158] CONFORMAL REPRESENTATION. 313 



1 4- r 2 /I l r 2\ 2 



(9) * + f + 2a* -j + a? {^J 



1 > = 



we have 



r 2 + 1 r + l/r 2ar 2a 



a = a 7 = & -- ST-j ^~~y - 7 = -- 7T~- 



r- 2 1 r 1/r r 2 1 r 1/r 



from which 



< + c 2 - 2 

 <"> r= -^" 



Since ^2 and r must be positive, we take the upper signs in 

 (10) and (i i) for r > 1, which makes c > 0, and gives the circles on 

 the right, the lower for r < 1, which makes d < 0, and gives the 

 circles on the left. 



Now making use of the results of the last example, we have 

 for the functions V and SP 



V= log r = log (w 2 + t; 2 ), 



so that in the transformation, for r = i\, r = r 2 , 



T . , , c^ 



F! = log n = log - 



(I3) di 



F 2 =logr 2 = log- 



and the capacity of the pair of eccentric cylinders is 



1 



* 



(14) 



In case r 2 =l, c? 2 and R 2 become infinite, and we have for the 

 capacity of a single cylinder in presence of the infinite conducting 

 plane x 0, 



(15) K= 



The formula given above for the capacity of a pair of cylinders 

 of which one is internal to the other is not convenient in practice, 

 since we are given not the distances d lt d, 2 , but only their difference 



