STEAM

CHAP. I.—CONSIDERATIONS OF A GENERAL NATURE RE-
GARDING THE PROPERTIES, PHENOMENA, AND APPLICA-
TIONS OF STEAM.

STEAM is the name given in our language to the visible, moist vapour which arises from all bodies which contain juices easily expelled from them by heats not sufficient for their combustion. Thus we say, the steam of boiling water, of malt, of a tan-bed. It is distinguished from smoke by its not having been produced by combustion, by not containing any soot, and by its being condensable by cold into water, oil, inflammable spirits, or liquids composed of these. We see it rise in great abundance from bodies when they are heated, forming a white cloud, which diffuses itself and disappears at no very great distance from the body from which it was produced. In this case the surrounding air is found loaded with the water or moisture which seems to have produced it; and the steam seems to be completely soluble in air, composing, while thus united, a transparent elastic fluid. But in order to its appearance in the form of an opaque white cloud, the mixture with or dissemination in air seems necessary. If a tea-kettle boils violently, so that the steam is formed at the spout in great abundance, it may be observed that the visible cloud is not formed at the very mouth of the spout, but at a small distance before it, and that the vapour is perfectly invisible at its first emission. This is rendered still more evident by fitting to the spout of the tea-kettle a glass-pipe of any length, and of as large a diameter as we please. The steam is produced as copiously as without this pipe, but the vapour is transparent and colourless throughout the whole of the pipe; nay, if this pipe communicate with a glass vessel terminating in another pipe, and if the vessel be kept sufficiently hot, the steam will be as abundantly produced at the mouth of this second pipe as before, and the vessel will remain quite transparent. The visibility, therefore, of the matter which constitutes the steam is an accidental circumstance, and appears to require its dissemination in the air; and we know that one perfectly transparent body, when minutely divided and diffused among the parts of another transparent body, but not dissolved in it, makes a mass which is visible. Thus oil beaten up with water makes a white opaque mass.

When steam is produced, the water gradually wastes in the tea-kettle, and will soon be totally expended if we continue it on the fire. It is reasonable, therefore, to suppose that this steam is nothing but water, changed by heat into an aerial or elastic form. If so, we should expect that the privation of this heat would leave it in the form of water again. Accordingly, this is fully verified by experiment; for if the pipe fitted to the tea-kettle be surrounded with ice, or any cold substance, no steam will issue, but water will continually trickle from it in drops; and if the process be conducted with the proper precautions, the water which we thus obtain from the pipe will be found equal in quantity to that which disappears from the tea-kettle. Steam is therefore the matter of water converted by heat into an elastic vapour.

We are most familiar with steam when in the act of rising violently from heated water in the process of ebullition. The history of steam at this crisis is highly instructive, and its phenomena may be studied with advantage by examining it in a glass vessel placed over a strong lamp. When heat is first applied, a rapid circulation of the fluid ensues. The water on the bottom, being first heated and expanded, becoming lighter than the rest, rises

to the top, and is replaced by the current of colder water descending, to receive in its turn a further accession of heat. By and by small globules of steam, formed on the bottom and surrounded by a film of water, are observed adhering to the glass; as the heat increases they enlarge; in a short time several of them unite, form a bubble larger than the others, and, detaching themselves from the glass, rise upwards in the fluid. But they never reach the surface; they encounter currents of water still comparatively cold, and descending to receive from the bottom their supply of heat, and encountering them, the bubbles are robbed of their heat, shrivel up into their original bulk, and are lost among the other particles of water. In a short time the mass of the water becomes more uniformly heated; the bubbles, becoming larger and more frequent, are condensed with a loud crackling noise, and at last, when the heat of the whole mass reaches $212^\{\backslash circ\}$, the bubbles from the bottom rise without condensation through the water, swell and unite with others as they rise, and burst out upon the air in a copious volume of steam, of the same heat as the water from which they are formed, and pushing aside the air, make room for themselves. In this process, by continuing the application of heat, the whole of the water may be "boiled away" or converted into steam.

The singular sounds produced from a vessel of water exposed to heat previously to boiling, have attracted attention; the water is then vulgarly said to be simmering or singing, and, when this takes place, it is because the vessel is boiling at one place and comparatively cold at another. This noise is most distinctly heard when the fire or flame applied is small and its heat intense, when the vessel is large and the water deep; for in that case the entrance of the caloric will take place more rapidly than the circulation can convey it to the remote particles of fluid, and so bubbles of steam will form rapidly at one place and be rapidly condensed at another; the degree of velocity with which such bubbles succeed will determine the pitch of the singing tone. We have observed this phenomenon in greatest perfection when we have attached a slender pipe to a close boiler producing steam, and carried its open mouth, of the diameter of $\backslash frac\{1\}\{4\}$ or $\backslash frac\{1\}\{8\}$ths of an inch, down below the surface of cold water in a glass jar. When the mouth of the steam-pipe is held just below the surface of the water, the steam issues with great rapidity in small bubbles, producing an acute tone; and, on the other hand, when the pipe is held at a considerable depth, the concussions become more violent and louder, their intervals of succession greater, the tone is lowered, and finally, the shocks become detached, and so violent as to shake the glass and surrounding objects with much force. On this subject Professor Robison observes that a violent and remarkable phenomenon appears if we suddenly plunge a lump of red-hot iron into a vessel of cold water, taking care that no red part be near the surface. If the hand be now applied to the side of the vessel, a most violent tremor is felt, and sometimes strong thumps; these arise from the collapsing of very large bubbles. If the upper part of the iron be too hot, it warms the surrounding water so much, that the bubbles from below come up through it uncondensed, and produce ebullition without concussion. The great resemblance of this tremor to the sensation which we experience during the shock of an earthquake has led many to suppose that the latter is produced in the same way; and their hypothesis is by no means unfeasible. Any obstruction on the bottom of a boiler on the inside, as a piece of metal or stone introduced among the water, may

Considerations regarding Steam. produce a succession of smart concussions by the sudden condensation of gas collected under it.

The permanence of the boiling point is one of the most remarkable of the phenomena of ebullition. When water has once been brought to boil in an open vessel, it is not possible to make the water sensibly hotter, however strongly the fire may be urged or its intensity increased. This circumstance is very striking, because we know that heat continues to be thrown in exactly as fast as before the boiling point; and that, in that case, the heat rose rapidly, whereas now it has altogether ceased to increase. If a thermometer of mercury, air, oil, or metal be placed among the water, the temperature will constantly increase, and expand the matter of the thermometer until the water boils; and then, whether it boil slowly or rapidly, with a strong fire or a gentle one, the thermometer will continue to stand at the same point. This point is so well defined as to furnish our standard for the comparison of temperature, and is the same on all thermometers, being called the boiling point, although it is differently numbered on each, being called $212^\{\backslash circ\}$ on our common thermometer, or Fahrenheit's, $80^\{\backslash circ\}$ on Reaumur's, and $100^\{\backslash circ\}$ on the centigrade thermometer.

It is also to be remarked that the temperature of the steam issuing from boiling water is the same with the temperature of the water itself, and remains equally invariable; so that all the steam produced from water boiling at $212^\{\backslash circ\}$ is itself at $212^\{\backslash circ\}$. This remark will assist us in accounting for the disposal of the heat which the fire gives out during the time of ebullition; for it is manifest that the heat is all the while carried off by the large volumes of steam, at a temperature of $212^\{\backslash circ\}$, that are diffused through the air; and so it happens that an increase of heat in the fire, instead of increasing the heat of the water, only increases the volumes of the steam thrown off and the quantity of heat carried away. This view of the subject is confirmed by a simple experiment. Take a strong glass flask, place water in it, and a thermometer among the water, and let it be held over a lamp until the water boil, and the thermometer will be observed rising till it reach $212^\{\backslash circ\}$, when the steam will begin to escape rapidly from the neck of the flask. Let it now be corked tightly, and the heat continually applied; and it will be observed that the thermometer does not now stand at $212^\{\backslash circ\}$, but rises rapidly from that point up to $220^\{\backslash circ\}$ and $230^\{\backslash circ\}$, showing that the free escape of the steam into the open air is necessary to the permanence of the boiling point. If the heat be still applied, the experiment may be rendered still more instructive by suddenly pulling out the cork of the flask, when the vapour will instantly rush out in a large volume, and the thermometer sink down to $212^\{\backslash circ\}$, showing that all the excess of heat has been carried off by the steam into the air.

It has thus been seen that a large quantity of heat may be given out in the particles of a certain quantity of water, converting them into steam; and yet that the thermometer shall afford no indication of this quantity. As soon as water boils the whole mass is heated up to $212^\{\backslash circ\}$; and although the same heat that produced the ebullition be still continually applied, and although we know that this heat must be continually entering into the water, still it is not detected, or in any way exhibited by the thermometer. On this account the heat given to water during ebullition is said to become latent, or lie hid from the thermometer; and, indeed, the thermometer merely indicates the intensity of heat; the calorimeter alone can measure its quantity. The quantity of heat given out to water after it has begun to boil is more than fivefold that which is sufficient to bring it from the freezing up to the boiling point; for, if we continue the fire with the same intensity that was used in bringing it to boil, it will require more than fivefold that duration and quantity of fuel to boil all the water away, or convert it all into steam of $212^\{\backslash circ\}$ of heat. Thus the sen-

sible heat, added from $32^\{\backslash circ\}$, will be $180^\{\backslash circ\}$, and that latent in the steam is more than fivefold; or, in other words, the insensible caloric in steam is fivefold its sensible heat; or the same quantity of matter in the condition of steam at $212^\{\backslash circ\}$, and of water at $212^\{\backslash circ\}$, will hold different quantities of caloric, in the proportion of above 6 to 1. This is called the greater capacity of steam for caloric than of water for that substance; and it is in part accounted for by the greater distances of the particles of the matter of steam and water from each other in the former than the latter condition. Dr Black was the discoverer of the admirable doctrine of latent heat.

Dr Dalton has thus illustrated the doctrine of latent heat, and of the increased capacity of a liquid for holding caloric when it passes into the condition of vapour. The liquid and its vapour may be considered as two reservoirs of caloric, capable of holding different quantities of that fluid. Let fig. 1 represent to us such an arrangement: the internal cylinder of smaller capacity; the external one of enlarged capacity, surrounding and extending far above it; and a small open tube of glass, communicating freely at the bottom with the internal cylinder. Let us now conceive water to be poured into the internal cylinder; the water will manifestly flow into the slender tube till it stand on the same level in the tube as in the cylinder. If any additional quantity be now poured into the internal cylinder, the rise of water in the slender glass tube will serve as an index of the quantity of added fluid; and when it is filled to

the top, the fluid will stand at the height marked $212^\{\backslash circ\}$, and will still be a correct index of the addition of fluid. But if more water be now added to it, it will not make its appearance in the slender tube, but will simply overflow from the internal cylinder over into that of enlarged capacity, so that while a large quantity is passing into the vessel and gradually filling it up to $212^\{\backslash circ\}$, no additional rise takes place until the whole of the outer cylinder become filled to that point, after which any further addition will again become sensible by a corresponding rise in the tube. This process is in precise analogy to the succession of circumstances in heating a liquid and converting it into a steam. The internal cylinder represents the liquid; the external one, the vapour of greater capacity; and the slender glass tube at the side, the thermometer placed in communication with them. When heat flows into the liquid, it passes equally into the thermometer; and each increment of the one produces an equal increment in the other, until the liquid reaches the limit of its capacity, when it suddenly begins to enlarge its bulk and take the form of steam; but the quantity of heat required to fill up this enlarged capacity is so great as to require about $5\backslash frac\{1\}\{2\}$ times as much to fill it as was contained in the whole liquid before, so that all this time the thermometer is standing still, and it is not until the whole of the steam is thus supplied with $212^\{\backslash circ\}$ of caloric that the thermometer will begin to show any further elevation; after which, any increment of heat thrown into the steam will make its appearance on the thermometer, and proceed, as formerly, by simultaneous increments.

