PSINCIPLE OP THE CONSERVATIOlf OF ENERGY. 595 



Therefore ^ ( t - — j = ^ (r, r). 



But, by section (6), k (- — -\ = iV (r, r). 



.-.1 + 1 = '/ 

 bra 



•9. — When P and Q are in Positions op Rest, whether Superior or Inferior, the 

 Quantities of the Negative and Positive Energies in the System are Equal. 



In the inferior positions of rest, the quantity of energy in the 

 negative jar is JSf (b, b) or iV (r, r). Hence, F (r, r) is the quantity 

 of positive energy that has been expended while P and Q were 

 passmg from F and G to A' and £'. From A' and B' the particles 

 are driven apart till the limit of the sphere of repulsion is again 

 reached ; and, in effecting this, an additional quantity, JV {r, r), of 

 positive energy is expended ; so that the whole positive^ energy ex- 

 pended while the particles continue within the sphere of repulsion is 

 2 j iV (r, r) I _ But, as the reign of attraction ended, and that of 

 repulsion began, with the circumstance of the negative jar being empty, 

 so, if the two sorts of energy have a parallel relation to one another, we 

 must suppose that the reign of repulsion ends, and that of attraction 

 re-commences, with the cii'cumstance of the positive jar being empty.* 

 If this be so, then 2 \ JV (r, r) | represents the entire energy of the 

 system ; and therefore iV (r, r)= P (r, r), and N (b, b) ^= P {b, b). 



10. — Higher Law under which the Law of Gravitation is Contained, and of which it is 

 AN Expression within Certain Limits. 



The conclusions at which we have arrived imply that the law of 

 gravitation does not prevail universally, biit that it is only the form 

 which a higher law takes within certain limits. That this higher 

 law may receive convenient algebraical expression, let E^ denote the 

 effective energy when the particles, having passed the positions of rest 

 in the sphere, whether of attraction or of repulsion, in which they 

 are moving, are at the distance x from one another. The equation, 

 which represents the motions of P and Q in the spheres of attraction 

 and repulsion alike, is 



the positive sign of k being taken within the sphere of attraction, 

 and the negative within the sphere of repulsion. It would be easy 



* This is tantamonnt to sayiug that perfect elasticity prevails between the partici^Bpand 

 this, on the supposition on which the paper proceeds, namely, that none of the energ^JS the 

 system is dissipated, while it is not moreased ah extra, is involved in the principle of Con- 

 servation. • 



