Chapter V: Appendix (4)

7. Since the time when Carnot thus expressed himself, the necessity of a most careful examination of the entire experimental basis of the theory of heat has become more and more urgent. Especially all those assumptions depending on the idea that heat is a _substance_, invariable in quantity; not convertible into any other element, and incapable of being _generated_ by any physical agency; in fact the acknowledged principles of latent heat,—would require to be tested by a most searching investigation before they ought to be admitted, as they usually have been, by almost every one who has been engaged on the subject, whether in combining the results of experimental research, or in general theoretical investigations.

8. The extremely important discoveries recently made by Mr. Joule of Manchester, that heat is evolved in every part of a closed electric conductor, moving in the neighborhood of a magnet,[38] and that heat is _generated_ by the friction of fluids in motion, seem to overturn the opinion commonly held that heat cannot be _generated_, but only produced from a source, where it has previously existed either in a sensible or in a latent condition.

In the present state of science, however, no operation is known by which heat can be absorbed into a body without either elevating its temperature or becoming latent, and producing some alteration in its physical condition; and the fundamental axiom adopted by Carnot may be considered as still the most probable basis for an investigation of the motive power of heat; although this, and with it every other branch of the theory of heat, may ultimately require to be reconstructed upon another foundation, when our experimental data are more complete. On this understanding, and to avoid a repetition of doubts, I shall refer to Carnot’s fundamental principle, in all that follows, as if its truth were thoroughly established.

9. We are now led to the conclusion that the origin of motive power, developed by the alternate expansions and contractions of a body, must be found in the agency of heat entering the body and leaving it; since there cannot, at the end of a complete cycle, when the body is restored to its primitive physical condition, have been any absolute absorption of heat, and consequently no conversion of heat, or caloric, into mechanical effect; and it remains for us to trace the precise nature of the circumstances under which heat must enter the body, and afterwards leave it, so that mechanical effect may be produced. As an example, we may consider that machine for obtaining motive power from heat with which we are most familiar—the steam-engine.

10. Here, we observe, that heat enters the machine from the furnace, through the sides of the boiler, and that heat is continually abstracted by the water employed for keeping the condenser cool. According to Carnot’s fundamental principle, the quantity of heat thus discharged, during a complete revolution (or double stroke) of the engine, must be precisely equal to that which enters the water of the boiler;[39] provided the total mass of water and steam be invariable, and be restored to its primitive physical condition (which will be the case rigorously, if the condenser be kept cool by the external application of cold water instead of by injection, as is more usual in practice), and if the condensed water be restored to the boiler at the end of each complete revolution. Thus we perceive that a certain quantity of heat is _let down_ from a hot body, the metal of the boiler, to another body at a lower temperature, the metal of the condenser; and that there results from this transference of heat a certain development of mechanical effect.

11. If we examine any other case in which mechanical effect is obtained from a thermal origin, by means of the alternate expansions and contractions of any substance whatever, instead of the water of a steam-engine, we find that a similar transference of heat is effected, and we may therefore answer the first question proposed, in the following manner:

_The thermal agency by which mechanical effect may be obtained is the transference of heat from one body to another at a lower temperature._

11. On the measurement of Thermal Agency, considered with reference to its equivalent of mechanical effect.

12. A _perfect_ thermodynamic engine of any kind is a machine by means of which the greatest possible amount of mechanical effect can be obtained from a given thermal agency; and, therefore, if in any manner we can construct or imagine a perfect engine which may be applied for the transference of a given quantity of heat from a body at any given temperature to another body at a lower given temperature, and if we can evaluate the mechanical effect thus obtained, we shall be able to answer the question at present under consideration, and so to complete the theory of the motive power of heat. But whatever kind of engine we may consider with this view, it will be necessary for us to prove that it is a perfect engine; since the transference of the heat from one body to the other may be wholly, or partially, effected by conduction through a solid,[40] without the development of mechanical effect; and, consequently, engines may be constructed in which the whole or any portion of the thermal agency is wasted. Hence it is of primary importance to discover the criterion of a perfect engine. This has been done by Carnot, who proves the following proposition:

13. _A perfect thermodynamic engine is such that, whatever amount of mechanical effect it can derive from a certain thermal agency, if an equal amount be spent in working it backwards, an equal reverse thermal effect will be produced._[41]

14. This proposition will be made clearer by the applications of it which are given later (§ 29), in the cases of the air-engine and the steam-engine, than it could be by any general explanation; and it will also appear, from the nature of the operations described in those cases, and the principles of Carnot’s reasoning, that a perfect engine may be constructed with any substance of an indestructible texture as the alternately expanding and contracting medium. Thus we might conceive thermodynamic engines founded upon the expansions and contractions of a perfectly elastic solid, or of a liquid; or upon the alterations of volume experienced by substances in passing from the liquid to the solid state,[42] each of which being perfect, would produce the same amount of mechanical effect from a given thermal agency; but there are two cases which Carnot has selected as most worthy of minute attention, because of their peculiar appropriateness for illustrating the general principles of his theory, no less than on account of their very great practical importance: the steam-engine, in which the substance employed as the transferring medium is water, alternately in the liquid state and in the state of vapor; and the air-engine, in which the transference is effected by means of the alternate expansions and contractions of a medium always in the gaseous state. The details of an actually practicable engine of either kind are not contemplated by Carnot in his general theoretical reasonings, but he confines himself to the ideal construction, in the simplest possible way in each case, of an engine in which the economy is perfect. He thus determines the degree of perfectibility which cannot be surpassed; and by describing a conceivable method of attaining to this perfection by an air-engine or a steam-engine, he points out the proper objects to be kept in view in the practical construction and working of such machines. I now proceed to give an outline of these investigations.

CARNOT’S THEORY OF THE STEAM-ENGINE.

15. Let _CDF_{2}E_{2}_ be a cylinder, of which the curved surface is perfectly impermeable to heat, with a piston also impermeable to heat, fitted in it; while the fixed bottom _CD_, itself with no capacity for heat, is possessed of perfect conducting power. Let _K_ be an impermeable stand, such that when the cylinder is placed upon it the contents below the piston can neither gain nor lose heat. Let _A_ and _B_ be two bodies permanently retained at constant temperatures, _S°_ and _T°_, respectively, of which the former is higher than the latter. Let the cylinder, placed on the impermeable stand, _K_, be partially filled with water, at the temperature _S_, of the body _A_, and (there being no air below it) let the piston be placed in a position _EF_, near the surface of the water. The pressure of the vapor above the water will tend to push up the piston, and must be resisted by a force applied to the piston,[43] till the commencement of the operations, which are conducted in the following manner:

(1) The cylinder being placed on the body _A_, so that the water and vapor may be retained at the temperature _S_, _let the piston rise any convenient height EE_{1}, to a position E_{1}F_{1}, performing work by the pressure of the vapor below it during its ascent_.

