M. CLAPEYRON ON THE AXOTIVE POWER OF HEAT. 365 



body by a quantity b c, in such a manner that all the heat developed 

 by the diminution of volume may be absorbed by the body B, and the 

 temperature remain equal to its primitive value T. The volume V also 

 again becoming the same as it was at the commencement of the opera- 

 tion, it is certain that the pressure will return to its primitive value b d, 

 as will also the absolute quantity of heat Q. 



If we now connect the four points/, g, e, d by right linfes we shall form 

 a quadrilateral figure, the area of which will measure the quantity of 

 action developed during the operation described. Now it is easy to see 

 that fg and dc are two elements infinitely near, described upon two 

 curves infinitely near, the equations of which will be T + c? T = const., 

 and T = const. They ought therefore to be considered as parallel ; the 

 two ordinates which terminate the quadrilateral figure in the other di- 

 rection being also parallel, the figure is parallelogrammical, and mea- 

 sures bey. df. 



Now fd is nothing but the increase experienced by the pressure^?, 

 the volume v remaining constant, and T becoming T + d T. We have 

 therefore 



whence 



rf/=^rfT, 



fd^±^d1. 



dp 



And h c being the increase of volume d v 



/.,.., dv. dT 



fdy.bc = -^. 



dp 



It only remains to determine the heat consumed in the production of 

 this quantitj*^ of mechanical action. 



We have first raised the temperature of the body subjected to expe- 

 riment by the quantity d T without changing its primitive volume v ; 

 afterwards, when it had become v + dv, we have lowered its tempera- 

 ture by the same quantity d T without varying its primitive volume 

 V + dv. Now it may easily be seen that this double ojDeration can 

 be effected without loss of heat; let us suppose that n being a number 

 indefinitely great, the interval of temperature dT he divided into a 



number n of new intervals , and that we have n + I sources of 



n 



heat maintained at the temperatures T, T + IZ., T + ?^^ , 



n n 



T-h^^I-ll^andT + cfT. 



To raise the temperature of the body upon which we are operating 

 2c2 



