172 Prof. Thomson on the Di/namical Theonj of Heat. 



the hypothesis that its density is strictly subject to the " gaseous 

 laws/^ we should have, by Boyle and Mariotte's law of com- 

 pression, 



^--P. (20)- 



and by Dalton and Gay-Lussac's law of expansion, 



dv Ei; 



It ~ l+Eif 



from which we deduce 



dp _ Ep 



dt ~ l+Ei' 

 Equation (14) will consequently become 



(23); 



J-^ P\ 



d^ l/Lt(l+E/) JJ 



(2^), 



dv dt 



a result peculiar to the dynamical theory and equation (16), 



^-''=£^m ■ ■ ■ ■ <'''■ 



which agrees with the result of § 53 of my former paper. 



If V be taken to denote the volume of the gas at the tempe- 

 rature 0° under unity of pressure, (25) becomes 



E^V 



K-N=^:(f^ (««)• 



52. All the conclusions obtained by Clausius, with reference 

 to air or gases, are obtained immediately from these equations 

 by taking 



^— 'T+eF' 



which will make -y- =0, and by assuming, as he does, that N, 



thus found to be independent of the density of the gas, is also 

 independent of its temperature. 



53. As a last application of the two fundamental equations of 

 the theory, let the medium with reference to which ]M and N 

 are defined consist of a weight \—x of a certain substance in 

 one state, and a weight x in another state at the same tempera- 

 tiu-e, containing more latent heat. To avoid circumlocution and 

 to fix the ideas, in what follows we may suppose the former state 

 to be liquid and the latter gaseous ; but the investigation, as 

 will be seen, is equally applicable to the case of a solid in con- 

 tact with the same substance in the liquid or gaseous form. 



54. The volume and temperature of the whole medium being, 