It appears, therefore, that the cause why water boiling under the open air does not reach a higher temperature than $212^\{\backslash circ\}$ is, that the steam which is raised by any additional heat carries that additional quantity of heat along with it into the air. But here a question occurs at once

Considerations regarding Steam. to the inquirer into these phenomena, viz. Why does water require to be heated up to $212^\{\backslash circ\}$ before it will throw off its increments of heat and vapour into the air? Why does not steam rise equally strongly from water at $200^\{\backslash circ\}$ or $180^\{\backslash circ\}$? The categorical reply is, that the elastic force of the heat is not sufficient to enable the steam to force its way against the pressure of the air until it reaches this point. In order to understand the means by which we arrive at this conclusion, it is necessary to know that, when the pressure of air on the surface of the water is artificially diminished, the steam does actually rise, and the water bubbles and boils with great violence, at temperatures far below $212^\{\backslash circ\}$. It is only when the surface of the water is exposed to the full pressure of the air in a common vessel that it is prevented from rising in vapour, at temperatures lower than the usual boiling point. If the surface of the hot water be protected from the pressure of the air, by being placed under a glass shade, and the air removed from the inside of it by an air-pump, the water may be made to boil at all temperatures below $212^\{\backslash circ\}$.

Conceive a vessel of water first of all boiling at $212^\{\backslash circ\}$ in the open air, as the vessel A in fig. 2, the thermometer 1

being placed in it. After allowing the water to cool to $200^\{\backslash circ\}$, let the vessel of water and the immersed thermometer be now placed on the plate-stand P of an air-pump, and covered over with a strong glass receiver R; and let a portion of the enclosed air be now withdrawn by the pump from the inside of the receiver by the pipe r; and suppose that there are in all 30 cubic inches, or other volumes, of air in the receiver at first; then the water being at $200^\{\backslash circ\}$, when about 7 out of the 30 parts of the air have been withdrawn, leaving only about 23 parts out of 30 pressing on the water, it will be observed instantly to commence giving off steam in rapid ebullition. If next, the process be repeated, only allowing the water to cool to $190^\{\backslash circ\}$, the ebullition will not commence at this lower temperature till about 12 out of the 30 volumes of air have been withdrawn; and if, in a third experiment, the water be cooled down to $180^\{\backslash circ\}$, the elastic force communicated by this degree of heat will not be capable of overcoming the resistance arising from the pressure of the air until one-half of the original pressure of 30 has been removed. To this process there is no limit, for as we go on lowering the temperature, we can always find a point at which the water will boil, provided the counteracting pressure be sufficiently diminished.

Distillation is a method of separating a liquid from extraneous matter, by first of all converting it into steam, and then condensing that steam so as to form the liquid. Dif-

ferent substances take the liquid form at various temperatures; and, therefore, the heat may be so regulated that only one substance of a mixture shall take the form of vapour, and being conveyed by a pipe through a vessel of cold water, or otherwise exposed to the cooling process, the vapour being condensed will give the pure liquid. A great improvement upon the process of separating liquids has been successfully introduced by Mr Howard. It consists of distillation or evaporation in vacuo, and has been most usefully employed in the refining process of sugar. When sugar is dissolved in water, it requires a much higher temperature than $212^\{\backslash circ\}$ to boil the mixture, or to convert the water into steam and separate it from the solid; and as the process goes on, and the solution comes to hold less and less water, the requisite degree of heat is further augmented, until the temperature becomes so high as to injure the colour and otherwise deteriorate the article of merchandise in its crystallized state. Instead of this increased temperature, Mr Howard places the syrup in vacuo, and thus boils it at a low and innocuous heat.

The pulse-glass, an invention attributed to Dr Franklin, is an apparatus illustrating beautifully the process of ebullition in vacuo at low temperatures. If two glass-balls, A and B (fig. 3), be connected by a slender tube, and one of them A be filled with water, a small opening or pipe b being left at the top of the other, and this be made to boil, the vapour produced by it will drive all the air out of the other, and will at last come out

itself, producing steam at the mouth of the pipe. When the ball B is observed to be occupied by transparent vapour, we may conclude that the air is completely expelled. Now, shut the pipe by sticking into it a piece of tallow or wax, the vapour in B will soon condense, and there will be a vacuum. The flame of a lamp and blow-pipe being directed to the little pipe b, will immediately cause it to close and seal hermetically. We have now a pulse-glass. Grasp the ball A in the hollow of the hand; the heat of the hand will immediately expand the bubble of vapour which may be in it, and this vapour will drive the water into B, and then will blow up through it for a long while, keeping it in a state of violent ebullition, as long as there remains a drop or film of water in A. But care must be taken that B is all the while kept cold, that it may condense the vapour as fast as it rises through the water. Touching B with the hand, or breathing warm on it, will immediately stop the ebullition. When the water in A has thus been dissipated, grasp B in the hand; the water will be driven into A, and the ebullition will take place there as it did in B. Putting one of the balls into the mouth will make the ebullition more violent in the other, and the one in the mouth will feel very cold. This is a pretty illustration of the rapid absorption of the heat by the particles of water, which are thus converted into elastic vapour. We have seen this little toy suspended by the middle of the tube like a balance, and thus placed in the inside of a window, having two holes a, b cut in the pane, in such a situation, that, when A is full of water and preponderates, B is opposite to the hole b. Whenever the room became sufficiently warm, the vapour was formed in A and immediately brought the water into B, which was kept cool by the air coming into the room through the hole b. By this means B was made to preponderate in its turn, and A was then opposite to the hole a, and the process was now repeated in the opposite direction. This amusement continued as long as the room was warm enough. Instead of water, alcohol or ether may be substituted, and will act more readily.

The following experiment, where ebullition is produced by the application of cold, is instructive. A Florence flask

Consideration regarding Steam.

a lamp E until the water boils; and when the steam has been rising for a short time violently from the neck of the vessel, the cork s is to be applied as a stopper, and must fit with great accuracy. The flask thus closed is to be set aside for a few minutes till it have cooled considerably, and is then to be suddenly placed on a stand in the cold water w, contained in the glass reservoir R. The ebullition in the flask will recommence with a degree of violence proportioned to the coldness of the water w. The theory of this action is simple. When the flask is plunged in the cold water, two-fifths of its contents are steam: the chill water condenses it into water, it shrinks up into $\backslash frac\{1\}\{1642\}$ part of its bulk, and would leave 1641 parts out of 1642 vacuous; but the warm water being now in vacuo, throws up in rapid ebullition copious volumes of vapour of its own temperature, which is, again, by coming into contact with the sides of the vessel and by directly giving off its heat to the water, chilled into water; and so in succession all the vapour thus sent up is in turn reconverted into water and the vacuum sustained, until at last, the equilibrium between the temperature of the water, within and around the flask, having been established, the interchange of caloric ceases; and even now, if the flask were plunged into freezing water, the ebullition would recommence as violently as before.

We have already noticed the fact, that, when water is confined in a close vessel, and heat is applied to it, the water will not boil even at a temperature of $212^\{\backslash circ\}$. If heat be continually thrown into the water in this state, the particles will acquire a very high temperature; and, at the same time, the tendency of the enclosed fluid to burst the vessel will become very great. The following experiment upon this subject is one of the most interesting and the earliest of which we are in possession. It was published in 1663 by the Marquis of Worcester, and we give it in his own words:—"I have taken a piece of a whole cannon, whereof the end was burst, and filled it [with water] three-quarters full, stopping and screwing up the broken end as also the touch-hole, and making a constant fire under it; within twenty-four hours it burst, and made a great crack."

It is in virtue of the great elastic force by which water, when heated, tends to expand into many times its bulk, in the form of steam, that this element has become a mechanical mover, subject to the control of man. There are two great principles of classification upon which such machines are constructed; the one commonly called high-pressure or non-condensing steam-engines, and the other low-pressure or condensing steam-engines.

In order, however, to its successful application as a mechanical power, and its profitable use in each of the various functions which it is capable of performing, it is necessary to study its various phenomena in greater detail; to obtain

an intimate acquaintance with its properties; to determine its laws in the various relations of space, time, and quantity; how much heat it requires; what fuel it consumes; what force it exerts; how fast it will move; how it will condense, expand, and contract; and what relation it bears to the fluid from which it is derived. These inquiries, and the manner in which these objects may be most satisfactorily obtained, is the subject of this article, and of the following articles on the steam-engine in its various applications.

CHAP. II.—HISTORICAL NOTICE OF EXPERIMENTS ON THE PROPERTIES OF STEAM.

The earliest researches into the phenomena of steam, undertaken with the philosophical purpose of obtaining experimental data for the scientific investigation of its properties and relations, are to be met with in a scarce work, printed at Basle in 1769, written by Jo. Henrico Ziegler. Unhappily, he lived too remote from the scene of the philosophical discoveries of that period to adopt the precautions necessary to give value to his experiments. He allowed atmospheric air to mingle with the steam to such an extent as greatly to vitiate the results.

M. Betancourt, about the end of last century, undertook a series of experiments on the force of the vapour of water, alcohol, or other liquids, at various temperatures. His apparatus was tolerably perfect; and the precautions which he adopted for the removal of atmospheric air from intermixture with the vapour, gave his experiments considerable value and precision. Some of his experiments were made in vacuo; and he seems to have been one of the first philosophers who examined the production of steam at temperatures below the ordinary point of ebullition, under the pressure of the atmosphere. His experiments extend from $32^\{\backslash circ\}$ up to $279^\{\backslash circ\}$, being $67^\{\backslash circ\}$ above the ordinary boiling point.

Of British philosophers, Dr Robison was one of the first to make accurate and systematic experiments on the phenomena of the temperature and elastic force of steam; they appear to have been made in 1778. Previously, however, Mr Watt had been led, in the course of his invention of the steam-engine, in 1764-65, to make experiments on the elastic force of steam. From the data he then obtained, he laid down a curve, in which he says, "the abscissæ represented the temperatures, and the ordinates the pressures, and thereby found the law by which they were governed, sufficiently near for my then purpose." In 1773-74, he resumed his experimental researches on the relative temperatures and pressures of saturated steam up to about 40 lb. total pressure per square inch. Having been dissatisfied with the irregularities of the results he obtained, Mr Southern and Mr Creighton, at his request, repeated his experiments in 1803, with the view of ascertaining the density of steam raised from water under different pressures above, as well as below, that of the atmosphere; they extended from a pressure of $\backslash frac\{1\}\{4\}$ inch to 240 inches of mercury, or 8 atmospheres.

Dr Dalton appears to have been the first to escape from the natural enough error of assuming that the vapour of water at $32^\{\backslash circ\}$ Fahr., the freezing point, would be represented by 0. This apparatus was the most simple and refined of any that had been employed for temperatures below $212^\{\backslash circ\}$, and his experiments continued for many years to possess the greatest authority. His experiments were first published in 1793; and afterwards, when more extended, in 1802. Dr Ure, in 1817, and subsequently Mr Philip Taylor, and Professor Arsberger, of Vienna, made experiments on high pressure steam through an extensive range of temperatures.

In 1823, the government of France having resolved to

legislate on the means for obtaining security in the use of steam-engines, consulted the Academy of Sciences upon the mode of most effectually promoting the public safety, without placing useless restrictions on commercial enterprise and manufacturing industry. The examination into the state of knowledge concerning the phenomena of vapour at elevated temperatures, which resulted from this application, having brought the imperfections of this part of science prominently into notice, the Academy were induced to undertake a long and laborious inquiry, not entirely free from personal danger, into the law connecting temperature with the pressure of steam. The commission consisted of the illustrious members of the Academy, Baron de Prony, Arago, Girard, and Dulong; and the results of their investigations, finished in 1829, were published in the Memoirs of the Academy of Sciences in 1831. The experiments were conducted principally by MM. Arago and Dulong, and on a scale of magnitude and expense suited to the munificence of the French government and the resources of the Academy; they were carried as high as to 24 atmospheres of pressure, or about 360 lb. per square inch.

Towards the end of 1830, a committee of the Franklin Institute, of the State of Pennsylvania, United States, was appointed to examine into the causes of the explosion of steam-boilers, and to devise the most effectual means of preventing the accidents. The committee experimented on the elastic force and temperature of steam, at pressures varying from $1\backslash frac\{1\}\{2\}$ to 10 atmospheres.

