90 Systems of Conductors [OH. iv 



potential of every conductor must be intermediate between the highest and 

 lowest potentials in the field, a conclusion which is obviously absurd, as 

 it prevents every conductor from having either the highest or the lowest 

 potential. It follows that the potentials of all the conductors must be equal, 

 so that again there can be no lines of force and no charges at any point, i.e., 

 o- = a-' everywhere. 



It is clear from this that the distribution of electricity in the field is fully 

 specified when we know either 



(i) the total charge on each conductor, 

 or (ii) the potential of each conductor. 



SUPERPOSITION OF EFFECTS. 



100. Suppose we have two equilibrium distributions : 



(i) A distribution of which the surface density is o- at any point, 



giving total charges E lt E 2 , ... on the different conductors, and potentials 



V V 



r i, v o, . ... 



(ii) A distribution of surface density cr', giving total charges E^, E^ ... 

 and potentials F/, F 2 ', 



Consider a distribution of surface density a + a'. Clearly the total 

 charges on the conductors will be E^E^, E 2 + E 2 ', ..., and if V P is the 

 potential at any point P, 



where the notation is the same as before. If P is o% the first conductor, 

 however, we know that 



/ / 



<7 

 T 



^dS=V l ', 

 r 



so that Fp=F I +F 1 '; and similarly when P is on any other conductor. 

 Thus the imaginary distribution of surface density is an equilibrium dis- 

 tribution, since it makes the surface of each conductor an equipotential, 



and the potentials are 



F.+ F/, F 2 +F 2 ', .... 



The total charges, as we have seen, are E l + E^, E 2 + E 2 , . . . , and from 

 the proposition previously proved, it follows that the distribution of surface- 

 density a- + a-' is the only distribution corresponding to these charges. 



We have accordingly arrived at the following proposition : 



// charges E l} E 2 , ... give rise to potentials Fj, F 2 , ..., and if charges 



