190 



R. Clausius on the Application of the 



creased by d T, the quantity of heat necessary, will be repre- 

 sented generally by 



TT dT ' 



This quantity of heat consists of three portions — 1. The fluid 

 portion, M—m of the whole mass, must be warmed by d T 1 for 

 which purpose, if c denotes the specific heat of the liquid, the 

 quantity of heat (M—m)cdT is necessary. 



2. The portion m in the state of vapor must in like manner 

 be heated by d T, but will thereby at the same time be so much 

 compressed, that for the increased temperature T+d T, it is 

 again at a maximum density. The quantity of heat which 

 must be communicated to a unit of mass of vapor during its 

 compression, in order that it shall have at every density pre- 

 cisely the temperature for which this density is a maximum, we 

 shall denote for an increase of temperature of d T, in general by 

 hd Tin which h is a magnitude which is previously unknown as 

 to its value, and even as to its sign. The quantity of heat neces- 

 sary for our case, will hence be represented by mhdT. 



3. In the process of heating, a small quantity of the previously 

 fluid portion, passes into the state of vapor, which is represented 



generally by d T, and which consumes the quantity of heat 



In this, according to equation (7) 



dm v - Mv du M dcr 

 d~f~ u*~'TT~ u'dT 

 m du M d a 

 ~~ u~'df~~ u'dT' 

 by which the previous expression becomes 



Jm du M du\ 

 [u 'dT^u'dT) 1 ' 



If we add these three quantities of heat together, and put their 

 sum equal to ^-Qd T we have 



, . dO / r da\ , / r du\ 



13. The first of these expressions for ^-^ and must now 

 r dv dT 



also, as is signified in equation (in), be differentiated, the first 

 with respect to T, and the last with respect to v. If we consider 

 moreover that the quantity M is constant, the quantities u, <?, r, 