In 1844, Professor Gustav Magnus published, in Pogendorff's Annalen, No. 2, a memoir on the expansive force of steam, in which he notices the defects in former methods, arising from the difficulty of determining with accuracy the true temperature of the vapour, and also the correct pressure, owing to the unequal heating and consequent partial expansion of the mercurial column; and explains the method he adopted to obviate these difficulties.

In July 1844, M. V. Regnault published his very valuable memoir on the Elastic Force of Aqueous Vapour, in the Annales de Chimie et de Physique, and since, more fully, in the Memoirs of the Institute, in 1847. "To establish a physical fact," says this ingenious and accurate physicist, "we must not confine ourselves to a single method of investigation. It is necessary to employ various methods, and even to repeat those made use of by former experimenters, unless they are absolutely faulty; and we must show that all, when used with proper precaution, conduct to the same result; or, if this be not the case, we must point out by direct experiment the cause of error in the defective methods." Acting on this principle, M. Regnault repeated the methods of Dalton, Ure, Magnus, Dulong, and Arago, with such modifications as the improved state of experimental science, and his own skill and experience suggested; and he has pointed out the defects under which they labour, and the limits within which their results may be relied on.

Mr Dalton's method may be described as consisting essentially in determining the heights of the mercurial column in two barometer tubes, the chamber of one being occupied with vapour and the other being a vacuum. In this method the temperature of the vapour is determined by that of a water-bath surrounding the chamber, and either the whole or a part of the mercurial column is maintained by the same means at the same temperature. M. Biot has remarked, that the chief defect in this method arises from the fact, that it is impossible to maintain the water surrounding the tubes at a constant temperature through all its depth, if this depth is considerable, and its temperature differs much from that of the surrounding medium. Mr Ure attempted to remedy this defect by limiting the space occupied by the vapour, and thus re-

ducing the depth of the bath, and in this respect, certainly, his modification of Mr Dalton's method was a decided improvement. M. Regnault has shown that, if the whole of the tubes be surrounded by water, for the purpose of maintaining the columns at the same known temperature, Mr Dalton's method is capable of giving accurate results between the limits $+10^\{\backslash circ\}\backslash text\{C\}$, and $+30^\{\backslash circ\}\backslash text\{C\}$, provided the water be incessantly and rapidly agitated, the agitation being merely interrupted for a moment to observe the heights of the mercurial column. Above the higher limit, however, the separation of the liquid into strata of unequal temperature commences the instant the agitation ceases, and the observations are accordingly rendered uncertain. The following account of the apparatus and method of experiment adopted by Regnault, is derived from Dixon's Treatise on Heat.

Where it was intended only to raise the chambers and a part of the mercurial column to the temperature of the vapour, M. Regnault made use of the following form of apparatus:—Two barometers A, B, as similar as possible, of about 14 millimetres internal diameter, are arranged, side by side, on a frame (fig. 5). These barometers pass

through two tubular openings in the bottom of a vessel V of galvanized sheet-iron, and are secured by means of caoutchouc collars. The vessel V, shown in section and plan in fig. 6, has, on one side, a rectangular aperture, round which is fixed an iron frame. A plate of glass, with perfectly parallel faces, is secured to this frame by means of a similar frame attached to the former by screws. A slip of caoutchouc, of the form of the contour of the aperture, is placed between the glass and the frame, and renders the joint perfectly water-tight. The two barometers are plunged in the same reservoir U. The capacity of the vessel is about 45 litres. This vessel is filled with water, which is continually agitated, and its temperature is given by a very sensitive mercurial thermometer immersed in it, which is observed by means of a small horizontal telescope L. The height of the column in the barometer is read off by means of a kathetometer, the agitation of the water being stopped for an instant at the time of the observation of each column. Observations are made with great precision at the temperature of the surrounding air; to observe the force at higher temperature, a small quantity of water is removed from the vessel by means of a syphon, and replaced with a corresponding quantity of hot-water. A spirit-lamp is then placed under the vessel, and its distance from

it, as well as the height of its wick, is so arranged that the temperature of the water, which is still kept in a state of brisk agitation, finally becomes constant. This condition is easily obtained after some trials, and if the temperature does not surpass $50^\{\backslash circ\}\backslash text\{C\}$, it may be maintained stationary and uniform for any length of time, provided only that the agitation of the water is brisk and constant. Three or four observations were made every time that the temperature was rendered stationary, an interval of eight or ten minutes being left between each. In this method of operating, the portions of the columns outside the vessel are in circumstances completely identical, and the difference of height of the portions within, which are at the temperature of the bath, being reduced to $0^\{\backslash circ\}$, measures exactly the tension of the vapour, allowance, of course, being made for the pressure of the film of water. It is unnecessary to point out how much more accurate this method of ascertaining the temperature corresponding to observed forces is, than either M. Betancourt's or Mr Dalton's.

A second series of experiments M. Regnault made with the following apparatus:—A balloon $\backslash lambda$ (figs. 5 and 6), whose capacity equals 500 cubic centimeters, contains a little glass vessel full of water recently boiled. The balloon is soldered to a curved tube, cemented into a tubular piece of copper, with three branches, $d,\; e,\; f$. In the branch $e$ is cemented a tube soldered to the upper part of the barometer $h$; and in the branch $f$ a tube, communicating with an air-pump, by means of the desiccating apparatus $mn$, filled with powdered pumice, steeped in sulphuric acid. The tube $o$ is a perfect barometer as before. The apparatus being arranged, as in the figures, a vacuum was made forty or fifty times successively; and each time the air slowly re-admitted; by this means the interior of the balloon and barometric chamber was completely dried. The vacuum was then finally made as perfectly as possible, and the tube $f$ sealed by a blow-pipe. At first M. Regnault was unable to reduce the vacuum below two millimetres, but his air-pump having been cleaned, he subsequently succeeded in bringing it below one millimetre. The balloon was then surrounded with melting ice, and the difference of the columns in the two tubes gave the elastic force at $0^\{\backslash circ\}$ of the remaining air. The ice having been removed, the vessel $v$ was filled with water, and its temperature being sufficiently raised, the little glass vessel was burst by the expansion of the water it contained, and the balloon and chamber filled with aqueous vapour, whose force at the corresponding temperatures was observed as before.

This method answers very well for temperatures below that of the surrounding medium, and for $10^\{\backslash circ\}$ or $15^\{\backslash circ\}\backslash text\{C\}$. above it; it also answers for determining the force of aqueous vapours in air of any density within those limits of temperature.

Those two forms of apparatus do not answer for tensions above 200 millimetres; beyond this force M. Regnault employed the following, which in principle is similar to Professor Magnus's, and which he also employed in the case of liquids more volatile than water.

A syphon-shaped tube $abce$ (fig. 5), of about 15 millimetres internal diameter, terminates in a fine curved tube $ee$. The closed branch $ab$ is filled with mercury, which is carefully boiled to expel all air and moisture. When the mercury is cool, a small quantity of volatile liquid is introduced into the branch $be$, and boiled for some minutes; the tube is then inclined, and a little of the liquid, yet hot, is passed up into the branch $ab$; the branch $be$ is then completely dried. The tube is now fixed in a perfectly vertical position in the vessel $v$, in front of the glass-plate. The tube $ee$ is cemented into one branch of a piece of copper $edf$, whose other branches communicate, one $d$, with a manometric apparatus with a stop-cock $r$, and the other $f$ with an air-pump, if necessary.

The tubes $hi,\; kl$, are first completely filled with mercury, Historical Notice of Experiments on Steam.

the air being expelled through $fg$; the tube $fg$ is then closed with the blow-pipe and a portion of the mercury being allowed to flow out through the stop-cock $r$, the pressure of the air in the branch $nee$ is so far diminished as ultimately to become nearly equal to the pressure of the vapour in $am$, when the mercury in the two branches of the syphon-shaped tube falls nearly to the same level. The force of the vapour is then measured by the atmospheric pressure, diminished by the column $rs$ in the manometer, and the column $mn$ in the tube $ab$, both these columns, whose temperatures are known, being reduced to their heights at $0^\{\backslash circ\}$. The temperature of the vapour is ascertained as in the preceding experiment.

None of the foregoing methods answer for temperatures above $60^\{\backslash circ\}$ or $70^\{\backslash circ\}\backslash text\{C\}$. At higher degrees the water in the vessel $v$ separates so promptly into strata of different temperatures as to require constant agitation to prevent this result from taking place. For temperatures above $100^\{\backslash circ\}\backslash text\{C\}$, moreover, those methods become impracticable from other causes. For higher degrees, therefore, M. Regnault had recourse to the well-known method employed by Mr Dalton, and other physicists subsequently, of ascertaining the temperature of the vapour of water boiling under determined pressures.

In order to obtain results of the degree of accuracy which this method is capable of giving, it is necessary to boil the water in a vessel communicating freely with a space of tolerable capacity, in which we can dilate or condense at will; and by this means form an artificial atmosphere, which exerts a determined pressure on the surface of the heated liquid. We thus obtain a temperature of ebullition as perfectly stationary as that of water boiling in free air, and we can maintain this temperature stationary as long as we will. The apparatus employed for this purpose by M. Regnault is represented in fig. 7. It consists of a

retort of red copper $A$, closed with a cover. This cover carries four iron tubes closed below; of these, two descend to the bottom of the retort, the others only reach half-way down. These tubes, which are 7 millimetres in internal diameter, and about 1 millimetre thick, are surrounded by a case of very thin copper attached to the cover, and having apertures in its upper part. They are filled with mercury to within a few centimetres of their upper edge, and hold four mercurial thermometers, whose bulbs descend to the bottom of the tubes. From the arrangement of the tubes, it appears that two of the thermometers are plunged in vapour and two in water, as shown in section. The neck of the retort is connected with a tube $tr$, about 1 metre

in length, surrounded by a copper cylinder, through which flows a constant current of cold water, supplied by a reservoir r. This tube communicates with a copper balloon b of about 24 litres in capacity, contained in a vessel v full of water at the temperature of the surrounding medium. To the balloon is attached a pipe, with two branches, one of which is cemented to the tube egh of the apparatus represented in fig. 6, when the experiments are made at pressures inferior to that of the atmosphere, or with the tube pq of the apparatus in fig. 7, for greater pressures. The second branch is connected by means of a lead tube tt, with an exhausting or condensing air-pump.

For pressures inferior to the atmospheric, the air in the balloon having been rarefied, the water in the retort is heated until ebullition commences. The vapour, according as it forms, is condensed in the refrigerator tr, and falls back into the retort. The pressure is measured by the difference of heights of the mercury in the barometric tubes. It may be remarked that the column in the barometric tube connected with the balloon, is never absolutely stationary; the amplitude of its oscillations, however, when the fire is properly regulated, is very small, not exceeding one-tenth of a millimetre. The mercury in the barometer o, on the other hand, remains perfectly stationary. The difference of the heights of these columns is observed by means of a kathetometer; and an assistant, at the same instant, notes the height of the thermometers. Several observations were made, at intervals of eight or ten minutes, under the same pressure; and it was thus easy to perceive the perfect constancy of the temperatures indicated by the thermometer for the same pressure, and to show that the least change in the latter was followed by a corresponding change in the former.

The height of the kathetometers not exceeding one metre, when greater differences of level in the apparatus required to be measured, it was necessary to employ two of them. In order to ascertain if the divisions of the scales of the two instruments were identical, M. Regnault read off the divisions of one, in lengths of centimetres, by means of the other; and such was the accuracy of their graduation, that he, in no case, encountered a difference of more than one-twentieth of a millimetre. To attain such a degree of precision in the measures, it is evident that the instruments must be constructed with the greatest accuracy; the telescopes must not have too great a focal length ($0^\{\backslash text\{m\}\}.30$), and the levels, in particular, must possess extreme sensibility. Those in the kathetometers constructed by M. Gambey indicated inclinations of one second. The verniers gave directly one-fiftieth of a millimetre, and easily admitted of the estimation of one-hundredth.

