Reflections on the motivepower of heat, and on machines fitted to develop that power
by (Sadi) (Carnot), 1824
translated by R. H. Thurston, 1897
(\Huge{Everyone}) knows that heat can produce motion. That it possesses vast motivepower no one can doubt, in these days when the steamengine is everywhere so well known.
To heat also are due the vast movements which take place on the earth. It causes the agitations of the atmosphere, the ascension of clouds, the fall of rain and of meteors, the currents of water which channel the surface of the globe, and of which man has thus far employed but a small portion. Even earthquakes and volcanic eruptions are the result of heat.
From this immense reservoir we may draw the moving force necessary for our purposes. Nature, in providing us with combustibles on all sides, has given us the power to produce, at all times and in all places, heat and the impelling power which is the result of it. To develop this power, to appropriate it to our uses, is the object of heat engines.
The study of these engines is of the greatest interest, their importance is enormous, their use is continually increasing, and they seem destined to produce a great revolution in the civilized world.
Already the steamengine works our mines, impels our ships, excavates our ports and our rivers, forges iron, fashions wood, grinds grains, spins and weaves our cloths, transports the heaviest burdens, etc. It appears that it must some day serve as a universal motor, and be substituted for animal power, waterfalls, and air currents.
Over the first of these motors it has the advantage of economy, over the two others the inestimable advantage that it can be used at all times and places without interruption.
If, some day, the steamengine shall be so perfected that it can be set up and supplied with fuel at small cost, it will combine all desirable qualities, and will afford to the industrial arts a range the extent of which can scarcely be predicted. It is not merely that a powerful and convenient motor that can be procured and carried anywhere is substituted for the motors already in use, but that it causes rapid extension in the arts in which it is applied, and can even create entirely new arts.
The most signal service that the steamengine has rendered to England is undoubtedly the revival of the working of the coalmines, which had declined, and threatened to cease entirely, in consequence of the continually increasing difficulty of drainage, and of raising the coal. ^{1} We should rank second the benefit to iron manufacture, both by the abundant supply of coal substituted for wood just when the latter had begun to grow scarce, and by the powerful machines of all kinds, the use of which the introduction of the steamengine has permitted or facilitated.
Iron and heat are, as we know, the supporters, the bases, of the mechanic arts. It is doubtful if there be in England a single industrial establishment of which the existence does not depend on the use of these agents, and which does not freely employ them. To take away today from England her steamengines would be to take away at the same time her coal and iron. It would be to dry up all her sources of wealth, to ruin all on which her prosperity depends, in short, to annihilate that colossal power. The destruction of her navy, which she considers her strongest defence, would perhaps be less fatal.
The safe and rapid navigation by steamships may be regarded as an entirely new art due to the steamengine. Already this art has permitted the establishment of prompt and regular communications across the arms of the sea, and on the great rivers of the old and new continents. It has made it possible to traverse savage regions where before we could scarcely penetrate. It has enabled U8 to carry the fruits of civilization over portions of the globe where they would else have been wanting for years. Steam navigation brings nearer together the most distant nations. It tends to unite the nations of the earth as inhabitants of one country. In fact, to lessen the time, the fatigues, the uncertainties, and the dangers of travelis not this the same as greatly to shorten distances? ^{2}
The discovery of the steamengine owed its birth, like most human inventions, to rude attempts which have been attributed to different persons, while the real author is not certainly known. It is, however, less in the first attempts that the principal discovery consists, than in the successive improvements which have brought steamengines to the condition in which we find them today. There is almost as great a distance between the first apparatus in which the expansive force of steam was displayed and the existing machine, as between the first raft that man ever made and the modern vessel.
If the honor of a discovery belongs to the nation in which it has acquired its growth and all its developments, this honor cannot be here refused to England. Savery, Newcomen, Smeaton, the famous Watt, Woolf, Trevithick, and some other English engineers, are the veritable creators of the steamengine. It has acquired at their hands all its successive degrees of improvement. Finally, it is natural that an invention should have its birth and especially be developed, be perfected, in that place where its want is most strongly felt.
Notwithstanding the work of all kinds done by steamengines, notwithstanding the satisfactory condition to which they have been brought today, their theory is very little understood, and the attempts to improve them are still directed almost by chance.
The question has often been raised whether the motive power of heat ^{3} is unbounded, whether the possible improvements in steamengines have an assignable limit,a limit which the nature of things will not allow to be passed by any means whatever; or whether, on the contrary, these improvements may be carried on indefinitely. We have long sought, and are seeking today, to ascertain whether there are in existence agents preferable to the vapor of water for developing the motive power of heat; whether atmospheric air, for example, would not present in this respect great advantages. We propose now to submit these questions to a deliberate examination.
The phenomenon of the production of motion by heat has not been considered from a sufficiently general point of view. We have considered it only in machines the nature and mode of action of which have not allowed us to take ill the whole extent of application of which it is susceptible. In such machines the phenomenon is, in a way, incomplete. It becomes difficult to recognize its principles and study its laws.
In order to consider in the most general way the principle of the production of motion by heat, it must be considered independently of any mechanism or any particular agent. It is necessary to establish principles applicable not only to steamengines ^{4} but to all imaginable heatengines, whatever the working substance and whatever the method by which it is operated.
Machines which do not receive their motion from heat, those which have for a motor the force of men or of animals, a waterfall, an aircurrent, etc., can be studied even to their smallest details by the mechanical theory. All cases are foreseen, all imaginable movements are referred to these general principles, firmly established, and applicable under all circumstances. This is the character of a complete theory. A similar theory is evidently needed for heatengines. We shall have it only when the laws of Physics shall be extended enough, generalized enough, to make known beforehand all the effects of heat acting in a determined manner on anybody.
We will suppose in what follows at least a superficial knowledge of the different parts which compose an ordinary steamengine; and we consider it unnecessary to explain what are the furnace, boiler, steamcylinder, piston, condenser, etc.
The production of motion in steamengines is always accompanied by a circumstance on which we should fix our attention. This circumstance is the reestablishing of equilibrium in the caloric; that is, its passage from a body in which the temperature is more or less elevated, to another in which it is lower. What happens in fact in a steamengine actually in motion? The caloric developed in the furnace by the effect of the combustion traverses the walls of the boiler, produces steam, and in some way incorporates itself with it. The latter carrying it away, takes it first into the cylinder, where it performs some function, and from thence into the condenser, where it is liquefied by contact with the cold water which it encounters there. Then, as a final result, the cold water of the condenser takes possession of the caloric developed by the combustion. It is heated by the intervention of the steam as if it had been placed directly over the furnace. The steam is here only a means of transporting the caloric. It fills the same office as in the heating of baths by steam, except that in this case its motion is rendered useful.
We easily recognize in the operations that we have just described the reestablishment of equilibrium in the caloric, its passage from a more or less heated body to a cooler one. The first of these bodies, in this case, is the heated air of the furnace; the second is the condensing water. The reestablishment of equilibrium of the caloric takes place between them, if not completely, at least partially, for on the one hand the heated air, after having performed its function, having passed round the boiler, goes out through the chimney with a temperature much below that which it had acquired as the effect of combustion; and on the other hand, the water of the condenser, after having liquefied the steam, leaves the machine with a temperature higher than that with which it entered.
The production of motive power is then due in steamengines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body, that is, to its reestablishment of equilibrium  an equilibrium considered as destroyed by any cause whatever, by chemical action such as combustion, or by any other, We shall see. shortly that this principle is applicable to any machine set in motion by heat.
According to this principle, the production of heat alone is not sufficient to give birth to the impelling power: it is necessary that there should also be cold; without it, the heat would be useless. And in fact, if we should find about us only bodies as hot as our furnaces, how can we condense steam? What should we do with it if once produced? We should not presume.that we might discharge it into the atmosphere, as is done in some engines;^{5} the atmosphere would not receive it. It does receive it under the actual condition of things, only because it fulfils the office of a vast condenser, because it is at a lower temperature; otherwise it would soon become fully charged, or rather would be already saturated. ^{6}
Wherever there exists a difference of temperature, wherever it has been possible for the equilibrium of the caloric to be reestablished, it is possible to have also the production of impelling power. Steam is a means of realizing this power, but it is not the only one. All substances in nature can be employed for this purpose, all are susceptible of changes of volume, of successive contractions and dilatations, through the alternation of heat and cold. All are capable of overcoming in their changes of volume certain resistances, and of thus developing the impelling power. A solid body  a metallic bar for example  alternately heated and cooled increases and diminishes in length, and can move bodies fastened to its ends. A liquid alternately heated and cooled increases and diminishes in volume, and can overcome obstacles of greater or less size, opposed to its dilatation. An aeriform fluid is susceptible of considerable change of volume by variations of temperature. If it is enclosed in an expansible space, such as a cylinder provided with a piston, it will produce movements of great extent. Vapors of all substances capable of passing into a gaseous condition, as of alcohol, of mercury, of sulphur, etc., may fulfil the same office as vapor of water. The latter, alternately heated and cooled, would produce motive power in the shape of permanent gases, that is, without ever returning to a liquid state. Most of these substances have been proposed, many even have been tried, although up to this time perhaps without remarkable success.
We have shown that in steamengines the motivepower is due to a reestablishment of equilibrium in the caloric; this takes place not only for steamengines, but also for every heatengine  that is, for every machine of which caloric is the motor. Heat can evidently be a cause of motion only by virtue of the changes of volume or of form which it produces in bodies.
These changes are not caused by uniform temperature, but rather by alternations of heat and cold. Now to heat any substance whatever requires a body warmer than the one to he heated; to cool it requires a cooler body. We supply caloric to the first of these bodies that we may transmit it to the second by means of the intermediary substance. This is to reestablish, or at least to endeavor to reestablish, the equilibrium of the caloric.
It is natural to ask here this curious and important question: Is the motive power of heat invariable in quantity, or does it vary with the agent employed to realize it as the intermediary substance, selected as the subject of action of the heat?
It is clear that this question can be asked only in regard to a given quantity of caloric,^{7} the difference of the temperatures also being given. We take, for example, one body $A$ kept at a temperature of 100° and another body B kept at a temperature of 0°, and ask what quantity of motive power can be produced by the passage of a given portion of caloric $($for example, as much as is necessary to melt a kilogram of ice$)$ from the first of these bodies to the second. We inquire whether this quantity of motive power is necessarily limited, whether it varies with the substance employed to realize it, whether the vapor of water offers in this respect more or less advantage than the vapor of alcohol, of mercury, a permanent gas, or any other substance. We will try to answer these questions, availing ourselves of ideas already established.
We have already remarked upon this selfevident fact, or fact which at least appears evident as soon as we reflect on the changes of volume occasioned by heat: wherever there exists a difference of temperature, motivepower can be produced. Reciprocally, wherever we can consume this power, it is possible to produce a difference of temperature, it is possible to occasion destruction of equilibrium in the caloric. Are not percussion and the friction of bodies actually means of raising their temperature, of making it reach spontaneously a higher degree than that of the surrounding bodies, and consequently of producing a destruction of equilibrium in the caloric, where equilibrium previously existed? It is a fact proved by experience, that the temperature of gaseous fluids is raised by compression and lowered by rarefaction. This is a sure method of changing the temperature of bodies, and destroying the equilibrium of the caloric as many times as may be desired with the same substance. The vapor of water employed in an inverse manner to that in which it is used in steamengines can also be regarded as a means of destroying the equilibrium of the caloric. To be convinced of this we need but to observe closely the manner in which motive power is developed by the action of heat on vapor of water. Imagine two bodies A and B, kept each at a constant temperature, that of A being higher than that of B. These two bodies, to which we can give or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of caloric. We will call the first the furnace and the second the refrigerator.
If we wish to produce motive power by carrying a certain quantity of heat from the body A to the body B we shall proceed as follows:
$($1$)$ To borrow caloric from the body A to make steam with itthat is, to make this body fulfil the function of a furnace, or rather of the metal composing the boiler in ordinary engineswe here assume that the steam is produced at the same temperature as the body A.
$($2$)$ The steam having been received in a space capable of expansion, such as a cylinder furnished with a piston, to increase the volume of this space, and consequently also that of the steam. Thus rarefied, the temperature will fall spontaneously, as occurs with all elastic fluids; admit that the rarefaction may be continued to the point where the temperature becomes precisely that of the body B.
$($3$)$ To condense the steam by putting it in contact with the body B, and at the same time exerting on it a constant pressure until it is entirely liquefied. The body B fills here the place of the injectionwater in ordinary engines, with this difference, that it condenses the vapor without mingling with it, and without changing its own temperature.^{8}
The operations which we have just described might have been performed in an inverse direction and order. There is nothing to prevent forming vapor with the caloric of the body B, and at the temperature of that body, compressing it in such a way as to make it acquire the temperature of the body A, finally condensing it by contact with this latter body, and continuing the compression to complete liquefaction.
By our first operations there would have been at the same time production of motive power and transfer of caloric from the body A to the body B. By the inverse operations there is at the same time expenditure of motive power and return of caloric from the body B to the body A. But if we have acted in each case on the same quantity of vapor, if there is produced no loss either of motive power or caloric, the quantity of motive power produced in the first place will be equal to that which would have been expended in the second, and the quantity of caloric passed in the first case from the body A to the body B would be equal to the quantity which passes back again in the second from the body B to the body A; so that an indefinite number of alternative operations of this sort could be carried on without in the end having either produced motive power or transferred caloric from one body to the other.
Now if there existed any means of using heat preferable to those which we have employed, that is, if it were possible by any method whatever to make the caloric produce a quantity of motive power greater than we have made it produce by our first series of operations, it would suffice to divert a portion of this power in order by the method just indicated to make the caloric of the body B return to the body A from the refrigerator to the furnace, to restore the initial conditions, and thus to be ready to commence again an operation precisely similar to the former, and so on : this would be not only perpetual motion, but an unlimited creation of motive power without consumption either of caloric or of any other agent whatever. Such a creation is entirely contrary to ideas now accepted, to the laws of mechanics and of sound physics. It is inadmissible. ^{9} We should then conclude that the maximum of motive power resulting from the employment of steam is also the maximum of motive power realizable by any means whatever. We will soon give a second more rigorous demonstration of this theory. This should be considered only as an approximation. $($See page 59.$)$