[During this operation a certain quantity, _H_, of heat, the amount of
latent heat in the fresh vapor which is formed, is abstracted from the
body _A_.]

(2) The cylinder being removed, and placed on the impermeable stand _K, let the piston rise gradually, till, when it reaches a position E_{2}F_{2}, the temperature of the water and vapor is T, the same as that of the body B_.

[During this operation the fresh vapor continually formed requires
heat to become latent; and, therefore, as the contents of the cylinder
are protected from any accession of heat, their temperature sinks.]

(3) The cylinder being removed from _K_, and placed on _B, let the piston be pushed down, till, when it reaches the position E_{3}F_{3}, the quantity of heat evolved and abstracted by B amounts to that which, during the first operation, was taken from A_.

[Note of Nov. 5, 1881. The specification of this operation, with a view to the return to the primitive condition, intended as the conclusion to the four operations, is the only item in which Carnot’s temporary and provisional assumption of the materiality of heat has effect. To exclude this hypothesis, Prof. James Thomson has suggested the following corrected specification for the third operation: _Let the piston be pushed down, till it reaches a position E_{3}F_{3}, determined so as to fulfil the condition, that at the end of the fourth operation the primitive temperature S shall be reached_:[44]]

[During this operation the temperature of the contents of the cylinder
is retained constantly at _T°_, and all the latent heat of the vapor
which is condensed into water at the same temperature is given out to
_B_.]

(4) The cylinder being removed from _B_, and placed on the impermeable stand, _let the piston be pushed down from E_{3}F_{3} to its original position EF_.

[During this operation, the impermeable stand preventing any loss of
heat, the temperature of the water and air must rise continually, till
(since the quantity of heat evolved during the third operation was
precisely equal to that which was previously absorbed) at the
conclusion it reaches its primitive value, _S_, in virtue of Carnot’s
fundamental axiom.]

[Note of Nov. 5, 1881. With Prof. James Thomson’s correction of
operation (3), the words in virtue of “Carnot’s Fundamental Axiom”
must be replaced by “the condition fulfilled by operation (3),” in the
description of the results of operation (4).]

16. At the conclusion of this cycle of operations[45] the total thermal agency has been the _letting down_ of _H_ units of heat from the body _A_, at the temperature _S_, to _B_, at the lower temperature _T_; and the aggregate of the mechanical effect has been a certain amount of _work produced_, since during the ascent of the piston in the first and second operations, the temperature of the water and vapor, and therefore the pressure of the vapor on the piston, was on the whole higher than during the descent, in the third and fourth operations. It remains for us actually to evaluate this aggregate amount of work performed; and for this purpose the following graphical method of representing the mechanical effect developed in the several operations, taken from Mons. Clapeyron’s paper, is extremely convenient.

17. Let _OX_ and _OY_ be two lines at right angles to one another. Along _OX_ measure off distances _ON_{1}_, _N_{1}N_{2}_, _N_{2}N_{3}_, _N_{3}O_, respectively proportional to the spaces described by the piston during the four successive operations described above; and, with reference to these four operations respectively, let the following constructions be made:

(1) Along _OY_ measure a length _OA_, to represent the pressure of the saturated vapor at the temperature _S_; and draw _AA_{1}_ parallel to _OX_, and let it meet an ordinate through _N_{1}_, in _A_{1}_.

(2) Draw a curve _A_{1}PA_ such that, if _ON_ represent, at any instant during the second operation, the distance of the piston from its primitive position, _NP_ shall represent the pressure of the vapor at the same instant.

(3) Through _A__{2} draw _A_{2}A_{3}_ parallel to _OX_, and let it meet an ordinate through _N_{3}_ in _A_{3}_.

(4) Draw the curve _A_{3}A_ such that the abscissa and ordinate of any point in it may represent respectively the distances of the piston from its primitive position, and the pressure of the vapor, at each instant during the fourth operation. The last point of this curve must, according to Carnot’s fundamental principle, coincide with _A_, since the piston is, at the end of the cycle of operations, again in its primitive position, and the pressure of the vapor is the same as it was at the beginning.

18. Let us now suppose that the lengths, _ON_{1}_, _N_{1}N_{2}_, _N_{2}N_{3}_, and _N_{3}O_, _represent numerically_ the volumes of the spaces moved through by the piston during the successive operations. It follows that the mechanical effect obtained during the first operation will be _numerically represented_ by the area _AA_{1}N_{1}O_; that is, the number of superficial units in this area will be equal to the number of “foot-pounds” of work performed by the ascending piston during the first operation. The work performed by the piston during the second operation will be similarly represented by the area _A_{1}A_{2}N_{2}N_{1}_. Again, during the third operation a certain amount of work is spent on the piston, which will be represented by the area _A_{2}A_{3}N_{3}N_{2}_; and lastly, during the fourth operation, work is spent in pushing the piston to an amount represented by the area _A_{3}AON_{3}_.

19. Hence the mechanical effect (represented by the area _OAA_{1}A_{2}N_{2}_) which was obtained during the first and second operations, exceeds the work (represented by _N_{2}A_{2}A_{3}AO_) spent during the third and fourth, by an amount represented by the area of the quadrilateral figure _AA_{1}A_{2}A_{3}_; and, consequently, it only remains for us to evaluate this area, that we may determine the total mechanical effect gained in a complete cycle of operations. Now, from experimental data, at present nearly complete, as will be explained below, we may determine the length of the line _AA_{1}_ for the given temperature _S_, and a given absorption _H_, of heat, during the first operation; and the length of _A_{2}A_{3}_ for the given lower temperature _T_, and the evolution of the same quantity of heat during the fourth operation: and the curves _A_{1}PA_{2}_, _A_{3}P′A_ may be drawn as graphical representations of actual observations. The figure being thus constructed, its area may be measured, and we are, therefore, in possession of a graphical method of determining the amount of mechanical effect to be obtained from any given thermal agency. As, however, it is merely the area of the figure which it is required to determine, it will not be necessary to be able to describe each of the curves _A_{1}PA_{2}_, _A_{3}P′A_, but it will be sufficient to know the difference of the abscissas corresponding to any equal ordinates in the two; and the following analytical method of completing the problem is the most convenient for leading to the actual numerical results.

