82, 83] NEWTONIAN POTENTIAL FUNCTION. 163 



in the whole space r, so that (i) becomes 



The surface integrals are to be taken over S, over the small sphere, 

 and over the infinite sphere. For a sphere with center at P, 



a! a 1 



the upper or lower sign being taken according as the sphere is 

 the inner or outer boundary of T ; 



and for r = <x> 



V vanishes, hence this integral vanishes. Also 



rr iaF ^e Vr 18F ^ /r aF ,7 



(3) 1 1 - -5- dtf = I - ^- f 2 aft> = - r K- c?o>. 



Jj r 3n JJ r 3?^ JJ 9n 



Now at infinity, is of order , and being multiplied by r, 

 on T 



still vanishes. Accordingly the infinite sphere contributes nothing. 

 For the small sphere the case is different. The first integral 



becomes, as the radius e of the sphere diminishes, 

 (4) -F P jTdft> = -47rFp. 



The second part 



[[dV 



-ell^-do), 

 JJ dn 



however, since 5 is finite in the sphere, vanishes with e. Hence 

 on 



there remain on the left side of the equation only 4-TrFp and the 

 integral over $. We obtain therefore 



II 



112 