We have a right to ask, in regard to the proposition just enunciated, the following questions: What is the sense of the word maximum here? By what sign can it be known that this maximum is attained? By what sign can it be known whether the steam is employed to greatest possible advantage in the production of motive power?
Since every reestablishment of equilibrium in the caloric may be the cause of the production of motive power, every reestablishment of equilibrium which shall be accomplished without production of this power should be considered as an actual loss. Now, very little reflection would show that all change of temperature which is not due to a change of volume of the bodies can be only a useless reestablishment of equilibrium in the caloric.^{10} The necessary condition of the maximum is, then, that in the bodies employed to realize the motive power of heat there should not occur any change of temperature which may not be due to a change of volume. Reciprocally, every time that this condition is fulfilled the maximum will be attained. This principle should never be lost sight of in the construction of heatengines; it is its fundamental basis. If it cannot be strictly observed, it should at least be departed from as little as possible.
Every change of temperature which is not due to a change of volume or to chemical action $($an action that we provisionally suppose not to occur here$)$ is necessarily due to the direct passage of the caloric from a more or less heated body to a colder body. This passage occurs mainly by the contact of bodies of different temperatures; hence such contact should be avoided as much as possible. It cannot probably be avoided entirely, but it should at least be so managed that the bodies brought in contact with each other differ as little as possible in temperature. When we just now supposed, in our demonstration, the caloric of the body A employed to form steam, this steam was considered as generated at the temperature of the body A ; thus the contact took place only between bodies of equal temperatures; the change of temperature occurring afterwards in the steam was due to dilatation, con sequently to a change of volume. Finally, condensation took place also without contact of bodies of different temperatures. It occurred while exerting a constant pressure on the steam brought in contact with the body B of the same temperature as itself. The conditions for a maximum are thus found to be fulfilled. In reality the operation cannot proceed exactly as we have assumed. To determine the passage of caloric from one body to another, it is necessary that there should be all: excess of temperature in the first, but this excess may be supposed as slight as we please. We can regard it as insensible in theory, without thereby destroying the exactness of the arguments.
A more substantial objection may be made to our demonstration, thus: When we borrow caloric from the body A to produce steam, and when this steam is afterwards condensed by its contact with the body B, the water used to form it, and which we considered at first as being of the temperature of the body A, is found at the close of the operation at the temperature of the body B. It has become cool. If we wish to begin again an operation similar to the first, if we wish to develop a new quantity of motive power with the same instrument, with the same steam, it is necessary first to reestablish the original condition  to restore the water to the original temperature. This can undoubtedly be done by at once putting it again in contact with the body A ; but there is then contact between bodies of different temperatures, and loss of motive power ^{11}. It would be impossible to execute the inverse operation, that is, to return to the body A the caloric employed to raise the temperature of the liquid.
This difficulty may be removed by supposing the difference of temperature bet:ween the body A and the body B indefinitely small. The quantity of heat necessary to raise the liquid to its former temperature will be also indefinitely small and unimportant relatively to that which is necessary to produce steam  a quantity always limited.
The proposition found elsewhere demonstrated for the case in which the difference between the temperatures of the two bodies is indefinitely small, may be easily extended to the general case. In fact, if it operated to produce motive power by the passage of caloric from the body A to the body Z, the temperature of this latter body being very different from that of the former, we should imagine a series of bodies B, C, D ... of temperatures intermediate between those of the bodies A, Z, and selected so that the differences from A to B, from B to C, etc., may all be indefinitely small. The caloric coming from A would not arrive at Z till after it had passed through the bodies B, C, D, etc., and after having developed in each of these stages maximum motive power. The inverse operations would here be entirely possible, and the reasoning of page 52 would be strictly applicable.
According to established principles at the present time, we can compare with sufficient accuracy the. motive power of heat to that of a waterfall. Each has a maximum that we cannot exceed, whatever may be, on the one hand, the machine which is acted upon by the water, and whatever, on the other hand, the substance acted upon by the heat. The motive power of a waterfall depends on its height and on the quantity of the liquid; the motive power of heat depends also on the quantity of caloric used, and on what may be termed, on what in fact we will call, the height of its fall, ^{12} that is to say, the difference of temperature of the bodies between which the exchange of caloric is made. In the waterfall the motive power is exactly proportional to the difference of level between the higher and lower reservoirs. In the fall of caloric the motive power undoubtedly increases with the difference of temperature between the warm and the cold bodies; but we do not know whether it is proportional to this difference. We do not know, for example, whether the fall of caloric from 100 to 50 degrees furnishes more or less motive power than the fall of this same caloric from 50 to zero. It is a question which we propose to examine hereafter.
We shall give here a second demonstration of the fundamental proposition enunciated on page 56, and present this proposition under a more general form than the one already given. When a gaseous fluid is rapidly compressed its temperature rises. It falls, on the contrary, when it is rapidly dilated. This is one of the facts best demonstrated by experiment. We will take it for the basis of our demonstration.^{13}
If, when the temperature of a gas has been raised by compression, we wish to reduce it to its former temperature without subjecting its volume to new changes, some of its caloric must be removed. This caloric might have been removed in proportion as pressure was applied, so that the temperature of the gas would remain constant. Similarly, if the gas is rarefied we can avoid lowering the temperature by supplying it with a certain quantity of caloric. Let us call the caloric employed at such times, when no change of temperature occurs, caloric due to change of volume. This denomination does not indicate that the caloric appertains to the volume: it does not appertain to it any more than to pressure, and might as well be called caloric due to the change of pressure. We do not know what laws it follows relative to the variations of volume: it is possible that its quantity changes either with the nature of the gas, its density, or its temperature. Experiment has taught us nothing on this subject. It has only shown us that this caloric is developed in greater or less quantity by the compression of the elastic fluids.
This preliminary idea. being established, let us imagine an elastic fluid, atmospheric air for example, shut up in a cylindrical vessel, abcd $($Fig. 1$)$, provided with a movable diaphragm or piston, cd. Let there be also two bodies, A and B, kept each at a constant temperature, that of A being higher than that of B. Let us picture to ourselves now the series of operations which are to be described :
$($1$)$ Contact of the body $A$ with the air enclosed in the space $abcd$ or with the wall of this space  a wall that we will suppose to transmit the caloric readily. The air becomes by such contact of the same temperature as the body A; cd is the actual position of the piston.
$($2$)$ The piston gradually rises and takes the position ef. The body A is all the time in contact with the air, which is thus kept at a constant temperature during the rarefaction. The body A furnishes the caloric necessary to keep the temperature constant.
$($3$)$ The body A is removed, and the air is then no longer in contact with any body capable of furnishing it with caloric. The piston meanwhile continues to move, and passes from the position ef to the position gh. The air is rarefied without receiving caloric, and its temperature falls. Let us imagine that it falls thus till it becomes equal to that of the body B; at this instant the piston stops, remaining at the position gh.
$($4$)$ The air is placed in contact with the body B; it is compressed by the return of the piston as it is moved from the position gh to the position cd. This air remains, however, at a constant temperature because of its contact with the. body B, to which it yields its caloric.
$($5$)$ The body B is removed, and the compression of the air is continued, which being then isolated, its temperature rises. The compression is continued till the air acquires the temperature of the body A. The piston passes during this time from the position cd to the position ik.
$($6$)$ The air is again placed in contact with the body A. The piston returns from the position ik to the position ef; the temperature remains unchanged.
$($7$)$ The step described under number 3 is renewed, then successively the steps 4, 5, 6, 3, 4, 5, 6, 3, 4, 5; and so on.
In these various operations the piston is subject to an effort of greater or less magnitude, exerted by the air enclosed in the cylinder; the elastic force of this air varies as much by reason of the changes in volume as of changes of temperature. But it should be remarked that with equal volumes, that is, for the similar positions of the piston, the temperature is higher during the movements of dilatation than during the movements of compression. During the former the elastic force of the air is found to be greater, and consequently the quantity of motive power produced by the movements of dilatation is more considerable than that consumed to produce the movements of compression. Thus we should obtain an excess of motive power an excess which we could employ for any purpose whatever. The air, then, has served as a heatengine; we have, in fact, employed it in the most advantageous manner possible, for no useless reestablishment of equilibrium has been effected in the caloric.
All the abovedescribed operations may be executed in an inverse sense and order. Let us imagine that, after the sixth period, that is to say the piston having arrived at the position ef, we cause it to return to the position ik, and that at the same time we keep the air in contact with the body A. The caloric furnished by this body during the sixth period would return to its source, that is, to the body A, and the conditions would then become precisely the same as they were at the end of the fifth period. If now we take away the body A, and if we cause the piston to move from ef to cd, the temperature of the air will diminish as many degrees as it increased during the fifth period, and will become that of the body B. We may evidently continue a series of operations the inverse of those already described. It is only necessary under the same circumstances to execute for each period a movement of dilatation instead of a movement of compression, and reciprocally.
The result of these first operations has been the production of a certain quantity of motive power and the removal of caloric from the body A to the body B. The result of the inverse operations is the consumption of the motive power produced and the return of the caloric from the body B to the body A; so that these two series of operations annul each other, after a fashion, one neutralizing the other.
The impossibility of making the caloric produce a greater quantity of motive power than that which we obtained from it by our first series of operations, is now easily proved. It is demonstrated by reasoning very similar to that employed at page 56; the reasoning will here be even more exact. The air which we have used to develop the motive power is restored at the end of each cycle of operations exactly to the state in which it was at first found, while, as we have already remarked, this would not be precisely the case with the vapor of water.^{14}
We have chosen atmospheric air as the instrument which should develop the motive power of heat, but it is evident that the reasoning would have been the same for all other gaseous substances, and even for all other bodies susceptible of change of temperature through successive contractions and dilatations, which comprehends all natural substances, or at least all those which are adapted to realize the motive power of heat. Thus we are led to establish this general proposition:
The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of the caloric.
We must understand here that each of the methods of developing motive power attains the perfection of which it is susceptible. This condition is found to be fulfilled if, as we remarked above, there is produced in the body no other change of temperature than that due to change of volume, or, what is the same thing in other words, if there is no contact between bodies of sensibly different temperatures.
Different methods of realizing motive power mar be taken, as in the employment of different substances, or in the use of the same substance in two different states for example, of a gas at two different densities.
This leads us naturally to those interesting researches on the aeriform fluids researches which lead us also to new results in regard to the motive power of heat, and give us the means of verifying, in some particular cases, the fundamental proposition above stated. ^{15}
We readily see that our demonstration would have been simplified by supposing the temperatures of the bodies A and B to differ very little. Then the movements of the piston being slight during the periods 3 and 5, these periods might have been suppressed without influencing sensibly the production of motive power. A very little change of volume should suffice in fact to produce a very slight change of temperature, and this slight change of volume may be neglected in presence of that of the periods 4 and 6, of which the extent is unlimited.
If we suppress periods 3 and 5, in the series of operations above described, it is reduced to the following:
$($1$)$ Contact of the gas confined in abcd $($Fig. 2$)$ with the body A, passage of the piston from cd to ef.
$($2$)$ Removal of the body A, contact of the gas confined in abel with the body B, return of the piston from ef to cd.
$($3$)$ Removal of the body B, contact of the gas with the body A, passage of the piston from cd to ef, that is, repetition of the first period, and so on.
The motive power resulting from the ensemble of operations 1 and 2 will evidently be the difference between that which is produced by the expansion of the gas while it is at the temperature of the body A, and that which is consumed to compress this gas while it is at the temperature of the body B.
Let us suppose that operations 1 and 2 be performed on two gases of different chemical natures but under the same pressureunder atmospheric pressure, for example. These two gases will behave exactly alike under the same circumstances, that is, their expansive forces, originally equal, will remain always equal, whatever may be the variations of volume and of temperature, provided these variations are the same in both. This results obviously from the laws of Mariotte and MM. GayLussac and Daltonlaws common to all elastic fluids, and in virtue of which the same relations exist for all these fluids between the volume, the expansive force, and the temperature.
Since two different gases at the same temperature and under the same pressure should behave alike under the same circumstances, if we subjected them both to the operations above described, they should give rise to equal quantities of motive power.
Now this implies, according to the fundamental proposition that we have established, the employment of two equal quantities of caloric; that is, it implies that the quantity of caloric transferred from. the body A to the body B is the same, whichever gas is used.
The quantity of caloric transferred from the body A to the body B is evidently that which is absorbed by the gas in its expansion of volume, or that which this gas relinquishes during compression. We are led, then, to establish the following proposition:
When a gas passes without change of temperature from one definite volume and pressure to another volume and another pressure equally definite, the quantity of caloric absorbed or relinquished is always the same, whatever may be the nature of the gas chosen as the subject of the experiment.
Take, for example, 1 litre of air at the temperature of 100° and under the pressure of one atmosphere. If we double the volume of this air and wish to maintain it at the temperature of 100°, a certain quantity of heat must be supplied to it. Now this quantity will be precisely the same if, instead of operating on the air, we operate upon carbonicacid gas, upon nitrogen, upon hydrogen, upon vapor of water or of alcohol, that is, if we double the volume of 1 litre of these gases taken at the temperature of 100° and under atmospheric pressure.
It will be the same thing in the inverse sense if, instead of doubling the volume of gas, we reduce it one half by compression. The quantity of heat that the elastic fluids set free or absorb in their changes of volume has never been measured by any direct experiment, and doubtless such an experiment would be very difficult, but there exists a datum which is very nearly its equivalent. This has been furnished by the theory of sound. It deserves much confidence because of the exactness of the conditions which have led to its establishment. It consists in this:
Atmospheric air should rise one degree Centigrade when by sudden compression it experiences a reduction of volume of $\frac{1}{116}$.^{16}
Experiments on the velocity of sound having been made in air under the pressure of 760 millimetres of mercury and at the temperature of 6°, it is only to these two circumstances that our datum has reference. We will, however, for greater facility, refer it to the temperature 0°, which is nearly the same.
Air compressed $\frac{1}{116}$, and thus heated one degree, differs from air heated directly one degree only in its density. The primitive volume being supposed to be V, the compression of $\frac{1}{116}$ reduces it to V  $\frac{1}{116}$V.
Direct heating under constant pressure should, according to the rule of M. GayLussac, increase the volume of air $\frac{1}{267}$ above what it would be at 0°: so the air is, on the one hand, reduced to the volume V  $\frac{1}{116}$V; on the other, it is increased to V + $\frac{1}{267}$V.
The difference between the quantities of heat which the air possesses in both cases is evidently the quantity employed to raise it directly one degree; so then the quantity of heat that the air would absorb in passing from the volume V  $\frac{1}{116}$V to the volume V + $\frac{1}{267}$V is equal to that which is required to raise it one degree.
