Mechanical Theory of Heat to the Steam Engine. 191 



€ and h, only functions of T, and the quantity m only a function 

 of T and v we obtain 



d tdQ\ 1 dr r du 

 dT\Tv)~~u'df^u^'df 



dv 



or, if we put for its value 

 dv u 



d /dQ\ / r du\dm 



MdT)-[ u'df)~d^' 



(m d ( d Q\_ h ~ c r £ 



* } dv\dTJ~~ u u 2 ' d 



du 

 ~T' 



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

 (ill) and (iv) we obtain the sought equations, which represent 

 the two principal theorems of the mechanical theory of heat for 

 vapors at a maximum density, namely 



/ \ d v dp 



(v.) ~+ e -h=A.u. m . 



(vi.) r=A.Tu d -? T . 

 and from the combination of the two, we also obtain 



(12) %+^-^' T - 



14. "With the help of these equations we will now consider a 

 case which will so often occur in what follows, that it is advan- 

 tageous to fix, a priori, the results which refer to it. 



Let it namely be assumed that the previously considered vessel 

 with its contents of partly fluid and partly vaporized mass, 

 changes its volume, without any heat being added to or taken 

 from the mass. Then together with the volume, the temperature 

 and the quantity of that portion of the mass which is present in 

 the form of vapor will change, and besides, a positive or negative 

 external work will be done by the heat which produces the pres- 

 sure of the vapor, since in the change of volume the pressure of 

 the enclosed vapor which is exerted in the expansion overcomes 

 an external force, and in the compression is overcome by an ex- 

 ternal force. 



Under these circumstances, the quantity of the portion m, in 

 the form of vapor, the volume v and the work "FT are to be de- 

 termined as functions of the temperature T. 



15. If the volume and the temperature are changed by the 



arbitrary infinitely small quantities dv and d T, the quantity of 



heat, which for this purpose must be communicated to the mass, 



will be expressed according to the foregoing by the sum 



dm , r . _ _ . , dm -, , J, 

 r— dv4- [ (M-m)c+mh±r \d T. 

 dv u d T J 



