59] WORK AND ENERGY. 113 



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

 m r as for m s . For the force F exerted on m r by m g is equal and 

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

 of ( os r ) that it is of x 8) therefore 





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

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

 function of all the coordinates, containing sc g) y g , z g as U 8 does, then 

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



In particular, let the force of repulsion vary as the product of 

 the masses of the particles divided by the square of their distance 



apart (r rs ) =- r ~> Such forces are called Newtonian forces, the 



fra 



most familiar examples of which are the mutual attractions of the 

 sun and the planets. Then 



, , m r m s n m r m s 



(30) </> (r n ) = j- , U n = - , 



(31) P.=- 



^ ' IS '28 



and the symmetrical function U will be 



u=-^\ r ^ 2 + ~ ri3 +., 



/ x a! 32 t 3n 



...... 



or more briefly 



r =l s =i T rs 



understanding that terms in which r = s are to be omitted. 

 w. E. 8 



