MOTIVE POWER OF HEAT. 155 



in the third operation must be also equal to dv, or 

 only differ from it by an infinitely small quantity of 

 the second order. During the second operation we 

 may suppose the volume to be increased by an in- 

 finitely small quantity 0; which will occasion a 

 diminution of pressure and a diminution of tem- 

 perature, denoted respectively by GJ and r. During 

 the fourth operation there will be a diminution of 

 volume and an increase of pressure and temperature, 

 which can only differ, by infinitely small quantities 

 of the second order, from the changes in the other 

 direction, which took place in the second operation, 

 and they also may, therefore, be denoted by 0, GO, 

 and r, respectively. The alteration of pressure 



function, of two independent variables v and t, is merely 

 an analytical expression of Carnot's fundamental axiom, as 

 applied to a mass of air. The general principle may be 

 analytically stated in the following terms : If Mdv denote 

 the accession of heat received by a mass of any kind, not 

 possessing a destructible texture, when the volume is in- 

 creased by dv, the temperature being kept constant, and if 

 Ndt denote the amount of heat which must be supplied to 

 raise the temperature by dt, without any alteration of vol- 

 ume ; then Mdv -\-Ndt must be the differential of a func- 

 tion of v and t. [Note of Nov. 5, 1881. In the corrected 

 theory it is (M Jp) dv -\- Ndt t th&t is a complete differential, 

 not Mdv + Ndt. See Dynamical Theory of Heat (Art. XLVIII. , 

 below), 20. J 



