Prof. Thomson on the Dynamical Theory of Heat. 529 



N = N 0+ ^ log^ ^ dt . . . . { m, 



^{— -(1 + E/)V 

 K = N 0+jftPo log-— ^ + /;(I ^ ) (14). 



o 

 The first of these equations shows, that unless Mayer's hypo- 

 thesis be true, there is a difference in the thermal capacities in 

 constant volume, of the same gas at the same temperatures for 

 different densities, proportional in amount to the difference of 

 the logarithms of the densities. The second, compared with the 

 first, leads to an expression for the difference between the thermal 

 capacities of a gas in constant volume, and under constant pres- 

 sure, agreeing with results arrived at formerly. [Account of 

 Carnot's Theory, Appendix III. ; and Dynamical Theory of 

 Heat, § 48.] 



93. It may be that more or less information, regarding expli- 

 citly the pressure and thermal capacities of the fluid, may have 

 been had as the data for determining the mechanical energy ; 

 but these converse deductions are still interesting, as showing 

 how much information regarding its physical properties is com- 

 prehended in a knowledge of the mechanical energy of a fluid 

 mass, and how useful a table of the values of this function for 

 different temperatures and volumes, or an empirical function of 

 two variables expressing it, would be, whatever be the experi- 

 mental data from which it is deduced. It is not improbable 

 that such a table or empirical function, and a similar representa- 

 tion of the pressure, may be found to be the most convenient 

 expression for results of complete observations on the compres- 

 sibility, the law of expansion by heat, and the thermal capacities 

 of a vapour or gas. 



94. The principles brought forward in a former communica- 

 tion " On a Means of discovering experimentally, &c." (which is 

 now referred to as Part IV. of a series of papers on the Dyna- 

 mical Theory of Heat), may be expressed in a more convenient 

 and in a somewhat more comprehensive manner than in the 

 formula? contained in that paper, by introducing the notations 

 and principles which form the subject of the present communi- 

 cation. Thus, let t be the temperature, and u the volume of a 

 pound of air flowing gently in a pipe (under very high pressure 

 it may be) towards a very narrow passage (a nearly closed stop- 

 cock, for instance), and let p be its pressure. Let l', v! , and y/ 

 be the corresponding qualities of the air flowing gently through 

 a continuation of the pipe, after having passed the " rapids" in 



Phil. May. S. 4. No. G2. Suppl. Vol. 9. 2 M 



