596 RELATION OF THE LAW OF GRAVITATION, ETC. 



to show that this is merely a summation of the results established in 

 the previous sections of the paper ; but, instead of doing this, we 

 shall point out how, assuming the wider generalization embodied in 

 the equation given — a generalization which, apart from its greater 

 width, has the advantage, over the law of gravitation, of exhibiting 

 the motion of P and Q in its relation to the quantity of effective 

 energy — the law of gravitation can be deduced as valid withm certain 

 limits, and as undergoing transformation, beyond these limits, into a 

 law of repulsion. 



First, let the particles be considered when they are moving within 

 the sphere of attraction. Then the effective energy is the negative. 

 Hence, E^ is the value of N {r, x) after the positions of rest have 

 been passed. But, between the time when the particles were in the 

 positions of rest and the instant under consideration, an expenditure 

 of negative energy, equal in amount to -| (mw^ -|- nu^), has taken 



place. Hence, 



E^ = N (r, r) — |- {mv^ -f- nu'^). 



Substitute for E^ its value in the assumed equation, taking the upper 

 sign of k, and for | [mv^ -f- nu^) its value, as foiuid in section (6), 



^('•.'•)-i(,-^)('sy=*(s-^> 



™, „ mn d'^x k 



Therefore, 



Or, putting c for 



Ic (ni -\- n) 



mn 



d'^x c 



df'' a;^ ' 



Next, let the particles be considered when they are moving within 

 the sphei'e of repulsion, iato which they must of necessity enter. 

 Then the effective energy is the positive. Hence E^ is the value of 

 F {h, x), after the inferior positions of rest, whose distance from one 

 another is h, have been passed. That is, 



^, = P (6, h) - 1 {mv-^ + nu% 

 Therefore, taking now the lower sign of k in the assumed value of E^,^ 



p /, 7\ I mn /dx\^ 7/1 1\ 



^ ' ^ 2 ^j^Tip^ I, (It/ ^^ \a x)' 



ihereiore --— = — - . 



ar x^ 



I||a subsequent paper, I shall point out the effect of the introduc- 

 tion of foreign energy into the system. 



