380 Mr Searle, On the Calculation of Capacities 



If, when [1] is raised to unit potential, and the n — 1 remain- 

 ing conductors are at zero potential, the charge on [2] is q 21 and 

 if, when [2] is raised to unit potential and the n — 1 remaining 

 conductors are at zero potential, the charge on [1] is q 12 , it can be 

 shewn that 



2l2 = ?21 (2). 



On account of this relation, q 12 or q 21 may be called the Coeffi- 

 cient of Mutual Induction between [1] and [2]. To facilitate the 

 mathematical operations upon the determinants, which will appear 

 in the course of the work, it will be often convenient to treat 

 q 12 and q 21 as separate quantities. If we wish to do so, we can 

 substitute q 12 for q 21 in the final results. 



The relations between the charges e 1} e 2 ,...e n , and the 

 potentials v 1} v 2 , ... v n of the n conductors can now be expressed, as 

 Maxwell shews, by means of the n equations, 



q-a Vi 4 qn v 2 4 . . . + q m v n = e 1 \ 



q 21 v 1 + q 22 v 2 + ...+ q 2n v n = e 2 



.(3). 



^ni^i 4 qwiVi T • ■ • 4 qnn^n — &n 



Maxwell says that " a coefficient in which the two suffixes are 

 the same is called the Electric Capacity of the conductor to which 

 it belongs," but, in view of the extended definitions of capacity 

 which are given in the present paper, it is necessary to abandon 

 Maxwell's term and, in its stead, to speak of the coefficient of self 

 induction of the conductor. It must be remembered that when 

 there are n conductors in the field, every coefficient, whether of 

 self or of mutual induction, is, in general, a function of the form 

 and position of every one of the n conductors. 



Capacity of [1] in the presence of [2], [3], ... [»], when 

 [2], . . . [m] are insulated and [m + 1], ... [n\ are kept at 

 zero potential. 



§ 4. If v 1} v 2 , ... v n be the potentials of the conductors after 

 [1] has received the charge e 1} as in Definition A, then, since, 



e 2 — e 3 = ... = e m = i), 



and v m+1 = v m+2 = . . . = v n = 0, 



the first m of Maxwell's equations become 



9n v x + q 12 v 2 + ... + q lm v m = e 1 \ 



ffai 0i 4 #22 v 2 + ... + q 2m v m = [ ( 4 y 



SWi^i 4 qm,2 v 2 4 ••• 4 qmm^m — 



