Equipotential Curves and Surfaces, §*c. 19 



potential curves. If, then, at two points A and B, on a circular disk HEK, 

 we have two equal and opposite battery-electrodes, the equipotential 

 curves will be the same as the equipotential curves in an infinite sheet 

 which are due to those two electrodes, together with equal and like elec- 

 trodes at their electrical images with regard to the circle *. 



[Extension of the Theory of Electrical Images. 



"We have seen that when the boundary-line of a disk is made up of 

 arcs of circles passing through the two battery-electrodes, then the forms 

 of the lines of flow will be the same as in an infinite sheet ; and conse- 

 quently the forms of the equipotential curves are arcs of circles. 



We have also seen that when the boundary of the disk is a circle, and 

 the two electrodes lie within it, the forms of the lines of flow will be 

 altered because of the presence of the edge of the disk ; and the change 

 in the electrical distribution will be that due to equal quantities of elec- 

 tricity in the position of the electrical images of the electrodes formed 

 by the edges of the disk. If from a circular disk with two equal battery- 

 electrodes lying within it, we cut off a portion bounded by the arc of that 



* [July 23. — That the circle HEK is a line of flow for the four elec- 

 trodes may be readily seen if we resolve the rate of flow at any point, Q, 

 of the circumference of this circle in the direction of the radius at that 

 point. As shown by Prof. W. R. Smith, the rate of flow arising from 

 each electrode is inversely as the distance from the electrode. 



Let CA=a 1 , CA 1 =a 1 , AQ=r, A 1 Q=r', and p the radius; then, ac- 

 cording to Prof. Smith's notation, the resolved part along the radius from 

 the two like electrodes A and A L will be 



E y/ + P 2 - a2 + E y / 2 +(**-"* _ 

 2irr 2rp 2irr 2r'p 



E 

 or - — 



2irp <±np 



but r n = P^IL=P±l\ 



r a x -p a x +p 



so that r l = £li" ; therefore ^ "'f +£=^ = 0. 



r a i — 9 r r 



Hence the rate of flow along the radius, viz. - — , is the same as if 



2irp 



there were an equal electrode at C. 



In the same way it may be shown that the rate of flow along the 

 radius arising from the other two electrodes is the same in quantity, but 

 in the opposite direction. 



Hence there is no flow along the radius at the point Q when there are 

 two equal positive electrodes at A and A v and two other equal negative 

 electrodes at B and B r 



So that at every point, Q, of the circle HEK the circumference of the 

 circle must be the line of flow.] 



o2 



4™ I r z r' 2 J 



