112 THEORY OF NEWTONIAN FORCES. [PT. I. CH. II. 



Now since 



rf . = (cc s - # r ) 2 + ( y s - 2/ r ) 2 + (z 9 - z r )\ 

 differentiating partially by or 8 , 



i-,K\ Sr n _x,-x r dr rs _y s -y r Sr n _z e -z r 



fo~ r n ' dy~ s r n Sz s ~ r n 

 and accordingly 



fa,' F. M dy,' F.v-di,,' 





If we put U rs such a function of r r ^ that 

 dU 



dr r8 

 dU rs dr 



rsr8 



dr 



dr r8 <jz 8 dz 8 



If now we find the resultant F s of all the forces acting on m s 

 due to the repulsions by all the particles m r , we shall have 



r H 



m 



~r ~^ -- r H 



s "<5 ~r ~^ -- r x -z , 



oz s oz 8 dz s oz 8 



if we write U s = U 18 + U 28 ...+ U ns . Thus U 8 satisfies the con- 

 ditions for a force-function as far as concerns the point m 8 . In the 

 summation 5 does not occur as the first index. 



