132 134] SIGN OF ELECTRIFICATION. 261 



THEOREM I. If the system consists of a single electrified 

 conductor, the distribution is monogenic. For on the conductor 

 the potential is constant, while at infinity it is zero. In free 

 space, being harmonic, it has neither maximum nor minimum 

 ( 34), hence all the equipotential surfaces are closed surfaces 

 surrounding the conductor, and the tubes of force proceed from 



dV . 

 the conductor to infinity. Thence ^ is of the same sign all over 



dn e 



the surface of the conductor, and the theorem is proved. 



THEOREM II. If the system is composed of two conductors, 

 the distribution of at least one of them is monogenic. For the 

 greatest and least values of the potential are two of the three 

 values of the potential on the two conductors, and at infinity. 

 The potential on one of the conductors is accordingly an extreme 

 value, so that the derivative has the same sign over its surface. 



THEOREM III. If an insulated conductor with zero charge is 

 placed in presence of a charged conductor, the charge of the 

 former is amphigenic, of the latter monogenic. For since the 

 charge of the first is zero, the surface-density and hence the 



derivative = must be positive in some regions, negative in 



on e 



others, consequently its potential lies between the extreme values, 

 which are accordingly on the second conductor and at infinity. 



FIG. 57. 



