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 air 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 tts 
primitive value S, in virtue of 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 during 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 from A, and Jetting 
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 will be con- 
venient to consider the alterations of volume of the mass of air in the several 
operations as extremely small. 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 dq be the quantity of heat absorbed during the first operation, which 
is evolved again during the third; and let dv 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 dv, or only differ from it by an infinitely small 
ee lhnie. = will be the partial differential coefficient, with respect to v of that function of 
Vv 
v and ¢, 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 temperature zero, and under the atmospheric pressure), to bring it to the 
temperature t, and the volume v. That there is sucha function, of two independent variables v and ¢, 
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 M dv 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 N dt denote the amount of heat which 
must be supplied to raise the temperature by d ¢, without any alteration of volume; then M dvu+N dt 
must be the differential of a function of v and ¢. 
