Electricity in a uniform plane conducting Surface. 393 



Q ; R 5 or RR,, and since these quantities are equal to each 

 other, it follows that the Hues F Q' R', P Q R, and P y Q y R, are 

 consecutive flow-lines of a system which divides the conducting 

 sheet into portions each of which conveys an equal current. 



It is important to observe that the reasoning employed here 

 is general, and not limited to the special case to which it has been 

 applied. The general conclusion to which it leads may be thus 

 stated : — If similar * systems of lines of flow be drawn, corre- 

 sponding to each of two separate systems of sources and sinks, 

 the lines of flow which would result from the combined action 

 of the sources and sinks of both systems will be obtained by 

 drawing curves through the alternate angles of the quadrilaterals 

 produced by the intersections of the two primary systems of 

 flow-lines, in directions concurrent with both the primary flow- 

 lines that intersect each other at each angle. 



11. The method which, as we have seen, allows the flow-lines 

 for two equal opposite poles to be drawn, also enables us to 

 deduce very easily their general form. Let a be the constant 

 angle between consecutive flow-lines of the pencil diverging 

 from A and of that converging to B. Then, evidently, 



ZAPB=ZAQB=ZARB= . . . =wa, 

 where n is a constant integer. Also 



ZAP'B=ZAQ'B= . . . =(»+l) a , 

 and 



ZAP,B=ZAQ i B= . . . =(n-l)«. 



Hence the lines of flow due to one source and one sink of equal 

 strength are arcs of circles passing through the poles, each one dif- 

 fering from the next by a constant change ( =a) in the angle which 

 the radii vectores from the poles make with each other ; or, what 

 comes to the same thing, they are arcs of circles cutting each 

 other at the poles with a constant difference of angle equal to 

 the constant difference of angle (=a) between the rectilinear 

 flow-lines which either the source or the sink would produce by 

 itself. 



2tt 



12. The whole number of flow-lines is therefore — , or the 



a 



same as the number of lines leaving the source or entering the 

 sink when either of them is by itself in the sheet. This is evi- 

 dent also if we consider that infinitely near to either pole the 

 effect of the other vanishes in comparison, and therefore the 



* By similar systems is here to be understood systems such that the 

 total flow between any pair of consecutive lines of the one set is the same 

 as that between any two consecutive lines of the other set. 



