Mechanical Theory of Heat to the Steam Engine. 201 



signifies herein for our case, the heat communicated in the 

 boiler to the mass M, and we have therefore 



In determining the integral f the two single quantities of 



0 



heat contained in Q u Mc (T A — T 0 ) andm^, must be particu- 

 larly considered. In order to execute the integration for the 

 first, we may write the element of heat dQ in the form McdT } 

 then this portion of the integral becomes 



Tl d T T 

 McJ* -^ — Mc log-i. 



To ° 



During the communication of the last quantity of heat, the 

 temperature is constantly equal to T x) and the portion of the in- 



171 V 



tegral relating to this quantity of heat is therefore simply -^ T — \ 



By substituting these values, the above expression for W be- 

 comes the following. 



W = I [m t r t +Me(T t -T 0 )-T 0 + Mc log 



= A [ m « r > ^TT° +Mc(T l -T 0 +T a log -^)] 



and this is the same expression as that contained in equation 

 (xi), which we have previously found by the successive deter- 

 mination of the single quantities of work done during the circu- 

 lar process. 



24. Hence it follows that if the temperatures at which the 

 substance conveying the action of the heat takes up the heat delivered 

 by the source or gives out heat outwardly, are considered as previously 

 given, then the steam engine, under the suppositions made in 

 deducing equation (xi) is a perfect machine, inasmuch as for a 

 definite quantity of heat communicated to it, it does as much 

 work as, according to the mechanical theory of heat, is possible 

 at the same temperatures. 



The matter is otherwise however if we do not regard these 

 temperatures as given a priori, but consider them as a variable 

 element which must be taken into consideration in judging the 

 machine. In consequence of the fact that the liquid, during its 



SECOND SERIES, VOL. XXII, NO. 65. — SEPT., 1856. 



26 



