Electricity in a uniform plane conducting Surface. 395 



where the values 0, 7r, — 7r, and 2tt are represented by the 

 straight line through A and B, and negative angles indicate flow- 

 lines passing in the lower side of A B. But, as is evident from 

 the figure, the same circle gives the two flow-lines whose charac- 

 teristic angles are net and — (7T— na) ; hence the number of circles 

 to be drawn or of centres to be found is equal to half the number 

 of flow-lines ; and we need only consider those characterized by 

 angles between and it, of which the complements are 



nr it ~ fir \ 7r 



*' * ' * \2~°7' 



2' 2 "' •'•">"> ■*■•'• \2 "/' 2" 



Consequently the required centres are obtained by drawing lines 

 from A making the above angles with A B, and letting them in- 

 tersect the perpendicular to A B through ; or, without mea- 

 suring angles, by laying off from in both directions along the 

 perpendicular to A B lengths proportional to 



o— «)> fltan— . 



Plate IX. shows a system of lines of flow for which a =20°. 



15. The strength of the current at any point P due to a source 

 at A and an equal sink at B is represented in the construction 

 given in § 13 by the length of the line P T (fig. 2) ; but by the 

 similarity of the triangles P B T and A P B, 



PT _AB 

 BT ~ PB ; 



or, putting AB = 2a, BT=~, and PB = r', the strength of 

 the current is 



-£- PT =£v- • • • • • 0) 



Hence, for given poles at a given distance apart, the strength 

 of the current is the same at all points of the sheet for which 

 the product of the distances from the two poles is constant ; or 

 the loci of equal flow are a system of lemniscates*. 



* The strength of the current at various parts of a circular disk with 

 equal opposite poles on the edge was examined experimentally by Kirch- 

 hoff, by suspending a small magnet, with a mirror attached, close above 

 the disk, and was found to agree closely with the results of calculation. 

 Mach has since given a more direct proof that the lines passing through 

 points of equal flow are lemniscates. A disk of silver-leaf was coated with 

 a thin film of wax ; and on allowing a strong current to traverse it for a few 

 moments, the wax was melted within a space bounded by a lemniscate and 

 the edge of the disk (Carl's Repertorium f. experim. Physik, vol. vi. p. 11, 

 1870). 



