62] 



Charges + e, +e 



49 



from a small sphere of radius r drawn about A, and at every point of 

 this sphere the intensity is e/r 2 normal to the sphere. The surface again 

 cuts off an area 



27T 1 - COS 



from a sphere of very great radius R drawn about C, and at every point 

 of this sphere the intensity is 2e/R 2 . Hence, applying Gauss' Theorem 

 to the part of the field enclosed by the two spheres of radii r and R, 

 and the surface formed by the revolution of the line of force about AB y 

 we obtain 



2-7T (1 - cos 6) r 2 x^ 2 - 27r(l- cos 0) R 2 x = 0, 



from which follows the relation 



sin \6 = \/2 sin J<. 



In particular, the line of force which leaves A in a direction perpendicular 

 to AB is bent through an angle of 45 before it reaches its asymptote at 

 infinity. 



The sections of the equipotentials made by the plane of xy for this case 

 are shewn in fig. 16 which is drawn on the same scale as fig. 15. The equa- 

 tions of these curves are of course 



curves of the sixth degree. The equipotential which passes through C is 

 of interest, as it intersects itself at the point C. This is a necessary conse- 



FIG. 16. 



quence of the fact that C is a point of equilibrium. Indeed the conditions 

 for a point of equilibrium, namely 



may be interpreted as the condition that the equipotential (V constant) 

 through the point should have a double tangent plane or a tangent cone at 

 the point. 



J. 



