162 PARTICULAR CASES OF EQUILIBRIUM. 



the resultant density is 



m - m m 



+ = 



4 7rR 



The density will be zero for all points of the small circle perpen- 

 dicular to the axis defined by the equation 



The plane of this small circle cuts the axis OA on the left of 

 the point A', since we have r> \/*/ 2 -R 2 . It is the neutral line 

 which separates the positive from the negative zone. It is a line 

 of equilibrium the force and the density there are null. It is the 



intersection, by the sphere, of the equipotential surface V = ; we 



know, moreover, that the two surfaces intersect at a right angle. 



The density will be null on the small circle, formed by the con- 

 tact of the tangent cone to the sphere, and having its apex at the 

 point A, the plane of which circle passes through the point A', pro- 

 vided that 



M d* - R 2 m 



m 



4 7rR 

 and therefore 



M R 



m 



174. The action of the insulated sphere S, electrified by induc- 

 tion from the mass m^ on all external points, may be replaced by that 



of a mass - m' = - , placed at A'. In like manner the uninsulated 



k 



surface S' acted on by induction from -m', is equivalent for all 

 internal points to the mass m = km' placed at A. 



175. IMAGE OF ANY GIVEN SYSTEM. The principle of images 

 in reference to a sphere may be extended to any system whatever for 

 instance, to an electrified layer. For each element of the systems 

 develops by induction, on the sphere, a layer whose action on external 

 points is identical with that of the corresponding image. As each of 



