Equipotential Curves and Surfaces, fyc. 



27 



Hence for the line of flow through (xy) which will cut this curve at 

 right angles, 



or y 2 te - O + a)yly _ y 2 hx - (a? - a)yly _ 



r* r 3 



Zee _(x + a) {(x -f a) Bx-\-ydy} _Sx (x — a)\(x—a)?ix + ydy} _~ 



r x r* r r 3 



Integrating, we get 



x+a x—a , . , , 



— ! — — =m, where m is constant. 



r r 



The curve cuts the axis of y at a point where 

 a a m 



If a is the angle which the two radii drawn to the same point make 

 with the axis when they are equal to one another, then wi=2cos a. 



If <p and 6 be the angles which the two radii drawn to any point in the 

 curve make with the axis, then the equation to a line of flow may be 

 written in the form 



cos 0— cos 0=2 cos a. 



This will be the equation to the line of flow in any plane through the 

 two electrodes. 



These lines of flow coincide with the lines of force in non-conducting 

 space with equal and opposite charges in the positions of the two elec- 

 trodes. They also coincide with the magnetic lines of force when the 

 two poles are in the same positions as the electrodes. From this equa- 

 tion the form of the lines of flow may be readily drawn. 



Let AB be the two electrodes, O the middle point of AB, and OE 

 bisect AB at right angles. 



Let E be a point on the line of force. Describe a semicircle on AB 

 cutting AE in the point F. 



Take AH equal to twice AF. Place AH in any position ANH cut- 

 ling the circle in N, and take a chord BGr equal to NH ; BGr produced 

 will meet AH on the line of force. 



Fig. 6. 



