468 Messrs. G. C. Foster and 0. J. Lodge on the Flow 

 V - V ' = iks P(Q1°S' J ) -S(Qlogr)] 



since, for each pole, there are the corresponding terms Q n log r n 

 and Q, n \ogr' n . If there are altogether K poles, k being of one 

 sign and k! of the opposite sign, where k is not less than k' } 

 and if, further, they are all of the same strength, the quantity 

 of electricity crossing each complete equipotential line in unit 

 of time is kQ, ; consequently the resistance of the portion of the 

 sheet lying between the equipotential lines which pass through 

 the given points is 



This formula is in principle quite general; but the practical 

 application of it in actual cases requires that we should know 

 the position of the poles from which the distances r v r 2 , ... are 

 to be measured ; and these cannot be ascertained (or at least not 

 by elementary methods), except for comparatively few and simple 

 cases ; for although it is comparatively easy to determine the 

 equipotential lines for a given set of poles, the inverse problem, 

 of finding the distribution of poles required to produce equipo- 

 tential lines coinciding with two given curves on the conducting 

 sheet, presents in general very great mathematical difficulties, 

 and has hitherto received only partial solutions. 



38. We will give here, in the first place, the approximate 

 application of the general formula to the case of two sources and 

 two equal sinks at the angles of a quadrilateral inscribed in a 

 circle and so placed that unlike poles are diagonally opposite 

 each other. This is the arrangement shown in figs. 8 and 9, where 

 A and A' may be taken as sources and B and B' as sinks. It is 

 evident that the equipotential lines very near the poles will each 

 consist of two branches, one of them surrounding one source (or 

 sink), and the other surrounding the other source (or sink), and 

 also that they will be very approximately circles having the poles 

 at their centres. Hence, if the sources and sinks are formed by 

 four circular electrodes, whose common radius p is a small frac- 

 tion of the distance between any two of them, and if they are so 

 placed that the distances of their centres A, A', B, and B' from 

 a fixed point C fulfil the condition CA . CA'=CB . CB' (which 

 is equivalent to saying that their centres are on the circumference 

 of one and the same circle), we may without serious error regard 

 the circles of contact between the electrodes and the conducting 

 sheet as forming together a pair of equipotential lines due to such 



