40 GENERAL THEOREMS. 



Making it equal to zero, we get 



an equation which represents n planes passing through the z 



axis, and the successive angles of which are equal to - . 



n 



50. THERE is ONLY ONE STATE OF EQUILIBRIUM. It may be 

 observed, in the first place, that the superposition of two states of 

 equilibrium is itself a state of equilibrium. 



For in each of the two states of equilibrium the potential is 

 constant on all the conductors. The superposition of the two 

 systems of electrical layers produces, at each point, a potential 

 equal to the sum of the potentials relative to the two primitive 

 states. The potential is constant, therefore, on each of the 

 conductors, and equilibrium exists. 



It follows from this, that if we change the electrical density 

 at each point in a constant ratio, a new state of equilibrium will 

 be formed, for the operation amounts to superposing two or more 

 identical states of equilibrium. 



51. A system of conductors A p A 2 , A 3 ..., whose electrical 

 charges are separately null, is necessarily in the neutral state. 



Let Vp V 2 , V 3 denote the potentials of these various con- 

 ductors, and let Vj be the greatest. 



There can be no point in the dielectric where the potential is 

 higher than V v since there is no maximum of potential outside the 

 acting masses. The potential sinks, therefore, in all directions from 

 the conductor A x ; all the lines of force start from this conductor, 

 and none terminate there. As the sum of the flows of force must 

 be zero (for by hypothesis the total charge of A x is zero) it is seen 

 that all the elementary flows of forces are zero. The density is 

 therefore zero over the whole surface, and therefore the conductor 

 is not electrified. 



The conductor A 1? being in the neutral state, may be suppressed, 

 and the same reasoning applied to the next conductor ; it may thus 

 be shown successively that all the conductors are in the neutral 

 state. 



The conductors A v A 2 , A 3 . . . , having charges M 15 M 2 , M 3 . . . , 

 differing from zero, let us now suppose that two states of equilibrium 

 are possible, such that the densities on A 1? A 2 , A 3 are o- 1} o- 2 , o- 3 . . . in 

 the first case, and <r'j, o-' 2 , </ 3 . . . , in the second. 



Changing the signs of all the electrical masses of the second state, 



