carnot's theory of the motive power of heat. 551 



water in the liquid state, at the beginning and end of a cycle of operations. The 

 four successive operations are conducted in the following manner : — 



(1.) The cylinder is laid on the body A, so that the air in it is kept at the 

 temperature S ; and the piston is allowed to rise, performing work. 



(2.) The cylinder is placed on the impermeable stand K, so that its contents 

 can neither gain nor lose heat, and the piston is allowed to rise farther, still per- 

 forming work, till the temperature of the air sinks to T. 



(3.) The cylinder is placed on B, so that the an* is retained at the tempera- 

 ture T, and the piston is pushed down till the air gives out to the body B as much 

 heat as it had taken in from A, during the first operation. 



(4.) The cylinder is placed on K, so that no more heat can be taken in or 

 given out, and the piston is pushed down to its primitive position. 



23. At the end of the fourth operation the temperature must have reached its 

 primitive value S, in virtue o/" Carnot's axiom. 



24. Here, again, as in the former case, we observe that work is performed 

 by the piston during the first two operations ; and, during the third and fourth, 

 work is spent upon it, but to a less amount, since the pressure is on the whole less 

 during the third and fourth operations than diu-ing the first and second, on ac- 

 count of the temperature being lower. Thus, at the end of a complete cycle of 

 operations, mechanical effect has been obtained ; and the thermal agency from 

 which it is drawn is the taking of a certain quantity of heat fi'om A, and letting 

 it down, through the medium of the engine, to the body B at a lower temperature. 



25. To estimate the actual amount of effect thus obtained, it wUl be con- 

 venient to consider the alterations of volume of the mass of air in the several 

 operations as extremely smaU. We may afterwards pass by the integral calcu- 

 lus, or, practically, by summation, to determine the mechanical effect whatever 

 be the amplitudes of the different motions of the piston. 



26. Let dqh& the quantity of heat absorbed during the first operation, which 

 is evolved again during the third ; and let di; be the corresponding augmentation 

 of volume which takes places while the temperature remains constant, as it 

 does during the first operation.* The diminution of volume in the third ope- 

 ration must be also equal to d v, or only differ from it by an infinitely small 



* Thus, — will be the partial differential coefficient, with respect to v of that function of 

 d V 



V and t, which expresses the quantity of heat that must be added to a mass of air when in a " stan- 

 dard" state (such as at the temperatm-e zero, and under the atmospheric pressure), to bring it to the 

 temperature t, and the volume v. That there is such a function, of two independent variables v and (, 

 is merely an analytical expression of Caenot's fundamental axiom, as applied to a mass of air. The 

 general principle may be analytically stated in the following terms : — If M d ii denote the accession of 

 heat received by a mass of any kind, not possessing a destructible texture, when the volume is in- 

 creased by d V, the temperature being kept constant, and if N d t denote the amount of heat which 

 must be supplied to raise the temperature hy dt, without any alteration of volume ; then M d i/ + N rf t 

 must be the differential of a function of v and t. 



