594. RELATION OF THE LAW OF GRAVITATION TO THE 



We may call A', B\ the inferior, and A, B, the superior, positions of' 

 rest. Since, between the positions F, G, and the positions A', B'\ 

 the force is repulsive, we have 



d'^x c 



I assume that c is the same as in the sphere of attraction. Unless 

 there were some reason to suppose it not the sam,e, the law of Parci- 

 mony would lead us to take for granted that no change occurs ; and 

 not only does there seem to be no reason to assume a change in this 

 respect, but it is difficult to imagine that a constant, which indicates 

 the amount of action between the particles at a given distance, alters 

 its value per salfuni. At the crisis, when aj = a, there may be, 

 indeed (as I have shown) there must be, reasons determining this 

 mutual action to take the character of repulsion instead of that of 

 attraction ; but that the quantity of the reciprocal action should 

 suddenly leap from one value to another appears to be at variance 

 with the law of Continuity. Denoting by F (b, x) the effective 

 positive energy at the position in the sphere of repulsion where 

 the distance of the particles, still in their descending course, is x, we 

 obtain, by the same reasoning as in section (5), 



F{b,x)-^l=F{b,b)+l. 



With respect to the sign of the terms in this equation containing k, 

 I may observe that, the force being repulsive, this consideration, if 

 taken alone, would have given k a different sign from what it has in 

 the equation deduced in section (5) ; but,, to counterbalance this, 

 F (6, X) is greater than F (b, b), whereas iV" (r, x) was less than 

 iV" (r, r). By putting x =^ a, 



F{b,a)^l=P{b,b)+\. 



But F (b, a), at the limit of the sphere of repulsion, coincides with 

 F (r, a) at the limit of the sphere of attraction ; and since, at that 

 limit, the whole energy of the system is found in the positive jar, 

 F (r, a) = q. Therefore F (b, a) = q. Also, since the Kinetic 

 Energy of the system has, on the whole, been neither increased nor 

 diminished in the passage of F and Q from the superior to the infe- 

 rior positions of rest, no increase or diminution has taken place in 

 the quantity of energy in either jar. Therefore 



^ (r, r) = JV {b, b), and, F (r, r) = F (b, b). 



.-. g +^= P (r, r) +^= ^ - iV-(r, r) +f. 