The thermometers, says M. Regnault, employed in the experiments, made at pressures inferior to the atmospheric, ranged from $0^\{\backslash circ\}$ to $100^\{\backslash circ\}$ C.; they had from six to eight divisions in $1^\{\backslash circ\}$ C.; it was, consequently, easy to read with certainty the one-sixtieth of a degree. Those employed for higher pressures had a range from $0^\{\backslash circ\}$ to $240^\{\backslash circ\}$, and $1^\{\backslash circ\}$ C. contained 2.5 or three divisions of their scale. All these instruments were graduated and verified with the greatest care. In estimating the temperature from the indication of the thermometers, a correction required to be made for the portion of mercury in the stem unassimilated in the temperature. The temperature of this portion was ascertained by means of a sensitive mercurial thermometer suspended between the four stems. It may be remarked, that the temperature indicated by the thermometer in the liquid was always higher, at low temperatures, than that indicated by those in the vapour; the difference in some cases amounted to $0^\{\backslash circ\}.7$. As the pressure approached the atmospheric, this difference became less, and, at high temperatures, was quite insensible.

The method by which M. Regnault investigated the force

of vapours at high temperatures was similar to that last described; and the apparatus which he employed for this purpose differed from the preceding merely in the size and strength of the parts. The boiler was made of copper, about 5 millimetres thick, and had a total capacity of 70 litres. The reservoir B forming the artificial atmosphere was also of copper, 13 millimetres thick, and contained about 280 litres. The manometer destined to measure the pressure of the artificial atmosphere, and thus of the vapour in the boiler, consisted of a tube, or rather a series of tubes, placed vertically over one another, and attached to the walls of the building in the vicinity of which the experiments were performed, and continued along a strong pole or mast firmly secured to the top of the wall. This system of tubes was open above, and contained the column of mercury, which measured the tension in B. Its total height was 24 metres, and it was accordingly able to measure a pressure of 30 atmospheres.

The experiments were conducted as follows:—The water in the boiler having been raised nearly to its boiling point, the air in the reservoir B was compressed, until it had reached the pressure under which it was desired to make the experiment. A column of mercury of the corresponding height was next forced up the manometer from below by means of a force-pump, and connection was then made between the manometer and the reservoir. Meanwhile, the temperature of the water in A was rising, and continued to increase until it reached its boiling point under the pressure to which it was exposed. It was then kept in a state of ebullition for at least half an hour, and no observation was made until it was ascertained that the mercurial thermometers connected with the boiler were perfectly stationary. The temperatures indicated by these thermometers were then observed, and also the indications of an air thermometer, which, to insure greater accuracy, was employed in addition to the former. At the same time, the difference of level of the mercury in the branches of the manometer was noted. M. Regnault's experiments with this apparatus extended to a pressure of about 28 atmospheres, corresponding to a temperature of $230^\{\backslash circ\}.56$ C. measured on the air thermometer.

CHAP. III.—THE MECHANICAL THEORY OF HEAT.

An important and interesting inquiry relative to steam and its operation in the steam-engine, is that which traces the connection between the heat expended and the dynamical effect, or work, produced. The method of separate condensation, and the application of the force of expanding steam, changed to an important extent the accepted relations of heat to power, and added remarkably to the dynamical effect of the fuel; and though the steam-engine has been progressively improved by the continual elaboration of small economies, there is yet good reason to believe that the field of improvement is wide, and that the labourer in that field has the prospect of a good return. The inquiries of scientific men on the subject of the relation of heat to mechanical effect, have resulted in the establishment of the principle that heat and mechanical force are identical and convertible, and that the action of a given quantity of heat may be represented by a constant quantity of mechanical work performed. "Motion and force," says Professor Rankine, "being the only phenomena of which we thoroughly and exactly know the laws, and mechanics the only complete physical science, it has been the constant endeavour of natural philosophers, by conceiving the other phenomena of nature as modifications of motion and force, to reduce the other physical sciences to branches of mechanics. Newton expresses a wish for the extension of this kind of investigation. The theory of radiant heat and light having been reduced to a branch of mechanics by means of the hypo-

Mechanical theory of undulations, it is the object of the hypothesis of molecular vortices"—oscillation or vibratory motion—"to reduce the theory of thermometric heat, and elasticity also, to a branch of mechanics, by so conceiving the molecular structure of matter that the laws of these phenomena shall be the consequences of those of motion and force. This hypothesis, like all others, is neither demonstrably true nor demonstrably false, but merely probable in proportion to the extent of the class of facts with which its consequences agree." It must, however, be remarked that, whether the hypothesis of molecular motion be probable or improbable, the theoretical and practical results arrived at in regard to the mechanical action of heat remain unaffected, being deduced from principles which have been established by experiment and demonstration. From these principles, Professor Rankine announced the specific heat of air before it was otherwise known,—the accuracy of his deductions having since been verified to within less than 1 per cent. by the experiments of Regnault. The best experiments, previous to those made by Regnault, in regard to the specific heat of air, were those of Delaroche and Berard, from which they deduced a specific heat of 266; but, arguing from the mechanical theory of heat, Professor Rankine declared that this value must be erroneous, and that the specific heat of air could not exceed 240. It has been found accordingly, by Regnault, since the statement was made, as the result of a hundred experiments, that the specific heat of air was 238, and that it is constant for all pressures from one to ten atmospheres, or at least differs almost inappreciably. This coincidence of theoretical prediction with experimental evidence, it has been well observed, should have something like the same tendency in strengthening our belief of the theory upon which Professor Rankine's estimate was based, as the discovery of an unknown planet, previously indicated by Le Verrier and Adams, had in confirming our faith in the science of astronomy.

The principle of the dynamical or mechanical theory of heat, as already stated, is that, independently of the medium through which heat may be developed into mechanical action, the same quantity of heat converted is invariably resolved into the same total quantity of mechanical action. For the exact expression of this relation, of course, units of measure are established:—in terms of the English foot, as the measure of space; the pound avoirdupois, as the measure of weight, pressure, elasticity; and the degree of Fahrenheit's scale, as the measure of temperature and heat. Work done consists of the exertion of pressure through space, and the English unit of work is the exertion of 1 lb. of pressure through 1 foot, or the raising of 1 lb. weight through a vertical height of 1 foot: briefly, a foot-pound. The unit of heat is that which raises the temperature of 1 lb. of ordinary cold water by 1 degree Fahr. If 2 lb. of water be raised 1 degree, or 1 lb. be raised 2 degrees in temperature, the expenditure of heat is, equally in both cases, two units of heat. Similarly, if 1 lb. weight be raised through 1 foot, or 2 lb. weight be raised through 2 feet, the power expended, or work done, is equally in both cases two units of work, or two foot-pounds. From these definitions, then, the comparison lies between the unit of heat, on the one part, and the unit of work, or the foot-pound, on the other.

M. Clapeyron, in his treatise on the moving power of heat, and M. Noltzman, of Manheim, in 1845, who availed himself of the labours of M. Clapeyron and M. Carnot in the same field, grounding their investigations on the received laws of Boyle, or Marriotte, and Gay Lussac, which express the observed relations of heat, elasticity, and volume, in steam and other gaseous matter, concluded that the unit of heat was capable of raising a weight, between the limits of 626 lb. and 782 lb., 1 foot high; that is to say, that one unit of heat was equivalent to from 626 to 782 foot-pounds.

By this mode of investigation, they suppose a given weight of steam, or gaseous matter, to be contained in a vertical cylinder formed of non-conducting material, in which is fitted an air-tight but freely-moving piston, which is pressed downwards by a weight equal to the elasticity of the gas. Now, the weight, initial temperature, pressure, and volume being known, a definite quantity of heat from without is supposed to be imparted to the vapour; and the result is partly an elevation of the temperature of the vapour, and partly a dilation or increase of volume, or, in other words, an exertion of pressure through space—the elasticity remaining the same. But the result may be represented entirely by dilation, so that there shall not be any final alteration of temperature; and for this purpose, it is only necessary to allow the vapour to dilate without any loss of its original or imparted heat until it re-acquires its initial temperature. In this case, the ultimate effect is purely dilatation, or motion against pressure; and the work done is represented by the product of that pressure into the space moved through.

Mr Joule, of Manchester, in 1843–47, proceeded, by entirely different, independent, and, in fact, purely experimental methods, to investigate the relation of heat and work:—1st, By observing the calorific effects of magnetoelectricity. He caused to revolve a small compound electromagnet, immersed in a glass vessel containing water, between the poles of a powerful magnet: heat was proved to be excited by the machine, by the change of temperature in the water surrounding it, and its mechanical effect was measured by the motion of such weights as by their descent were sufficient to keep the machine in motion at any assigned velocity. 2d, By observing the changes of temperature produced by the rarefaction and condensation of air. In this case, the mechanical force producing compression being known, the heat excited was measured by observing the changes of temperature of the water in which the condensing apparatus was immersed. 3d, By observing the heat evolved by the friction of fluids:—a brass paddle-wheel, in a copper can containing the fluid, was made to revolve by descending weights. Sperm oil and water yielded the same results. Mr Joule considered the third method the most likely to afford accurate results; and he arrived at the conclusion that one unit of heat was capable of raising 772 lb. 1 foot in height; or, that the mechanical equivalent of heat was expressible by 772 foot-pounds for 1 unit of heat,—known as "Joule's equivalent."

The following are the values of Joule's equivalent for different thermometric scales, and in English and French units:—

The mechanical theory of heat rests upon a wide basis, and proofs in verification of the theory are constantly accumulating. When the weight of any liquid whatever is known, with the comparative weight of its vapour at different pressures, the latent heat at the different pressures is readily estimated from the theory; and this method of estimation agrees with the best experimental results, as may afterwards be shown; and when the latent heat is also known, the specific heat of the liquid can be determined by means of the same theory: in other words, the quantity of work, in foot-pounds, may be determined, which would, by agitating the liquid or by friction, be required to raise the temperature of any given quantity of the liquid by, say, one degree, altogether independently of Joule's experiments. The theory enables us to discover the utmost power it is possible to realize from the combination of any given weight of carbon and oxygen, or other elementary substances, with

Mechanical Theory of Heat. nearly as much precision as we can estimate the utmost quantity of work it is possible to obtain from a known weight of water falling through a given height. It is not difficult to comprehend, then, that the theory of the mechanical equivalent of heat proves of great practical utility.

According to the mechanical theory of heat, in its general form, heat, mechanical force, electricity, chemical affinity, light, sound, are but different manifestations of motion. Dulong and Gay Lussac proved, by their experiments on sound, that the greater the specific heat of a gas, the more rapid are its atomic vibrations. Elevation of temperature does not alter the rapidity, but increases the length of their vibrations, and in consequence produces "expansion" of the body. All gases and vapours are assumed to consist of numerous small atoms, moving or vibrating in all directions with great rapidity; but the average velocity of these vibrations can be estimated when the pressure and weight of any given volume of the gas is known, pressure being, as explained by Joule, the impact of those numerous small atoms striking in all directions, and against the sides of the vessel containing the gas. The greater the number of these atoms, or the greater their aggregate weight, in a given space, and the higher the velocity, the greater is the pressure. A double weight of a perfect gas, when confined in the same space, and vibrating with the same velocity—that is, having the same temperature—gives a double pressure; but the same weight of gas, confined in the same space, will, when the atoms vibrate with a double velocity, give a quadruple pressure. An increase or decrease of temperature is simply an increase or decrease of molecular motion. The truth of this hypothesis is very well established, as already intimated, by the numerous experimental facts with which it is in harmony.

When a gas is confined in a cylinder under a piston, so long as no motion is given to the piston, the atoms, in striking, will rebound from the piston after impact with the same velocity with which they approached it, and no motion will be lost by the atoms. But when the piston yields to the pressure, the atoms will not rebound from it with the same velocity with which they strike, but will return after each succeeding blow, with a velocity continually decreasing as the piston continues to recede, and the length of the vibrations will be diminished. The motion gained by the piston will, it is obvious, be precisely equivalent to the energy, heat, or molecular motion lost by the atoms of the gas. Vibratory motion, or heat, being converted into its equivalent of onward motion, or dynamical effect, the conversion of heat into power, or of power into heat, is thus simply a transference of motion; and it would be as reasonable to expect one billiard-ball to strike and give motion to another without losing any of its own motion, as to suppose that the piston of a steam-engine can be set in motion without a corresponding quantity of energy being lost by some other body.

