CONDUCTORS AT CONSTANT POTENTIAL. 85 



increase in . the energy, it is borrowed from the sources which keep 

 the potentials constant. The sources yield then, at every moment, 

 a quantity of energy which is divided into two equal parts; one 

 serves to perform the work dT of the electrical forces, the other 

 goes to increase by dW a the electrical energy of the system. 



In this case the energy of the system tends towards a maximum. 



98. We shall proceed to apply these theorems to the following 

 problem, which may serve as basis of the theory of symmetrical 

 electrometers. 



Let us suppose that a system of conductors is formed of two 

 fixed unlimited cylinders A and B (Fig. 21) having a common axis, 



Fig. 21. 



and of a cylinder concentric with the preceding ones, movable 

 along this axis, the length of the inner cylinder C being, moreover, 

 so great that the density at each end only depends on that of the 

 nearest fixed conductor. Let V 15 V 2 , and V be the potentials of 

 these three bodies, and A , B , and C the charges which they 

 possess when the movable cylinder is in a position symmetrical with 

 the two others. 



If the cylinder C is displaced by a small quantity x, towards 

 the right for instance, the distribution of electricity on the various 

 surfaces near the opening and at the ends is not modified ; we have 

 merely on this side increased, by a quantity proportional to x, the 

 surface on which the electrical density is uniform and proportional 

 to the difference of potentials of the adjacent conductors The right 

 half of the movable cylinder will have gained a quantity of elec- 

 tricity proportional to x, and the fixed cylinder B an equal quantity 

 of the contrary electricity ; the opposite effect will be produced on 

 the other side. 



Thus, calling A , B , C the initial charges on the three cylinders, 

 A, B, and C the new charges, and a the capacity for unit length of 

 the inner cylinder at some distance from the middle and from the ends, 



= B -cu;(V-V 2 ), 

 = A + ca;(V-V 1 ). 



