Prof. Thomson on the Dynamical Theory of Heat. 431 



the vokime, v, of the substance between the pistons, kept at a 

 constant temperature, t, as has been used uniformly in this and 

 the preceding paper; we shall have, for the quantity of heat 

 absorbed during the motion of the piston, 



*j u 



or, by the second fundamental equation of the theory, (3) of § 21 

 of the preceding paper, 



-Jt'^'' 



where ■or denotes the actual pressure (intermediate between p 

 and ju') of the substance when its volume is v. Again, the work 

 done by the pistons will be given by the equation 



1 /*" 



y^ 



^=/ 'HTdv+pu—p'a' . . . (e). 



•-^ 7/. 



If now the transference of the substance from the one portion 

 of the tube, where the pressure is p, to the other, where the 

 pressure is p', take place through a small orifice, exactly that 

 amount, W, of work will be lost as external mechanical effect, 

 and will go to generate thermal vis viva: The quantity of heat 

 thus produced will be 



j-< I mdv-^jiu—p'u' y. 



Hence the total quantity of heat emitted will be the excess of 

 this above the amount previously found to be absorbed when the 

 mechanical effect is all external ; and therefore we have* 



Whatever changes of temperature there may actually be of the 

 air in or near the orifice, this expression will give rigorously the 

 total quantity of heat emitted by that portion of tube wTiich 

 contains the orifice and the whole of the second spiral during 

 the passage of a volume u through the first spiral, or u' through 

 any portion of t1ie second spiral where the temperature is sensibly/. 

 75. To apply this result to the case of a gas fulfilling the 

 gaseous laws, we may put 



j)U=-p'u!. 



* A more comprehensive investigation, including a proof of this result, 

 is given in a subsequent communication (Royal Soc. Edinb. Dec. 15, 1851), 

 constituting jiart 5 of tlic present series of articles, which will be re- 

 published in an early Number of this Journal. 



