158 THOMSON ON CARNOTS 



for the amounts gained during the first and second, 

 and spent during the third and fourth operations ; 

 and hence, by addition and subtraction, we find 



, ,dv , ,.dv 



codv p4> , or (VGJ pep) , 



for the aggregate amount of mechanical effect 

 gained during the cycle of operations. It only re- 

 mains for us to express this result in terms of dq 

 and r, on which the given thermal agency depends. 

 For this purpose we remark that and GO are al- 

 terations of volume and pressure which take place 

 along with a change of temperature r, and hence, 

 by the laws of compressibility and expansion, we 

 may establish a relation* between them in the fol- 

 lowing manner : 



Let p 9 be the pressure of the mass of air when 

 reduced to the temperature zero, and confined 

 in a volume v ; then, whatever be v , the product 

 p Q v Q will, by the law of compressibility, remain con- 

 stant ; and, if the temperature be elevated from 

 to t + *> and the gas be allowed to expand freely 

 without any change of pressure, its volume will be 



* We might also investigate another relation, to express 

 the fact that there is no accession or removal of heat during 

 either the second or the fourth operation; but it will be 

 seen that this will not affect the result in the text, although 

 it would enable us to determine both and GO in terms of T. 



