\j)OimN- Theory of Heat to the Steam-engine. 253 



dm . , 1 

 -7 - its value -, 

 dv u 



or, substituting tor -7- its value -, ,1? ..; o'j TV T 



By substituting the expressions given in (10), (11), and (8) 

 in (III) and (IV), we obtain the required equations, which repre- 

 sent the two principal theorems of the mechanical theory of heat 

 as applied to vapours at their maximum density. These are 



J^+o-A = A.4, (V) 



T=k.iu^r: . , ;/. . . {VI) 



and from a combination of both we have 



14. By means of these equations we will now treat a case, 

 which in the following will so frequently occur, as to render it 

 desirable at once to establish the results which have reference 

 thereto. 



Let us suppose that the vessel before considered, containing 

 the liquid and vaporous parts of the mass, changes its volume 

 without heat being imparted to, or withdrawn from, the mass. Then, 

 simultaneously with the volume, the temperature and magnitude 

 of the vaporous part of the mass will change ; and besides this — 

 seeing that during the change of volume the pressure of the 

 enclosed vapour is active, which pressure during expansion over- 

 comes, and during contraction is overcome by an external force — 

 a positive' or negative amount of external work will be done by 

 the heat which produces the pressure. 



Under these circumstances, the magnitude of the vaporous part m, 

 the volume v, and the work W shall be determined as functions of 

 the temperature T. 



15. It has already been shown, that, in order that the volume 

 and temperature may suffer any infinitely small increments dv 

 and dT, a quantity of heat expressed by the sum . 



r 



dm 



^£dv + [{M-^m)c-\-mh-\-r^'\dll 



must be imparted to the mass. In consequence of the present 

 condition, according to which heat is neither imparted to, nor 

 abstracted from the mass, this sum must be set equal to zero. 



