Electrostatics and the Stead// Flow of Electricity. 573 



§ 3. Charged insulated conductors. — Suppose that our 

 system of conductors is charged and insulated, and under 

 the influence of fixed external charges which create at any 

 point p a potential TJ(p). Let W(p) be the potential due to 

 the equilibrium distribution of the electricity on the con- 

 ductor which consists of volume and surface distributions, 

 7) and /A respectively, as before. For any point ;> of a 

 conductor 



U(p)+W(p)=P* (12) 



where P t is the constant potential of that conductor. The 

 potential WQo) is given for an inner or boundary point /> by 



and for an outer point a by 



W(«)= Sg(*q)n(q)dq+ $g(at)fi(t)dt. 



Putting in this last equation the values of //(«<7) and //(«£) 

 given by (10), we find 



W(«)=- jV(**) [$9(.qt)v(q)dq+ $g(St)rf?)d^dt 



which may be written 



W(-)=-JW(0H +1 (te>« (13) 



This is the result previously found elsewhere*, giving the 

 value at an external point a of the solution W(p) of (2) in 

 terms of its boundary values W(t). 



Using the relation (12) we may write the last result 



W(«)=fU(0H(/«)^-JL > .H(^)^, . . (14) 



which expresses the potential due to the electricity on the 

 conductor in terms of that, due to the fixed external charges 

 and the constant potentials of the conductors. 



In the particular case when there are no external charges, 

 U(p) =0, so that the relation becomes 



W(a) = -JP.H(*«>fc, 



riving the potential at an external point in terms of the 

 constant potentials of the conductors. 



§ 4. Second Problem. — Suppose the same system of con- 

 ductors as in the first problem, with the same known potentials, 



* Obtained from equations (3) and (4) above by putting X=l. 



