of Electricity in a uniform plane conducting Surface, 467 



If, fu 

 L . CA', 



Fig. 8. 



equal to them at B and B'. If, further, the points B and B' be 

 taken so that CB. CB'=CA. CA', the circle of radius CP will 



be a common line of flow in each of the systems due to C, A, 

 and A' and to C, B, and B', and will therefore remain a line of 

 flow in the resultant system (see fig. 9) due to A, A', B, and B'. 

 Moreover, since these four points are placed so that one circle can 

 be drawn through them all, it is easy to see that this circle will be 

 another flow-line of the resultant system ; for the four poles may 

 be grouped in two ways into two pairs of equal opposite poles 

 (either A, B and A', B', or A, B' and A', B), each of which 

 would separately give this circle as a line of flow. If the straight 

 lines AB and A'B' are drawn and produced to their intersection 

 at D (fig. 8), a circle drawn about this point as centre with radius 

 = \/DB .DA= VDB' .DA' will be an equipotential line com- 

 mon to the systems due respectively to the source and sink at 

 A and B, and to those at A' and B' ; consequently this circle is 

 also an equipotential line of the resultant system due to all four 

 poles. Another equipotential line of the resultant system would 

 be the circle (in this case imaginary) drawn with the point of 

 intersection of AB' and A'B as centre so as to cut each of these 

 straight lines in points harmonically situated with respect to 

 their extremities. 



Again, if four equal poles be taken situated as in the last 

 example, but so that poles of the same sign are diagonally oppo- 

 site each other — in other words, if the signs of either A and B 

 or of A' and B' be interchanged, the circle passing through 

 the four poles will still be a flow-line of the resultant system, 

 but C, as well as D, will now be the centre of an equipotential 

 circle, while the imaginary circle will be a line of flow. 



37. Resistance, — Equation (17) gives, for the difference of 

 potential of any two points whose respective distances from the 

 several poles are r v r 2 , r 3 , . . . , and r\, r'^ . . . the value 



2K2 