Let us suppose now that, instead of heating one degree the air subjected to a constant pressure and able to dilate freely, we inclose it within an invariable space, and that in this condition we cause it to rise one degree in temperature. The air thus heated one degree will differ from the air compressed $\frac{1}{116}$ only by its $\frac{1}{116}$ greater volume. So then the quantity of heat that the air would set free by a reduction of volume of $\frac{1}{116}$ is equal to that which would be required to raise it one degree Centigrade under constant volume. As the differences between the volumes V  $\frac{1}{116}$V, V, and V + $\frac{1}{267}$V are small relatively to the volumes themselves, we may regard the quantities of heat absorbed by the air in passing from the first of these volumes to the second, and from the first to the third, as sensibly proportional to the changes of volume. We are then led to the establishment of the following relation:
The quantity of heat necessary to raise one degree air under constant pressure is to the quantity of heat necessary to raise one degree the same air under constant volume, in the ratio of the numbers
\begin{equation} \frac{1}{116} + \frac{1}{267} \phantom{XX} \text{to} \phantom{XX} \frac{1}{116}V \end{equation}or, multiplying both by 116 $\times$ 267, in the ratio of the numbers 267 + 116 to 267.
This, then, is the ratio which exists between the capacity of air for heat under constant pressure and its capacity under constant volume. If the first of these two capacities is expressed by unity, the other will be expressed by the number $\frac{267}{267 + 116}$ or very nearly 0.700; their difference, 1  0.700 or 0.300, will evidently express the quantity of heat which will produce the increase of volume in the air when it is heated one degree under constant pressure.
According to the law of MM. GayLussac and Dalton, this increase of volume would be the same for all other gases; according to the theory demonstrated on page 87, the heat absorbed by these equal increases of volume is the same for all the elastic fluids, which leads to the establishment of the following proposition:
The difference between specific heat under constant pressure and specific heat under constant volume is the same for all gases.
It should be remarked here that all the gases are considered as taken under the same pressure, atmospheric pressure for example, and that the specific heats are also measured with reference to the volumes.
It is a very easy matter now for us to prepare a table of the specific heat of gases under constant volume, from the knowledge of their specific heats under constant pressure. Here is the table:
The first column is the result of the direct experiments of MM. Delaroche and Berard on the specific heat of the gas under atmospheric pressure, and the second column is composed of the numbers of the first diminished by 0.300.
The numbers of the first column and those of the second are here referred to the same unit, to the specific heat of atmospheric air under constant pressure.
The difference between each number of the first column and the corresponding number of the second being constant, the relation between these numbers should be variable. Thus the relation between the specific heat of gases under constant pressure and the specific heat at constant volume, varies in different gases.
We have seen that air when it is subjected to a sudden compression of $\frac{1}{116}$ of its volume rises one degree in temperature. The other gases through a similar compression should also rise in temperature. They should rise, but not equally, in inverse ratio with their specific heat at constant volume. In fact, the reduction of volume being by hypothesis always the same, the quantity of heat due to this reduction should likewise be always the same, and consequently should produce an elevation of temperature dependent only on the specific heat acquired by the gas after its compression, and evidently in inverse ratio with this specific heat. Thus we can easily form the table of the elevations of temperature of the different gases for a compression of $\frac{1}{116}$.
A second compression of $\frac{1}{116}$ $($of the altered volume$)$, as we shall presently see, would also raise the temperature of these gases nearly as much as the first; but it would not be the same with a third, a fourth, a hundredth such compression. The capacity of gases for heat changes with their volume. It is not unlikely that it changes also with the temperature.
We shall now deduce from the general proposition stated on page 68 a second theory, which will serve as a corollary to that just demonstrated.
Let us suppose that the gas enclosed in the cylindrical space abcd $($Fig. 2$)$ be transported into the space a'b'c'd' $($Fig. 3$)$ of equal height, but of different base and wider. This gas would increase in volume, would diminish in density and in elastic force, in the inverse ratio of the two volumes abcd, a'b'c'd'. As to the total pressure exerted in each piston cd, c'd', it would be the same from all quarters, for the surface of these pistons is in direct ratio to the volumes.
Let us suppose that we perform on the gas inclosed in a'b'c'd' the operations described on page 70, and which were taken as having been performed upon the gas inclosed in abcd; that is, let us suppose that we have given to the piston c'd' motions equal to those of the piston cd, that we have made it occupy successively the positions c'd' corresponding to cd, and e'f' corresponding to ef, and that at the same time we have subjected the gas by means of the two bodies A and B to the same variations of temperature as when it was inclosed in abcd The total effort exercised on the piston would be found to be, in the two cases, always the same at the corresponding instants. This results solely from the law of Mariotte ^{17}. In fact, the densities of the two gases maintaining always the same ratio for similar positions of the pistons, and the temperatures being always equal in both, the total pressures exercised on the pistons will always maintain the same ratio to each other. If this ratio is, at any instant whatever, unity, the pressures will always be equal.
As, furthermore, the movements of the two pistons have equal extent, the motive power produced by each will evidently be the same; whence we should conclude, according to the proposition on page 68, that the quantities of heat consumed by each are the same, that is, that there passes from the body A to the body B the same quantity of heat in both cases.
The heat abstracted from the body A and communicated to the body B, is simply the heat absorbed during the rarefaction of the gas, and afterwards liberated by its compression. We are therefore led to establish the following theorem:
When an elastic fluid passes without change of temperature from the volume U to the volume V, and when a similar ponderable quantity of the same gas passes at the same temperature from the volume U' to the volume V', if the ratio of U' to V' is found to be the same as the ratio of U to V, the quantities of heat absorbed or disengaged in the two cases will be equal.
This theorem might also be expressed as follows:
When a gas varies in volume without change of temperature, the quantities of heat absorbed or liberated by this gas are in arithmetical progression, if the increments or the decrements of volume are found to be in geometrical progression.
When a litre of air maintained at a temperature of ten degrees is compressed, and when it is reduced to one half a litre, a certain quantity of heat is set free. This quantity will be found always the same if the volume is further reduced from a half litre to a quarter litre, from a quarter litre to an eighth, and so on.
If, instead of compressing the air, we carry it successively to two litres, four litres, eight litres, etc., it will be necessary to supply to it always equal quantities of heat in order to maintain a constant temperature.
This readily accounts for the high temperature attained by air when rapidly compressed. We know that this temperature inflames tinder and even makes air luminous. If, for a moment, we suppose the specific heat of air to be constant, in spite of the changes of volume and temperature, the temperature will increase in arithmetical progression for reduction of volume in geometrical progression.
Starting from this datum, and admitting that one degree of elevation in the temperature corresponds to a compression of $\frac{1}{116}$, we shall readily come to the conclusion that air reduced to $\frac{1}{14}$ of its primitive volume should rise in temperature about 300 degrees, which is sufficient to inflame tinder. ^{18}
The elevation of temperature ought, evidently, to be still more considerable if the capacity of the air for heat becomes less as its volume diminishes. Now this is probable, and it also seems to follow from the experiments of MM. Delaroche and Bérard on the specific heat of air taken at different densities. $($See the Memoire in the Annales de Chimie, t. Ixxxv. pp. 72,224$)$. The two theorems explained on pp. 72 and 81 suffice for the comparison of the quantities of heat absorbed or set free in the changes of volume of elastic fluids, whatever may be the density and the chemical nature of these fluids, provided always that they be taken and maintained at a certain invariable temperature. But these theories furnish no means of comparing the quantities of heat liberated or absorbed by elastic fluids which change in volume at different temperatures. Thus we are ignorant what relation exists between the heat relinquished by a litre of air reduced one half, the temperature being kept at zero, and the heat relinquished by the same litre of air reduced one half, the temperature being kept at 100°. The knowledge of this relation is closely connected with that of the specific heat of gases at various temperatures, and to some other data that Physics as yet does not supply.
The second of our theorems offers us a means of determining according to what law the specific heat of gases varies with their density.
Let us suppose that the operations described on p. 70, instead of being performed with two bodies, A, B, of temperatures differing indefinitely small, were carried on with two bodies whose temperatures differ by a finite quantity one degree, for example. In a complete circle of operations the body A furnishes to the elastic fluid a certain quantity of heat, which may be divided into two portions: $($1$)$ That which is necessary to maintain the temperature of the fluid constant during dilatation; $($2$)$ that which is necessary to restore the temperature of the fluid from that of the body B to that of the body A, when, after having brought back this fluid to its primitive volume, we place it again in contact with the body A. Let us call the first of these quantities a and the second h. The total caloric furnished by the body A will be expressed by a + b.
The caloric transmitted by the fluid to the body B may also be divided into two parts: one, b', due to the cooling of the gas by the body B; the other, a', which the gas abandons as a result of its reduction of volume. The sum of these two quantities is a' + b'; it should be equal to a + b, for, after a complete cycle of operations, the gas is brought back exactly to its primitive state. It has been obliged to give up all the caloric which has first been furnished to it. We have then
a + b = a' + b';
or rather,
a  a' = b'  b.
Now, according to the theorem given on page 81, the quantities a and a' are independent of the density of the gas, provided always that the ponderable quantity remains the same and that the variations of volume be proportional to the original volume. The difference a  a' should fulfil the same conditions, and consequently also the difference b'  b, which is equal to it. But b' is the caloric necessary to raise the gas enclosed in abcd $($Fig. 2$)$ one degree; b' is the caloric surrendered by the gas when, enclosed in abef, it is cooled one degree. These quantities may serve as a measure for specific heats. We are then led to the establishment of the following proposition:
The change in the specific heat of a gas caused by change of volume depends entirely on the ratio between the original volume and the altered volume. That is, the difference of the specific heats does not depend on the absolute magnitude of the volumes, but only on their ratio.
This proposition might also be differently expressed, thus:
When a gas increases in volume in geometrical progression, its specific heat increases in arithmetical progression
Thus, a being the specific heat of air taken at a given density, and a + h the specific heat for a density one half less, it will be, for a density equal to one quarter, a + 2h; for a density equal to one eighth, a + 3h; and so on.
The specific heats are here taken with reference to weight. They are supposed to be taken at an invariable volume, but, as we shall see, they would follow the same law if they were taken under constant pressure.
To what cause is the difference between specific heats at constant volume and at constant pressure really due? To the caloric required to produce in the second case increase of volume. Now, according to the law of Mariotte, increase of volume of a gas should be, for a given change of temperature, a determined fraction of the original volume, a. fraction independent of pressure. According to the theorem expressed on page 76, if the ratio between the primitive volume and the altered volume is given, that determines the heat necessary to produce increase of volume. It depends solely on this ratio and on the weight of the gas. We must then conclude that:
The difference between specific heat at constant pressure and specific heat at constant volume is always the same, whatever may be the density of the gas, provided the weight remains the same.
These specific heats both increase accordingly as the density of the gas diminishes, but their difference does not vary.^{19}
Since the difference between the two capacities for heat is constant, if one increases in arithmetical progression the other should follow a similar progression: thus one law is applicable to specific heats at constant pressure.
We have tacitly assumed the increase of specific heat with that of volume. This increase is indicated by the experiments of MM. Delaroche and Bérard: in fact these physicists have found 0.967 for the specific heat of air under the pressure of 1 metre of mercury $($see Mémoire already cited$)$, taking for the unit the specific heat of the same weight of air under the pressure of 0$^m$.760.
According to the law that specific heats follow with relation to pressures, it is only necessary to have observed them in two particular cases to deduce them in all possible cases: it is thus that, making use of the experimental result of MM. Delaroche and Bérard which has just been given, we have prepared the following table of the specific heat of air under different pressures:
The first column is, as we see, a geometrical progression, and the second an arithmetical progression.
We have carried out the table to the extremes of compression and rarefaction. It may be believed that air would be liquefied before acquiring a density 1024 times its normal density, that is, before becoming more dense than water. The specific heat would become zero and even negative on extending the table beyond the last term. We think, furthermore, that the figures of the second column here decrease too rapidly. The experiments which serve as a basis for our calculation have been made within too contracted limits for us to expect great exactness in the figures which we have obtained, especially in the outside numbers.
Since we know, on the one hand, the law according to which heat is disengaged in the compression of gases, and on the other, the law according to which specific heat varies with volume, it will be easy for us to calculate the increase of temperature of a gas that has been compressed without being allowed to lose heat. In fact, the compression may be considered as composed of two successive operations: $($1$)$ compression at a constant temperature; $($2$)$ restoration of the caloric emitted. The temperature will rise through the second operation in inverse ratio with the specific heat acquired by the gas after the reduction of volume, specific heat that we are able to calculate by means of the law demonstrated above. The heat set free by compression, according to the theorem of page 81, ought to be represented by an expression of the form
\begin{equation} s = A + B \log v, \end{equation}s being this heat, v the volume of the gas after compression, A and B arbitrary constants dependent on the primitive volume of the gas, on its pressure, and on the units chosen.
The specific heat varying with the volume according to the law just demonstrated, should be represented by an expression of the form
\begin{equation} z = A' + B' \log v, \end{equation}A' and B' being the different arbitrary constants of A and B.
The increase of temperature acquired by the gas, as the effect of compression, is proportional to the ratio $\frac{s}{z}$ or to the relation $\frac{A + B\log v}{A' + B' \log v}$. It can be represented by this ratio itself; thus, calling it t, we shall have
\begin{equation} t = \frac{A + B\log v}{A' + B' \log v} \end{equation}If the original volume of the gas is 1, and the original temperature zero, we shall have at the same time t = 0, $\log v$ = 0, whence A = 0; t will then express not only the increase of temperature, but the temperature itself above the thermometric zero.
We need not consider the formula that we have just given as applicable to very great changes in the volume of gases. We have regarded the elevation of temperature as being in inverse ratio to the specific heat; which tacitly supposes the specific heat to be constant at all temperatures. Great changes of volume lead to great changes of temperature in the gas, and nothing proves the constancy of specific heat at different temperatures, especially at temperatures widely separated. This constancy is only an hypothesis admitted for gases by analogy, to a certain extent verified for solid bodies and liquids throughout a part of the thermometric scale, but of which the experiments of MM. Dulong and Petit have shown the inaccuracy when it is desirable to extend it to temperatures far above 100°.^{20}
According to a law of MM. Clement and Desormes, a law established by direct experiment, the vapor of water, under whatever pressure it may be formed, contains always, at equal weights, the same quantity of heat; which leads to the assertion that steam, compressed or expanded mechanically without loss of heat, will always be found in a saturated state if it was so produced in the first place. The vapor of water so made may then be regarded as a permanent gas, and should observe all the laws of one. Consequently the formula
\begin{equation} t = \frac{A + B\log v}{A' + B' \log v} \end{equation}should be applicable to it, and be found to accord with the table of tensions derived from the direct experiments of M. Dalton.
We may be assured, in fact, that our formula, with a convenient determination of arbitrary constants, represents very closely the results of experiment. The slight irregularities which we find therein do not exceed what we might reasonably attribute to errors of observation.^{21}
We will return, however, to our principal subject, from which we have wandered too far the motive power of heat.