In expanding air spontaneously to a double volume, delivering it, say into a vacuous space, it has been proved repeatedly that the air does not fall appreciably in temperature, no external work being performed; but, on the contrary, if the air, at a temperature, say of 230° Fahr., be expanded under pressure or resistance, as against the piston of a cylinder, giving motion to it, raising a weight, or otherwise doing work, by giving motion to some other body, the temperature will fall nearly 170° when the volume is doubled, that is, from 230° to about 60°; and, taking the initial pressure at 40 lb., the final pressure would be 15 lb. per square inch.

When a pound weight of air, in expanding, at any temperature or pressure, raises 130 lb. 1 foot high, it loses 1° in temperature; in other words, this pound of air would lose as much molecular energy as would equal the energy

acquired by a weight of 1 lb. falling through a height of 130 feet. It must, however, be remarked, that but a small portion of this work, 130 foot-pounds, can be had as available work, as the heat which disappears does not depend on the amount of work or duty realised, but upon the total of the opposing forces, including all resistance from any external source whatever. When air is compressed, the atmosphere descends and follows the piston, assisting in the operation with its whole weight; and when air is expanded, the motion of the piston is, on the contrary, opposed by the whole weight of the atmosphere, which is again elevated. Although, therefore, in expanding air, the heat which disappears is in proportion to the total opposing force, it is much in excess of what can be rendered available; and, commonly, where air is compressed, the heat generated is much greater than that which is due to the work which is required to be expended, the weight of the atmosphere assisting in the operation.

Let a pound of water, at a temperature of 212° Fahr., be injected into a vacuous space or vessel, having 26.36 cubic feet of capacity—the volume of 1 pound of saturated steam at that temperature—and let it be evaporated into such steam, then 893.8 units of heat would be expended in the process. But, if a second pound of water, at 212°, be injected and evaporated at the same temperature, under a uniform pressure of 14.7 lb. per square inch due to the temperature, the second pound must dislodge the first, by repelling that pressure, involving an amount of labour equal to 55,800 foot-pounds (that is, 14.7 lb. $\backslash times$ 144 square inches $\backslash times$ 26.36 cubic feet), and an additional expenditure of 72.3 units of heat (that is, 55,800 $\backslash div$ 772), making a total for the second pound of 965.1 units.

Similarly, when 1408 units of heat are expended in raising the temperature of air at constant pressure, 1000 of these units increase the velocity of the molecules, or produce a sensible increment of temperature; while the remaining 408 parts which disappear as the air expands, are directly expended in repelling the external pressure.

Again, if steam be permitted to flow from a boiler into a comparatively vacuous space, without giving motion to another body, the temperature of the steam entering this space would rise much higher than that of the steam in the boiler. Or, suppose two vessels, side by side, one of them vacuous, and the other filled with air at, say two atmospheres, a communication being opened between the vessels, the pressure would become equal in the two vessels; but the temperature would fall in one vessel and rise in the other; and although the air is expanded in this manner to a double volume, there would not on the whole be any appreciable loss of heat, for if the separate portions of air be mixed together, the resulting average temperature of the whole would be very nearly the same as at first. It has been proved experimentally, corroborative of this argument, that the quantity of heat required to raise the temperature of a given weight of air, to a given extent, was the same, irrespective of the density or volume of the air. Regnault and Joule found that, to raise the temperature of a pound weight of air, 1 cubic foot in volume, or 10 cubic feet, the same quantity of heat was expended.

In rising against the force of gravity, steam becomes colder, and partially condenses while ascending, in the effort of overcoming the resistance of gravity, by the conversion of heat into water. For instance, a column of steam weighing, on a square inch of base, 250.3 lb., that is, a pressure of 250.3 lb. per square inch, would, at a height of 275,000 feet, be reduced to a pressure of 1 lb. per square inch, and, in ascending to this height, the temperature would fall from 401° to 102° Fahr., while, at the same time, nearly 25 per cent. of the whole vapour would be precipitated in the form of water, if not supplied with heat while ascending.

If a body of compressed air be allowed to rush freely into the atmosphere, the temperature falls in the rapid part of the current, by the conversion of heat into motion, but the heat is almost all reproduced when the motion is quite subsided; and from recent experiments, it appears that nearly similar results are obtained from the emission of steam under pressure.

When water falls through a gaseous atmosphere, its motion is constantly retarded as it is brought into collision with the particles of that atmosphere; and by this collision it is partly heated and partly converted into vapour.

If a body of water descends freely through a height of 772 feet, it acquires from gravity a velocity of 223 feet per second; and if suddenly brought to rest when moving with this velocity, it would be violently agitated, and raised one degree in temperature. But suppose a water-wheel, 772 feet in diameter, into the buckets of which the water is quietly dropped, when the water descends to the foot of the fall, and is delivered gently into the tail-race, it is not sensibly heated. The greatest amount of work it is possible to obtain from water falling from one level to another lower level, is expressible by the weight of water multiplied by the height of the fall.

The objects of these illustrative exhibitions of the nature and reciprocal action of heat and motive power, with their relations, are,—first, to familiarise the reader with the doctrine of the mechanical equivalence of heat; second, to show that the nature and extent of the change of temperature of a gas while expanding, depends nearly altogether upon the circumstances under which the change of volume takes place.

CHAP. IV.—GENERAL RELATIONS OF GASEOUS BODIES.

Gases are divided into the two classes,—permanent gases, and vapours. The former were originally so called, under the impression that they existed permanently in the gaseous state, and could not possibly be reduced to the liquid form; while those which could be so reduced, and could be reconverted to the state of gas, were called vapours. It has, however, been shown by Sir Humphrey Davy and Mr Faraday, that by the conjoined effects of great pressure, and of a high degree of cold, most of the permanent gases may be liquefied. The under-mentioned still retained the gaseous state at the annexed temperatures and pressures:—

Several of the liquefied gases are further capable of being reduced to the solid state. Thus, sulphurous acid becomes solidified at -105°; sulphuretted hydrogen at -122°; carbonic acid at -72°; ammonia at -103°. The difference, then, between the permanent gases and vapours, is merely one of degree, and depends upon the temperature at which the change from the fluid to the gaseous state occurs. Those which exist in the fluid state under ordinary temperatures and pressures are called vapours; those which require strong pressure and extremely low temperatures to reduce them to the liquid form, are called permanent gases.

Steam, as the elastic vapour of water, is amenable to the laws of gaseous fluids; and, according to these laws, the pressure, the density or the volume, and the temperature, bear fixed relations to each other. The influence of temperature on the expansion of permanent gases under constant pressures is such, that, for equal increments of temperature, the increments of volume by expansion are also equal, and they are nearly the same for different gases.

The expansion of air by increase of temperature may be assumed to represent that of other gases; and, it may be added, the most exact measure of real temperature is to be found in the expansion of air, or any other perfect gas. By real or absolute temperature is signified the measure of the whole of the heat of a body; and at the absolute zero point of the scale, all gases would cease to have elasticity or molecular motion. As the expansion of air under constant pressure is found experimentally to be uniform for uniform increments of temperature, it is inferred, conversely, that it would contract uniformly, under uniform reduction of temperature, until, on arriving at a temperature 461° below zero of Fahrenheit's scale, or, exactly -461.2°, the air would be in a state of collapse, without appreciable elasticity. This point has, therefore, been adopted as that of absolute zero, standing at the foot of the natural scale of temperature. For example, let a volume of air, 673 cubic inches in bulk, at a temperature of 212° Fahr., be confined at a constant pressure in a cylinder, under a piston movable without friction. If the gas be cooled 10°, the piston will descend through 10 cubic inches; if cooled 100°, the piston will descend, and the air will contract, through 100 cubic inches; and so on, in the same ratio; so that, by lowering the temperature 673°, the air would not possess appreciable volume; and 673 - 212 = 461° below the artificial zero of Fahr., would, therefore, be arrived at as the point of absolute zero.

Again, if a given weight of air at 0° Fahr. be raised in temperature to 461° under a constant pressure, its volume will be doubled by expansion; and, if heated to 461 × 2 = 922°, its volume will be trebled; in short, for every increment of one degree of temperature, its volume will be enlarged by equal increments uniformly $\backslash frac\{1\}\{461\}$ part of the volume at 0°.

The following, then, are the established relations of the properties of permanent gases:—

With a constant temperature, the pressure varies simply as the density, or inversely as the volume. This is known as Boyle's or Marriotte's law.

With a constant pressure, expansion is uniform under a uniform accession of heat or rise of temperature, at the rate of $\backslash frac\{1\}\{461\}$ part of the volume at 0° Fahr. for each degree of heat. If, then, 461° be added to the indicated temperature by Fahrenheit's scale, the sum, or absolute temperature, varies directly as the total volume, expanding or contracting, and inversely as the density. This is known as the law of Gay Lussac.

With a constant volume, or density, the increase of pressure is uniformly at the rate of $\backslash frac\{1\}\{461\}$ part of the pressure at 0° Fahr. for each degree of temperature acquired. Adding 461° to the indicated temperature, the sum, or absolute temperature, varies directly as the total pressure.

In brief, 1st, the pressure varies inversely as the volume when the temperature is constant; 2d, the volume varies as the absolute temperature when the pressure is constant; 3d, the pressure varies as the absolute temperature when the volume is constant.

The foregoing enunciation of the relations of temperature, pressure, and density, should be qualified by the remark, that the more easily condensable gases, as they approach the liquefying point, become sensibly more compressible than air; and that they do not strictly conform to the relations of pressure and volume belonging to the permanent gases. It has been found that, as far as 100 atmospheres, oxygen, nitrogen, hydrogen, nitric oxide, and carbonic oxide, follow the same law of compression as atmospheric air, these being amongst the incompressible gases; and that sulphurous acid, ammoniac gas, carbonic acid, and protoxide of nitrogen—proved to be condensable—commence to be sensibly more compressible than air when they have been reduced to one-third or one-fourth of their

General Relations of Saturated Steam. original volume. Carbonic acid, for example, in place of following the simple ratio of the pressure and density for a constant temperature, increased in density in a greater ratio than the pressure, as indicated in the following table, showing, in the third column, the volume of carbonic acid and increasing pressures relative to that of air, which is expressed by unity:—

Table of the Compressibility of Carbonic Acid, as referred to Air. Temperature, $10^\{\backslash circ\}$ Centig., or $50^\{\backslash circ\}$ Fahr.

The deviations from unity in the last column express the deviations from the law of Boyle, as applicable to dry air at a constant temperature; and they show that, under a pressure of 40 atmospheres, carbonic acid, near the condensing point, occupied rather less than three-fourths of the volume which would have been occupied by air under the same circumstances. This accelerated density, or incipient condensation, characteristic of carbonic acid and other condensible gases, in approaching the point of liquefaction, foretells the approaching change. It is, nevertheless, established that all gases, at some distance from the point of maximum density for the pressure, do virtually follow the law of Boyle, according to which the pressure and the density vary directly as each other when the temperature is constant; and, on such conditions, they rank as perfect gases.

CHAP. V.—GENERAL RELATIONS OF ORDINARY OR SATURATED STEAM.

The accelerated reduction of volume and increase of density, observable in the condensible gases, as they approach their condensing points, hold likewise with steam. Steam produced in an ordinary boiler, over water, is generated at its maximum density and pressure for the temperature, whatever this may be. In this condition of maximum density, steam is said to be saturated, being incapable of vaporizing or absorbing more water into its substance, or increasing its pressure, so long as the temperature remains the same. Nor, on the contrary, will steam be generated with less than the maximum constituent quantity of water which it is capable of appropriating from the liquid out of which it ascends. It stands both at the condensing point and at the generating point; so that a change in any one of the three elements of pressure, density, or temperature, is necessarily accompanied by a change of the two others. One density, one pressure, and one temperature, unalterably occur in conjunction: the same density is invariably accompanied by the same pressure and temperature.

If a part of the heat of saturated steam be withdrawn, the pressure will fall, and also the density, by the precipitation of a part of the steam in the liquid form.

If, while the temperature remains constant, the volume of steam over water be increased, then, as long as there is liquid in excess to supply fresh vapour to occupy the increased space opened for its reception, the density will not be diminished, but will, with the pressure, remain constant,

—the maximum density and pressure due to the temperature being maintained.

If, when all the liquid is evaporated, the fire or source of heat be removed, the pressure and density diminish when the volume is increased, as in permanent gases; and, if the volume be again reduced, the pressure and density increase until the latter returns to the maximum due to the temperature—that is to say, reaches the condensing point; and the effect of any further diminution of volume, or attempt to further increase the density at the same temperature, is simply attended by the precipitation of a portion of the vapour to the liquid state,—the density remaining the same.

