Equipotential Curves and Surfaces, fyc. 29 



The potential at any point is of the form 

 l-fi+fa-fs-f&c. 



r r x r 2 r 3 



If the currents flowing out and in at the several points are all equal, 

 then the potential is of the form 



\r r x r 2 r 3 J 



If the electrodes are all in one plane, then this plane may be taken as 

 one of the bounding surfaces of the conducting liquid, without altering 

 the forms of the equipotential surfaces, which cut the plane always at 

 right angles. 



If r, r v r 2 , r 3 are the distances of a point from the electrodes, then, 

 with four electrodes, the equation to an equipotential surface will be 



1-1+1-1=1 



Fig. 7. 



Suppose the electrodes to be in the same straight line at A, B, C, D ; and 

 let AB=2a, and CD = 26; the positive electrodes at A and C, and the 

 negative electrodes at B and D ; then the equation to a section of an 

 equipotential surface through the axis referred to rectangular coordinates, 

 as before, will be 



{(x-ay+fl'i-\(.v + ay + f}-i+{(.v-by + f}-i 



c l 

 Differentiating, we get for a consecutive point 



x-a+y 



dy 

 dx 



+ ^)+; 



dy 



^ + &c = 0. 



Hence for a line of flow which cuts this at right angles, 



U 



(*-a)%-y K"+«)g-* 



{(*-«) 2 +2/ 2 }^ {(xJray+tf}% 



+ &c.=0. 