We have shown that the quantity of motive power developed by the transfer of caloric from one body to another depends essentially upon the temperature of the two bodies, but we have not shown the relation between these temperatures and the quantities of motive power produced. It would at first seem natural enough to suppose that for equal differences of temperature the quantities of motive power produced are equal; that is, for example, the passage of a given quantity of caloric from a body, A, maintained at 100°, to a body, B, maintained at 50°, should give rise to a quantity of motive power equal to that which would be developed by the transfer of the same caloric from a body, B, at 50°, to a body, C, at zero. Such a law would doubtless be very remarkable, but we do not see sufficient reason for admitting it á priori. We will investigate its reality by exact reasoning.
Let us imagine that the operations described on p. 70 be conducted successively on two quantities of atmospheric air equal in weight and volume, but taken at different temperatures. Let us suppose, further, the differences of temperature between the bodies A and B equal, so these bodies would have for example, in one of these cases, the temperatures 100° and 100°  h $($h being indefinitely small$)$, and in the other 1° and 1°  h. The quantity of motive power produced is, in each case, the difference between that which the gas supplies by its dilatation and that which must be expended to restore its primitive volume. Now this difference is the same in both cases, as anyone can prove by simple reasoning, which it seems unnecessary to give here in detail; hence the motive power produced is the same.
Let us now compare the quantities of heat employed in the two cases. In the first, the quantity of heat employed is that which the body $A$ furnishes to the air to maintain it at the temperature of 100° during its expansion. In the second, it is the quantity of heat which this same body should furnish to it, to keep its temperature at one degree during an exactly similar change of volume. If these two quantities of heat were equal, there would evidently result the law that we have already assumed. But nothing proves that it is so, and we shall find that these quantities are not equal.
The air that we shall first consider as occupying the space abcd $($Fig. 2$)$, and having 1 degree of temperature, can be made to occupy the space abef, and to acquire the temperature of 100 degrees by two different means:
$($1$)$ We may heat it without changing its volume, then expand it, keeping its temperature constant.
$($2$)$ We may begin by expanding it, maintaining the temperature constant, then heat it, when it has acquired its greater volume. Let a and b be the quantities of heat employed successively in the first of the two operations, and let b' and a' be the quantities of heat employed successively in the second. As the final result of these two operations is the same, the quantities of heat employed in both should be equal. We have then
\begin{equation} a + b = a' + b', \end{equation}whence
\begin{equation} a'  a = b b'. \end{equation}a' is the quantity of heat required to cause the gas to rise from 1° to 100° when it occupies the space abef.
a is the quantity of heat required to cause the gas to rise from 1° to 100° when it occupies the space abcd.
The density of the air is less in the first than in the second case, and according to the experiments of MM. Delaroche and Bérard, already cited on page 87, its capacity for heat should be a little greater.
The quantity a' being found to be greater than the quantity a, b should be greater than b'. Consequently, generalizing the proposition, we should say:
The quantity of heat due to the change of volume of a gas is greater as the temperature is higher.
Thus, for example, more caloric is necessary to maintain at 100° the temperature of a certain quantity of air the volume of which is doubled, than to maintain at 1° the temperature of this same air during a dilatation exactly equal.
These unequal quantities of heat would produce, however, as we have seen, equal quantities of motive power for equal fall of caloric taken at different heights on the thermometric scale; whence we draw the following conclusion:
The fall of caloric produces more motive power at inferior than at superior temperatures.
Thus a given quantity of heat will develop more motive power in passing from a body kept at 1 degree to another maintained at zero, than if these two bodies were at the temperature of 101° and 100°.
The difference, however, should be very slight. It would be nothing if the capacity of the air for heat remained constant, in spite of changes of density. According to the experiments of MM. Delaroche and Bérard, this capacity varies little so little even, that the differences noticed might strictly have been attributed to errors of observation or to some circumstances of which we have failed to take account.
We are not prepared to determine precisely, with no more experimental data than we now possess, the law according to which the motive power of heat varies at different points on the thermometric scale. This law is intimately connected with that of the variations of the specific heat of gases at different temperatures a law which experiment has not yet made known to us with sufficient exactness.^{22}
We will endeavor now to estimate exactly the motive power of heat, and in order to verify our fundamental proposition, in order to determine whether the agent used to realize the motive power is really unimportant relatively to the quantity of this power, we will select several of them successively: atmospheric air, vapor of water, vapor of alcohol.
Let us suppose that we take first atmospheric air. The operation will proceed according to the method indicated on page 70. We will make the following hypotheses: The air is taken under atmospheric pressure. The temperature of the body A is $\frac{1}{1000}$ of a degree above zero, that of the body B is zero. The difference is, as we see, very slight a necessary condition here.
The increase of volume given to the air in our operation will be $\frac{1}{116} + \frac{1}{267}$ of the primitive volume; this is a very slight increase, absolutely speaking, but great relatively to the difference of temperature between the bodies A and B.
The motive power developed by the whole of the two operations described $($page 70$)$ will be very nearly proportional to the increase of volume and to the difference between the two pressures exercised by the air, when it is found at the temperatures 0°.001 and zero.
This difference is, according to the law of M. GayLussac, $\frac{1}{267000}$ of the elastic force of the gas, or very nearly $\frac{1}{267000}$ of the atmospheric pressure.
The atmospheric pressure balances at 10.40 metres head of water; $\frac{1}{267000}$ of this pressure equals $\frac{1}{267000} \times 10^{\text{m}}.40$ of head of water.
As to the increase of volume, it is, by supposition, $\frac{1}{116} + \frac{1}{267}$ of the original volume, that is, of the volume occupied by one kilogram of air at zero, a volume equal to 0$^{\text{mc}}$.77, allowing for the specific weight of the air. So then the product,
\begin{equation} \left( \frac{1}{116} + \frac{1}{267} \right)\times 0.77 \times \frac{1}{267000}\times 10.40 \end{equation}will expres the motive power developed. This power is estimated here in cubic metres of water raised one metre. If we carry out the indicated multiplications, we find the value of the product to be 0.000000372.
Let us endeavor now to estimate the quantity of heat employed to give this result; that is, the quantity of heat passed from the body A to the body B.
The body A furnishes:
$($1$)$ The heat required to carry the temperature of one kilogram of air from zero to 0°.001;
$($2$)$ The quantity necessary to maintain at this temperature the temperature of the air when it experiences a dilatation of
\begin{equation} \frac{1}{116} + \frac{1}{267} \end{equation}The first of these quantities of heat being very small in comparison with the second, we may disregard it. The second is, according to the reasoning on page, 4, equal to that which would be necessary to increase one degree the temperature of one kilogram of air subjected to atmospheric pressure.
According to the experiments of MM. Delaroche and Bérard on the specific heat of gases, that of air is, for equal weights, 0.267 that of water. If, then, we take for the unit of heat the quantity necessary to raise 1 kilogram of water 1 degree, that which will be required to raise 1 kilogram of air 1 degree would have for its value 0.267. Thus the quantity of heat furnished by the body A is
\begin{equation} 0.267 \text{ units.} \end{equation}This is the heat capable of producing 0.000000372 units of motive power by its fall from 0°.001 to zero.
For a fall a thousand times greater, for a fall of one degree, the motive power will be very nearly a thousand times the former, or
\begin{equation} 0.000372. \end{equation}If, now, instead of 0.267 units of heat we employ 1000 units, the motive power produced will be expressed by the proportion
\begin{equation} \frac{0.267}{0.000000372} = \frac{1000}{x} \text{ whence } x = \frac{372}{267} = 1.395 \end{equation}Thus 1000 units of heat passing from a body maintained at the temperature of 1 degree to another body maintained at zero would produce, in acting upon the air,
\begin{equation} 1.395 \text{ units of motive power.} \end{equation}We will now compare this result with that furnished by the action of heat on the vapor of water,
Let us suppose one kilogram of liquid water enclosed in the cylindrical vessel abcd $($Fig. 4$)$, between the bottom ab and the piston cd. Let us suppose, also, the two bodies A, B maintained each at a constant temperature, that of A being a very little above that of B. Let us imagine now the following operations:
$($1$)$ Contact of the water with the body A, movement of the piston from the position cd to the position ef, formation of steam at the temperature of the body A to fill the vacuum produced by the extension of volume. We will suppose the space abef large enough to contain all the water in a state of vapor.
$($2$)$ Removal of the body A, contact of the vapor with the body B, precipitation of a part of this vapor, diminution of its elastic force, return of the piston from ef to ab, liquefaction of the rest of the vapor through the effect of the pressure combined with the contact of the body B.
$($3$)$ Removal of the body B, fresh contact of the water with the body A, return of the water to the temperature of this body, renewal of the former period, and so on.
The quantity of motive power developed in a complete cycle of operations is measured by the product of the volume of the vapor multiplied by the difference between the tensions that it possesses at the temperature of the body A and at that of the body B. As to the heat employed, that is to say, transported from the body A to the body B, it is evidently that which was necessary to turn the water into vapor, disregarding always the small quantity required to restore the temperature of the liquid water from that of B to that of A.
Suppose the temperature of the body A 100 degrees, and that of the body B 99 degrees: the difference of the tensions will be, according to the table of M. Dalton, 26 millimetres of mercury or 0$^{\text{m}}$.36 head of water.
The volume of the vapor is 1700 times that of the water. If we operate on one kilogram, that will be 1700 litres, or 1$^\text{mc}$.700.
Thus the value of the motive power developed is the product
\begin{equation} 1.700 \times 0.36 = 0.611 \text{ units}, \end{equation}of the kind of which we have previously made use.
The quantity of heat employed is the quantity required to turn into vapor water already heated to 100°. This quantity is found by experiment. We have found it equal to 550°, or, to speak more exactly, to 550 of our units of heat.
Thus 0.611 units of motive power result from the employment of 550 units of heat. The quantity of motive power resulting from 1000 units of heat will be given by the proportion
\begin{equation} \frac{550}{0.611} = \frac{1000}{x}, \phantom{XX}\text{ whence }\phantom{XX} x = \frac{611}{550} = 1.112 \end{equation}Thus 1000 units of heat transported from one body kept at 100 degrees to another kept at 99 degrees will produce, acting upon vapor of water, 1.112 units of motive power.
The number 1.112 differs by about $\frac{1}{4}$ from the number 1.395 previously found for the value of the motive power developed by 1000 units of heat acting upon the air; but it should be observed that in this case the temperatures of the bodies A and B were 1 degree and zero, while here they are 100 degrees and 99 degrees. The difference is much the same; but it is not found at the same height in the thermometric scale. To make an exact comparison, it would have been necessary to estimate the motive power developed by the steam formed at 1 degree and condensed at zero. It would also have been necessary to know the quantity of heat contained in the steam formed at one degree.
The law of MM. Clement and Desormes referred to on page 92 gives us this datum. The constituent heat of vapor of water being always the same at any temperature at which vaporization takes place, if 550 degrees of heat are required to vaporize water already brought up to 100 degrees, 550 + 100 or 650 will be required to vaporize the same weight of water taken at zero.
Making use of this datum and reasoning exactly as we did for water at 100 degrees, we find, as is easily seen,
\begin{equation} 1.290 \end{equation}for the motive power developed by 1000 units of heat acting upon the vapor of water between one degree and zero. This number approximates more closely than the first to
\begin{equation} 1.395. \end{equation}It differs from it only $\frac{1}{13}$, an error which does not exceed probable limits, considering the great number of data of different sorts of which we have been obliged to make use in order to arrive at this approximation. Thus is our fundamental law verified in a special case.^{23}
We will examine another case in which vapor of alcohol is acted upon by heat. The reasoning is precisely the same as for the vapor of water. The data alone are changed. Pure alcohol boils under ordinary pressure at 78°.7 Centigrade. One kilogram absorbs, according to MM. Delaroche and Bérard, 207 units of heat in undergoing transformation into vapor at this same temperature, 78°.7.
The tension of the vapor of alcohol at one degree below the boilingpoint is found to be diminished $\frac{1}{25}$. It is $\frac{1}{25}$ less than the atmospheric pressure; at least, this is the result of the experiment of M. Bétancour reported in the second part of l' Architecture hydraulique of M. Prony, pp. 180, 195.^{24}
If we use these data, we find that, in acting upon one kilogram of alcohol at the temperatures of 78°.7 and 77°.7, the motive power developed will be 0.251 units.
This results from the employment of 207 units of heat. For 1000 units the proportion must be
\begin{equation} \frac{207}{0.254} = \frac{1000}{x}, \phantom{XX} \text{ whence } \phantom{XX} x = 1.230 \end{equation}This number is a little more than the 1.112 resulting from the use of the vapor of water at the temperatures 100° and 99°; but if we suppose the vapor of water used at the temperatures 78° and 77°, we find, according to the law of MM. Clement and Desorme, 1.212 for the motive power due to 1000 units of heat. This latter number approaches, as we see, very nearly to 1.230. There is a difference of only $\frac{1}{50}$.
We should have liked to be able to make other approximations of this sort to be able to calculate, for example, the motive power developed by the action of heat on solids and liquids, by the congelation of water, and so on; but Physics as yet refuses us the necessary data.^{25}
The fundamental law that we propose to confirm seems to us to require, however, in order to be placed beyond doubt, new verifications. It is based upon the theory of heat as it is understood today, and it should be said that this foundation does not appear to be of unquestionable solidity. New experiments alone can decide the question. Meanwhile we can apply the theoretical ideas expressed above, regarding them as exact, to the examination of the different methods proposed up to date, for the realization of the motive power of heat.
It has sometimes been proposed to develop motive power by the action of heat on solid bodies. The mode of procedure which naturally first occurs to the mind is to fasten immovably a solid body  metallic bar, for example by one of its extremities; to attach the other extremity to a movable part of the machine; then, by successive heating and cooling, to cause the length of the bar to vary, and so to produce motion. Let us try to decide whether this method of developing motive power can be advantageous. We have shown that the condition of the most effective employment of heat in the production of motion is, that all changes of temperature occurring in the bodies should be due to changes of volume. The nearer we come to fulfilling this condition the more fully will the heat be utilized. Now, working in the manner just described, we are very far from fulfilling this condition: change of temperature is not due here to change of volume; all the changes are due to contact of bodies differently heated to the contact of the metallic bar, either with the body charged with furnishing heat to it, or with the body charged with carrying it off.
The only means of fulfilling the prescribed condition would be to act upon the solid body exactly as we did on the air in the operations described on page 92. But for this we must be able to produce, by a single change of volume of the solid body, considerable changes of temperature, that is, if we should want to utilize considerable falls of caloric. Now this appears impracticable. In short, many considerations lead to the conclusion that the changes produced in the temperature of solid or liquid bodies through the effect of compression and rarefaction would be but slight.
$($1$)$ We often observe in machines $($particularly in steamengines$)$ solid pieces which endure considerable strain in one way or another, and although these efforts may be sometimes as great as the nature of the substances employed permits, the variations of temperature are scarcely perceptible.
$($2$)$ In the action of striking medals, in that of the rollingmill, of the drawplate, the metals undergo the greatest compression to which we can submit them, employing the hardest and strongest tools. Nevertheless the elevation of temperature is not great. If it were, the pieces of steel used in these operations would soon lose their temper.
$($3$)$ We know that it would be necessary to exert on solids and liquids a very great strain in order to produce ill them a reduction of volume comparable to· that which they experience in cooling $($cooling from 100° to zero, for example$)$. Now the cooling requires a greater abstraction of caloric than would simple reduction of volume. If this reduction were produced by mechanical means, the heat set free would not then be able to make the temperature of the body vary as many degrees as the cooling makes it vary. It would, however, necessitate the employment of a force undoubtedly very considerable.
Since solid bodies are susceptible of little change of temperature through changes of volume, and since the condition of the most effective employment of heat for the development of motive power is precisely that all change of temperature should be due to a change of volume, solid bodies appear but ill fitted to realize this power.
