ELECTRIFICATION OF A SPHERE BY A POINT. \ 159 



_ ' X. 



In this case equation (32) becomes 



COS to 



it represents a curve formed of two branches, one proceeding from 

 the point A, the other from the point A' (Fig. 46). 



The branch T proceeding from A is at first vertical at this point. 

 For points at a considerable distance, the angles co and w' tend 

 towards equality, and we have 



cos <o cos o>' cos W cos to' sin o> . 8r 20 sin 2 o> 



from which 



sin 2 a>_r(i->E 2 ) 



COS to 20 



The second member increases to infinity with r. The angle o> tends 



77 



then towards -, and the curve has a vertical asymptote which evi- 



dently passes through the point O. 



The second branch is a closed curve T'j it passes through the 

 point A', and through the point I. 



172. ELECTRIFICATION OF A SPHERE UNDER THE INFLUENCE OF 

 A POINT. We know from the theorems already proved (61), that we 

 can replace the mass - m' by an equal layer in equilibrium on any 

 one of the equipotential surfaces which surround the point A', com- 

 prising the sheet S'^ of the surface with two sheets. In like manner 

 we* may replace the mass m by an equal layer on one of the equi- 

 potential surfaces which surround A, including the surface S^ The 

 two masses m and - m' may, lastly, always be replaced, for external 

 points, by one mass m - m' on one of the surfaces which surround 

 the two points, including again the surface S^. 



If, in particular, we consider the sphere S of potential zero, which 

 surrounds the point A', we can replace m' by an equal mass in 

 equilibrium on the sphere. Nothing will be changed for external 

 points ; but for points in the interior the potential will be constant 

 and equal to the value which it has on this surface that is to say, 

 zero. For points inside the surface S, the mass m may be replaced 

 by a mass + m' in equilibrium on this surface, and thus the potential 

 will everywhere be zero on the outside. 



