84 WORK OF ELECTRICAL FORCES. 



from which is deduced 



Lastly, the total charge M + CV for any system consisting of a 

 conductor and the corresponding condenser is constant; we have 

 then 



and therefore 



which gives for the conductor and condenser together 



Taking into account this latter relation, equation (2) may be 

 written 



ii ii 



22 22 



We have then 



(3) 



^ 



This equation holds, whatever be the capacities of the condenser. 

 There is nothing to prevent our considering the capacities as being 

 infinitely large in reference to those of the conductors, so that the 

 variations of potential dV v dV 2 . . and the variations of energy 

 Mj^Vj, M 2 ^V 2 . . . are absolutely negligable. We come then to the 

 case of conductors kept at constant potentials by external sources, 

 and equation (3) reduces to 



whence 



which gives finally, from equation (i), 



Thus, when conductors are kept respectively at constant poten- 

 tentials, the energy of the system, for a given deformation, increases 

 by a quantity equal to the work of the electrical forces. This 

 work is positive if the system is left to itself; like the corresponding 



