134, 135] GENERAL PROBLEM. 263 



For the potential of the conductor having the greatest absolute 

 value is, if positive, the highest, and if negative, the lowest value 

 occurring, so that in the former case a- is positive, in the latter, 

 negative. 



THEOREM VIII. On each of two conductors whose potentials 

 are of opposite signs the distribution is monogenic. For the 

 potentials of the conductors are the highest and lowest oc- 

 curring. 



THEOREM IX. If one of two conductors has the potential zero, 

 the other a potential not zero, the distribution is monogenic on 

 both, and on the second the density has the sign of the potential, 

 on the first the contrary sign. For this is a limiting case of the 

 preceding, as the potential of one of the conductors approaches 

 zero. 



THEOREM X. On a conductor connected to earth, a charge 

 concentrated at a point causes a monogenic charge of sign 

 opposite to its own. For this is a particular case of the preceding 

 theorem. 



Theorem I may be generalized as 



THEOREM XI. In a system formed of any number of con- 

 ductors, the distribution on at least one is monogenic. For the 

 highest or lowest value of the potential must be on one of the 

 conductors. 



135. General Problem of Electrostatics. If we have a 

 number of conductors in a state of equilibrium, of which some 

 are insulated and charged with quantities e s , others connected 

 to earth, or kept, by means to be hereafter described, at given 

 constant potentials V s , and influenced by certain bodies D rigidly 

 electrified with density p, the problem to be solved consists in 

 finding a potential function V which, 1, is constant in each con- 

 ductor, taking the values V s in those conductors for which the 

 constant is given, 2, in the bodies D satisfies the equation 



AF=-47T/3, 



and 3, in the rest of space is harmonic, 



AF=0. 



We can satisfy these conditions if we can solve n + 1 inde- 

 pendent problems, n being the number of conductors. 



