﻿64 Dr. W. J. M. Rankine on the Hypothesis of 



7. As to my having in my paper of October "represented 

 the piston as imparting vis viva to the particles without losing 

 any of its own/' I consider that I was warranted in doing 

 so, because I provided, in the investigation, for the action of 

 a pressure on the outside of the piston sufficient to balance 

 that produced by the impulse of the particles against the in- 

 side, and therefore to impart to the piston exactly the same 

 quantity of energy which it imparts to the particles. 



It is easy, however, if desired, to provide for the possible re- 

 tardation or acceleration of the piston through any inequality of 

 the pressures exerted on its outer and inner faces, as follows. 



Let u be the mean velocity of the piston during the small in- 

 terval of time dt, and — du the retardation during that interval. 

 Let —v be the normal component of the velocity, relatively to 

 the vessel, with which a set of particles are moving towards the 

 piston. The mean normal velocity of these particles, relatively 

 to the piston, is — (v + u) ; and the mean normal velocity, re- 

 latively to the piston, with which they rebound, is +v + u; and 

 relatively to the vessel, v + 2u. 



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

 the piston in the interval dt. Their mean total change of velo- 

 city is 2(v + u) ; therefore the outward pressure exerted by them 

 on the piston, and the inward pressure exerted by the piston on 

 them, are each equal to 



Q=2m{v + u) (1) 



The pressure exerted on the outside of the piston is 



*-«-™ <*) 



M being the mass of the piston ; and as the piston moves in- 

 wards through the distance udt, the work done by that pressure 

 on the piston is 



Tudt=Q / udt—M.udu (3) 



The energy lost by the piston through its retardation is M.udu, 

 which being added to the preceding quantity of work, gives for 

 the work done by the piston on the confined particles, 



Yudt + Mudu = Qudt=2mu(v + u)dt = 2m(uv + u 2 )dt. . (4) 



The increase of energy of the mass mdt of moving particles is 



^{(v + 2u) 9 -v*}=2mdt\uv + u*), . . (5) 



being exactly equal to the sum of the work done by the external 

 force on the piston, and the energy lost by the piston through 



