58 APPLICATION OF THE PRECEDING RESULTS 



But, since it is evident that p is the density corresponding 

 to the potential function V, we shall have for any point on the 

 segment s, treated as a plane, 



_-leZF 



P ~ 27T dw ' 



as it is easy to see, from what has been before shown (art. 4) ; 

 dw being perpendicular to the surface, and directed towards the 

 centre of the sphere ; the horizontal line always serving to in- 

 dicate quantities belonging to the surface. "When the point P 

 is very near the plane s, and z is a perpendicular from P 

 upon s, z will be a very small quantity, of which the square 

 and higher powers may be neglected. Thus b = a z, and by 

 substitution 



the integral extending over the surface of the small plane s, and 

 f being, as before, the distance P, do: Now 



= 



dw ~~ dz 



at the surface of s, and ^ = ~~X f> nence 



^dV'_~ldV'_--l d_ [zdo- = 1 d* [dv = 

 2^~dw"-~*jr^z~~te?dz) f "47T 2 d#]J 



provided we suppose z at the end of the calculus. Now the 



p 

 density -- H /o, upon the surface of the orifice 5, is equal to 



zero, and therefore we have for the whole of this surface 



F 



Hence by substitution 



F (12). 



l ' ( } ' 



the integral extending over the whole of the plane s, of which 

 da- is an element, and z being supposed equal to zero, after all 

 the operations have been effected. 



