92 Systems of Conductors [OH. iv 



Green's Reciprocation Theorem. 



102. Let us suppose that charges e p , e Q , ... on elements of conducting 

 surfaces at P, Q, ... produce potentials V P , V Q , ... at P, Q, ..., and that 

 similarly charges e p ', e Q ', ... produce potentials V P ', V Q ', Then Green's 

 Theorem states that 



2e p Fp' = 2e P 'Vp, 



the summation extending in each case over all the charges in the field. 

 To prove the theorem, we need only notice that 



the summation extending over all charges except e p , so that in *2<e P V P the 

 coefficient of 

 e Q 'V Q . Thus 



coefficient of -p-~ is e P 6Q from the term e P V P , and e P e<J from the term 



~ PQ 



= SepFp', from symmetry. 



103. The following theorem follows at once : 



If total charges E lt E 2 on the separate conductors of a system produce 

 potentials Fi, V 2 , ..., and if charges E^, E%, ... produce potentials V, 

 F 2 ', ...,then 



2EV'=2E'V .............................. (33), 



the summation extending in each case over all the conductors. 



To see the truth of this, we need only divide up the charges E lt E 2 , ... 

 into small charges e p , e Q , ... on the ditferent small elements of the surfaces 

 of the conductors, and the proposition becomes identical with that just 

 proved. 



104. Let us now consider the special case in which 



^ = 1, E, = E 9 = Et=...=0, 



so that F 1 = p 11 , F 2 =p 1 . 2 , etc,; 



and #/ = (), AV=1, # 3 / = #4 / =...=0, 



so that VI^PK, ^z=P-22, e ^c. 



Then 2EV =p 2l and 2 t E'V=p l2 , so that the theorem just proved becomes 



Pi* = P*i- 



In words: The potential to which (1) is raised by putting unit charge on 

 (2), all the other conductors being uncharged, is equal to the potential to 

 which (2) is raised by putting unit charge on (1), all the other conductors 

 being uncharged. 



