370 



VIII. NEWTONIAN POTENTIAL FUNCTION. 



128. Green's Formulae. Let us apply Green's theorem to 

 two functions, of which one, V, is the potential function due to 



any distribution of matter, and the other, U = > where r is the 



distance from a fixed point P, lying in the space x over which we 



take the integral. Let the space t 

 concerned he that hounded hy a 

 closed surface S, a small sphere H 

 of radius s about P, and, if P is 

 without S, a sphere of infinite radius 

 with center P. 



Now the theorem was stated in 

 115 ; 22) for the normal drawn in 



toward T, which means outward from S and 27, and inward from 



the infinite sphere, as 



. 184. 



and since 



in the whole space r, so that 1) 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, 



-5 = - -5 = __ , 

 dn ~ dr r r* 



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

 inner or outer boundary of T; 



and for 



r = oo 



V vanishes, hence this integral vanishes. Also 



O T7" j 



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



