Moreover, it rises to S exactly, by the fundamental axiom, because the volume occu pied by the water and steam is the same as before the first operation, and the quantity of caloric they contain is also the same—as much having been abstracted in the third operation as was communicated in the first—while in the second and fourth operations the contents of the cylinder neither gain nor lose caloric, as they are surrounded by non conductors.
Now, during the first two operations, work was done by the steam on the piston; during the last two, work was done against the steam; on the whole, the work done by the steam exceeds that done upon it, since evidently the temperature of the contents, for any position of the piston in its ascent, was greater than for the same position in the descent, except at the initial and final positions, where it is the same. Hence the pres sure also was greater at each stage in the ascent than at the corresponding stage in the descent; from which the theorem is evident.
Hence, on the whole, a certain amount of work has been communicated by the motion of the piston to external bodies; and the contents of the cylinder having been exactly restored to their primitive condition, we are entitled to regard this work as due to the caloric employed in the process. This, we see, was taken from A, and wholly transferred to B. It thus appears that caloric does work by being let down from a higher to a lower temperature. And the reader may easily see that if we knew the laws wl fch connect the pressure of saturated steam, and the amount of caloric it contains, with its volume and temperature, it would be possible to apply a rigorous calculation to the various processes of the cycle above explained, and to express by formula the amount of work gained on the whole in the series of operations, in terms of the temperatures (S and T) of the boiler and condenser of a steam-engine, and the whole amount of caloric which passes from one to the other.
Though the above process is exceedingly ingenious and important, it is to a consid• erable'extent vitiated by the assumption of the materiality of heat which is made through out. To show this, it is only necessary to consider the second operation, where work is supposed to be done by the contents of the cylinder expanding without loss or gain of caloric, a supposition which our present knowledge of the nature of heat shows to be incorrect. But it is quite easy, as seems to have been first remarked by J. Thomson in 1849, to put Carnot's statement in a form which is rigorously correct, whatever be the nature of heat. J. Thomson says: " We should not say, in the third operation, Compress till the same amount of heat is given out as was taken in during the first.' But we should
say, ' Compress till we have let out so much heat that the further compression (during the fourth stage) to the original volume may give back the original temperature.'" It is but bare justice, however, to acknowledge that Carrot himself was by no means satin fled with the caloric hypothesis, and that he insinuates, as we have already seen, more than a mere suspicion of its correctness.
If we carefully examine the above cycle of operations, we easily see that they are reversible, i.e., that the transferrence of the given amount of caloric back again from B to A, by performing the same operations in the opposite order, requires that we expend on the piston, on the whole, as much work as was gained during the direct operations. This most important idea is due to Carnot, and from it he deduces his test of a perfect engine, or one which yields from the transferrence of a given quantity of caloric from one body to another (each being at a given temperature) the greatest possible amount of work. And the test is simply that the cycle of operations must be reversible.
To prove it, we need only consider that, if a heat-engine, 3I, could be made to give more work by transferring a given amount of caloric from A to B, than it reversible engine, N, does, we may set 3/ and N to work in combination, 31 driven by the transfer of heat, and in turn driving N, which is employed to restore the heat to the source. The compound system would thus in each cycle produce an amount of work equal to the excess of that done by Al over that expended on N, without on the whole any transfer rence of heat; which is of course absurd.
The application of the true theory of heat to these propositions was made in 1849, 1850, and 1851 respectively, by Rankine, Clausius, and sir W. Thomson. Rankine employed a hypothesis as to the nature of the motion of which heat consists, from which he deduced a great many valuable results. Clausius supplied the defects of Carnot's beautiful reasoning; accommodating it to the dynamical theory by a very simple change, and evolving a great number of important consequences. But by far the simplest, though at the same time the most profound, writings on this subject, are those of sir W. Thomson, to be found in the Transactions of the Royal Society of Edinburgh; and these must be consulted by any reader who desires to have a clear statement and proof of thermodynamical laws, not complicated by unnecessary hypotheses or formulm, and yet perfectly general in its application. See also Tait's (2d ed. 1877).