PUINCIPLE OF THE CONSERVATION OF ENERGY. 593 



a Value of a, say a, such that 



- = i\^ (r, r) 4- -) or, k (- — - ) = iV^ (r, r), 

 a \ / ' ^ \a r/ 



and therefore, N (r, a) = 0. 

 1 call a the critical value of x. The negative jar is then empty, and 

 the positive jar is charged with the entire energy of the system. 



Let us for a moment consider what has been happening since the 

 last crisis, when the entire energy of the system was collected in the 

 negative jar. That energy has been expending itself in diminishing 

 the distance between P and Q, the amount expended being trans- 

 ferred to the positive jar, where it has lain in a state of latency, til) 

 iiow the order of things is reversed; the negative jar is empty; the 

 positive energy becomes free', and begins to operate ; and the portion 

 of it which is expended in doing the appropriate work of positive 

 energy passes over into the negative jar, where it lies latent till the 

 iiext crisis. 



7. — At the Crisis, WBen TSe Negative Jae has become Empty, the Law of Gran'itation 



TJNDERGOES A TrANSFOJEIMATION FROM ATTRACTION TO REPOLSION. 



When X = a, though the negative jar is empty, the particles P 

 and Q have acquired velocities, in vii'tue of which they sweep onwards 

 towards one another across the critical positions. Now, at the ciisis 

 the law of the reciprocal action of the particles changes from a law 

 of attraction to one of repulsion. For suppose, if possible, that it 

 continues as a law of attraction. Then the equation, 



iV (r, x) 4- - = i\^ Cr, r) 4- -, 



Still holds. But X is now less than a : therefore - is greater than - 



X a 



or N' (r, r) -|- - ; hence iV (r, x) is negative : which implies that 



P (r, x), the latent energy in the positive jar, exceeds the entire 

 energy of the system. This, on the principle of Conservation, is 

 impossible. Therefore, the force of gravity cannot continue to act as 

 a,n attractive force subsequently to the crisis. The enei'gy in the 

 positive jar becomes effective, and repulsion is the result. 



8. — The DisTai^ce between P and Q aT the Crisis is the Harmonical Mean between- 

 THEIR Distances at the Superior and Inferior Positions of Rest. 



At the crisis, let the positions of P and Q he F and G. Then 

 FG- = a. When the distance becomes less than a, the particles, 

 having entered the sphere of repulsion, are gradually retarded, and 

 at length brought to rest at A' and £', where their distance is b. 