On the contrary, if when all the liquid is evaporated, the application of heat be continued, the state of saturation ceases, the temperature and pressure are increased, whilst the density remains the same: the steam is said to be superheated, or surcharged with heat, and it becomes more perfectly gaseous. And were it, whilst in this condition, to be replaced in contact with water of the original temperature, it would evaporate a part of the water, transferring to it the surcharge of heat, and would resume its normal state of saturation.

Further, let the space for steam over the water remain unaltered, then, if the temperature is raised by addition of heat, the density of the vapour is increased by fresh vaporization, and the elastic force is consequently increased in a much more rapid ratio than it would be in a permanent gas by the same change of temperature. Conversely, if the temperature be lowered, a part of the vapour is condensed, the density is diminished, and the elastic force reduced more rapidly than in a permanent gas.

An account of the special results of M. Regnault's experiments, and of the investigations and deductions of himself and others based upon them, is given in detail in the following sections.

CHAP. VI.—RELATION OF THE PRESSURE AND TEMPERATURE OF SATURATED STEAM.

The admirable investigations of the constants relating to the economical employment of steam as a motive agency, conducted by M. Regnault, may be fairly considered as affording conclusive data of all the phenomena included within the range examined, until some new discovery in science, of a fundamental character, shall offer additional facilities of research. The direct methods of trial and observation may, in the meantime, be regarded as exhausted, and to have yielded the full measure of accuracy of which they are susceptible. It, therefore, only remained to give effect to the results obtained by reducing them to rules of calculation, of ready application, and the most simple of which the relations admit.

One of the most important of those relations is that subsisting between the temperature and the pressure, or elasticity of the steam in contact with the fluid from which it is generated. As yet this relation has only been expressed approximately, and by empirical formulas. The true law of connection has hitherto eluded analysis; and one is compelled to rely in most important calculations on rules which represent the law more or less distinctly, and usually over a very small portion of the curve graphically representing the pressures. There are many such rules, and some of them represent very exactly the data on which they are founded; but as these data are much less complete than those obtained from the elegant and extended researches of Regnault, it becomes necessary, even supposing the forms the most convenient, to lay aside the constants they contain, and to derive them anew from the more recent data.

There are two qualities required in a formula of this kind,—accuracy and simplicity. The first is obtainable by such a form of equation as that suggested by Laplace, which

expresses the expansive force by a series arranged according to the ascending powers of the temperature. This suggestion was afterwards modified by Biot, whose form has been adopted, in the main, by Regnault, as the basis of his principal and most approved and exact formulas. The general form given by Regnault is the following:—

in which $\backslash theta$ is a function of the thermometrical temperature; the other literal quantities are constants, to be determined from the series of experiments which the formula is intended to represent.

Egen's formula is also susceptible of accuracy. It is, in some measure, the inverse of that of Biot, and expresses the temperature by a series arranged according to the ascending powers of the logarithms of the elasticity.

Formulas, according to these models, may include any number of points of the curve of pressures, and may therefore be made to express any required degree of exactness. But such formulas become exceedingly unwieldy and inconvenient for the ordinary purposes of calculation; and they, moreover, do not admit of direct inversion. The formula of Dr Thomas Young, on which those of Creighton, Southern, Tredgold, Mellet, Coriolis, the Commission of the French Academy, the Committee of the Franklin Institute, and others, are founded, is comparatively simple in form; but it does not admit of very great exactness over any considerable extent of the curve. The expression in its most general form is

This equation passes the curve through three given points, and when these are taken at no great distance apart, it may be employed to interpolate; but it cannot with safety be extended to any considerable distance beyond the assumed limits.

Another class of formulas is founded on that proposed by Professor Roche in 1828, from theoretical considerations. It expresses the elasticity by a constant number multiplied by a second constant raised to a power of which the exponent is a fraction, having the temperature in the nominator, and some function of the temperature in the denominator, thus —

This form has been virtually adopted by August and Strehlke, Von Wrede, Magnus, Holtzmann, and Shortrede. It is greatly superior, as a formula of interpolation, to that of Dr Young in extent and accuracy, and to that of Biot in point of simplicity. It approaches more nearly to the double condition of accuracy and simplicity than any other expression which has yet been proposed; and, in fact, as a practical formula applicable to calculations relative to the steam-engine, leaves little to be regretted that it is not absolute. The most simple and convenient form to which this expression is reducible is, for the elastic force,

and the inverse formula for finding the temperature, when the pressure is given, is, accordingly,

The late Mr W. M. Buchanan, of Glasgow, adopted this general equation as the basis of his formula, of which he published an account, in 1850, in the Practical Mechanics Journal, and he tested it by a number of very careful determinations of the constants, from the graphic curve of pressures constructed by Regnault to represent the mean results of his experiments. He was led to conclude that

no three points of that curve, which can be taken as data for the values of the constants, render the expression satisfactory throughout the entire range, experimentally represented. That range, however, extends over a space of 262° of the centigrade scale, equal to 471.6° of Fahrenheit's thermometer; namely, from 25.6° below 0° Fahr., at which the pressure is less than 0.006 lb. on the square inch of surface, to 446° Fahr., at which the pressure is over 400 lb. on the square inch. Both extremes of this range are at present much beyond the limits at which a practical formula is required for calculations relating to the steam-engine. The lower limit, especially, is obviously of no moment for such an object, however important it may be for other scientific purposes. Bearing in mind these considerations, Mr Buchanan adopted a temperature of 120° Fahr. as the lower limit of temperature at which it is practically necessary to consider the elasticity of steam as a motive-power, and he determined the constants from that limit to the higher extremity of the given curve, for the results obtained, both by the air and the mercurial thermometer. The values of these constants are arranged in the following statement:—

The formulas for $p$ lb. pressure on the square inch, by the two modes of measuring the temperature are, therefore—

For the Air Thermometer.

For the Mercurial Thermometer.

Mr Buchanan observes, that "the indications by the air thermometer are greatly more to be relied upon than those of the mercurial. The air thermometer is not only more sensitive, but likewise admits of the employment of a relatively larger volume of the expanding fluid, compared with the volume of the glass envelope in which it is enclosed. The errors arising from the different expansibilities of different qualities of glass are, in consequence, much reduced relatively in amount; and, besides, the expansion of the fluid is very nearly uniform for equal increments of temperature. It is, however, the mercurial thermometer which is ordinarily employed in the measurement of temperatures, and accordingly it is of importance that the indications of the ordinary instrument should be represented by an appropriate formula. This formula, it is true, cannot possess

more than an average approximation to the measurement by any particular instruments; for all thermometers made from different qualities of glass, and even when the usual fixed points are exactly and accurately determined, differ from one another at the higher temperatures. This is fully illustrated by the comparisons given by M. Regnault, in his memoir on the measurement of temperature, which has been justly characterized as one of the most elegant and successful examples we possess of the combination of experimental adaptation with inductive application of the results obtained. Let us take a single line of one of the many tables furnished; it compares four of the mercurial thermometers used with each other, and with the temperature indicated by the standard air thermometer. Take the temperature of 250° C. by the standard: in the medium having that temperature, the mercurial thermometer of

At 100° higher, namely, 350° C., by the air thermometer, the first of these four thermometers gave 360.5°; and the second, 354°, as the temperature of the same medium. In this, it is to be remarked, that the deviations of the ordinary glass thermometer are the least; and this is true throughout the whole extent of the table,—a circumstance which ought to attract the attention, especially of the makers of these instruments. The wide differences thus shown to exist among thermometrical instruments of the very best description, render it little surprising that there should have existed very considerable discrepancies among the results obtained by different experimenters, in investigations involving the measurement of temperature. Both Regnault and Magnus have fortunately avoided this source of uncertainty in their researches relative to the elasticity of gaseous fluids, and accordingly their results agree with remarkable nearness.

The formulas employed by Regnault to connect the temperature with the pressure of steam in a state of saturation, chiefly constructed on the model of Biot's equation, though greatly more laborious, do not appear to be much, if in any degree, more exact than those constructed on Professor Roche's model. The wide range over which Buchanan's rules extend, based on Roche's model, and the great accuracy which they exhibit within the limits for which they are determined, seem to indicate that they contain at least the first terms of the absolute law. This supposition is further countenanced by the circumstance referred to by Mr Buchanan, that the same form expresses, better than any other, the tension of the vapours of some other liquids, as ether and alcohol; and he suggests that the formula ought to contain higher powers of the temperature than the first; that it ought to take some such form as the following:—

M. Bary applied the formula in this form, continued to the third power of $t$, to vapours.

Professor Rankine, of Glasgow, in 1849, published a formula for vapours in general, as follows:—

in which $\backslash log\; p$ represents the logarithm of the pressure of vapour at saturation; $t$, the absolute temperature; $a,\; b,\; c$, three constants, to be determined from three experimental data for each fluid. When the pressure is expressed in inches of mercury, and the temperature in degrees of Fahrenheit, the values of these constants, for steam, are as follows:—

The inverse formula, for calculating the temperature from the pressure, is—

in which $\backslash frac\{b\}\{2c\}\; =\; 0.0035163$, $\backslash frac\{b^2\}\{4c^2\}\; =\; 0.000012364$.

The operations of this formula are considerable, but in point of accuracy it is generally very satisfactory. Extending from $-22^\backslash circ$ to $446^\backslash circ$ of Fahrenheit's scale, it is the most exact of all the formulae hitherto proposed for the same width of range. It is, however, much more tedious, especially in the inverse form, and is at least not more exact than Mr Buchanan's formula, between the same limits.

CHAP. VII.—CONSTITUENT HEAT OF SATURATED STEAM.

The relation of the sensible temperature, measured by the thermometer, and the pressure of saturated steam, having been approximately determined and formulated, the next stage of the inquiry is the relation which the sensible temperature bears to the total heat of saturated steam. The total heat of steam comprises the latent heat, in addition to the sensible heat or temperature; that which is not directly measurable by the thermometer, and therefore called latent, together with that which is directly sensible to, and measurable by it. The total heat of steam would appear, at first sight, to be in some way related to, if not identical with, total or absolute temperature. The latter is, however, a speculative quantity, employed in the consideration of gaseous bodies, for the convenient expression of their known properties. The total heat of steam, according to the general acceptance, as defined by M. Regnault, is that quantity of heat which would be transferred to some other body in condensing the steam at the same temperature and pressure as those at which it was generated, and in cooling the condensed steam, or water, down to the freezing point. That is to say, conversely, if water be supplied at the freezing point of temperature, $0^\backslash circ$ centigrade, or $32^\backslash circ$ Fahrenheit, for evaporation into steam, the total quantity of heat applied to the water, and consumed in generating steam of any pressure and temperature from it, is said to be the total heat of the steam of the given pressure and temperature; and in general, whatever may be the actual temperature of the water from which the steam is generated, the total heat of the steam is reckoned from the freezing point. The adoption of the freezing-point as the zero for total heat, as well as for that of the sensible temperature in the case of the centigrade thermometer, is not done, of course, with any purpose of fixing an absolute datum for total constituent heat; but for convenience, being situated sufficiently low in the scale of temperature to underlie all the ordinary calculations about steam.

