28] NEWTONIAN FORCE -FUNCTION. 



If we put U r , such a function of r r , that 



75 



dU 



dr 



dr 



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

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



" 



. 



- 



00 



32 ) 



9U 



dU nt SU, 



-t- o - = - 



dz cz 



if we write U s = U Ls +U 2s ---- h U ns . Thus U s satisfies the con- 

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

 summation s does not occur as the first index. 



It is evident that the function U rs serves the same purpose 

 for m r as for m s . For the force JP r W exerted on m r hy m s is equal 

 and opposite to that exerted on m s .by m r . But r rs is the same 

 function of ( x r ) that it is of x s , therefore 



dx 



dx 



and 



We may add to U s terms independent of x s , y s , z s , without 

 affecting the values of X s , Y S) Z t . If we make U a symmetrical 

 function of all the coordinates, containing x s , y s , z s , as U s does, 

 then U will serve as the force -function for all the coordinates. 



In particular, let the force of repulsion vary according to the 

 Newtonian law of gravitation. Then 



33) 



m^m 