20. Draw any line _PP′_ parallel to _OX_, meeting the curvilinear sides of the quadrilateral in _P_ and _P′_. Let ξ denote the length of this line, and _p_ its distance from _OX_. The area of the figure, according to the integral calculus, will be denoted by the expression

∫_{_p_{3}_} ^{_p_{1}_} ξ_dp_,

where _p_{1}_ and _p_{3}_ (the limits of integration indicated according to Fourier’s notation) denote the lines _OA_ and _N_{3}A_{3}_, which represent respectively the pressures during the first and third operations. Now, by referring to the construction described above, we see that ξ is the difference of the volumes below the piston at corresponding instants of the second and fourth operations, or instants at which the saturated steam and the water in the cylinder have the same pressure _p_, and consequently the same temperature, which we may denote by _t_. Again, throughout the second operation the entire contents of the cylinder possess a greater amount of heat by _H_ units than during the fourth; and, therefore, at any instant of the second operation there is as much more steam as contains _H_ units of latent heat than at the corresponding instant of the fourth operation. Hence if _k_ denote the latent heat in a unit of saturated steam at the temperature _t_, the volume of the steam at the two corresponding instants must differ by (_H_)/(_k_). Now, if σ denote the ratio of the density of the steam to that of the water, the volume (_H_)/(_k_) of steam will be formed from the volume σ (_H_)/(_k_) of water; and consequently we have, for the difference of volumes of the entire contents at the corresponding instants,

ξ = (1 - σ)(_H_)/(_k_).

Hence the expression for the area of the quadrilateral figure becomes

∫^{_p_{1}_}_{_p_{3}_}(1 - σ)(_H_)/(_k_)_dp_.

Now, σ, _k_, and _p_, being quantities which depend upon the temperature, may be considered as functions of _t_; and it will be convenient to modify the integral so as to make _t_ the independent variable. The limits will be from _t_ = _T_ to _t_ = _S_, and, if we denote by _M_ the value of the integral, we have the expression

_M_ = _H_ ∫_{_T_}^{_S_}(1 - σ)((_dp_/_dt_)/_k_)_dt_. (1)

for the total amount of mechanical effect gained by the operations described above.

21. If the interval of temperatures be extremely small,—so small that (1 − σ)(_dp_)/(_dt_/_k_) will not sensibly vary for values of _t_ between _T_ and _S_,—the preceding expression becomes simply

_Μ_ = (1 - σ)(_dp_)/(_dt_)/(_k_). _Η_(_S_ - _Τ_). (2)

This might, of course, have been obtained at once by supposing the breadth of the quadrilateral figure _AA_{1}A_{2}A_ to be extremely small compared with its length, and then taking for its area, as an approximate value, the product of the breadth into the line _AA_{1}_, or the line _A_{3}A_{2}_, or any line of intermediate magnitude.

The expression (2) is rigorously correct for any interval _S_ − _T_, if the mean value of (1 − σ)((_dp_/_dt_)/_k_) for that interval be employed as the coefficient of _H_(_S_ − _T_).

CARNOT’S THEORY OF THE AIR-ENGINE.

22. In the ideal air-engine imagined by Carnot four operations performed upon a mass of air or gas enclosed in a closed vessel of variable volume constitute a complete cycle, at the end of which the medium is left in its primitive physical condition; the construction being the same as that which was described above for the steam-engine, a body _A_, permanently retained at the temperature _S_, and _B_ at the temperature _T_; an impermeable stand _K_; and a cylinder and piston, which in this case contains a mass of air at the temperature _S_, instead of water in the liquid state, at the beginning and end of a cycle of operations. The four successive operations are conducted in the following manner:

(1) The cylinder is laid on the body _A_, so that the air in it is kept at the temperature _S_; and the piston is allowed to rise, performing work.

(2) The cylinder is placed on the impermeable stand _K_, so that its contents can neither gain nor lose heat, and the piston is allowed to rise farther, still performing work, till the temperature of the air sinks to _T_.

(3) The cylinder is placed on _B_, so that the air is retained at the temperature _T_, and the piston is pushed down till the air gives out to the body _B_ as much heat as it had taken in from _A_, during the first operation.

[Note of Nov. 5, 1881. To eliminate the assumption of the materiality
of heat, make Professor James Thomson’s correction here also; as above
in § 15; or take Maxwell’s rearrangement of the cycle described in the
foot-note to § 15, p. 144.]

(4) The cylinder is placed on _K_, so that no more heat can be taken in or given out, and the piston is pushed down to its primitive position.

23. _At the end of the fourth operation the temperature must have reached its primitive value S, in virtue of_ CARNOT’S _axiom_.

24. Here, again, as in the former case, we observe that work is performed by the piston during the first two operations; and during the third and fourth work is spent upon it, but to a less amount, since the pressure is on the whole less during the third and fourth operations than during the first and second, on account of the temperature being lower. Thus, at the end of a complete cycle of operations, mechanical effect has been obtained; and the thermal agency from which it is drawn is the taking of a certain quantity of heat from _A_, and _letting it down_, through the medium of the engine, to the body _B_ at a lower temperature.

25. To estimate the actual amount of effect thus obtained, it will be convenient to consider the alterations of volume of the mass of air in the several operations as extremely small. We may afterwards pass by the integral calculus, or, practically, by summation to determine the mechanical effect whatever be the amplitudes of the different motions of the piston.

26. Let _dq_ be the quantity of heat absorbed during the first operation, which is evolved again during the third; and let _dv_ be the corresponding augmentation of volume which takes place while the temperature remains constant, as it does during the first operation.[46] The diminution of volume in the third operation must be also equal to _dv_, or only differ from it by an infinitely small quantity of the second order. During the second operation we may suppose the volume to be increased by an infinitely small quantity φ; which will occasion a diminution of pressure and a diminution of temperature, denoted respectively by ω and τ. During the fourth operation there will be a diminution of volume and an increase of pressure and temperature, which can only differ, by infinitely small quantities of the second order, from the changes in the other direction, which took place in the second operation, and they also may, therefore, be denoted by φ, ω, and τ, respectively. The alteration of pressure during the first and third operations may at once be determined by means of Mariotte’s law, since in them the temperature remains constant. Thus, if, at the commencement of the cycle, the volume and pressure be _v_ and _p_, they will have become _v_ + _dv_ and _pv_/(_v_ + _dv_) at the end of the first operation. Hence the diminution of pressure during the first operation is _p_ − _pv_/(_v_ + _dv_) or _pdv_/(_v_ + _dv_) and therefore, if we neglect infinitely small quantities of the second order, we have _pdv_/_v_ for the diminution of pressure during the first operation; which to the same degree of approximation, will be equal to the increase of pressure during the third. If _t_ + τ and _t_ be taken to denote the superior and inferior limits of temperature, we shall thus have for the volume, the temperature, and the pressure at the commencements of the four successive operations, and at the end of the cycle, the following values respectively:

(1) _v_, _t_ + τ, _p_;
(2) _v_ + _dv_, _t_ + τ, _p_(1 − (_dv_)/(_v_));
(3) _v_ + _dv_ + φ, _t_, _p_(1 − (_dv_)/(_v_)) − ω;
(4) _v_ + φ, _t_, _p_ − ω;
(5) _v_, _t_ + τ, _p_.

