nNn$sts>iv^^o^y of Heat to the Steam-engine. 263 



the original temperature Tq. The matter here' passes through 

 the same conditions as before, but in an opposite order. All 

 communications and abstractions of heat take place in opposite 

 order, but in the same quantity and at the same temperature of 

 the mass; all quantities of work have opposite signs, but the 

 same numerical value. 



Hence it follows, that in this case no uncompensated trans- 

 formation is involved in the circular process, and we must con- 

 sequently set N = in equation (2), by which we obtain the fol- 

 lowing equation, — already given in (3), with the exception that 

 W is here put in the place of W,- 



-=k«.-4:t)- 



In our present case, Q, denotes the quantity of heat imparted to 

 the mass M in the boiler, that is, 



Qi=mirj+Mc(Ti-To). 



In determining the integral j -7=p, the two quantities of heat 



Mc(Ti — Tq) and m^r^ contained in Qi must be separately con- 

 sidered. In order to execute the integration extending over the 

 first quantity, we have but to give to the element of heat dQ, the 

 form Mc^T, and this part of the integral is at once expressed by 



McJ^ _=Mclog^. 



During the communication of the latter quantity of heat, the 

 temperature is constant and equal to Tj, and consequently the 

 part of the integral referring to this quantity is simply 



By substituting these values, the foregoing expression for W 

 becomes 



W'= i-[m,., + Mc(T, -To) -To(^+ Mc log|l)] 



and this is the same expression as that contained in equation (XI), 

 which was before obtained by the successive determination of the 

 several quantities of work done during the circular process. 



24. From this it follows, that if the temperatures at which the 

 matter manifesting the action of heat receives heat from the source 

 of heat, or imparts heat to some external object, are considered as 

 previously given, then the steam-engine, under the conditions 

 made in deducing the equation (XI), is 2, perfect machine; that 



