theory so that further 



c dt 



As regards the vector A 2 , this is a new vector potential whose curl is required in 

 our subsequent discussions. For this we have 



Curl B = Curl H + 47T Curl I = - 4 Curl A 2 , 



c dt 



whilst if we use C e as the total current of true electric flux we have, by AMPERE'S 

 equation 



c dt 

 Thus, if we use 



C' e = C e + 

 we have 



The main part of Curl A 2 is therefore represented by the electric force : there is 

 however in addition a local term E depending on the time integral of the current 

 density at the point. We can thus write 



Curl A 3 = E + E . 



If we use the values thus determined for the various undetermined multipliers 

 introduced at the outset, the remaining terms of the variation give for the motion of 

 the material and electrical elements equations of the type 



8L /T,3E\ + / I |B\l/ rp 



SxJ \ dxj c 



