Mechanical Theory of Heat to the Steam Engine. 189 



depends only on the temperature T. It only remains to decide 

 whether the quantity of the two parts which are present in dif- 

 ferent conditions is determined. For this purpose the condition 

 is given, that these two parts must together exactly fill up the 

 content of the vessel. If we therefore denote the volume of the 

 unit of weight of steam, at its maximum density, at the tempera- 

 ture T by s, and that of a unit of weight of fluid by or, we must 

 have : 



v = m . s + (M—m)v 

 = m(s— o) + M<*. 

 The quantity s occurs in what follows, only in the combination 

 (s - cr), and we will therefore introduce a special letter for this 

 difference, putting 



(5) u=s—or, 



by which the previous equation becomes 



(6) v—rnu+Mv, 



and hence 



v -Ma 



(7) ra= . 



By this equation, m is determined as a function of T and v, since 

 u and v are functions of T. 



12. In order now to be able to apply equations (in) and (iv) to 



our case, we must first determine the quantities and 



Let us first assume that the vessel expands so much that its 

 content increases by dv, then a quantity of heat must be thereby 

 communicated to the mass, which will in general, be represented 



by %- dv. 



Now since this quantity of heat is only consumed in the forma- 

 tion of vapor which takes place during the expansion, it may 

 also be represented, if the heat of evaporation be denoted for 

 the unit of mass by r, by the expression 



dm _ 

 r -z-av. 

 dv 



and we may also put 



dQ dm 

 dv dv 

 whence, since according to (7), 



dm 1 . 



dv u 1 



we find (8) 4^=-. 



v J dv u 



If we assume in the second place, that the temperature of the 

 mass, while the content of the vessel remains constant, is in- 



