194 R. Clausius on the Application of the 



in expanding remains exactly at a maximum density, without 

 partially precipitating, (2.) that it obeys the laws of Mariotte 

 and Gray Lussac. According to these assumptions we should 

 have 



v T 



t 



3 50° 



125° 



100° 



75° 



50° 



25° 



V 



p' T 

 P 



1 



1-88 



3-90 



923 



25-7 



88-7 



1 



1-98 



4-16 



10-21 



297 



1071 



17. It remains finally to determine also the work done during 

 the change of volume. For this purpose we have generally the 

 equation 



Now according to equation (6) if o be regarded as constant : 



dv — d(mu) 

 whence pdv=pd(mu) 

 for which we may also write 



d 'o 



(17.) p dv — d (mup)-~ mu-j-f^d T. 



We might put in this for m u < ~ 1 the expression given by equa- 

 tion (vili) and then execute the integration. We obtain the re- 

 sult however at once in a rather more convenient form by the 

 following substitution. According to (vi) we have 

 dp 1 mr 



mU dT dT =A--f dT > 

 and from this by employing equation (14) : 



mu-^d T= ~ [d (mr) -\-McdT], 



Hence (17) becomes 



pdv=zd (mup) — [ d (mr) -f Mc d T"] , 



and by integrating this equation we obtain 



(ix.) W— mup^m 1 u 1 p 1 + ^-[?n 1 r x - mr -J- Mc (T \- T)~\ 



whence W may be calculated, since the quantities mr and mu 

 are already known from the foregoing equations. 



