The question now arises, Whether the quantity of heat converted into work, or generated out of work, stands in a generally constant proportion to the quantity which passes over from the hotter to the colder body, or vice versa; or whether the proportion existing between them varies according to the nature of the variable body, which is the medium of the transfer.

§ 4. Carnot's view as to the work performed during a Cyclical Process.

Carnot, who was the first to remark that in the production of mechanical work heat passes from a hotter into a colder body, and that conversely in the consumption of mechanical work heat can be brought from a colder into a hotter body, and who also conceived the simple cyclical process above described (which was first represented graphically by Clapeyron), took a special view of his own as to the fundamental connection of these processes*.

In his time the doctrine was still generally prevalent that heat was a special kind of matter, which might exist within a body in greater or lesser quantity, and thereby occasion. differences of temperature. In accordance with this doctrine it was supposed that heat might change the character of its distribution, in passing from one body into another, and further that it could exist in different conditions, which were denominated respectively 'free' and 'latent'; but that the whole quantity of heat existing in the universe could neither be increased nor diminished, inasmuch as matter can neither be created nor destroyed.

Carnot shared these views, and accordingly treated it as self-evident that the quantities of heat, which the variable body in the course of the cyclical process receives from and gives out to the surrounding space, are equal to each other, and consequently cancel each other. He lays this down very distinctly in § 27 of his work, where he says: "we shall assume that the quantities of heat absorbed and emitted in these different transformations compensate each other exactly. This fact has never been held in doubt; admitted at first without reflection, it has since been verified in many instances by

*

Reflexions sur la puissance motrice du feu. Pa: is, 1824.

experiments with the calorimeter. To deny it would be to subvert the whole theory of heat, which rests on it as its basis."

Now since on this assumption the quantity of heat existing in the body was the same after the cyclical process as before it, and yet a certain amount of work had been achieved, Carnot sought to explain this latter fact from the circumstance of the heat falling from a higher to a lower temperature. He drew a comparison between this descending passage of heat (which is especially striking in the steam-engine, where the fire gives off heat to the boiler, and conversely the cold water of the condenser absorbs heat) and the falling of water from a higher to a lower level, by means of which a machine can be set in motion, and work done. Accordingly in § 28, after making use of the expression 'fall of water,' he applies the corresponding expression 'fall of caloric' to the sinking of heat from a higher to a lower temperature.

Starting from these premises, he laid down the principle that the quantity of work done must bear a certain constant relation to the passage of heat,' i.e. the quantity of heat passing over at the time, and to the temperature of the bodies. between which it passes; and that this relation is independent of the nature of the substance which serves as a medium for the performance of work and passage of heat. His proof of the necessary existence of this constant relation. rests on the principle "That it is impossible to create moving force out of nothing," or in other words, "That perpetual motion is an impossibility."

This mode of dealing with the question does not accord with our present views, inasmuch as we rather assume that in the production of work a corresponding quantity of heat is consumed, and that in consequence the quantity of heat given out to the surrounding space during the cyclical process is less than that received from it. Now if for the production of work heat is consumed, then, whether at the same time. with this consumption of heat there takes place the passage of another quantity of heat from a hotter to a colder body, or not, at least there is no ground whatever for saying that the work is created out of nothing. Accordingly not only must the principle enunciated by Carnot receive some modification, but a different basis of proof from that used by him must be discovered.

§ 5. New Fundamental Principle concerning Heat.

Various considerations as to the conditions and nature of heat had led the author to the conviction that the tendency of heat to pass from a warmer to a colder body, and thereby equalize existing differences of temperature (as prominently shewn in the phenomena of conduction and ordinary radiation), was so intimately bound up with its whole constitution that it must have a predominant influence under all conceivable circumstances. He thereupon propounded the following as a fundamental principle: "Heat cannot, of itself, pass from a colder to a hotter body."

