114 Systems of Conductors [CH. iv 



successively two positions, one practically wholly within A, the other within B, the 

 positions being similar and such that the coefficients of potential of C in either position 

 are p, q, r in ascending order of magnitude. In each position C is in turn connected with 

 the conductor surrounding it, put to earth, and then insulated. Determine the charges 

 on the conductors after any number of cycles of such operations, and shew that they 

 ultimately lead to the ratios 



l:-/3:/3 2 -l, 

 where /3 is the positive root of 



rx z -qx+p-r=0. 



27. Two conductors are of capacities C l and <7 2 , when each is alone in the field. 

 They are both in the field at potentials V\ and F 2 respectively, at a great distance r 

 apart. Prove that the repulsion between the conductors is 



As far as what power of - is this result accurate ? 



28. Two equal and similar insulated conductors are placed symmetrically with regard 

 to each other, one of them being uncharged. Another insulated conductor is made to 

 touch them alternately in a symmetrical manner, beginning with the one which has a 

 charge. If e it e% be their charges when it has touched each once, shew that their charges, 

 when it has touched each r times, are respectively 



. ^ . . . and 



2<?i i 



29. Three conductors AI, A 2 and A 3 are such that A 3 is practically inside A 2 . A l is 

 alternately connected with A 2 and A 3 by means of a fine wire, the first contact being with 

 A 3 . AI has a charge E initially, A 2 and A 3 being uncharged. Prove that the charge on 

 AI after it has been connected n times with A 2 is 



Eft f 



a+/3 1 * 



where a, /3, y stand for pn-pu, Pw-pu and p 33 -pi 2 respectively. 



30. Two spheres, radii a, b, have their centres at a distance c apart. Shew that 



neglecting (a/c) 6 and (&/c) 6 , 



1 6 3 1 la 3 



