Theory of Heat to the Steam-engine. 247 



when the gas or vapour does not, during its expansion, encounter 

 all the resistance it is capable of overcoming ; when, for instance, 

 it issues from a vessel in which the pressure is greater than in 

 the one into which it enters. In this case a compression, under 

 circumstances similar to those accompanying expansion, is im- 

 possible. 



By equation (II) we can determine the sum of all the uncom- 

 pensated transformations in a circular process. As, however, a 

 circular process may consist of several changes of condition in 

 the given matter, of which some have occurred in a reversible, 

 and others in an irreversible manner, it is often interesting to 

 know how much of the whole sum of uncompensated transforma- 

 tions has resulted from each of the latter. For this purpose let 

 us conceive the matter, after the modification which has to be 

 examined in this manner, reduced to its original condition by 

 any reversible operation. We shall thereby obtain a small circular 

 process, to which the equation (II) will be just as applicable as 

 to the whole. Consequently, if we know the quantities of heat 

 which the matter has received during the process, and the tem- 



peratures which correspond thereto, the negative integral ~ i ttt 



will give the uncompensated transformation involved therein. 

 But as the uncompensated transformation involved in the given 

 change of condition could not have been increased by the above 

 reduction, which was executed in a reversible manner, it will be 

 fully represented by the above expression. 



Having thus investigated all the parts of the whole circular 

 process which are not reversible, and found the values Nj, N^, 

 &c., which must all be positive, their sum will give the magni- 

 tude N corresponding to the whole circular process, without its 

 being necessary to take into consideration those parts which are 

 known to be reversible. 



9. If we now apply the equations (I) and (II) to the circular 

 process which occurs during a period in a thermo-dynamic 

 machine, it will be at once evident, that, the whole quantity of 

 heat communicated during this period to the matter in the 

 machine being given, the corresponding amount of work can be 

 immediately determined from the first equation without its being 

 necessary to know the nature of the operations constituting the 

 circular process. 



In an equally general manner the work may be determined 

 from other data by a combination of both equations. 



We will assume that the quantities of heat successively im- 

 parted to the changing material, as well as the temperatures at 

 the times of reception, are given, and that only one temperature, 

 Tq, remains at which a certain as yet unknown quantity of heat 



