Prof. Thomson on the Dynamical Theory of Heat. 173 

 as before, denoted respectively by v and t, we shall have 



\{\-x) + -jx=v (27), 



if X and 7 be the volumes of unity of weight of the substance in 

 the liquid and the gaseous states respectively : and/j, the pres- 

 sure, may be considered as a function of t, depending solely on 

 the nature of the substance. To express M and N for this mixed 

 medium, let L denote the latent heat of a unit of weight of the 

 vapour, c the specific heat of the liquid, and h the specific heat 

 of the vapour when kept in a state of saturation. We shall have 



av 

 'Ndt=c{l-x)dt-]-hxdt + Lpdt. 

 Now, by (27), we have 



(V-^)§ = 1 (28). 



"^^ M=^-i^ (30), 



N=c(l-a-) + Aa?-L '^ — (31). 



55. The expression of the second fundamental proposition in 

 this case becomes, consequently, 



f^= — j; ^^^^' 



which agrees with Carnot's original result, and is the formula 

 that has been used (referred to above in § 31) for determining 

 /A by means of llegnault's observations on steam. 



5G. To express the conclusion derivable from the first funda- 

 mental proposition, we have, by differentiating the preceding 

 expressions for M and N with reference to t and v respectively, 

 JM_ _1_ ^ dL _ L djy-X ) 

 dt ~ y—\' dt (7— A)-' dt 

 dy dX\ 

 dt dt \dx 

 7 — X / dv 



L \ d{y-X) 



{y-XfJ dt • 



