592 DELATION OF THE LAW Of GRAVlTATlOS" TO 1SE 



5i — Reiation beiwebn x Aira N (r, x). 



Attraction according to the inverse square of tlie distance being' 

 accepted as a fact when the particles are at ordinary sen&ible dis- 

 tances, we have 



d'^x c 



where c is constant for the same particles. Therefore 



Vdt) ^ V r/' 



and, iV" (r, x) = iV" (r, r) — {- — '- | ; 



^ ' ' \ ' / m -{- n \x r) 



the particles being supposed to be in their descending course. Or^ 

 puttmsf k lor V 



6. — The Critical Value of x. 



The quantity of energy represented by N [r, r) will afterwards be' 

 found to be one half of the entire energy of the System ; but at pre- 

 sent I merely say that it is a positive quantity distinct from zero, 

 i'or Suppose, if possible, that it is zero. This ineaiis that, when P 

 and Q are in the positions A and B, there is no energy in the nega- 

 tive jar ', the entire energy of the system is collected in the positive 

 jar. But, when P and Q have descended to and D, the positive 

 energy is greater than it was when they were at A and B ; and 

 thei^efore there is now latent in the positive jar more energy than the 

 entire energy of the system. This, however, is opposed to the prin- 

 ciple of conservation, which, as was pointed out in section 1, implies' 

 that, in a finite system such as we are now considering, neither jar 

 can ever contain more than the fixed maximum q. Hence, N (r, r) 

 is not zero- Nor is it negative ; for then the energy, P (r, r), latent 

 in the positive jar, would exceed q. 



Since N (r, r) is a positive quantity distinct from zero, it follows 



h h 



that, when x is made equal to r, N (r, r) + - is greater than - ; while, 



on the other hand, as x is taken indefinitely small, - becomes greater 



than iV (r, r) -\- -. Consequently, between r and zero, there must be 



