RECIPROCAL ACTION OF TWO ELECTRICAL CONDUCTORS. 75 



Reversing the functions of the conductors, we shall obtain the 

 masses C 6 V & on B and - C' b V' b on A, corresponding to a new state 

 of equilibrium, with zero potential on A, and potential V & on B. 



The superposition of these two states of equilibrium gives a new 

 state of equilibrium with the addition of the potentials on each 

 of the conductors that is to say, the potential V a on A and V 6 on 

 B. The total charges of the two conductors are these 



These equations enable us to calculate the total masses of the 

 two conductors when the potentials are known. 



In like manner the potentials may be deduced as functions of 

 the masses, which gives 



V =^ 

 a r r 



87. RECIPROCAL ACTION OF Two ELECTRIFIED CONDUCTORS. 

 The preceding method enables us to determine the distribution of 

 electricity on the two conductors, for the final density at each point 

 is the sum of the densities relative to the various superposed layers, 

 and by hypothesis we know the law of distribution for each. We 

 have then all the elements needed for calculating the action exerted 

 between the two bodies ; the problem only presents then difficulties 

 of calculation. 



This force consists of the action of each of the two layers C a V a 

 and - C' 6 V & of the body A, on the two layers C 6 V & and - C' a V a of 

 the body B. The potentials being supposed positive, the action 

 f CaYa ls m ade up of two terms one repulsive, proportional to the 

 product V a V & of the two potentials, and the other attractive pro- 

 portional to V* . 



The action of - C' b V b comprises also two terms, one attractive 

 proportional to V^ and the other repulsive proportional to the product 



v.v 6 . 



Calling a, Z>, and c coefficients which depend on the form of the 

 body and on their distance, the reciprocal action R, considered as 

 repulsive, has an expression of the form 



