135, 136] SYSTEMS OF CONDUCTORS. 265 



There is one such equation for each conductor. These n equa- 

 tions determine the charges in terms of the potentials, and if 

 the potentials of some of the conductors are given, and the charges 

 of the rest, all the remaining charges and potentials are deter- 

 mined. Q s is the charge of the conductor K B by induction from D 

 when all the conductors (including K s ) are connected to earth, and 

 consequently 



F, = F 2 =...= F n = 0. 



136. Coefficients of Induction. Reciprocal Relation. 



We shall now suppose the system of conductors to be under the 

 action only of their own field, so that Q = 0. Then we have 



0i = tfn "Pi + 7 21 F 2 ... -f q m V n , 



i , e 2 = q 12 V 1 + q^Vz + qwVn, 



(4) 



The constants q rs are called coefficients of induction, and any 

 q rs is defined as the charge induced on the conductor K s when 

 it and all the others are earthed, except K r which is brought 

 to potential 1. Any coefficient with double suffix q ss is the 

 charge of K 8 when it is at potential unity, and all the other 

 conductors are earthed. It is called the capacity of the con- 



ductor K s . The dimensions of the q's are Tr = [L]. We shall 



now show that the order of the suffixes in q rs is immaterial. 

 Applying Green's theorem in the second form to the functions v g 

 and v r) we have 



IL (*%, - * 



The volume integral being taken throughout the space ex- 

 ternal to the conductors where v r and v 8 are both harmonic, 

 vanishes, and since v r vanishes on all conductors except K r where 

 it is constant and equal to unity, and v s vanishes on all conductors 

 except K s where it is equal to unity, (5) becomes 



(6) 



--Lff ^.Iff p. 

 *Tr]] Kg c>n e torJJfrjfa* 