The words of itself,' here used for the sake of brevity, require, in order to be completely understood, a further explanation, as given in various parts of the author's papers. In the first place they express the fact that heat can never, through conduction or radiation, accumulate itself in the warmer body at the cost of the colder. This, which was already known as respects direct radiation, must thus be further extended to cases in which by refraction or reflection the course of the ray is diverted and a concentration of rays thereby produced. In the second place the principle must be applicable to processes which are a combination of several different steps, such as e.g. cyclical processes of the kind described above. It is true that by such a process (as we have seen by going through the original cycle in the reverse direction) heat may be carried over from a colder into a hotter body our principle however declares that simultaneously with this passage of heat from a colder to a hotter body there must either take place an opposite passage of heat from a hotter to a colder body, or else some change or other which has the special property that it is not reversible, except under the condition that it occasions, whether directly or indirectly, such an opposite passage of heat. This simultaneous passage of heat in the opposite direction, or this special change entailing an opposite passage of heat, is then to be treated as a compensation for the passage of heat from the colder to the warmer body; and if we apply this conception we may replace the words "of itself" by "without coinpensation," and then enunciate the principle as follows:

"A passage of heat from a colder to a hotter body cannot take place without compensation."

This proposition, laid down as a Fundamental Principle by the author, has met with much opposition; but, having repeatedly had occasion to defend it, he has always been able to shew that the objections raised were due to the fact that the phenomena, in which it was believed that an uncompensated passage of heat from a colder to a hotter body was to be found, had not been correctly understood. To state these objections and their answers at this place would interrupt too seriously the course of the present treatise. In the discussions which follow, the principle, which, as the author believes, is acknowledged at present by most physicists as being correct, will be simply used as a fundamental principle; but the author proposes to return to it further on, and then to consider more closely the points of discussion which have been raised upon it.

§ 6. Proof that the relation between the quantity of heat carried over, and that converted into work, is independent of the nature of the matter which forms the medium of the change.

Assuming the foregoing principle to be correct, it may be proved that between the quantity of heat Q, which in a cyclical process of the kind described above is transformed into work (or, where the process is in the reverse order, generated by work), and the quantity of heat Q2, which is transferred at the same time from a hotter to a colder body (or vice versâ), there exists a relation independent of the nature of the variable body which acts as the medium of the transformation and transfer; and thus that, if several cyclical processes are performed, with the same reservoirs of heat K, and K„, but with different variable bodies, the ratio will be the same for 4.2

all. If we suppose the processes so arranged, according to their magnitude, that the quantity of heat Q, which is transformed into work, has in all of them a constant value, then we have only to consider the magnitude of the quantity of heat Q, which is transferred, and the principle which is to be proved takes the following form: "If where two different variable bodies are used, the quantity of heat Q transformed into work is the same, then the quantity of heat Q2, which is transferred, will also be the same."

2

Let there, if possible, be two bodies C and C' (e.g. the perfect gas and the combined mass of liquid and vapour, described above) for which the values of Q are equal, but those of the transferred quantities of heat are different, and let these different values be called Q, and Q', respectively: ', being the greater of the two. Now let us in the first place subject the body C to a cyclical process, such that the quantity of heat Q is transformed into work, and the quantity Q, is transferred from K, to K. Next let us subject C' to a cyclical process of the reverse description, so that the quantity of heat is generated out of work, and the quantity Q', is transferred from K, to K,. Then the above two changes, from heat into work, and work into heat, will cancel each other; since we may suppose that when in the first process the heat Q has been taken from the body K, and transformed into work, this same work is expended in the second process in producing the heat Q, which is then returned to the same body K1. In all other respects also the bodies will have returned, at the end of the two operations, to their original condition, with one exception only. The quantity of heat Q',, transferred from K, to K,, has been assumed to be greater than the quantity Q, transferred from K, to K2. Hence these two do not cancel each other, but there remains at the end a quantity of heat, represented by the difference Q-Q,, which has passed over from K, to K,. Hence a passage of heat will have taken place from a colder to a warmer body without any other compensating change. But this contradicts the fundamental principle. Hence the assumption that ', is greater than Q, must be false.

2

2

2

Again, if we make the opposite assumption, that Q', is less than Q2, we may suppose the body C to undergo the cyclical process in the first, and C in the reverse direction. We then arrive similarly at the result that a quantity of heat Q-Q, has passed from the colder body K, to the hotter K1, which is again contrary to the principle.

2

2

Since then Q, can be neither greater nor less than Q2 it must be equal to Q,; which was to be proved.

We will now give to the result thus obtained the mathematical form most convenient for our subsequent reasoning.

Since the quotient

Q

Q2

is independent of the nature of the