266 



ELECTROSTATICS. 



[PT. II. CH. VI. 



We may accordingly state the reciprocal theorem : 



The quantity of electricity induced upon a conductor A of a 

 system when another conductor E is brought to a given po- 

 tential V and all the others including A are earthed, is the same 

 as the quantity induced on B when it and the others are earthed, 

 except A, which is brought to the same potential V. 



137. Energy of System. The energy of the system is 



(7) 





V n e n ), 



or introducing the values of e s and bearing in mind the relation 



(8) 



That is, the energy is a homogeneous quadratic function of the 

 potentials of the n conductors, the coefficients being the coefficients 

 of capacity and induction. The energy expressed in this form will 

 be denoted by W v . 



We have 



(9) ~-ff = q u r l + q m V t + + 9n ,F M = e s> 



or the charge of any conductor is obtained by differentiating the 

 energy-function expressed in terms of the potentials partially with 

 respect to the potential of the given conductor. 



138. Coefficients of Potential. Solving the linear equa- 

 tions (4) we get n equations 



do : . 



' n = Pin @i T PMI &2 ~\~pnn &n > 



where any coefficient p rs is the minor of q rs divided by A in the 

 determinant 



r i 2 q~> q 



