155] GREEN'S FUNCTION. 297 



If we should place a charge e at the outside point A, and a 

 charge e at the inside B, the potential on the surface of the 



e e' 



sphere would be V - + . which, if we make e = Re/ a becomes 

 r r 



zero. The action of the charge e at B in portions of space 

 outside of the sphere may thus be exactly replaced by making the 

 sphere conducting and removing the charge B. Accordingly the 

 charges at A and B are electrical images of each other in the 

 sphere. 



Suppose now that the sphere, instead of being connected to 

 earth, is insulated and charged to a potential V, then beside the 

 induced charge it will have a uniformly distributed charge VR of 

 density V/4i7rR, so that the whole charge of the sphere is now 



( I3 ) Jf 



The surface density 



4-7T (R ' Rr* 



vanishes along the circle Fr 3 = a 2 R 2 , which divides the surface 

 into two parts oppositely electrified. If however 



or 



(a - R)* 



the surface density is of the same sign all over the sphere. Since 

 the action of the induced charge on external points is the same as 

 would be that of a charge e' at B, and the action of the uniform 

 charge is the same as that of a charge VR at the center, the 

 repulsion of the whole charge of the sphere on the charge 

 e at A is 



FZte e'e ^(V ea 



(15) -TiT-+7^ i^ = e ' 



= \& e 



This is negative, so that there is an attraction, when V= 0, or 

 E = 0, or a R is small; that is if the sphere is connected to earth, 

 if it is insulated without charge, or in any case if the charged 

 point A is very near to the sphere. On the other hand, by 

 making V or E of the same sign as e and great enough in absolute 



