and conductors 393 



and if the second condenser, whose capacity when its electrodes are in contact 

 is / + 2m + , is placed at a distance R from the first and connected to earth, 



its charge will be 



- P (I + 2m + n) = Q. 



This charge of the second condenser will produce a potential QR" 1 at a 

 distance R, and will therefore alter the potentials of A and B by this quantity, 

 so that the potentials of A and B will be x + QR~ l and y + QR~ l respectively. 



To find the capacity of A as altered by the presence of the second con- 

 denser, we must make the potential of A = i and that of B o, which gives 



x - {Lx + M (x + y) + Ny} (I + 2m + n) R-* = i, 

 y - {Lx + M (x + y) + Ny} (I + 2m + n) R~ z = o. 

 Hence x = y + i, 



and y = {L + M + (L + 2M + N) y} (I + 2m -| n) R-' 2 , 



(L + M)(l+2m + n} R~* 

 i - (L + 2M + N) (1+ 2m + n) R~ 2 ' 

 and the capacity of A is Lx + My or L + (L + M) y, or 



(L + M) 2 (I + 2m + n) 



[AA] = L 

 of B is M 



[AB] = M + 

 af a and b 



[Aa] = - 



2 - (L + 2M + N) (I + 2m + n) ' 

 The charge of B is MX + Ny or M + (M + N) y, or 



(Z. + M) (M + N) (I + 2tn + n) 

 R*-(L+2M + N) (l + 2m + n) ' 



The charges of a and b are (I + m) P and (m + n) P respectively, 01 



R (L + M)(l + m) 



[Ab] = 



A' 2 - (L + 2M + N) (I + 2m + n) 

 R (L + M) ( m + n) 



R* - (L + 2M + N) (I + 2m + n) ' 



In these expressions we must remember that M is a negative quantity, 

 that L + M and M + N can neither of them be negative, and that their sum 

 L + zM + N cannot be greater than the largest semidiameter of the condenser. 

 Hence if R is large compared with the dimensions of the condensers, the second 

 terms of the values of [A A] and [AB] will be quite insensible, and even if the 

 condensers are placed very near together these terms will be small compared 

 with L, M, or N. 



If a, instead of being part of a condenser, is a conductor at a considerable 

 distance from any other conductor, we may put m = n = o, and if A is also 

 a simple conductor, M = N = o, and we find 



L 2 / 

 [AA] = L + ^5- Tl , 



RLl 



~^' 



by which the capacities and mutual induction of two simple conductors at a 

 distance R can be calculated when we know their capacities when at a great 

 distance from other conductors. See Note 24. 