Taking the mean of the pressures at the beginning and end of each operation, we find

(1) _p_(1 − ½(_dv_)/(_v_)),

(2) _p_(1 − (_dv_)/(_v_)) − ½ω,

(3) _p_(1 − ½(_dv_)/(_v_))) − ω,

(4) _p_ − ½ω,

which, as we are neglecting infinitely small quantities of the second order, will be the expressions for the mean pressures during the four successive operations. Now, the mechanical effect gained or spent, during any of the operations, will be found by multiplying the mean pressure by the increase or diminution of volume which takes place; and we thus find

(1) _p_(1 − ½(_dv_)/(_v_))_dv_,

(2) {_p_(1 − (_dv_)/(_v_)) − ½ω}φ,

(3) {_p_(1 − ½(_dv_)/(_v_)) − ω}_dv_,

(4) (_p_ − ½ω)φ.

for the amounts gained during the first and second, and spent during the third and fourth operations; and hence, by addition and subtraction, we find

ω_dv_ − _p_φ(_dv_)/(_v_), or (_v_ω − _p_φ)(_dv_)/(_v_),

for the aggregate amount of mechanical effect gained during the cycle of operations. It only remains for us to express this result in terms of _dq_ and τ, on which the given thermal agency depends. For this purpose we remark that φ and ω are alterations of volume and pressure which take place along with a change of temperature τ, and hence, by the laws of compressibility and expansion, we may establish a relation[47] between them in the following manner:

Let _p_{0}_ be the pressure of the mass of air when reduced to the temperature zero, and confined in a volume _v_{0}_; then, whatever be _v_{0}_, the product _p_{0}v_{0}_ will, by the law of compressibility, remain constant; and, if the temperature be elevated from 0 to _t_ + τ, and the gas be allowed to expand freely without any change of pressure, its volume will be increased in the ratio of 1 to 1 + _E_(_t_ + τ), where _E_ is very nearly equal to .00366 (the Centigrade scale of the air-thermometer being referred to), whatever be the gas employed, according to the researches of Regnault and of Magnus on the expansion of gases by heat. If, now, the volume be altered arbitrarily with the temperature continually at _t_ + τ, the product of the pressure and volume will remain constant; and therefore we have

_pv_ = _p_{0}v_{0}_{1 + _E_(_t_ + τ)}.

Similarly,

(_p_ − ω)(_v_ + φ) = _p_{0}v_{0}_{1 + _Et_}.

Hence, by subtraction, we have

_v_ω − _p_φ + ωφ = _p_{0}v_{0}E_τ,

or, neglecting the product ωφ,

_v_ω − _p_φ = _p_{0}v_{0}E_τ.

Hence the preceding expression for mechanical effect, gained in the cycle of operations, becomes

_p_{0}v_{0}_. _E_τ . _dv_/_v_.

Or, as we may otherwise express it,

(_Ep_{0}v_{0}_)/(_vdq_/_dv_). _dq_. τ.

Hence, if we denote by _M_ the mechanical effect due to _H_ units of heat descending through the same interval τ, which might be obtained by repeating the cycle of operations described above, (_H_)/(_dq_) times, we have

_M_ = (_Ep_{0}v_{0}_)/(_vdq_/_dv_). _H_τ. (3)

27. If the _amplitudes_ of the operations had been finite, so as to give rise to an absorption of _H_ units of heat during the first operation, and a lowering of temperature from _S_ to _T_ during the second, the amount of work obtained would have been found to be expressed by means of a double definite integral thus:[48]

_M_ = ∫_{0}^{_H_} _dq_ ∫_{_T_}^{_S_} _dt_. (_Ep_{0}v_{0}_)/(_vdq_/_dv_), ⎫
or ⎬. (4)
_M_ = _Ep_{0}v_{0}_ ∫_{0}^{_H_} ∫_{_T_}^{_S_} (1)/(_v_) (_dv_)/(_dq_). _dtdq_; ⎭

this second form being sometimes more convenient.

28. The preceding investigations, being founded on the approximate laws of compressibility and expansion (known as the law of Mariotte and Boyle, and the law of Dalton and Gay-Lussac), would require some slight modifications to adapt them to cases in which the gaseous medium employed is such as to present sensible deviations from those laws. Regnault’s very accurate experiments show that the deviations are insensible, or very nearly so, for the ordinary gases at ordinary pressures; although they may be considerable for a medium, such as sulphurous acid, or carbonic acid under high pressure, which approaches the physical condition of a vapor at saturation; and therefore, in general, and especially in practical applications to real air-engines, it will be unnecessary to make any modification in the expressions. In cases where it may be necessary, there is no difficulty in making the modifications, when the requisite data are supplied by experiment.

29.[49] Either the steam-engine or the air-engine, according to the arrangements described above, gives all the mechanical effect that can possibly be obtained from the thermal agency employed. For it is clear that in either case the operations may be performed in the reverse order, with every thermal and mechanical effect reversed. Thus, in the steam-engine, we may commence by placing the cylinder on the impermeable stand, allow the piston to rise, performing work, to the position _E_{3}F_{3}_; we may then place it on the body _B_, and allow it to rise, performing work, till it reaches _E_{2}F_{2}_ after that the cylinder may be placed again on the impermeable stand, and the piston may be pushed down to _E_{1}F_{1}_; and, lastly, the cylinder being removed to the body _A_, the piston may be pushed down to its primitive position. In this inverse cycle of operations a certain amount of work has been spent, precisely equal, as we readily see, to the amount of mechanical effect gained in the direct cycle described above; and heat has been abstracted from _B_, and deposited in the body _A_, at a higher temperature, to an amount precisely equal to that which in the direct style was _let down_ from _A_ to _B_. Hence it is impossible to have an engine which will derive more mechanical effect from the same thermal agency than is obtained by the arrangement described above; since, if there could be such an engine, it might be employed to perform, as a part of its whole work, the inverse cycle of operations, upon an engine of the kind we have considered, and thus to continually restore the heat from _B_ to _A_, which has descended from _A_ to _B_ for working itself; so that we should have a complex engine, giving a residual amount of mechanical effect without any thermal agency, or alteration of materials, which is an impossibility in nature. The same reasoning is applicable to the air-engine; and we conclude, generally, that any two engines, constructed on the principles laid down above, whether steam-engines with different liquids, an air-engine and a steam-engine, or two air-engines with different gases, must derive the same amount of mechanical effect from the same thermal agency.

