FUNDAMENTAL FORMULATIONS dF ELECTRODYNAMICS. 237 



that is to be made null, afterwards determining the functions 0,, <j> 2 , A,, A a to satisfy 

 the restrictions which necessitated their introduction. The variation can now be 

 affected in the usual way, and the condition that it vanishes leads to the following 

 equations 



with 



Curl Aj-grad*,-- d ^ = 0, 



c at 



with three equations of each of the types 



d!/3L\ 3L /30, l tL4,,\ e< 



jj\5^)""a~ = ~ c (a + ~ ~^T ) + ~ 

 dt\oxj tix, \dx e c at j c 



The first and third of these equations show that <i and A, are the usual scalar and 

 vector potentials ; in fact from the third we have 



Curl A, = grad fa + '* 



(,- fA/if 



so that A! is the vector potential and then 0, is the scalar potential. 



The fourth equation thus determines the usual expression for the reaction forces on 

 the moving electron ; the fifth equation determines similarly the force on the moving 

 magnetic pole in the form 



m (grad 0,+ - 1 - ^ ) - ^ [>, Curl AJ = m (%-- [.', (.-'url AJ ). 

 V c cw / <5 v c 



We have yet to determine Curl A a : we have 



so that 

 Now 



Curl B = - ~ Curl A^- 

 c at 



^ r , n; i TJ. J dE 47r r - - l dE 



Curl B = C, + 4ir Curl IH --- 37 = ^ - -rr > 



c c at 