It was determined experimentally by Regnault, that the latent heat of saturated steam, at $0^\backslash circ$ C., was $606.5^\backslash circ$ C.; so that the latent heat of 1 lb. of steam, at $0^\backslash circ$ C., would raise the temperature of $606.5$ lb. of cold water through $1^\backslash circ$. The total heat of steam at $0^\backslash circ$ C. is the same as the latent heat, namely, $606.5^\backslash circ$ C.; and it was found that the total heat of saturated steam increased uniformly between the temperatures of $0^\backslash circ$ and $230^\backslash circ$ C., by $.305^\backslash circ$, with each increment of $1^\backslash circ$ of temperature. The specific heat of ordinary steam is thus $.305^\backslash circ$, that of water being $=\; 1$. The total heat $H$ of saturated steam of any temperature $t$, in centigrade degrees, is therefore expressed by the equation—

From this equation it appears, that, whilst the sensible heat or temperature rises $1^\backslash circ$, the total heat increases only $.305^\backslash circ$, or less than a third of a degree. The latent heat

must, therefore, necessarily be diminished as the temperature rises, other circumstances being the same, by as much as $.305^\backslash circ$ falls short of $1^\backslash circ$, or $1\; -\; .305\; =\; .695^\backslash circ$ for each degree of temperature; and the decreasing latent heat would be expressed by $606.5^\backslash circ\; -\; .695^\backslash circ\; t$. There is one slight disturbing element, however,—the specific heat of water, which is not constant for all temperatures, but is slightly increased by a rise of temperature; and by as much as the specific heat of the water is increased, the latent heat of the steam is still further diminished, and the true rate of reduction is expressed by a higher fraction than $.695^\backslash circ\; t$. In fact, if the specific heat of water, at temperatures between $0^\backslash circ$ and $30^\backslash circ$ C., be represented by an average of unity, it will be equal to $1.005$ between $30^\backslash circ$ and $120^\backslash circ$, and $1.013$ between $120^\backslash circ$ and $190^\backslash circ$ C., or $374^\backslash circ$ Fahr. M. Regnault embodies this slight rate of increase in the formula, $C\; =\; 1\; +\; .00004\; t\; +\; .000009\; t^2$, in which $C$ is the specific heat of water at any temperature $t$, that at $0^\backslash circ$ C., the freezing-point, being $=\; 1$. The introduction of this element into a general formula for the latent heat of steam, would complicate it too much for general use; and for present purposes, the equation employed by Clausius is preferred, namely,

in which $L$ is the latent heat due to the temperature $t$; and it may be noted that the co-efficient of $t$ is slightly increased above that which would be due to a constant specific heat of water, as the deduction due to a slightly increasing specific heat. That the results afforded by the simpler equation are sufficiently near correctness, appears by the following comparative instances of its application, at different temperatures, by Fahrenheit's scale, as against the use of Regnault's correct but more complicated process:—

In estimating the latent heat of steam at $100^\backslash circ$ C., or $212^\backslash circ$ Fahr., Regnault found, that on account of the slight variation of the specific heat of water, $100.5$ centigrade units, or $180.9$ Fahrenheit units, of heat, were required to raise the unit of water from $0^\backslash circ$ to $100^\backslash circ$ C., or through $180^\backslash circ$ Fahr.; and he found that the total heat of steam, at $100^\backslash circ$ C., was $636.67^\backslash circ$ C. From this deduct $100.5^\backslash circ$, and the difference, $636.67\; -\; 100.5\; =\; 536.17^\backslash circ$ C., represents the true latent heat of steam at $100^\backslash circ$ C. But as, in the compilation of his tables, Regnault started with the integral number $637^\backslash circ$, the latent heat of saturated steam, at $100^\backslash circ$ C., or $212^\backslash circ$ Fahr., is estimated by him at $536.5^\backslash circ$ C. = $965.7^\backslash circ$ Fahr.

To modify the formula for the total heat of steam, in terms of Fahrenheit degrees, $606.5^\backslash circ$ C. $\backslash times\; 9\; \backslash div\; 5\; =\; 1091.7^\backslash circ$ Fahr., is the total heat at $32^\backslash circ$ Fahr.; and as $t$ represents the indicated temperature, the total heat would be expressed by $1091.7\; +\; .305(t\; -\; 32)$, or, in a more general form, by $(1123.7\; -\; 32)\; +\; (.305t\; -\; 9.76)\; =\; (1113.94\; -\; 32)\; +\; .305t$. The first element in this expression should be reduced to $1113.4$, in order to produce exact conformity with the observed total heat at $212^\backslash circ$ Fahr., Regnault's starting-point; and the formula for the total heat, in terms of Fahrenheit degrees, becomes,

By this equation, the total heat of steam generated at $212^\backslash circ$ Fahr. is equal to $1113.4\; -\; 32\; +\; (.305\; \backslash times\; 212)\; =\; 1146^\backslash circ$ Fahr.; and this represents what would be consumption of heat in generating the steam, if the water were supplied at $32^\backslash circ$ Fahr. By means of the same form of equation, the total expenditure of heat consumed in raising steam from water supplied at ordinary temperatures may be calculated, by substituting for $32$ in the formula, the initial temperature of the water supplied for evaporation, subject to an allow-

ance, if deemed sufficiently important, for the slight increase of specific heat of the water of higher temperature. Thus, when the water is supplied at $62^\backslash circ$ Fahr., the average temperature of cold water, the extra specific heat may be neglected, and the heat expended in generating steam from the water is expressed by the equation,

If, as in condensing engines, the water be supplied at, say $100^\backslash circ$ Fahr., the heat expended in generating steam from it, again neglecting the specific heat, is expressed as follows:—

Again, if the water be supplied at a boiling temperature, $212^\backslash circ$ Fahr., the specific heat of the water at $212^\backslash circ$, as already noted, would be $.9$ unit, or degree, of heat in excess of that at $32^\backslash circ$, and $212\; +\; .9\; =\; 212.9^\backslash circ$ should be substituted. Hence, for an initial temperature of $212^\backslash circ$ Fahr., the expenditure of heat in generating steam would be

To convert Clausius's formula for the latent heat of steam, namely, $L\; =\; 607\; -\; .708t$, into Fahrenheit's measure, $607^\backslash circ$ C. $\backslash times\; 9\; \backslash div\; 5\; =\; 1092.6^\backslash circ$ Fahr., and for $t^\backslash circ$ C. substitute $(t\; -\; 32)$ Fahr., then $L\; =\; 1092.6^\backslash circ\; -\; .708(t\; -\; 32)$ Fahr., or finally, by the Fahrenheit scale,

It is convenient to bear in mind that the same figures which express in degrees the relations of the constituent heat of steam, as ratios simply, not as absolute quantities, express also positive values—in units of heat—when applied to 1 pound-weight of steam, in accordance with the definition of the heat-unit, or the thermal unit. Now, to trace the appropriation of all the heat which contributes to the formation of a pound of steam, in terms of thermal units, as well as of dynamic units or foot-pounds, take 1 pound of water at $32^\backslash circ$ Fahr., to be converted into saturated steam at $212^\backslash circ$. The first instalment of heat is provided to elevate the temperature to $212^\backslash circ$, through $180^\backslash circ$; in other words, to increase the molecular velocity and slightly expand the liquid, which appropriates $180.9$ units of heat, equivalent to $180.9\; \backslash times\; 772\; =\; 139655$ foot-pounds. Secondly, heat is absorbed in overcoming the molecular attraction, and separating the particles; that is, in the formation of steam, appropriating $892.8$ units of heat = $689,242$ foot-pounds. Thirdly, in repelling the incumbent pressure, whether of the atmosphere, or of the neighbouring steam; that is, to raise a load of $14.7$ lb. per square inch, or $2116.8$ lb. on a square foot, through a cubic space of $26.36$ cubic feet, which is the volume of 1 pound of saturated steam: equal to $55,815$ foot-pounds, or $72.3$ units of heat. Strictly, there is the initial volume of the original pound of water to be deducted from this total volume; but it is relatively small, and need not be further considered. The second of the above proportions of heat is formed by subtracting the sum of the first and third, which are both arrived at by direct observation, from the total heat. The first is the sensible heat, and the second and third together constitute the latent heat. With respect to the third constituent proportion of heat, it is simply an expression of the necessary mechanical labour of disengaging $26.36$ cubic feet of steam, and forcing its way into space against a pressure of $2116.8$ lb. per square foot; and these quantities being multiplied together, and divided by $772$, are equivalent to $72.3$ units of heat.

The proportions of the heat expended in generating saturated steam at $212^\backslash circ$ Fahr., and at $14.7$ lb. pressure per square inch, from water supplied at $32^\backslash circ$, may be exhibited thus:—

Supposing, however, that 1 lb. of water, at 32° Fahr., were injected into a vacuous space or vessel, having 26.36 cubic feet of capacity. If heat were applied to evaporate this water into steam of 212°, and 14.7 lb. per square inch pressure, so as to fill the whole space with saturated steam, the expenditure of heat would consist only of the sensible heat, to raise the temperature of the water 180.9 units, plus the latent heat for the formation of the steam, 892.8 units, = 1073.7 units, as in this case there would be no incumbent pressure to resist, and no extraneous work. But, again, let a second pound of water be injected into the same vessel, already full of steam, to be evaporated into steam of 14.7 lb. pressure per square inch, so that the vapour of the second pound of water must expel the first, a uniform pressure of 14.7 lb. per square inch being maintained within the vessel. The expenditure of heat in the generation of the second pound will be 72.3 units in excess of that required for the first pound, being the additional quantity required to repel the incumbent pressure; and the total expenditure will be 1146 units. The 72.3 units excess of heat expended on the second pound of steam disappears, or rather it does not appear as heat, but is transformed into the work of expelling the first pound of steam; and, after its production, the second pound contains just the same quantity of heat as the first, namely, 1073.7 units, which may be proved by condensing them both into water of 32° Fahr.

The latent heat of steam, then, is not, as is sometimes supposed, an expression of the total work or energy in the steam; but is the work expended in overcoming the attraction of the particles, forcing them asunder, together with the work expended in repelling the external pressure under which the steam is generated. As the temperature rises, the centrifugal velocity, or vibratory motion of the minute particles is accelerated, the liquid expands, and the attraction of the particles is consequently diminished. Hence that part of the latent heat, or work, expended in effecting an entire separation of the particles, diminishes as the temperature rises. When water is evaporated at a low temperature, it is obvious that the particles are held together by a greater force than if it were evaporated at a higher temperature, after heat has been expended in accelerating the velocity of the particles, and expanding the liquid; and less work is expended in effecting their separation. At high temperatures the particles are already in part separated; they have a less hold on each other, and consequently an entire separation is more easily completed at higher than at lower temperatures. On the contrary, the second, but inferior portion of latent heat expended in repelling the external resistance—the product of increasing pressure into diminishing volume—increases slowly as the temperature rises; but the increase in this respect is less than the decrease in respect of the chief duty of the latent heat. In this manner it is to be explained, that though the total constituent heat of steam slowly increases as the temperature rises, in consequence of the comparative rapidity with which the sensible heat increases, the latent heat slowly diminishes as the temperature is elevated.

On the principle of the mechanical equivalence of heat, according to which heat, and work or duty performed, are convertible into and representative of each other, the investigation of the properties of steam may be conveniently conducted in terms of one form of expression or the other—heat, or dynamic effect—as the nature of the experimental evidence may demand. The density of saturated steam is one of its properties which has not yet been accurately determined by direct experiment; nor, of course, has the relative volume, which is inversely as the density. The density of steam is expressed by the weight of a given constant volume—say one cubic foot; and the relative volume by the number of volumes of steam produced by one volume of water—hence called relative. The density and the relative volume are, however, most accurately determinable by means of the pressure, temperature, and latent heat of steam, all of which have been subjects of careful and comprehensive experiment. When steam is freely generated in contact with water in a boiler, the actual process of generation consists, first, in heating the water to the temperature due to the pressure under which the steam is generated; second, in the absorption of a large quantity of heat which becomes latent—not affecting the thermometer—and is replaced physically by a quantity of steam of the same temperature—the sensible heat of the water continuing sensible in the steam. It is properly argued, then, that the specific function of converting the water into steam, of changing a non-elastic into an elastic substance, of thus developing a reservoir of motive-power where none existed before, is performed by the latent heat; and that, inasmuch as the process is just the conversion or change of form, of heat into elastic force, the force or power so manifested is simply commensurate with the latent heat; and if the latent heat, the amount of which is known experimentally, be converted into foot-pounds, in terms of Joule's equivalent, it will constitute one side of an equation, which will show on the other side the dynamic expression of the relative temperature and pressure, which also are directly known by experiment.

