292 Dr. W, J. M. Rankine on Thermodynamics, 



but as regards the fundamental principles themselves there has 

 been no alteration. 



It is, of course, to be understood that I do not, in making the 

 preceding statement, contradict any allegation of Mr. Heath's as 

 to matters of fact ; for doubtless his information as to the history 

 and state of thermodynamics must have been derived from wri- 

 tings with which I am unacquainted. 



I have only to add the following remark as to Mr. Heath's ex- 

 planation of how he conceives that the capacity of a vessel con- 

 taining moving particles not exerting sensible attractions or re- 

 pulsions might be diminished without accelerating the motion of 

 the particles. He supposes the piston to be moved inwards du- 

 ring the intervals between the impulses of the moving particles 

 upon it, and to be at rest at the instant of each of those impulses. 

 According to this supposition there would, of course, be no addi- 

 tion to the energy of the particles; but at the same time there 

 would be no work done in moving the piston, for it would meet 

 with no resistance to its inward motion, and all the energy ex- 

 pended by the external force in setting the mass of the piston in 

 motion after each impulse of a particle would be obtained back 

 again in the act of stopping it before the impulse of the next 

 particle. 



As to the proposition that when work is done in moving the 

 piston inwards against the reactions due to the motions of 

 the particles alone, the additional energy due to the acceleration 

 of the particles is exactly equivalent to that work, it might be 

 treated simply as a particular case of the general principle of the 

 conservation of energy. For elementary purposes, however, it 

 may be desirable to use a special demonstration, such as the fol- 

 lowing. Let the piston be moving inwards with the uniform 

 velocity -\-u. At a given instant let — v be the normal compo- 

 nent of the velocity, relatively to the vessel, with which a set of 

 particles are moving towards the piston. The normal velocity 

 of those particles relatively to the piston is — (v + w). They 

 strike the piston and rebound with the velocity -\-v-\-u relatively 

 to it, so that after the collision their velocity relatively to the 

 vessel has become v + 2u instead of v, which it was before. 



Let m be the aggregate mass of the particles which act on the 

 piston in one second. Their total change of velocity is %{v + u)\ 

 therefore the outward pressure exerted by them on the piston, 

 and the inward pressure exerted by the external force which 

 pushes the piston inwards, are each equal to 2m(v-t-u). The 

 piston moves through the distance u in a second; therefore the 

 work done in driving the piston inwards in each second is 



2mu(v + u)=2m(uv + u 2 ) (1) 



