﻿66 On the Hypothesis of Molecular Motions in Thermodynamics. 



to every particle of the whole set. Then the dynamical equation 

 of the set of particles, relatively to forces parallel to a?, is as 

 follows : ,72 «, 



Q + R-S^-0 (7) 



Let v be the total or resultant velocity of any one particle of the 

 set, in Whatsoever direction it may be moving; then, when the 

 piston is shifted through the distance hoc, the following is the 

 variation of the actual energy, or energy of motion of the par- 

 ticles : 



S/AvS«=(Q + R)Sa?=2j*T-5-8a?. ... (8) 



11. It is obvious, from what has been stated in § 5, that if 

 the piston moves with a uniform velocity, we may substitute 

 VSx for 0,8% ; and the same is the case, indeed, if at the end of 

 the motion denoted by So? the piston returns to the velocity 

 which it had at the beginning of that motion, and also if the 

 piston starts from a state of rest at the beginning and returns to 

 a state of rest at the end of the motion 8tc. If, on the other 

 hand, there is a change 8u in the velocity of the piston during 

 that motion, we must make 



Q l 8x=¥8x-Mu8u, ' (9) 



in which u is to be taken to denote the mean between the velo- 

 cities of the piston at the beginning and at the end of the mo- 

 tion hjJO], and this equation may be used in the investigation of 

 the motion of a bullet, driven by the pressure Q from within, 

 and resisted by the pressure P from without. 



12. The reasoning which has been applied to particles moving 

 within a space of which a piston forms one boundary is evidently 

 applicable to particles moving within any space capable of en- 

 largement and contraction, and bounded by a surface of any 

 kind which is not crossed by the moving particles. The boun^ 

 dary of the space may be a mathematical surface, having at its 

 two sides systems of particles which meet and press or repel 

 each other at the surface, but do not cross it ; and then the con- 

 sideration of the mass of the piston may be omitted from the 

 problem. 



] 3. It can be shown that, for the particles at and near such a 



a oo 

 surface, the factor j-^- in the expression for the energy gained by 



the confined particles through compression varies proportionally 

 to v 2 , other circumstances being alike * and hence may be de- 

 duced the second law of thermodynamics ; but this has so often 

 been explained by different authors and according to different 

 methods, that it is unnecessary to enlarge upon it now. 