30. Hence, by comparing the amounts of mechanical effect obtained by the steam-engine and the air-engine from the letting down of the _H_ units of heat from _A_ at the temperature (_t_ + τ) to _B_ at _t_, according to the expressions (2) and (3), we have

_M_ = (1 − σ)(_dp_)/(_kdt_). _H_τ = (_Ep_{0}v_{0}_)/(_vdq_/_dv_). _H_τ. (5)

If we denote the coefficient of _Η_τ in these equal expressions by μ, which maybe called “Carnot’s coefficient,” we have

μ = (1 − σ)(_dp_)/(_kdt_) = (_Ep_{0}v_{0}_)/(_vdq_/_dv_), (6)

and we deduce the following very remarkable conclusions:

(1) For the saturated vapors of all different liquids, at the same temperature, the value of (1 − σ)(_dp_/_kdt_) must be the same.

(2) For any different gaseous masses, at the same temperature, the value of _Ep_{0}v_{0}_/(_vdq_/_dv_) must be the same.

(3) The values of these expressions for saturated vapors and for gases, at the same temperature, must be the same.

31. No conclusion can be drawn _a priori_ regarding the values of this coefficient μ for different temperatures, which can only be determined, or compared, by experiment. The results of a great variety of experiments, in different branches of physical science (Pneumatics and Acoustics), cited by Carnot and by Clapeyron, indicate that the values of μ for low temperatures exceed the values for higher temperatures; a result amply verified by the continuous series of experiments performed by Regnault on the saturated vapor of water for all temperatures from 0° to 230°, which, as we shall see later, give values for μ gradually diminishing from the inferior limit to the superior limit of temperature. When, by observation, μ has been determined as a function of the temperature, the amount of mechanical effect, _M_, deducible from _H_ units of heat descending from a body at the temperature _S_ to a body at the temperature _T_, may be calculated from the expression

_M_ = _H_ ∫_{_S_}^{_T_} μ_dt_, (7)

which is, in fact, what either of the equations (1) for the steam-engine, or (4) for the air-engine, becomes, when the notation μ, for Carnot’s multiplier, is introduced.

The values of this integral may be practically obtained, in the most convenient manner, by first determining, from observation, the mean values of μ for the successive degrees of the thermometric scale, and then adding the values for all the degrees within the limits of the extreme temperatures _S_ and _T_.[50]

32. The complete theoretical investigation of the motive power of heat is thus reduced to the experimental determination of the coefficient μ; and may be considered as perfect, when, by any series of experimental researches whatever, we can find a value of μ for every temperature within practical limits. The special character of the experimental researches, whether with reference to gases or with reference to vapors, necessary and sufficient for this object, is defined and restricted in the most precise manner, by the expressions (6) for μ, given above.

33. The object of Regnault’s great work, referred to in the title of this paper, is the experimental determination of the various physical elements of the steam-engine; and when it is complete, it will furnish all the _data_ necessary for the calculation of μ. The valuable researches already published in a first part of that work make known the latent heat of a given weight, and the pressure, of saturated steam for all temperatures between 0° and 230° Cent. of the air-thermometer. Besides these data, however, the density of saturated vapor must be known, in order that _k_, the latent heat of a unit of volume, may be calculated from Regnault’s determination of the latent heat of a given weight.[51] Between the limits of 0° and 100°, it is probable, from various experiments which have been made, that the density of vapor follows very closely the simple laws which are so accurately verified by the ordinary gases;[52] and thus it may be calculated from Regnault’s table giving the pressure at any temperature within those limits. Nothing as yet is known with accuracy as to the density of saturated steam between 100 and 230°, and we must be contented at present to estimate it by calculation from Regnault’s table of pressures; although, when accurate experimental researches on the subject shall have been made, considerable deviations from the laws of Boyle and Dalton, on which this calculation is founded, may be discovered.

34. Such are the experimental data on which the mean values of μ for the successive degrees of the air-thermometer, from 0 to 230°, at present laid before the Royal Society, is founded. The unit of length adopted is the English foot; the unit of weight, the pound; the unit of work, a “foot-pound;” and the unit of heat that quantity which, when added to a pound of water at 0°, will produce an elevation of 1° in temperature. The mean value of μ for any degree is found to a sufficient degree of approximation by taking, in place of σ, _dp_/_dt_ and _k_; in the expression

(1 − σ). (_dp_)/(_kdt_);

the mean values of those elements; or, what is equivalent to the corresponding accuracy of approximation, by taking, in place of σ and _k_ respectively, the mean of the values of those elements for the limits of temperature, and in place of _dp_/_dt_, the difference of the values of _p_, at the same limits.

35. In Regnault’s work (at the end of the eighth memoir), a table of the pressures of saturated steam for the successive temperatures 0°, 1°, 2°, ... 230°, expressed in millimetres of mercury, is given. On account of the units adopted in this paper, these pressures must be estimated in pounds on the square foot, which we may do by multiplying each number of millimetres by 2.7896, the weight in pounds of a sheet of mercury, one millimetre thick, and a square foot in area.

36. The value of _k_, the latent heat of a cubic foot, for any temperature _t_, is found from λ, the latent heat of a pound of saturated steam, by the equation

_k_ = (_p_)/(760). (1 + .00366 × 100)/(1 + .00366 × _t_). × .036869[53] . λ,

where _p_ denotes the pressure in millimetres, and λ the latent heat of a pound of saturated steam; the values of λ being calculated by the empirical formula[54]

λ = (606.5 + 0.305_t_) − (_t_ + .00002_t_^2 + 0.0000003_t_^3),

given by Regnault as representing, between the extreme limits of his observations, the latent heat of a unit weight of saturated steam.

EXPLANATION OF TABLE I.

37. The mean values of μ for the first, for the eleventh, for the twenty-first, and so on, up to the 231st[55] degree of the air-thermometer, have been calculated in the manner explained in the preceding paragraphs. These, and interpolated results, which must agree with what would have been obtained, by direct calculation from Regnault’s data, to three significant places of figures (and even for the temperatures between 0° and 100°, the experimental data do not justify us in relying on any of the results to a greater degree of accuracy), are exhibited in Table I.

