Electricity in a uniform plane conducting Surface. 397 



have recourse to a process of superposition of the same kind as 

 that employed (§ 10) for placing the lines of flow. 



We have seen (§ 5) that the equipotential lines for a single 

 pole are concentric circles, and that the radii of consecutive 

 circles form a geometrical progression. To find the system of 

 equipotential lines for two equal opposite poles, it is only needful 

 to draw for each pole separately a system of equipotential lines 

 with the same difference of potential between any one line and 

 the next as it is intended that there should be between the lines 

 of the combined system, and then to draw lines through the 

 intersections of the two overlapping systems of circles thus ob- 

 tained, taking care, in going from one intersection to the next, 

 that the changes of potential are in opposite directions for the 

 two primary systems taken separately. Thus, let the lines 1, 2, 

 3, 4 (fig. 4) represent portions of equipotential lines due to a 

 source at A ; and 1', 2 ! , 3', 4' portions of equipotential lines 

 due to a sink at B ; and let the potential of the line 1 be V, and 

 let that of the line 1' be V ; further, let the change of potential 

 in passing from any one line to the next in either system be v s 

 so that the potentials of the lines 2, 3, and 4 are Y —v, Y— 2v, 

 and V— Sv respectively, and the potentials of 2', 3', and 4' are 



V + v, V' + 2v, and V' + 3v respectively. Then, at the points 

 where 1 and 1', 2 and 2', 3 and 3', 4 and 4' respectively intersect 

 each other, the potentials will be the sums of the potentials of 

 the intersecting lines; and therefore the potential at all these 

 points is the same, namely V + V'. Consequently P, Q, R, and 

 S are points on the same equipotential line. Similarly it follows 

 that P x , Q x , and Rj are points on the line whose potential is 



V + V' + v; andQ', R', and S' points on the line whose potential 

 is Y + Y'—v. We thus get the potential of the resultant equi- 

 potential lines differing by the constant amount v, which is the 

 same as the difference of potential of the lines of the two primary 

 systems. 



It is evident from this that any two systems of equipotential 

 lines whatever, which have the same constant difference of poten- 

 tial, can be compounded, so as to give a single resultant system, 

 by tracing lines through alternate angles of the quadrilaterals 

 produced by the mutual intersection of the lines of the two 

 systems, and also that the constant difference of potential 

 between the lines of the resultant system will be the same as 

 that between the lines of each of the component systems. 



18. Let //, be the common ratio of the radii of the equipoten- 

 tial circles of the two primary systems considered in § 17, and 

 let the radius of the circle 1 be fju n and that of the circle 1' be fju m 

 (see § 8) ; the radii of the successive circles of the one set are 

 then i x n + 1 , //*+ 2 , . . . , and of those of the other fju m+1 , fju m+2 3 . . . 



Phil. May. S. 4. Vol. 49. No. 326. May 1875. 2 E 



