in terms of tlie Coefficients of Electrostatic Induction. 383 



If we solve these equations for e x and e. 2 , we shall find the charges 

 which must be given to the conductors [1] and [2] in order to 

 raise their potentials from zero to v x and v. 2 under the specified 

 conditions. Thus 



e x (A n A 22 - A, 2 A 21 ) = OiA.,, + v 2 A 21 ) A ) 



e 2 (A n A 22 - A 12 A 21 ) = («, A 12 + v 2 A n ) A J " 



Since a determinant is unchanged in value when its rows 

 are turned into columns and its columns into rows, and since 

 q>s = ([sr for every pair of conductors, it follows that A 21 = A 12 . 



§ 7. If A 11)22 denote the minor of A complementary to 



In 2i2 



^21 ^22 



then it is known* that 



AA 11>22 = A U A 22 -A 12 A 21 (13), 



and thus (12) may be written 



eiA 11)22 = v 1 A 22 + v. 2 A 2l 

 e 2 A 11)2 2 = i>i A 12 + v 2 A n 



.(14). 



We notice that equation (13) requires us to write A 11)22 = 1, when 

 A contains only two rows. 



Capacity of [1] and [2] "in parallel," when [3], ...[m] are 

 insulated and [wi+1], ...[/i] are kept at zero potential. 



§8. The words "in parallel" imply that [1] and [2] are 

 connected by a very fine wire, and the capacity in question is 

 the capacity, according to Definition A, of the single conductor 

 formed by joining [1] and [2]. It is further implied that the 

 relative positions of the n conductors are the same before and 

 after [1] and [2] are joined. Since the wire is very thin, the 

 charge upon it is negligible in comparison with the charges upon 

 the conductors themselves, and the only function of the wire, as 

 far as Maxwell's equations are concerned, is to make the potential 

 of [2] equal to that of [1]. 



It is easy to calculate, in terms of the n 2 coefficients "f* belonging 

 to the original n conductors, the (n — l) 3 coefficients for the n — 1 

 conductors, to which the system is reduced by joining [1] and [2], 

 and, when this has been done, the capacity of [1] and [2] "in 

 parallel" can be at once written down by means of equation (9). 



* Burnside and Panton, Theory of Equations, third edition, § 136. 

 t In this estimate, q rs and q^. are treated as distinct quantities. If we do not 

 distinguish q rs from q 3r the number of coefficients is only |n (n + 1). 