_To find the amount of mechanical effect due to a unit of heat, descending from a body at a temperature S to a body at T, if these numbers be integers, we have merely to add the values of μ in Table I. corresponding to the successive numbers._

_T_ + 1, _T_ + 2, ... _S_ − 2, _S_ − 1.

EXPLANATION OF TABLE II.

38. The calculation of the mechanical effect, in any case, which might always be effected in the manner described in § 37 (with the proper modification for fractions of degrees, when necessary), is much simplified by the use of Table II., where the first number of Table I., the sum of the first and second, the sum of the first three, the sum of the first four, and so on, are successively exhibited. The sums thus tabulated are the values of the integrals

∫_{0}^1 μ_dt_, ∫_{0}^2 μ_dt_, ∫_{0}^3 μ_dt_, ... ∫_{0}^{231} μ_dt_;

and, if we denote ∫_{0}^t μ_dt_ by the letter _M_, Table II. may be regarded as a table of the value of _M_.

_To find the amount of mechanical effect due to a unit of heat descending from a body at a temperature S to a body at T, if these numbers be integers, we have merely to subtract the value of M, for the number T, from the value for the number S, given in Table II._

TABLE I.[56]
MEAN VALUES OF Μ FOR THE SUCCESSIVE DEGREES OF THE AIR-THERMOMETER FROM
0° TO 230°.
───────────────────────────────────┬───────────────────────────────────
° │ μ
───────────────────────────────────┼───────────────────────────────────
1│ 4.960
2│ 4.946
3│ 4.932
4│ 4.918
5│ 4.905
6│ 4.892
7│ 4.878
8│ 4.865
9│ 4.852
10│ 4.839
11│ 4.826
12│ 4.812
13│ 4.799
14│ 4.786
15│ 4.773
16│ 4.760
17│ 4.747
18│ 4.735
19│ 4.722
20│ 4.709
21│ 4.697
22│ 4.684
23│ 4.672
24│ 4.659
25│ 4.646
26│ 4.634
27│ 4.621
28│ 4.609
29│ 4.596
30│ 4.584
31│ 4.572
32│ 4.559
33│ 4.547
34│ 4.535
35│ 4.522
36│ 4.510
37│ 4.498
38│ 4.486
39│ 4.474
40│ 4.462
41│ 4.450
42│ 4.438
43│ 4.426
44│ 4.414
45│ 4.402
46│ 4.390
47│ 4.378
48│ 4.366
49│ 4.355
50│ 4.343
51│ 4.331
52│ 4.319
53│ 4.308
54│ 4.296
55│ 4.285
56│ 4.273
57│ 4.262
58│ 4.250
59│ 4.239
60│ 4.227
61│ 4.216
62│ 4.205
63│ 4.194
64│ 4.183
65│ 4.172
66│ 4.161
67│ 4.150
68│ 4.140
69│ 4.129
70│ 4.119
71│ 4.109
72│ 4.098
73│ 4.088
74│ 4.078
75│ 4.067
76│ 4.057
77│ 4.047
78│ 4.037
79│ 4.028
80│ 4.018
81│ 4.009
82│ 3.999
83│ 3.990
84│ 3.980
85│ 3.971
86│ 3.961
87│ 3.952
88│ 3.943
89│ 3.934
90│ 3.925
91│ 3.916
92│ 3.907
93│ 3.898
94│ 3.889
95│ 3.880
96│ 3.871
97│ 3.863
98│ 3.854
99│ 3.845
100│ 3.837
101│ 3.829
102│ 3.820
103│ 3.812
104│ 3.804
105│ 3.796
106│ 3.788
107│ 3.780
108│ 3.772
109│ 3.764
110│ 3.757
111│ 3.749
112│ 3.741
113│ 3.734
114│ 3.726
115│ 3.719
116│ 3.712
117│ 3.704
118│ 3.697
119│ 3.689
120│ 3.682
121│ 3.675
122│ 3.668
123│ 3.661
124│ 3.654
125│ 3.647
126│ 3.640
127│ 3.633
128│ 3.627
129│ 3.620
130│ 3.614
131│ 3.607
132│ 3.601
133│ 3.594
134│ 3.586
135│ 3.579
136│ 3.573
137│ 3.567
138│ 3.561
139│ 3.555
140│ 3.549
141│ 3.543
142│ 3.537
143│ 3.531
144│ 3.525
145│ 3.519
146│ 3.513
147│ 3.507
148│ 3.501
149│ 3.495
150│ 3.490
151│ 3.484
152│ 3.479
153│ 3.473
154│ 3.468
155│ 3.462
156│ 3.457
157│ 3.451
158│ 3.446
159│ 3.440
160│ 3.435
161│ 3.430
162│ 3.424
163│ 3.419
164│ 3.414
165│ 3.409
166│ 3.404
167│ 3.399
168│ 3.394
169│ 3.389
170│ 3.384
171│ 3.380
172│ 3.375
173│ 3.370
174│ 3.365
175│ 3.361
176│ 3.356
177│ 3.351
178│ 3.346
179│ 3.342
180│ 3.337
181│ 3.332
182│ 3.328
183│ 3.323
184│ 3.318
185│ 3.314
186│ 3.309
187│ 3.304
188│ 3.300
189│ 3.295
190│ 3.291
191│ 3.287
192│ 3.282
193│ 3.278
194│ 3.274
195│ 3.269
196│ 3.265
197│ 3.261
198│ 3.257
199│ 3.253
200│ 3.249
201│ 3.245
202│ 3.241
203│ 3.237
204│ 3.233
205│ 3.229
206│ 3.225
207│ 3.221
208│ 3.217
209│ 3.213
210│ 3.210
211│ 3.206
212│ 3.202
213│ 3.198
214│ 3.195
215│ 3.191
216│ 3.188
217│ 3.184
218│ 3.180
219│ 3.177
220│ 3.173
221│ 3.169
222│ 3.165
223│ 3.162
224│ 3.158
225│ 3.155
226│ 3.151
227│ 3.148
228│ 3.144
229│ 3.141
230│ 3.137
231│ 3.134
───────────────────────────────────┴───────────────────────────────────

