138 140] SYSTEMS OF CONDUCTORS. 269 



Secondly, let all the conductors be at potential 0, except K S) 

 which is at potential unity. The energy is 



so that the capacity of any conductor is positive. The number of 

 unit tubes of force issuing from K 8 is proportional to the charge 

 q ss . Some of these extend to infinity, while others end on the 

 other conductors. At the latter the charges will be negative, but 

 the sum of all such charges is not as great as q ss . Accordingly, 

 every q with double suffixes is negative, and 



07) q ss > - (?if + ?...+ q s -i, s + q s +i, S '" + q ns )' 



If, however, K s is completely surrounded by a conductor K r , 



If a new Conductor be introduced into the field, the coefficient 

 of potential with the double suffix for any conductor is diminished. 

 For if any portion of the field be made suddenly conducting, elec- 

 tricity will move in it so as to make the energy less than before. 

 If K s was the only charged body, the energy ^p ss e z must be 

 diminished, but as the charge e has not changed, p ss must be 

 diminished. 



Introducing a new body into the field increases the capacity of 

 any conductor, and diminishes the absolute value of every co- 

 efficient of induction. For if the new conductor and all the others 

 be at potential 0, while K s is at potential unity, some of the tubes 

 of force which before ended on the other conductors, now end on 

 the new conductor, which receives a negative charge. This in- 

 duces a positive charge on K s , increasing its charge q ss , and positive 

 charges on the other conductors K r) diminishing their negative 

 charges q rs . 



140. Work done during displacement of conductors. 



Suppose that we deform or displace the conductors of the system, 

 thus changing the geometrical coefficients p and q. Suppose the 

 configuration of the system is specified by m parameters 



so that if the conductors are displaced as rigid bodies m = 6n. 

 Let the mechanical forces due to the electrification be denoted by 

 < so that the force tending to change the parameter <^ is fy. 



