Equipotential Curves and Surfaces, §c. 23 



is not a line of flow for such a system of electrodes, we cannot have this 

 system of equipotential curves and lines of flow in a circular disk. 



A particular solution of this case, the equipotential curve for zero 

 potential, is a rectangular hyperbola, having its centre and two foci at 

 the three electrodes (see Plate 1. fig. 5). 



In the case of a circular disk the electric image of the centre may be 

 regarded as a point at an infinite distance, so that the distance of the 

 image from any point of the circular disk remains constant. 



If, then, one electrode be at the centre and the other at the circum- 

 ference of a circular disk (Plate 1. fig. 6), the corresponding case in the 

 unlimited sheet is that where the current flowing in at the circumference 

 is double the current flowing out at the centre, we may conceive of it as 

 a case where there is one electrode at an infinite distance, equal to and 

 of like kind with the electrode at the centre. 



In such a case the potential at any point is expressed by 

 C+A log r—2Alogr 1 



Hence for an equipotential curve 



r 2 = 2par, 



where a is the radius of the circle (which is introduced for the sake of 

 making the equation homogeneous), and p is some ratio which is con- 

 stant for the same curve. 



With polar coordinates about the centre, taking a as the radius of the 

 circle, we get 



r 2 + a 2 — 2ar cos 6 = 2/jiar, 

 r 2 -2ar (jk + cos 0)+a 2 = O 



If fi— or >2, the curve is a closed curve, and, when r is small com- 

 pared with a, the curve nearly coincides with an ellipse of eccentricity 



- and mean distance ^ 



t 1 



2 -l 



2a J 1 + 1 cos d\ 



Hence — expresses the percentage of error in the value of r if the 



ellipse be taken instead of the curve. 



In a circular disk of 100 millims. in radius the distance between the 

 curve and the ellipse at a distance of 25 millims. from the centre of the 

 disk is only about 1*5 millim. 



"When fjL is less than 2 and greater than (V2— 1), the curves are of 

 double curvature, and have points of inflection. 



From the equation of the curve we obtain 



{r — a(fi-\-cos 0)} -j- + ar sin 0=0. 

 (id 