TABLE II.
MECHANICAL EFFECT IN FOOT-POUNDS DUE TO A THERMIC UNIT CENTIGRADE,
PASSING FROM A BODY, AT ANY TEMPERATURE LESS THAN 230° TO A BODY AT 0°.
───────────────────────────────────┬───────────────────────────────────
Superior Limit of Temperature. │ Mechanical Effect.
───────────────────────────────────┼───────────────────────────────────
° │ Ft.-Pounds.
│
1│ 4.960
2│ 9.906
3│ 14.838
4│ 19.756
5│ 24.661
6│ 29.553
7│ 34.431
8│ 39.296
9│ 44.148
10│ 48.987
11│ 53.813
12│ 58.625
13│ 63.424
14│ 68.210
15│ 72.983
16│ 77.743
17│ 82.490
18│ 87.225
19│ 91.947
20│ 96.656
21│ 101.353
22│ 106.037
23│ 110.709
24│ 115.368
25│ 120.014
26│ 124.648
27│ 129.269
28│ 133.878
29│ 138.474
30│ 143.058
31│ 147.630
32│ 152.189
33│ 156.736
34│ 161.271
35│ 165.793
36│ 170.303
37│ 174.801
38│ 179.287
39│ 183.761
40│ 188.223
41│ 192.673
42│ 197.111
43│ 201.537
44│ 205.951
45│ 210.353
46│ 214.743
47│ 219.121
48│ 223.487
49│ 227.842
50│ 232.185
51│ 236.516
52│ 240.835
53│ 245.143
54│ 249.439
55│ 253.724
56│ 257.997
57│ 262.259
58│ 266.509
59│ 270.748
60│ 274.975
61│ 279.191
62│ 283.396
63│ 287.590
64│ 291.773
65│ 295.945
66│ 300.106
67│ 304.256
68│ 308.396
69│ 312.525
70│ 316.644
71│ 320.752
72│ 324.851
73│ 328.939
74│ 333.017
75│ 337.084
76│ 341.141
77│ 345.188
78│ 349.225
79│ 353.253
80│ 357.271
81│ 361.280
82│ 365.279
83│ 369.269
84│ 373.249
85│ 377.220
86│ 381.181
87│ 385.133
88│ 389.076
89│ 393.010
90│ 396.935
91│ 400.851
92│ 404.758
93│ 408.656
94│ 412.545
95│ 416.425
96│ 420.296
97│ 424.159
98│ 428.013
99│ 431.858
100│ 435.695
101│ 439.524
102│ 443.344
103│ 447.156
104│ 450.960
105│ 454.756
106│ 458.544
107│ 462.324
108│ 466.096
109│ 469.860
110│ 473.617
111│ 477.366
112│ 481.107
113│ 484.841
114│ 488.567
115│ 492.286
116│ 495.998
117│ 499.702
118│ 503.399
119│ 507.088
120│ 510.770
121│ 514.445
122│ 518.113
123│ 521.174
124│ 525.428
125│ 529.075
126│ 532.715
127│ 536.348
128│ 539.975
129│ 543.595
130│ 547.209
131│ 550.816
132│ 554.417
133│ 558.051
134│ 561.597
135│ 565.176
136│ 568.749
137│ 572.316
138│ 575.877
139│ 579.432
140│ 582.981
141│ 586.524
142│ 590.061
143│ 593.592
144│ 597.117
145│ 600.636
146│ 604.099
147│ 607.656
148│ 611.157
149│ 614.652
150│ 618.142
151│ 621.626
152│ 625.105
153│ 628.578
154│ 632.046
155│ 635.508
156│ 638.965
157│ 642.416
158│ 645.862
159│ 649.302
160│ 652.737
161│ 656.167
162│ 659.591
163│ 663.010
164│ 666.424
165│ 669.833
166│ 673.237
167│ 676.636
168│ 680.030
169│ 683.419
170│ 686.803
171│ 690.183
172│ 693.558
173│ 696.928
174│ 700.293
175│ 703.654
176│ 707.010
177│ 710.361
178│ 713.707
179│ 717.049
180│ 720.386
181│ 723.718
182│ 727.046
183│ 730.369
184│ 733.687
185│ 737.001
186│ 740.310
187│ 743.614
188│ 746.914
189│ 750.209
190│ 753.500
191│ 756.787
192│ 760.069
193│ 763.347
194│ 766.621
195│ 769.890
196│ 773.155
197│ 776.416
198│ 779.673
199│ 782.926
200│ 786.175
201│ 789.420
202│ 792.661
203│ 795.898
204│ 799.131
205│ 802.360
206│ 805.585
207│ 808.806
208│ 812.023
209│ 815.236
210│ 818.446
211│ 821.652
212│ 824.854
213│ 828.052
214│ 831.247
215│ 834.438
216│ 837.626
217│ 840.810
218│ 843.990
219│ 847.167
220│ 850.340
221│ 853.509
222│ 856.674
223│ 859.836
224│ 862.994
225│ 866.149
226│ 869.300
227│ 872.448
228│ 875.592
229│ 878.733
230│ 881.870
231│ 885.004
───────────────────────────────────┴───────────────────────────────────

_Note on the curves described in Clapeyron’s graphical method of
exhibiting Carnot’s Theory of the Steam-Engine._

39. At any instant when the temperature of the water and vapor is _t_, during the fourth operation (see above, § 16, and suppose, for the sake of simplicity, that at the beginning of the first and at the end of the fourth operation the piston is absolutely in contact with the surface of the water), the latent heat of the vapor must be precisely equal to the amount of heat that would be necessary to raise the temperature of the whole mass, if in the liquid state, from _t_ to _S_.[57] Hence, if _v′_ denote the volume of the vapor, _c_ the mean capacity for heat of a pound of water between the temperatures _S_ and _t_, and _W_ the weight of the entire mass, in pounds, we have

_kv′_ = _c_(_S_ − _t_)_W_.

Again, the circumstances during the second operation are such that the mass of liquid and vapor possesses _H_ units of heat more than during the fourth; and consequently, at the instant of the second operation, when the temperature is _t_, the volume _v_ of the vapor will exceed _v′_ by an amount of which the latent heat is _H_, so that we have

_v_ = _v′_ + (_H_)/(_k_).

