and the Laws regarding the Nature of Heat, 15 



first case the quantity 



rdm 



of latent heat which has been extracted from the body A ; and 

 in the second case, the quantity 



hP') 



d'm 



of sensible heat which has been imparted to the body B. By 

 the other expansion and contraction heat is neither gained nor 

 lost ; hence at the end of the process we have 



The heat eocpended = rdm ~ ( ^ ~" jT ^M d^i^. . (6.) 



In this equation the differential d^m must be expressed through 

 dm and dt ; the conditions under which the second expansion 

 and the second contraction have been carried out enables us to 

 do this. Let the mass of vapour precipitated by the compression 

 from oh to oSj and which therefore would deyelope itself by expan- 

 sion from oe to oh, be represented by hm, and the mass developed 

 by the expansion from of to og by am ; then, as at the conclu- 

 sion of the experiment the original mass of fluid and of vapour 

 must be present, we obtain in the first place the equation 



dm + 8'm = d^m + hm. 



Further, for the expansion from oe to oh, as the temperature 

 of the fluid mass /a and the mass of vapour m must thereby be 

 lessened the quantity dt without heat escaping, we obtain the 

 equation 



rBm—fi . cdt—m . hdt = ; 



and in like manner for the expansion from of to og, as here we 

 have only to set fi—dm and m + ^m in the place of jm andm, and 

 h^m in the place of hm, we obtain 



rh^m — (ft — dm) cdt ■— (m + dm) hdt = 0. 



If from these three equations and equation (6.) the quantities 

 d^m, hm and S'm, be eliminated, and all diff'erentials of a higher 

 order than the second be neglected, we have 



The heat expended — ijr -{-c—h\dmdt. . . (7.) 



The formulae (7.) and (5.) must now be united, as in the case 

 of permanent gases, thus : 



e--) 



—a)^dmdt 



dmdt 

 =A 