The same remarks apply to liquids. The same reasons may be given for rejecting them.^{26}
We are not speaking now of practical difficulties. They will be numberless. The motion produced by the dilatation and compression of solid or liquid bodies would only be very slight. In order to give them sufficient amplitude we should be forced to make use of complicated mechanisms. It would be necessary to employ materials of the greatest strength to transmit enormous pressure; finally, the successive operations would be executed very slowly compared to those of the ordinary steamengine, so that apparatus of large dimensions and heavy cost would produce but very ordinary results.
The elastic fluids, gases or vapors, are the means really adapted to the development of the motive power of heat. They combine aU the conditions necessary to fulfil this office. They are easy to compress; they can be almost infinitely expanded; variations of volume occasion in them great changes of temperature; and, lastly, they are very mobile, easy to heat and to cool, easy to transport from one place to another, which enables them to produce rapidly the desired effects. We can easily conceive a multitude of machines fitted to develop the motive power of heat through the use of elastic fluids; but in whatever way we look at it, we should not lose sight of the following principles:
$($1$)$ The temperature of the fluid should be made as high as possible, in order to obtain a great fall of caloric, and consequently a large production of motive power.
$($2$)$ For the same reason the cooling should be carried as far as possible.
$($3$)$ It should be so arranged that the passage of the elastic fluid from the highest to the lowest temperature should be due to increase of volume; that is, it should be so arranged that the cooling of the gas should occur spontaneously as the effect of rarefaction. The limits of the temperature to which it is possible to bring the fluid primarily, are simply the limits of the temperature obtainable by combustion; they are very high.
The limits of cooling are found in the temperature of the coldest body of which we can easily and freely make use; this body is usually the water of the locality.
As to the third condition, it involves difficulties in the realization of the motive power of heat when the attempt is made to take advantage of great differences of temperature, to utilize great falls of heat. In short, it is necessary then that the gas, by reason of its rarefaction, should pass from a very high temperature to a very low one, which requires a great change of volume and of density, which requires also that the gas be first taken under a very heavy pressure, or that it acquire by its dilatation an enormous volume conditions both difficult to fulfil. The first necessitates the employment of very strong vessels to contain the gas at a very high temperature and under very heavy pressure. The second necessitates the use of vessels of large dimensions. These are, in a word, the principal obstacles which prevent the utilization in steamengines of a great part of the motive power of the heat. We are obliged to limit ourselves to the use of a slight fall of caloric, while the combustion of the coal furnishes the means of procuring a very great one.
It is seldom that in steamengines the elastic fluid is produced under a higher pressure than six atmospheresa pressure corresponding to about 160° Centigrade, and it is seldom that condensation takes place at a temperature much under 40°. The faU of caloric from 160° to 40° is 120°, while by combustion we can procure a fall of 1000° to 2000°.
In order to comprehend this more clearly, let us recall what we have termed the fall of caloric. This is the passage of the heat from one body, A, having an elevated temperature, to another, B, where it is lower. We say that the fall of the caloric is 100° or 1000° when the difference of temperature between the bodies A and B is 100° or 1000°.
In a steamengine which works under a pressure of six atmospheres the temperature of the boiler is 160°. This is the body A. It is kept, by contact with the furnace, at the constant temperature of 160°, and continually furnishes the heat necessary for the formation of steam. The condenser is the body B. By means of a current of cold water it is kept at a nearly constant temperature of 40°. It absorbs continually the caloric brought from the body $A$ by the steam. The difference of temperature between these two bodies is 160°  40°, or 120°. Hence we say that the fall of caloric is here 120°.
Coal being capable of producing, by its combustion, a temperature higher than 1000°, and the cold water, which is generally used in our climate, being at about 10°, we can easily procure a fall of caloric of 1000°, and of this only 120° are utilized by steamengines. Even these 120° are not wholly utilized. There is always considerable loss due to useless reestablishments of equilibrium in the caloric.
It is easy to see the advantages possessed by highpressure machines over those of lower pressure. This superiority lies essentially in the power of utilizing a greater fall of caloric. The steam produced under a higher pressure is found also at a higher temperature, and as, further, the temperature of condensation remains always about the same, it is evident that the fall of caloric is more considerable. But to obtain from highpressure engines really advantageous results, it is necessary that the fall of caloric should be most profitably utilized. It is not enough that the steam be produced at a high temperature: it is also necessary that by the expansion of its volume its temperature should become sufficiently low. A good steamengine, therefore, should not only employ steam under heavy pressure, but under successive and very variable pressures, differing greatly from one another, and progressively decreasing. ^{27}
In order to understand in some sort á posteriori the advantages of high pressure engines, let us suppose steam to be formed under atmospheric pressure and introduced into the cylindrical vessel abcd $($Fig. 5$)$, under the piston cd, which at first touches the bottom ab. The steam, after having moved the piston from ab to cd, will continue finally to produce its results in a manner with which we will not concern ourselves.
Let us suppose that the piston having moved to cd is forced downward to ef, without the steam being allowed to escape, or any portion of its caloric to be lost. It will be driven back into the space abef, and will increase at the same time in density, elastic force, and temperature. If the steam, instead of being produced under atmospheric pressure, had been produced just when it was being forced back into abef, and so that after its introduction into the cylinder it had made the piston move from ab to ef, and had moved it simply by its extension of volume, from ef to cd, the motive power produced would have been more considerable than in the first case. In fact, the movement of the piston, while equal in extent, would have taken place under the action of a greater pressure, though variable, and though progressively decreasing.
The steam, however, would have required for its formation exactly the same quantity of caloric, only the caloric would have been employed at a higher temperature.
It is considerations of this nature which have led to the making of doublecylinder enginesengines invented by Mr. Hornblower, improved by Mr. Woolf, and which, as regards economy of the combustible, are considered the best. They consist of a small cylinder, which at each pulsation is filled more or less $($often entirely$)$ with steam, and of a second cylinder having usually a capacity quadruple that of the first, and which receives no steam except that which has already operated in the first cylinder. Thus the steam when it ceases to act has at least quadrupled in volume. From the second cylinder it is carried directly into the condenser, but it is conceivable that it might be carried into a third cylinder quadruple the second, and in which its volume would have become sixteen times the original volume. The principal obstacle to the use of a third cylinder of this sort is the capacity which it would be necessary to give it, and the large dimensions which the openings for the passage of the steam must have.^{28} We will say no more on this subject, as we do not propose here to enter into the details of construction of steamengines. These details call for a work devoted specially to them, and which does not yet exist, at least in France.^{29}
If the expansion of the steam is mainly limited by the dimensions of the vessels in which the dilatation must take place, the degree of condensation· at which it is possible to use it at first is limited only by the resistance of the vessels in which it is produced, that is, of the boilers.
In this respect we have by no means attained the best possible results. The arrangement of the boilers generally in use is entirely faulty, although the tension of the steam rarely exceeds from four to six atmospheres. They often burst and cause severe accidents. It will undoubtedly be possible to avoid such accidents, and meantime to raise the steam to much greater pressures than is usually done.
Besides the highpressure doublecylinder engines of which we have spoken, there are also highpressure engines of one cylinder. The greater part of these latter have been constructed by two ingenious English engineers, Messrs. Trevithick and Vivian. They employ the steam under a very high pressure, sometimes eight to ten atmospheres, but they have no condenser. The steam, after it has been introduced into the cylinder, undergoes therein a certain increase of volume, but preserves always a pressure higher than atmospheric. When it has fulfilled its office it is thrown out into the atmosphere. It is evident that this mode of working is fully equivalent, in respect to the motive power produced, to condensing the steam at 100°, and that a portion of the useful effect is lost. But the engines working thus dispense with condenser and airpump. They are less costly than the others, less complicated, occupy less space, and can be used in places where there is not sufficient water for condensation. In such places they are of inestimable advantage, since no others could take their place. These engines are principally employed in England to move coalwagons on railroads laid either in the interior of mines or outside of them.
We have, further, only a few remarks to make upon the use of permanent gases and other vapors than that of water in the development of the motive power of heat.
Various attempts have been made to produce motive power by the action of heat on atmospheric air. This gas presents, as compared with vapor of water, both advantages and disadvantages, which we will proceed to examine.
$($1$)$ It presents, as compared with vapor of water, a notable advantage in that, having for equal volume a much less capacity for heat, it would cool more rapidly by an equal increase of volume. $($This fact is proved by what has already been stated.$)$ Now we have seen how important it is to produce by change of volume the greatest possible changes of temperature.
$($2$)$ Vapors of water can be formed only through the intervention of a boiler, while atmospheric air could be heated directly by combustion carried on within its own mass. Considerable loss could thus be prevented, not only in the quantity of heat, but also in its temperature. This advantage belongs exclusively to atmospheric air. Other gases do not possess it. They would be even more difficult to heat than vapor of water.
$($3$)$ In order to give to air great increase of volume, and by that expansion to produce a great change of temperature, it must first be taken under a sufficiently high pressure; then it must be compressed with a pump or by some other means before heating it. This operation would require a special apparatus, an apparatus not found in steamengines. In the latter, water is in a liquid state when injected into the boiler, and to introduce it requires but a small pump.
$($4$)$ The condensing of the vapor by contact with the refrigerant body is much more prompt and much easier than is the cooling of air. There might, of course, be the expedient of throwing the latter out into the atmosphere, and there would be also the advantage of avoiding the use of a refrigerant, which is not always available, but it would be requisite that the increase of the volume of the air should not reduce its pressure below that. of the atmosphere.
$($5$)$ One of the gravest inconveniences of steam is that it cannot be used at high temperatures without necessitating the use of vessels of extraordinary strength. It is not so with air for which there exists no necessary relation between the elastic force and the temperature. Air, then, would seem more suitable than steam to realize the motive power of falls of caloric from high temperatures. Perhaps in low temperatures steam may be more convenient. We might conceive even the possibility of making the same heat act successively upon air and vapor of water. It would be only necessary that the air should have, after its use, an elevated temperature, and instead of throwing it out immediately into the atmosphere, to make it envelop a steamboiler, as if it issued directly from a furnace.
The use of atmospheric air for the development of the motive power of heat presents in practice very great, but perhaps not insurmountable, difficulties. If we should succeed in overcoming them, it would doubtless offer a notable advantage over vapor of water.^{30}
As to the other permanent gases, they should be absolutely rejected. They have all the inconveniences of atmospheric air, with none of its advantages. The same can be said of other vapors than that of water, as compared with the latter.
If we could find an abundant liquid body which would vaporize at a higher temperature than water, of which the vapor would have, for the same volume, a less specific heat, which would not attack the metals employed in the construction of machines, it would undoubtedly merit the preference. But nature provides no such body.
The use of the vapor of alcohol has been proposed. Machines have even been constructed for the purpose of using it, by avoiding the mixture of its vapor with the water of condensation, that is, by applying the cold body externally instead of introducing it into the machine. It has been thought that a remarkable advantage might be secured by using the vapor of alcohol in that it possesses a stronger tension than the vapor of water at the same temperature. We can see in this only a fresh obstacle to be overcome. The principal defect of the vapor of water is its excessive .tension at an elevated temperature; now this defect exists still more strongly in the vapor of alcohol. As to the relative advantage in a greater production of motive power,an advantage attributed to it, we know by the principles above demonstrated that it is imaginary.
It is thus upon the use of atmospheric air and vapor of water that subsequent attempts to perfect heatengines should be based. It is to utilize by means of these agents the greatest possible falls of caloric that all efforts should be directed.
Finally, we will show how far we are from having realized, by any means at present known, all the motive power of combustibles.
One kilogram of carbon burnt in the calorimeter furnishes a quantity of heat capable of raising one degree Centigrade about 7000 kilograms of water, that is, it furnishes 7000 units of heat according to the definition of these units given on page 100.
The greatest fall of caloric attainable is measured by the difference between the temperature produced by combustion and that of the refrigerant bodies. It is difficult to perceive any other limits to the temperature of combustion than those in which the combination between oxygen and the combustible may take place. Let us assume, how ever, that 1000° may be this limit, and we shall certainly be below the truth. As to the temperature of the refrigerant, let us suppose it 0°. We estimated approximately $($page 104$)$ the quantity of motive power that 1000 units of heat develop between 100° and 99°. We found it to be 1.112 units of power, each equal to 1 metre of water raised to a height of 1 metre.
If the motive power were proportional to the fall of caloric, if it were the same for each thermometric degree, nothing would be easier than to estimate it from 1000° to 0°. Its value would be
\begin{equation} 1.112 \times 1000 = 1112. \end{equation}But as this law is only approximate, and as possibly it deviates much from the truth at high temperatures, we can only make a very rough estimate. We will suppose the number 1112 reduced onehalf, that is, to 560.
Since a kilogram of carbon produces 7000 units of heat, and since the number 560 is relatively 1000 units, it must be multiplied by 7, which gives
\begin{equation} 7 \times 560 = 3920. \end{equation}This is the motive power of 1 kilogram of carbon. In order to compare this theoretical result with that of experiment, let us ascertain how much motive power a kilogram of carbon actually develops in the bestknown steamengines.
The engines which, up to this time, have shown the best results are the large doublecylinder engines used in the drainage of the tin and copper mines of Cornwall. The best results that have been obtained with them are as follows:
65 millions of Ibs. of water have been raised one English foot by the bushel of coal burned $($the bushel weighing 88 Ibs.$)$. This is equivalent to raising, by a kilogram of coal, 195 cubic metres of water to a height of 1 metre, producing thereby 195 units of motive power per kilogram of coal burned:^{31}
195 unitss are only the twentieth of 3920, the theoretical maximum; consequently $\frac{1}{20}$ only of the motive power of the combustible has been utilized.
We have, nevertheless, selected our example from among the best steamengines known.
Most engines are greatly inferior to these. The old engine of Chaillot, for example, raised twenty cubic metres of water thirtythree metres, for thirty kilograms of coal consumed, which amounts to twentytwo units of motive power per kilogram, a result nine times less than that given above, and one hundred and eighty times less than the theoretical maximum.
We should not expect ever to utilize in practice all the motive power of combustibles. The attempts made to attain this result would be far more hurtful than useful if they caused other important considerations to be neglected. The economy of the combustible is only one of the conditions to be fulfilled in heatengines. In many cases it is only secondary. It should often give precedence to safety, to strength, to the durability of the engine, to the small space which it must occupy, to small cost of installation, etc. To know how to appreciate in each case, at their true value, the considerations of convenience and economy which may present themselves; to know how to discern the more important of those which are only accessories; to balance them properly against each other, in order to attain the best results by the simplest means: such should be the leading characteristics of the man called to direct, to coordinate among themselves the labors of his comrades, to make them cooperate towards one useful end, of whatsoever sort it may be.
Suggested reading. Clausius 1850FOOTNOTES