40. Now, at any instant, the volume between the piston and its primitive position is less than the actual volume of vapor by the volume of the water evaporated. Hence, if _x_ and _x′_ denote the abscissæ of the curve at the instants of the second and fourth operations respectively, when the temperature is _t_, we have

_x_ = _v_ − σ_v_, _x′_ = _v′_ − σ_v′_,

and, therefore, by the preceding equations,

_x_ = (1 − σ)/(_k_){_H_ + _c_(_S_ − _t_)_W_}, (_a_)
_x′_ = (1 − σ)/(_k_)_c_(_S_ − _t_)_W_. (_b_)
These equations, along with _y_ = _y′_ = _p_, (_c_)

enable us to calculate, from the data supplied by Regnault, the abscissa and ordinate for each of the curves described above (§ 17) corresponding to any assumed temperature _t_. After the explanations of §§ 33, 34, 35, 36, it is only necessary to add that _c_ is a quantity of which the value is very nearly unity, and would be exactly so were the capacity of water for heat the same at every temperature as it is between 0° and 1°; and that the value of _c_(_S_ − _t_), for any assigned values of _S_ and _t_, is found, by subtracting the number corresponding to _t_ from the number corresponding to _s_, in the column headed “_Nombre des unités de chaleur abandonnées par un kilogramme d’eau en descendant de T° à 0°_,” of the last table (at the end of the tenth memoir) of Regnault’s work. By giving _S_ the value 230°, and by substituting successively 220, 210, 200, etc., for _t_, values for _x_, _y_, _x′_, _y′_, have been found, which are exhibited in the table opposite.

─────────────┬─────────────────┬────────────────────────┬─────────────
Temperatures.│ Volumes to be │ Volumes from the │Pressures of
│described by the │ primitive position of │ saturated
│ piston, to │ the piston to those │ steam, in
│ complete the │occupied at instants of │pounds on the
│fourth operation.│ the second operation. │square foot.
_t_ │ _x′_ │ _x_ │_y_ = _y′_ =
│ │ │ _p_
─────────────┼─────────────────┼────────────────────────┼─────────────
0°│ 1269. _W_ │_x′_ + 5.409._H_ │ 12.832
10│ 639.6. _W_ │_x′_ + 2.847._H_ │ 25.567
20│ 337.3. _W_ │_x′_ + 1.571._H_ │ 48.514
30│ 185.5. _W_ │_x′_ + .9062._H_ │ 88.007
40│ 105.9. _W_ │_x′_ + .5442._H_ │ 153.167
50│ 62.62. _W_ │_x′_ + .3392._H_ │ 256.595
60│ 38.19. _W_ │_x′_ + .2188._H_ │ 415.070
70│ 21.94. _W_ │_x′_ + .1456._H_ │ 650.240
80│ 15.38. _W_ │_x′_ + .09962._H_ │ 989.318
90│ 10.09. _W_ │_x′_ + .06994._H_ │ 1465.80
100│ 6.744. _W_ │_x′_ + .05026._H_ │ 2120.11
110│ 4.578. _W_ │_x′_ + .03688._H_ │ 2999.87
120│ 3.141. _W_ │_x′_ + .02758._H_ │ 4160.10
130│ 2.176. _W_ │_x′_ + .02098._H_ │ 5663.70
140│ 1.519. _W_ │_x′_ + .01625._H_ │ 7581.15
150│ 1.058. _W_ │_x′_ + .01271._H_ │ 9990.26
160│ 0.7369. _W_ │_x′_ + .01010._H_ │ 12976.2
170│ 0.5085. _W_ │_x′_ + .008116._H_ │ 16630.7
180│ 0.3454. _W_ │_x′_ + .006592._H_ │ 21051.5
190│ 0.2267. _W_ │_x′_ + .005406._H_ │ 26341.5
200│ 0.1409. _W_ │_x′_ + .004472._H_ │ 32607.7
210│ 0.0784. _W_ │_x′_ + .003729._H_ │ 39960.7
220│ 0.3310. _W_ │_x′_ + .003130._H_ │ 48512.4
230│ 0 │_x′_ + .002643._H_ │ 58376.6
─────────────┴─────────────────┴────────────────────────┴─────────────

_Appendix._

(Read April 30, 1849.)

41. In p. 30 some conclusions drawn by Carnot from his general reasoning were noticed; according to which it appears, that if the value of μ for any temperature is known, certain information may be derived with reference to the saturated vapor of any liquid whatever, and, with reference to any gaseous mass, without the necessity of experimenting upon the specific medium considered. Nothing in the whole range of Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning. We have seen, however, that doubt may exist with reference to the truth of the axiom on which the entire theory is founded, and it therefore becomes more than a matter of mere curiosity to put the inferences deduced from it to the test of experience. The importance of doing so was clearly appreciated by Carnot; and, with such data as he had from the researches of various experimenters, he tried his conclusions. Some very remarkable propositions which he derives from his theory coincide with Dulong and Petit’s subsequently discovered experimental laws with reference to the heat developed by the compression of a gas; and the experimental verification is therefore in this case (so far as its accuracy could be depended upon) decisive. In other respects, the data from experiment were insufficient, although, so far as they were available as tests, they were confirmatory of the theory.

42. The recent researches of Regnault add immensely to the experimental data available for this object, by giving us the means of determining with considerable accuracy the values of μ within a very wide range of temperature, and so affording a trustworthy standard for the comparison of isolated results at different temperatures, derived from observations in various branches of physical science.

In the first section of this Appendix the theory is tested, and shown to be confirmed by the comparison of the values of μ found above, with those obtained by Carnot and Clapeyron from the observations of various experimenters on air, and the vapors of different liquids. In the second and third sections some striking confirmations of the theory arising from observations by Dulong, on the specific heat of gases, and from Mr. Joule’s experiments on the heat developed by the compression of air, are pointed out; and in conclusion, the actual methods of obtaining mechanical effect from heat are briefly examined with reference to their economy.

I. _On the values of μ derived by Carnot and Clapeyron from observations
on Air, and on the Vapors of various liquids._

43. In Carnot’s work, pp. 80–82, the mean value of μ between 0° and 1° is derived from the experiments of Delaroche and Bérard on the specific heat of gases, by a process approximately equivalent to the calculation of the value of (_Ep_{0}v_{0}_)/(_vdq_/_dv_) for the temperature ½°. There are also, in the same work, determinations of the values of μ from observations on the vapors of alcohol and water; but a table given in M. Clapeyron’s paper, of the values of μ derived from the data supplied by various experiments with reference to the vapors of ether, alcohol, water, and oil of turpentine, at the respective boiling-points of these liquids, affords us the means of comparison through a more extensive range of temperature. In the cases of alcohol and water, these results ought of course to agree with those of Carnot. There are, however, slight discrepancies which must be owing to the uncertainty of the experimental data.[58] In the opposite table, Carnot’s results with reference to air, and Clapeyron’s results with reference to the four different liquids, are exhibited, and compared with the values of μ which have been given above (Table I.) for the same temperatures, as derived from Regnault’s observations on the vapor of water.