146 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



If for any point so, y, z outside K, r^ is the shortest distance to 

 any point of K and r 2 the greatest distance, we have for any point 



r 2 > r > r lt 



111 

 < -< , 



r 2 r T! 



dm dm dm 

 < < ; 



r 2 r r-j 



dm dm f7 dm 



Since ^ and r 2 are constant 



Now since 1 1 1 dm = M } the whole mass of the body K, the 



above is 



M Jr M 



(6) <V<. 



r, r, 



Accordingly for an external point V is finite. 



If R is the distance of x, y, z from some point in or at a finite 

 distance from K, 



RM <RV< RM 

 r 2 n 



If now we move off x, y, z to an infinite distance we have 



,. R ,. R 

 lim = lim = 1, 



= oo ^ 2 R=C O Tj 



and accordingly since RV lies between two quantities having the 

 same limit 



(7) lim(#F) = Jf. 



5=00 



We say that V vanishes to the first order as R becomes 

 infinite. 



75. Derivatives. Consider the partial derivatives of V by 

 x, y, z. 



The element dm at a, 6, c, produces the potential 



dm 

 = - at x, y, z 



