254 Prof. Clausius on the Application of tlie Mechanical 



Accordingly, writing dm in place of 

 dm , dm ,_, 



we obtain the equation 



r(/wH-w(/i-c)^T+Mc^T=0 (13) 



But by (12), 



h— — ^^ ^ 



df 

 so that writing dr in the place of -pr^dl^, r being a function of T 



alone, we have 



rdm + mdr- -^dT-\-McdT=zO, 



or 



d(mr)-^dT-{-McdT=0 (14) 



This equation, divided by T, becomes 



or 



<;(^)+Mc^=0 (15) 



Inasmuch as the specific heat of a liquid changes only very 

 slowly with its temperature, we will in future always consider 

 the magnitude c as constant. In this case the above equation 

 can be immediately integrated, and gives 



mv 



-^ + Mc log T = const. ; 



or if T„ rj, and m^ be the initial values of T, r, and m, 



^ = ^-Mclogl (VII) 



If r may be considered as a known function of the tempera- 

 ture, as through Regnault's experiments it may be in the case of 

 steam, then by means of this equation m is also expressed as a 

 function of the temperature. 



In order to give some idea of the deportment of this function, 

 I have, for one ])articular case, collected together a few calculated 

 values in the following Table. For instance, it is assumed that 

 at the commencement the vessel contains no water in a liquid 

 state, but is exactly filled with vapour at a maximum density, so 

 that mi = M; and that an expansion of the vessel now takes 



