382 Mr Searle, On the Calculation of Capacities 



§ 5. As a simple example of the problem of § 4, I will take 

 the case in which all the conductors except [1] and [2] are kept at 

 zero potential. We then have 



A = 



$n ?u 



$21 $22 



— $11 $22 $i2$2i> 



and A u = $ 2 



Hence C 1 — 



$11 $22 $12 $21 _ $12 $21 



^22 $22 



$11 $22 $12 $21 $12 $21 



Similarly C 2 = ^ 22 tfl2 ^ 21 = a 2 , 



$n *" $n 



Since $ u and $ 22 are positive and $ 12 and $ 21 are negative, we 

 see that the capacity of [1] is smaller when [2] is insulated than 

 it is when [2] is kept at zero potential. 



Relations between the potentials and the charges of the insidated 

 conductors [1] and [2], when [3], ... [m] are insulated and 

 uncharged and [m + 1], ... [n] are kept at zero potential. 



§ 6. So far we have been concerned with the relation of the 

 potential of a single conductor to its charge under specified con- 

 ditions. It will, however, be convenient, for use in later problems, 

 to obtain the relations which exist between the potentials v\ and 

 v 2 and the charges e x and e 23 when the conductors [3], ... [m] are 

 insulated and [in 4- 1], ... [n] are kept at zero potential. 



In this case, the first tn of Maxwell's equations are 



.(10), 



q-miVi + q m . 2 v 2 + ...+ q mm v m = 0) 



the quantities e 3 , e 4 ...e m being all zero, since the corresponding 

 conductors are insulated and are initially without charges. As in 

 § 4, the remaining n — m equations will give, in terms of e x and e 2 , 

 the charges upon the conductors [m + 1], ... [n], which are kept at 

 zero potential. 



If A n , A 12 , A 21 and A 22 denote the minors of the determinant 

 (6), which are complementary to q n , $ 12 , $ 21 and $ 22 respectively, 

 we can write down the values of v x and v 2 in terms of e x and e 2 

 in the following forms : 



VjA= ejAu-flAa) (u , 



w 8 A = - 6^22 + 62^22) 



