of Electricity in a uniform plane conducting Surface. 465 



often as it contains the greatest common measure of the strengths 

 of the whole set. Moreover, if on any flow-line there is a point 

 of no flow, the line will intersect itself at this point one or more 

 times. Separate flow-lines cannot intersect each other anywhere 

 except at a pole. 



Equipotential lines form closed curves which always surround 

 one or more poles. At points where the strength of the current 

 is nothing, the same equipotential line cuts itself once or oftener ; 

 in such cases there is at least one pole within each loop formed 

 by the line. Separate equipotential lines never intersect each 

 other. 



The method of superposition, by which any system of flow- 

 lines or equipotential lines can be drawn, is easily carried out 

 by drawing, in rather strong lines on white paper, one of any two 

 systems that are to be compounded to a resultant system, and 

 drawing the other component system on tracing-paper. Placing 

 the two drawings one over Jthe other in any required manner, and 

 laying a second piece of tracing-paper on the top, the curves 

 which pass through the intersections of the two component 

 systems can be drawn at once. The resultant system may then, 

 if required, be recompounded with one of the components by 

 the same process. 



36. The general course of the lines of flow and equipotential 

 lines for a few of the simpler cases is shown in Plate X. 



Two equal similar poles (fig. 3) . — The equation to the lines of 

 flow for this case is 



0\ + 02=not, 

 which is equivalent to 



x<l "~ y 2 ~~ % x y c °t not = a% i 



if the straight line through the poles is taken as the axis of x, 

 and its middle point as the origin. They are a system of rectan- 

 gular hyperbolae cutting each other at the poles with a constant 

 difference of angle ( = «). The value na. = ir gives the two axes, 

 the origin where they intersect being a point of no flow. The 

 equipotential lines for the same case are a system of lemniscates, 

 r l rc, = fi n } and are identical with the loci of equal flow for two 

 equal opposite poles (§ 15). The strength of the current at any 



Q r 



point at a distance?* from the origin iss=— • ; and consequently 



7r rfo 



the loci of constant flow are the same as the equipotential lines 

 for three equal poles — two of the same sign, and one of the oppo- 

 site sign halfway between them, as shown in fig. 4. The strength 



of the current on the self-cutting curve is » — . For all other 



values of s the curve consists of two separate branches ; when s 

 Phil. Mag. S. 4. Vol. 49. No. 327. June 1875. 2 K 