It may be said that coalmining has increased tenfold in England since the invention of the steamengine. It is almost equally true in regard to the mining of copper, tin, and iron. The results produced in a halfcentury by the steamengine in the mines of England are today paralleled in the gold and silver mines of the New World mines of which the working declined from day to day, principally on account of the insufficiency of the motors employed in the draining and the extraction of the minerals.

We say, to lessen the dangers of journeys. In fact, although the use of the steamengine on ships is attended by some danger which has been greatly exaggerated, this is more than compensated by the power of following always an appointed and wellknown route, of resisting the force of the winds which would drive the ship towards the shore, the shoals, or the rocks.

We use here the expression motive power to express the useful effect that a motor is capable of producing. This effect can always be likened to the elevation of a weight to a certain height. It has, as we know, as a measure, the product of the weight multiplied by the height to which it is raised.

We distinguish here the steamengine from the heatengine in general. The latter may make use of any agent whatever, of the vapor of water or of any other, to develop the motive power of heat.

Certain engines at high pressure throw the steam out into the atmosphere instead of the condenser. They are used specially in places where it would be difficult to procure a stream of cold water sufficient to produce condensation.

The existence of water in the liquid state here necessarily assumed, since without it the steamengine could not be fed, supposes the existence of a pressure capable of preventing this water from vaporizing, consequently of a pressure equal or superior to the tension of vapor at that temperature. If such a pressure were not exerted by the atmospheric air, there would be instantly produced a quantity of steam sufficient to give use to that tension, and it would be necessary always to overcome this pressure in order to throw out the steam from the engines into the new atmosphere. Now this is evidently equivalent to overcoming the tension which the steam retains after its condensation, as effected by ordinary means.
If a very high temperature existed at the surface of our globe, as it seems certain that it exists in its interior, all the waters of the ocean would be in a state of vapor in the atmosphere, and no portion of it would be found in a liquid state.

It is considered unnecessary to explain here what is quantity of caloric or quantity of heat $($for we employ these two expressions indifferently$)$, or to describe how we measure these quantities by the calorimeter. Nor will we explain what is meant by latent heat, degree of temperature, specific heat, etc. The reader should be familiarized with these terms through the study of the elementary treatises of physics or of chemistry.

We may perhaps wonder here that the body $B$ being at the same temperature as the steam is able to condense it. Doubtless this is not strictly possible. but the slightest difference of temperature will determine the condensation, which suffices to establish the justice of our reasoning. It is thus that, in the differential calculus, it is sufficient that we can conceive the neglected quantities indefinitely reducible in proportion to the quantities retained in the equations, to make certain of the exact result.
The body B condenses the steam without changing its own temperature  this results from our supposition. We have admitted that this body may be maintained at a constant temperature. We take away the caloric as the steam furnishes it. This is the condition in which the metal of the condenser is found when the liquefaction of the steam is accomplished by applying cold water externally, as was formerly done in several engines. Similarly, the water of a reservoir can be maintained at a constant level if the liquid flows out at one side as it flows in at the other. One could even conceive the bodies A and B maintaining the same temperature, although they might lose or gain certain quantities of heat. If, for example, the body A were a mass of steam ready to become liquid, and the body $B$ a mass of ice ready to melt, these bodies might, as we know, furnish or receive caloric without thermometric change.

The objection may perhaps be raised here, that perpetual motion, demonstrated to be impossible by mechanical action alone, may possibly not be so if the power either of heat or electricity be exerted; but is it possible to conceive the phenomena of heat and electricity as due to anything else than some kind of motion of the body, and as such should they not be subjected to the general laws of mechanics? Do we not know besides, á posteriori, that all the attempts made to produce perpetual motion by any means whatever have been fruitless? that we have never succeeded in producing a motion veritably perpetual, that is, a motion which will continue forever without alteration in the bodies set to work to accomplish it? The electromotor apparatus $($the pile of Volta$)$ has sometimes been regarded as capable of producing perpetual motion; attempts have been made to realize this idea by constructing dry piles said to be unchangeable; but however it has been done, the apparatus has always exhibited sensible deteriorations when its action has been sustained for a time with any energy.
The general and philosophic acceptation of the words perpetual motion should include not only a motion susceptible of indefinitely continuing itself after a first impulse received, but the action of an apparatus, of any construction whatever, capable of creating motive power in unlimited quantity, capable of starting from rest all the bodies of nature if they should be found in that condition, of overcoming their inertia; capable, finally, of finding in itself the forces necessary to move the whole universe, to prolong, to accelerate incessantly, its motion. Such would be a veritable creation of motive power. If this were a possibility, it would be useless to seek in currents of air and water or in combustibles this motive power. We should have at our disposal an inexhaustible source upon which we could draw at will.

We assume here no chemical action between the bodies employed to realize the motive power of heat. The chemical action which takes place in the furnace is, in some sort, a preliminary action,an operation destined not to produce immediately motive power, but to destroy the equilibrium of the caloric, to produce a difference of temperature which may finally give rise to motion.

