98, 99] Systems of Conductors 89 



point of a conductor is 4-Tra-. At any point of a line of no electrification, 

 this intensity vanishes, so that every point of a " line of no electrification " 

 is also a point of equilibrium. 



At a point of equilibrium we have already seen that the equipotential 

 through the point cuts itself. A line of no electrification, however, lies 

 entirely on a single equipotential, so that this equipotential must cut itself 

 along the line of no electrification. Moreover, by 69, it must cut itself at 

 right angles, except when it consists of more than two sheets. 



99. We can prove the two following propositions : 



I. If the potential of every conductor in the field is given, there is only 

 one distribution of electric charges which will produce this distribution of 

 potential. 



II. If the total charge of every conductor in the field is given, there is 

 only one way in which these charges can distribute themselves so as to be in 

 equilibrium. 



If proposition I. is not true, let us suppose that there are two different 

 distributions of electricity which will produce the required potentials. Let 

 <r denote the surface density at any point in the first distribution, and cr' in 

 the second. Consider an imaginary distribution of electricity such that the 

 surface density at any point is cr cr'. The potential of this distribution 

 at any point P is 



where the integration extends over the surfaces of all the conductors, and 

 r is the distance from P to the element dS. If P is a point on the surface 

 of any conductor, 



f 



dS and 



are by hypothesis equal, each being equal to the given potential of the 

 conductor on which P lies. Thus 



so that the supposed distribution of density a & is such that the potential 

 vanishes over all the surfaces of the conductors. There can therefore be no 

 lines of force, so that there can be no charges, i.e., cr cr'= everywhere, so 

 that the two distributions are the same. 



And again, if proposition II. is not true, let us suppose that there are 

 two different distributions cr and a such that the total charge on each 

 conductor has the assigned value. A distribution a a' now gives zero 

 as the total charge on each conductor. It follows, as in 98, that the 