This kind of loss is found in all steamengines. In fact, the water destined to feed the boiler is always cooler than the water which it already contains. There occurs between them a useless reestablishment of equilibrium of caloric. We are easily convinced, à posteriori, that this reestablishment of equilibrium causes a loss of motive power if we reflect that it would have been possible to previously heat the feedwater by using it as condensingwater in a small accessory engine, when the steam drawn from the large boiler might have been used, and where the condensation might be produced at.a temperature intermediate between that of the boiler and that of the principal condenser. The power produced by the small engine would have cost no loss of heat, since all that which had been used would have returned into the boiler with the water of condensation.

The matter here dealt with being entirely new. we are obliged to employ expressions not in use as yet, and which perhaps are less clear than is desirable.

The experimental facts which best prove the change of temperature of gases by compression or dilatation are the following:
$($1$)$ The fall of the thermometer placed under the receiver of a pneumatic machine in which a vacuum has been produced: This faU is very sensible on the Bréguet thermometer: it may exceed 40° or 50°. The mist which forms in this case seems to be due to the condensation of the watery vapor caused by the cooling of the air.
$($2$)$ The inflammation of German tinder in the socalled pneumatic tinderboxes; which are, as we know, little pumpchambers in which the air is rapidly compressed.
$($3$)$ The fall of a thermometer placed in a space where the air has been first compressed and then allowed to escape by the opening of a cock.
$($4$)$ The results of experiments on the velocity of sound. M. de Laplace has shown that, in order to secure results accurately by theory and computation, it is necessary to assume the heating of the air by sudden compression.
The only fact which may be adduced in opposition to the above is an experiment of MM. GayLussac and Welter, described in the Annales de Chimie et de Physique. A small opening having been made in a large reservoir of compressed air, and the ball of a thermometer having been introduced into the current of air which passes out through this opening, no sensible fall of the temperature denoted by the thermometer has been observed.
Two explanations of this fact may be given: $($1$)$ The striking of the air against the walls of the opening by which it escapes may develop heat in observable quantity. $($2$)$ The air which has just touched the bowl of the thermometer possibly takes again by its collision with this bow I, or rather by the effect of the détour which it is forced to make by its rencounter, a density equal to that which it had in the receiver,much as the water of a current rises against a fixed obstacle, above its level.
The change of temperature occasioned in the gas by the change of volume may be regarded as one of the most important facts of Physics, because of the numerous consequences which it entails, and at the same time as one of the most difficult to illustrate, and to measure by decisive experiments. It seems to present in some respects singular anomalies.
Is it not to the cooling of the air by dilatation that the cold of the higher regions of the atmosphere must be attributed? The reasons given heretofore as au explanation of this cold are entirely insufficient; it has been said that the air of the elevated regions receiving little reflected heat from the earth, and radiating towards celestial space, would lose caloric, and that this is the cause of its cooling; but this explanation is refuted by the fact that, at an equal height, cold reigns with equal and even more intensity on the elevated plains than on the summit of the mountains, or in those portions of the atmosphere distant from the sun.

We tacitly assume in our demonstration, that when a body has experienced any changes, and when after a certain number of transformations it returns to precisely its original state, that is, to that state considered in respect to density, to temperature, to mode of aggregation let us suppose, I say, that this body is found to contain the same quantity of heat that it contained at first, or else that the quantities of heat absorbed or set free in these different transformations are exactly compensated. This fact has never been called in question. It was first admitted with out reflection. and verified afterwards in many cases by experiments with the calorimeter. To deny it would be to overthrow the whole theory of heat to which it serves as a basis. For the rest. we may say in passing, the main principles on which the theory of beat rests require the most careful examination. Many experimental facts appear almost inexplicable in the present state of this theory.

We will suppose, in what follows, the reader to be aucourant with the later progress of modern Physics in regard to gaseous substances and heat.

M. Poisson, to whom this figure is due, has shown that it accords very well with the result of an experiment of MM. Clement and Desormes on the return of air into a vacuum, or rather, into air slightly rarefied. It also accords very nearly with results found by MM. GayLussac and Welter. $($See note, p. 87$)$.

The law of Mariotte, which is here made the foundation upon which to establish our demonstration, is one of the best authenticated physical laws. It has served as a basis to many theories verified by experience, and which in turn verify all the laws on which they are founded. We can cite also, as a valuable verification of Mariotte's law and also of that of MM. GayLussac and Dalton, for a great difference of temperature, the experiments of MM. Dulong and Petit. $($See Annales de Chimie et de Physique, Feb. 1818, t. vii. p. 122.$)$
The more recent experiments of Davy and Faraday can also be cited.
The theories that we deduce here would not perhaps be exact if applied outside of certain limits either of density or temperature. They should be regarded as true only within the limits in which the laws of Mariotte and of MM. GayLussac and Dalton are themselves proven.

When the volume is reduced $\frac{1}{116}$, that is, when it becomes $\frac{115}{116}$ of what it was at first, the temperature rises one degree. Another reduction of $\frac{1}{116}$ carries the volume to $\left(\frac{115}{116}\right)^2$ and the temperature should rise another degree. After x similar reductions the volume becomes $\left(\frac{115}{116}\right)^x$ and the temperature should be raised x degrees. If we suppose $\left(\frac{115}{116}\right)^x$ = $\frac{1}{14}$, and if we take the logarithms of both, we find
x = about 300°
If we suppose $\left(\frac{115}{116}\right)^x$ = $\frac{1}{2}$, we find
x = about 80°
which shows that air compressed one half rises 80°.
All this is subject to the hypothesis that the specific heat of air does not change, although the volume diminishes. But if, for the reasons hereafter given $($pp. 86, 89$)$, we regard the specific heat of air compressed one half as reduced in the relation of 700 to 616, the number 80° must be multiplied by $\frac{700}{616}$ which raises it to 90°.

MM. GayLussac and Welter have found by direct experiments, cited in the Mécanique Céleste and in the Annales de Chimie et de Physique, July, 1822, p. 267, that the ratio between the specific heat at constant pressure and the specific heat at constant volume varies very little with the density of the gas. According to what we have just seen, the difference should remain constant, and not the ratio. As, further, the specific heat of gases for a given weight varies very little with the density, it is evident that the ratio itself experiences but slight changes.
The ratio between the specific heat of atmospheric air at constant pressure and at constant volume is, according to MM. GayLussac and Welter, 1.3748, a number almost constant for all pressures, and even for all temperatures. We have come, through other considerations, to the number $\frac{267 + 166}{267}$ = 1.44, which differs from the former $\frac{1}{20}$ and we have used this number to prepare table of the specific heats of gases at constant volume. So we need not regard this table as very exact, any more than the table given on p. 89. These tables are mainly intended to demonstrate the laws governing specific heats of aeriform fluids.

We see no reason for admitting, a priori, the constancy of the specific heat of bodies at different temperatures, that is, to admit that equal quantities of heat will produce equal increments of temperature, when this body changes neither its state nor its density; when, for example, it is an elastic fluid enclosed in a fixed space. Direct experiments on solid and liquid bodies have proved that between zero and 100°, equal increments in the quantities of heat would produce nearly equal increments of temperature. But the more recent experiments of MM. Dulong and Petit $($see Annales de Chimie et de Physique, February, March, and April, 1818$)$ have shown that this correspondence no longer continues at temperatures much above 100°, whether these temperatures be measured on the mercury thermometer or onthe air thermometer.
Not only do the specific heats not remain the same at different temperatures, but, also, they do not preserve the same ratios among themselves, so that no thermometric scale could establish the constancy of all the specific heats. It would have been interesting to prove whether the same irregularities exist for gaseous substances, but such experiments presented almost insurmountable difficulties. The irregularities of specific heats of solid bodies might have been attributed, it would seem, to the latent heat employed to produce a beginning of fusion  a softening which occurs in most bodies long before complete fusion. We might support this opinion by the following statement: According to the experiments of MM. Dulong and Petit, the increase of specific heat with the temperature is more rapid in solids than in liquids, although the latter possess considerably more dilatability. The cause of irregularity just referred to, if it is real, would disappear entirely in gases.

In order to determine the arbitrary constants A, B, A', B', in accordance with the results in M. Dalton's table, we must begin by computing the volume of the vapor as determined by its pressure and temperature, a result which is easily accomplished by reference to the laws of Mariotte and GayLussac, the weight of the vapor being fixed.
The volume will be given by the equation
\begin{equation} v = c \frac{267 + t}{p} \end{equation}in which v is this volume, t the temperature, p the pressure, and c a constant quantity depending on the weight of the vapor and on the units chosen. We give here the table of the volumes occupied by a gramme of vapor formed at different temperatures, and consequently under different pressures.
The first two columns of this table are taken from the Traité de Physique of M. Biot $($vol. i., p. 272 and 531$)$. The third is calculated by means of the above formula, and in accordance with the result of experiment, indicating that water vaporized under atmospheric pressure occupies a space 1700 times as great as in the liquid state.
By using three numbers of the first column and three corresponding numbers of the third column, we can easily determine the constants of our equation
\begin{equation} t = \frac{A + B\log v}{A' + B'\log v} \end{equation}We will not enter into the details of the calculation necessary to determine these quantities. It is sufficient to say that the following values,
\begin{equation*} \begin{aligned} A = 2268\phantom{XXXX} & A' = 19.64 \\ B = 1000 \phantom{XXXX}& B' = 3.30 \\ \end{aligned} \end{equation*}satisfy fairly well the prescribed conditions, 80 that the equation
\begin{equation} t = \frac{2268  1000 \log v}{19.64 + 3.30\log v} \end{equation}expresses very nearly the relation which exists between the volume of the vapor and its temperature. We may remark here that the quantity B' is positive and very small, which tends to confirm this propositionthat the specific heat of an elastic fluid increases with the volume, but follows a slow progression.

Were we to admit the constancy of the specific heat of a gas when its volume does not change, but when its temperature varies, analysis would show a relation between the motive power and the thermometric degree. We will show how this is, and this will also give us occasion to show how some of the propositions established above should be expressed in algebraic language. Let r be the quantity of motive power produced by the expansion of a given quantity of air passing from the volume of one litre to the volume of v litres under constant temperature. If v increasesby the infinitely small quantity dv, r will increase by the quantity dr, which, according to the nature of motive power, will be equal to the increase dv of volume multiplied by the expansive force which the elastic fluid then possesses; let p be this expansive force. We should have the equation
\begin{equation} dr = pdv \dots \dots \dots \dots (1) \end{equation}Let us suppose the constant temperature under which the dilatation takes place equal to t degrees Centigrade. If we call q the elastic force of the air occupying the volume 1 litre at the same temperature t, we shall have, according to the law of Mariotte,
\begin{equation} \frac{v}{1} = \frac{q}{p} \phantom{XXX} \text{ whence }\phantom{XX} p = \frac{q}{v} \end{equation}If now P is the elastic force of this same air at the constant volume 1, but at the temperature zero, we shall have, according to the rule of M. GayLussac,
\begin{equation} q = P + P\frac{t}{267} = \frac{P}{267}(267 + t); \end{equation}whence
\begin{equation} q = p = \frac{P}{267} \frac{267 + t}{v} \end{equation}If, to abridge, we call N the quantity $\frac{P}{267}$ the equation would become
\begin{equation} p = N \frac{t + 267 }{v} \end{equation}whence we deduce, according to equation $($1$)$,
\begin{equation} dr = N \frac{t + 267 }{v} dv \end{equation}Regarding t as constant, and taking the integral of the two numbers, we shall have
\begin{equation} r = N(t + 267) \log v + C. \end{equation}If we suppose r = 0 when v = 1, we shall have C=0; whence
\begin{equation} r = N(t + 267) \log v \dots \dots \dots \dots (2) \end{equation}This is the motive power produced by the expansion on of the air which, under the temperature t, has passed from the volume 1 to the volume v. If instead of working at the temperature t we work in precisely the same manner at the temperature t + dt, the power developed will be
\begin{equation} r + \delta r = N(t + dt + 267) \log v \end{equation}Subtracting equation $($2$)$, we have
\begin{equation} \delta r = N \log v dt \dots \dots \dots \dots (3) \end{equation}Let $e$ be the quantity of heat employed to maintain the temperature of the gas constant during its dilatation. According to the reasoning of page 69, $\delta r$ will be the power developed by the fall of the quantity e of heat from the degree t + dt to the degree t. If we call u the motive power developed by the fall of unity of heat from the degree t to the degree zero, as, according to the general principle established page 68, this quantity $u$ ought to depend solely on t, it could be represented by the function Ft, whence u = Ft.
When t is increased it becomes t + dt, u becomes u + du; whence
\begin{equation} u + du = F(t + dt) \end{equation}Subtracting the preceding equation, we have
\begin{equation} du = F(t + dt)  Ft = F' t dt \end{equation}This is evidently the quantity of motive power produced by the fall of unity of heat from the temperature t + dt to the temperature t.
If the quantity of heat instead of being a unit had been e, its motive power produced would have had for its value
\begin{equation} edu = e F' t dt \dots \dots \dots \dots (4) \end{equation}But edu is the same thing as $\delta r$; both are the power developed by the fall of the quantity $e$ of heat from the temperature t + dt to the temperature t; consequently,
\begin{equation} edu = \delta r, \end{equation}and from equations $($3$)$, $($4$)$,
\begin{equation} e F' t dt = N \log v dt; \end{equation}or, dividing by F'tdt,
\begin{equation} e = \frac{N}{F' t}\log v = T \log v \end{equation}Calling T the fraction $\frac{N}{F' t}$ which is a function of t only, the equation
\begin{equation} e = T \log v \end{equation}is the analytical expression of the law stated pp. 80, 81. It is common to all gases, since the laws of which we have made use are common to all.
If we call s the quantity of heat necessary to change the air that we have employed from the volume 1 and from the temperature zero to the volume v and to the temperature t, the difference between s and e will be the quantity of heat required to bring the air at the volume 1 from zero to t. This quantity depends on t alone; we will call it U. It will be any function whatever of t. We shall have
\begin{equation} s = e + U = T \log v + U. \end{equation}If we differentiate this equation with relation to t alone, and if we represent it by T' and U', the differential coefficients of T and U, we shall get
\begin{equation} \frac{ds}{dt} = T' \log v + U'; \dots \dots \dots \dots (5) \end{equation}$\frac{ds}{dt}$ is simply the specific heat of the gas under constant volume, and our equation $($1$)$ is the analytical expression of the law stated on page 86.
If we suppose the specific heat constant at all temperatures $($hypothesis discussed above, page 92$)$, the quantity $\frac{ds}{dt}$ will be independent of t; and in order to satisfy equation $($5$)$ for two particular values of v, it will be necessary that T' and U' be independent of t; we shall then have T' = C, a constant quantity. Multiplying T' and C by dt, and taking the integral of both, we find
\begin{equation} T= Ct+ C_1 \end{equation}but as $T= \frac{N}{F' t'}$, we have
\begin{equation} F' t = \frac{N}{T} = \frac{N}{Ct + C_1} \end{equation}Multiplying both by dt and integrating, we have
\begin{equation} Ft = \frac{N}{C}(Ct + C_1) + C_2 \end{equation}or changing arbitrary constants, and remarking further that Ft is 0 when t = 0°,
\begin{equation} Ft = \left( 1 + \frac{t}{B} \right) \dots \dots \dots \dots (6) \end{equation}The nature of the function Ft would be thus determined, and we would thus be able to estimate the motive power developed by any fall of heat. But this latter conclusion is founded on the hypothesis of the constancy of the specific heat of a gas which does not change in volume an hypothesis is which has not yet been sufficiently verified by experiment. Until there is fresh proof, our equation $($6$)$ can be admitted only throughout a limited portion of the thermometric scale.
In equation $($5$)$, the first member represents, as we have remarked, the specific heat of the air occupying the volume v. Experiment having shown that this heat varies little in spite of the quite considerable changes of volume, it is necessary that the coefficient T' of $\log v$ should be a very small quantity. If we consider it nothing, and, after having multiplied by dt the equation
\begin{equation} T'= 0 \end{equation}we take the integral of it, we find
T = 0, constant quantity;
but
\begin{equation} T= \frac{N}{F' t} \end{equation}whence
\begin{equation} F' t = \frac{N}{T} = \frac{N}{C} = A \end{equation}whence we deduce finally, by a second integration,
\begin{equation} Ft=At+B \end{equation}As Ft = 0 when t = 0, B is 0; thus
\begin{equation} Ft = At \end{equation}that is, the motive power produced would be found to be exactly proportional to the fall of the caloric. This is the analytical translation of what was stated on page 98.

We find $($Annales de Chimie et de Physique, July, 1818, p. 294$)$ in a memoir of M. Petit an estimate of the motive power of heat applied to air and to vapor of water. This estimate leads us to attribute a great advantage to atmospheric air, but it is derived by a method of considering the action of heat which is quite imperfect.

M. Dalton believed that he had discovered that the vapors of different liquids at equal thermometric distances from the boilingpoint possess equal tensions; but this law is not precisely exact; it is only approximate. It is the same with the law of the proportionality of the latent heat of vapors with their densities $($see Extracts from a Mémoire of M. C. Despretz, Annales de Chimie et de Physique, t. xvi. p. 105, and t. xxiv. p. 323$)$. Questions of this nature are closely connected with those of the motive power of heat. Quite recently MM. H. Davy and Faraday, after having conducted a series of elegant experiments on the liquefaction of gases by means of considerable pressure, have tried to observe the changes of tension of these liquefied gases on account of slight changes of temperature. They have in view the application of the new liquids to the production of motive power $($see Annales de Chimie et de Physique, January, 1824, p. 80$)$.
According to the abovementioned theory, we can foresee that the use of these liquids would present no advantages relatively to the economy of heat. The advantages would be found only in the lower temperature at which it would be possible to work, and in the sources whence, for this reason, it would become possible to obtain caloric.

Those that we need are the expansive force acquired by solids and liquids by a given increase of temperature, and the quantity of heat absorbed or relinquished in the changes of volume of these bodies.

The recent experiments of M. Oerstedt on the compressibility of water have shown that, for a pressure of five atmospheres, the temperature of this liquid exhibits no appreciable change $($See Annales de Chimie et de Physique, Feb. 1828, p. 192.$)$.

This principle, the real foundation of the theory of steamengines, was very clearly developed by M. Clement in a memoir presented to the Academy of Sciences several years ago. This Memoir has never been printed, and I owe the knowledge of it to the kindness of the author.Not only is the principle established therein, but it is applied to the different systems of steamengines actually in use. The motive power of each of them is estimated therein by the aid of the law cited page 92, and compared with the results of experiment.
The principle in question is so little known or so poorly appreciated, that recently Mr. Perkins, a celebrated mechanician of London, constructed amachine in which steam produced under the pressure of 35 atmospheres a pressure never before used is subjected to very little expansion of volume, as anyone with the least knowledge of this machine can understand. It consists of a single cylinder of very small dimensions, which at each stroke is entirely filled with steam, formed under the pressure of 35 atmospheres. The steam produces no effect by the expansion of its volume, for no space is provided in which the expansion can take place. It is condensed as soon as it leaves the small cylinder. It works therefore only under a pressure of 35 atmospheres, and not, as its useful employment would require, under progressively decreasing pressures. The machine of Mr. Perkins seems not to realize the hopes which it at first awakened. It has been asserted that the economy of coal in this engine was $\frac{9}{10}$ above the best engines of Watt, and that it possessed still other advantages $($see Annales de Chimie et de Physique, April, 1823, p. 429$)$. These assertions have not been verified. The engine of Mr. Perkins is nevertheless a valuable invention, in that it has proved the possibility of making use of steam under much higher pressure than previously, and because, being easily modified, it may lead to very useful results.
Watt, to whom we owe almost all the great improvements in steamengines, and who brought these engines to a state of perfection difficult even now to surpass, was also the first who employed steam under progressively decreasing pressures. In many cases he suppressed the introduction of the steam into the cylinder at a half, a third, or a quarter of the stroke. The piston completes its stroke, therefore, under a constantly diminishing pressure. The first engines working on this principle date from 1778. Watt conceived the idea of them in 1769, and took out a patent in 1782.
We give here the Table appended to Watt's patent. He supposed the steam introduced into the cylinder during the first quarter of the stroke of the piston; then, dividing this stroke into twenty parts, he calculated the mean pressure as follows:
On which he remarked, that the mean pressure is more than half the original pressure; also that in employing a quantity of steam equal to a quarter, it would produce an effect more than half.
Watt here supposed that steam follows in its expansion the law of Mariotte, which should not be considered exact, because, in the first place, the elastic fluid ill dilating falls in temperature, and in the second place there is nothing to prove that a part of this fluid is not condensed by its expansion. Watt should also have taken into consideration the force necessary to expel the steam which remains after condensation, and which is found in quantity as much greater as the expansion of the volume has been carried further. Dr. Robinson has supplemented the work of Watt by a simple formula to calculate the effect of the expansion of steam, but this formula is found to have the same faults that we have just noticed. It has nevertheless been useful to constructors by furnishing them approximate data practically quite satisfactory. We have considered it useful to recall these facts because they are little known, especially in France. These engines have been built after the models of the inventors, but the ideas by which the inventors were originally influenced have been but little understood. Ignorance of these ideas has often led to grave errors. Engines originally well conceived have deteriorated in the hands of unskilful builders, who, wishing to introduce in them improvements of little value, have neglected the capital considerations which they did not know enough to appreciate.

The advantage in substituting two cylinders for one is evident. In a single cylinder the impulsion of the piston would be extremely variable from the beginning to the end of the stroke. It would be necessary for all the parts by which the motion is transmitted to be of sufficient strength to resist the first impulsion, and perfectly fitted to avoid the abrupt movements which would greatly injure and soon destroy them. It would be especially on the working beam, on the supports, on the crank, on the connectingrod, and on the first gearwheels that the unequal effort would be felt, and would produce the most injurious effects. It would be necessary that the steamcylinder should be both sufficiently strong to sustain the highest pressure, and with a large enough capacity to contain the steam after its expansion of volume while in using two successive cylinders it is only necessary to have the first sufficiently strong and of medium capacity, which is not at all difficult, and to have the second of ample dimensions, with moderate strength.
Doublecylinder engines, although founded on correct principles, oft.en fail to secure the advantages expected from them. This is due principally to the fact that the dimensions of the different parts of these engines are difficult to adjust, and that they are rarely found to be in correct proportion. Good models for the construction of doublecylinder engines are wanting, while excellent designs exist for the construction of engines on the plan of Watt. From this arises the diversity that we see in the results of the former, and the great uniformity that we have observed in the results of the latter.

We find in the work called De la Richesse Minerale, by M. Heron de Villefosse, vol. iii. p. 50 and following, a good description of the steam.engines actually in use in mining. In England the steam.engine has been very fully discussed in the Encyclopedia Britannica. Some of the data here employed are drawn from the latter work.

Among the attempts made to develop the motive power of heat by means of atmospheric air, we should mention those of MM. Niepce, which were made in France several years ago, by means of an apparatus called by the inventors a pyréolophore. The apparatus was made thus: There was a cylinder furnished with a piston, into which the atmospheric air was introduced at ordinary density. A very combustible material, reduced to a condition of extreme tenuity, was thrown into it, remained a moment in suspension in the air, and then flame was applied. The inflammation produced very nearly the same effect as if the elastic fluid had been a mixture of air and combustible gas, of air and carburetted hydrogen gas, for example. There was a sort of explosion, and a sudden dilatation of the elastic fluid a dilatation that was utilized by making it act upon the piston. The latter may have a motion of any amplitude whatever, and the motive power is thus realized. The air is next renewed, and the operation repeated.
This machine, very ingenious and interesting, especially on account of the novelty of its principle, fails in an essential point. The material used as a combustible $($it was the dust of Lycopodium, used to produce flame in our theatres$)$ was so expensive, that all the advantage was lost through that cause; and unfortunately it was difficult to employ a combustible of moderate price, since a very finely powdered substance was required which would burn quickly, spread rapidly, and leave little or no ash.
Instead of working as did MM. Niepce, it would seem to us preferable to compress the air by means of pumps, to make it traverse a perfectly closed furnace into which the combustible had been introduced in small portions by a mechanism easy of conception, to make it develop its action in a cylinder with a piston, or in any other variable space; finally, to throw it out again into the atmosphere, or even to make it pass under a steamboiler in order to utilize the temperature remaining.
The principal difficulties that we should meet in this mode of operation would be to enclose the furnace in a sufficiently strong envelope, to keep the combustion meanwhile in the requisite state, to maintain the different parts of the apparatus at a moderate temperature, and to prevent rapid abrasion of the cylinder and of the piston. These difficulties do not appear to be insurmountable.
There have been made, it is said, recently in England, successful attempts to develop motive power through the action of heat on atmospheric air. We are entirely ignorant in what these attempts have consistedif indeed they have really been made.

The result given here was furnished by an engine whose large cylinder was 45 inches in diameter and 7 feet stroke. It is used in one of the mines of Cornwall called Wheal Abraham. This result should be considered as somewhat exceptional, for it was only temporary, continuing but a single month. Thirty millions of lbs. raised one English foot per bushel of coal of 88 lbs. is generally regarded as an excellent result for steamengines. It is sometimes attained by engines of the Watt type, but very rarely surpassed. This latter product amounts, in French measures, to 104,000 kilograms raised one metre per kilogram of coal consumed.
According to what is generally understood by one horsepower, in estimating the duty of steamengines, an engine of ten horsepower should raise per second 10 $\times$ 75 kilograms, or 750 kilograms, to a height of one metre, or more, per hour; 750 $\times$ 3600 = 2.700.000 kilograms to one metre. If we suppose that each kilogram of coal raised to this height 104.000 kilograms, it will be necessary, in order to ascertain how much coal is burnt in one hour by our tenhorsepower engine, to divide 2.700.000 by 104.000, which gives $\frac{2700}{104}$ = 26 kilograms. Now it is seldom that a tenhorsepower engine consumes less than 26 kilograms of coal per hour.
References
 Carnot, Sadi, Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance, A Paris, Chez Bachelier, Libraire Quai des Agustins, Nº 55.
 Carnot, Sadi; Thurston, Henry $($editor and translator$)$ $($1890$)$. Reflections on the Motive Power of Heat . New York: J. Wiley & Sons.
Notes
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