Suppose 1 pound of water in contact with other water, to be converted into 1 pound of steam within a boiler, and that the process of heating and conversion be commenced at the bottom of the scale of absolute temperature, at 461° below zero Fahrenheit. Whether it be possible or not, it is at least conceivable that the whole of the given weight of water may be in a state of vapour at 1° absolute temperature, of extreme tenuity, indefinitely large in volume, and indefinitely low in pressure; let it be supposed that this 1 pound of steam occupies the entire capacity of the boiler, and let the temperature be raised to 2°; another portion of vapour would be generated, occupying part of the capacity of the boiler, and forcing the prior steam into smaller compass, and thus increasing its density. If the temperature be thus continually elevated by degrees, the particular pound of steam under consideration would be continually reduced into smaller bulk, and its density would be increased, and likewise the pressure. It may properly be conceived, therefore, to undergo a process of compression from its conception to maturity—the increments of pressure accumulating with the increments of density and pressure. Now, the work or dynamic force accumulated in the given pound of steam, in virtue of the successive compression to which it may be supposed to have been subjected, is the same in quantity for each degree of temperature:—it is equal, in fact, to the product of the final increment of pressure multiplied into the final volume of the steam. Or—regarding the problem in another way—a pound of water is converted into a pound of steam, generating and occupying a certain volume, and this volume is consummated with a

Density of final increment of pressure for the final degree of tempera-
Saturated ture.

This final increment of pressure, then, represents, for this particular volume, one degree of temperature; and if multiplied into the volume, is an expression of the action for one degree of temperature. If further multiplied by the absolute temperature in degrees, the resulting product expresses the whole of the latent heat of evaporation inherent in the given weight of steam.

In strict argument, this mode of estimation, in terms of the whole volume of the steam, gives a result slightly in excess of the literal result, as the volume actually generated is not the whole final volume, but only the excess of this above the volume of the water from which it is generated.

To vary the form of the argument, suppose the final volume of the given pound-weight of steam to be erected into a vertical column on 1 square foot of base, the column would of course weigh 1 pound; and if the height be multiplied by the final increment of pressure in lbs. per square foot for one degree of temperature, the product would express the height of a vertical column of the steam on 1 square foot of base, equal in weight to the final increment of pressure. If the weight be again multiplied by the absolute temperature, the ultimate product will express the latent heat of 1 pound weight of steam in "feet of fall" of 1 pound, that is, foot-pounds; and, further, dividing by 772, Joule's equivalent, the quotient will be the equivalent value of the latent heat in units or degrees of Fahrenheit's scale.

This form of reasoning, no doubt, contains a principle of hypothetical origin, according to which the actual heat present in a substance is simply proportional to its temperature, measured from a certain point of absolute cold, and multiplied by a specific constant; and "although," as Professor Rankine observes, "existing experimental data may not be adequate to verify this principle precisely, they are still sufficient to prove, that it is near enough to the truth for all purposes connected with thermo-dynamic engines, and to afford a strong probability that it is an exact physical law."

Let $g$ = the increment of pressure for the final degree of temperature in lbs. on the square foot; 772 foot-pounds = the mechanical equivalent of one unit of heat, or so much heat as would raise the temperature of 1 pound of cold water 1°; $L$ = the latent heat of 1 pound of water in units, which is of course identical with the latent heat in degrees deduced from the temperature; $T$ = the absolute temperature; $V$ = the volume of 1 lb. of steam in cubic feet, and $v$ the volume of the water from which it is generated; then $V\; -\; v$ = the volume generated, and the contemplated equation would be as follows:—

or, for simplicity, let the volume of the water be neglected, as it is not practically important, then the equation would be—

From this equation, it follows that the latent heat of one pound of steam is

Consequently, also, the latent heat $l$, of one cubic foot of steam, dividing the above quantities by $V$, is

The volume of 1 lb. of steam in cubic feet, $L$ being expressed in units of heat, is

VOL. XX.

As the weight of 1 cubic foot of cold water is 62.3 lbs., it follows that 62.3V is the volume, in cubic feet, of the steam generated from 1 cubic foot of water. If $n$ be the relative volume, then $n\; =\; 62.3V$, and $V\; =\; n\; \backslash div\; 62.3$; and, by substitution,

from which the relative volume of the steam is

also, conversely, the latent heat, in Fahrenheit degrees, in terms of the relative volume, is

For illustration, let the temperature be raised from 211½ to 212½°, through one degree, the pressure will rise from 2094 lb. to 2136 lb. per square foot, making the increment of pressure $g\; =\; 42$ lb. per square foot. The mean temperature is 212°, and the absolute temperature $T\; =\; 461\; +\; 212\; =\; 673$°. The latent heat $Tg$ in foot-pounds, of 1 cubic foot of steam at 212°, and a pressure of 14.7 lb. per square inch, or 2116.8 lb. per square foot, is, therefore, $673\; \backslash times\; 42\; =\; 28,266$ foot-pounds. To determine next the value of $V$, the volume of 1 lb. of steam at 212°, and at atmospheric pressure, let the steam be gaseous, then, by the equation for gaseous steam (which will be afterwards explained), $V\; =\; 85.4\; T\; \backslash div\; P\; =\; 85.4\; \backslash times\; 673\; \backslash div\; 2116.8\; =\; 27.16$ cubic feet; and substituting numerical values, we have for the latent heat of 1 lb. of gaseous steam at 212°, and atmospheric pressure,

which is precisely the latent heat of gaseous steam at 212°, as deduced by Mr. Brownlee, in terms of Regnault's constant for the specific heat of gaseous steam, namely, .475.

According to the preceding equations, the volume of 1 lb. of saturated steam at 212°, is

and the relative volume of the same steam is

which has been, but erroneously, considered to be 1700 volumes.

The density or weight of 1 cubic foot of saturated steam is readily deducible from the equation for the volume in cubic feet of 1 pound of steam, in which $V\; =\; 772L\; \backslash div\; Tg$; as the weight of a cubic foot is simply the inverse of this equation. Thus, the density $D$, or the weight in pounds of 1 cubic foot = $\backslash frac\{1\}\{V\}$, or

Mr. James Brownlee has deduced a simple expression for the density of saturated steam in terms of the total pressure, thus—

in which $D$ is the weight of 1 cubic foot of steam, of the pressure $p$ Fahr. The results presented by means of this formula are very accurate; they do not differ from those

obtained in terms of the temperature and latent heat of steam for pressures from 1 lb. to 250 lb. per square inch by more than $\backslash frac\{1\}\{4\}$th per cent. The volume in cubic feet of 1 pound of saturated steam is of course expressed by the inverse of the weight in pounds of a cubic foot of the steam,

thus $V\; =\; \backslash frac\{1\}\{D\}$, consequently

Again, the relative volume of the steam is expressed by the ratio of 62.3 lb., the weight of a cubic foot of water, to $D$, the weight of a cubic foot of steam; hence, $62.3\; \backslash div\; \backslash frac\{p^\{.941\}\}\{330.36\}\; =\; 62.3\; \backslash times\; 330.36\; \backslash div\; p^\{.941\}$, = the relative volume.

Putting $n$ = the relative volume,

from which it appears that the relative volume of saturated steam of 14.7 lb. pressure per square inch, and $212^\backslash circ$ temperature, is 1642, the same as was found before in terms of the temperature and latent heat.

CHAP. IX.—GASEOUS STEAM.

If ordinary or saturated steam be superheated or "supercharged" with heat, it advances from the state of saturation into that of gaseity. The transition into the gaseous state involves a considerable elevation of temperature, by amounts which increase with the pressures; and steam, when thus sufficiently elevated in temperature above the saturation-point due to its density, is known as gaseous steam, distinctively from ordinary, or, as it may be called, imperfectly gaseous steam.

Regnault found, throughout the whole range of his observations, that the specific density of gaseous steam at all temperatures was .622; that is to say, the weights of equal volumes of air and sufficiently superheated or gaseous steam, having the same pressures and temperatures, were as 1 to .622, and that therefore the steam so treated was gaseous, as, the specific density being constant, the air and the gaseous steam, when taken at the same temperature, must expand alike when equally raised in temperature.

Confirmatory of Regnault's specific density of gaseous steam, the chemical union of oxygen and hydrogen, in the proportions to form steam, may be referred to. Two cubic feet of hydrogen and one of oxygen combining, will form two cubic feet of gaseous steam at the same temperature. The specific density of hydrogen is = .06926, and that of oxygen = 1.10563, and the density of the product is in the combined ratio of the densities and the uniting volumes.

being the same as was determined by Regnault from direct observation.

In accordance with the relations of perfect gases, the weight of a cubic foot of air, expressive of the density, $D$, at any pressure per square inch, $p$, and temperature, $t$, Fahrenheit, is expressed by the equation,

in which $144\; p$ expresses the pressure per square foot, $t\; +\; 461$ the absolute temperature, and 53.15 a constant determined for air. The same form will express the weight of steam by a modification of the constant in terms of the specific

density; thus, for gaseous steam, $53.15\; \backslash div\; .622\; =\; 85.4$ is the appropriate constant, and the weight of a cubic foot of gaseous steam is expressed by the equation,

in which $P\; =\; 144\; p$, and $T\; =\; t\; +\; 461$.

As the pressure, volume, and temperature of gaseous steam, and other gases, vary with each other in simple ratios—the pressure and the volume inversely with each other, and both of them directly with the absolute temperature—their mutual relation for any given constant weight of gas is such, that the pressure multiplied by the volume is equal to the absolute temperature multiplied by a constant number. For gaseous steam, as the weight in pounds of 1 cubic foot is equal to $P\; \backslash div\; 85.4\; T$; then, conversely, the volume in cubic feet of 1 pound of steam is $85.4\; T\; \backslash div\; P$, and, generally, $PV\; =\; 85.4\; T$, for gaseous steam. For air, and a few other gases, the following are the general equations for a given weight, 1 pound of gas:—

Constants for One Pound Weight of Gas.

In order to find the total heat of steam, it may be observed, that from some experiments made by Regnault, it appeared that ordinary steam is nearly gaseous at temperatures below $60^\backslash circ$ Fahr. Mr Brownlee has adopted a fundamental temperature of $40^\backslash circ$ Fahr. as that at which the saturated and the gaseous steams become identical in constitution. For gaseous steam, Regnault found the specific heat to be $.475$; that is, that the total heat of gaseous steam increases uniformly $.475$ for each degree of sensible temperature; and it follows that an equation on the model of that for saturated steam may be found to express the total heat of gaseous steam. Proceeding on this basis, Mr Brownlee finds, by the formula for saturated steam, that the total heat of steam at $40^\backslash circ$ Fahr., and at the pressure due to saturation, is $1113.4\; -\; 32\; +\; (.305\; \backslash times\; 40)\; =\; 1093.6^\backslash circ\; \backslash text\{\; F.\}$; and he substitutes for .305 the gaseous constant $.475$, and adjusts also the first quantity in the expression, reducing it to $1106.6$,—by as much, in fact, as the constant $.475$ adds to the first side of the equation. The expression, then, becomes $1106.6\; -\; 32\; +\; (.475\; \backslash times\; 40)\; =\; 1093.6^\backslash circ\; \backslash text\{\; F.\}$, showing the same total heat, $1093.6^\backslash circ$ Fahr., regarded as gaseous steam, as was found by the formula appropriate for saturated steam. The general formula for the total heat of gaseous steam, putting $t$ for the temperature in Fahrenheit degrees, is, then,

To raise the temperature of the vapour generated at $40^\backslash circ$ to $212^\backslash circ$ through $172^\backslash circ$, the pressure remaining the same, the additional heat is measured by $.475\; \backslash times\; 172\; =\; 81.7^\backslash circ$, and $1093.6\; +\; 81.7\; =\; 1175.3^\backslash circ$, the total heat. Or, employing the above equation in the calculation, $.475\; \backslash times\; 212\; =\; 100.7$, and $H\; =\; 1074.6\; +\; 100.7\; =\; 1175.3^\backslash circ$, as before.

As already shown for saturated steam, the foregoing equation for the total heat of gaseous steam may be employed to find the actual expenditure of heat in raising gaseous steam from water of any temperature higher than $32^\backslash circ$ Fahr. The latent heat of this steam may be estimated from the total heat by deducting $1809^\backslash circ$, the heat necessary to raise the temperature of water from $32^\backslash circ$ to $212^\backslash circ$; thus, $1175.3\; -\; 1809\; =\; 994.4^\backslash circ$, is the latent heat of any gaseous steam at $212^\backslash circ$.

As for saturated steam, so also for gaseous steam, the latent heat diminishes as the temperature rises, but not so rapidly in the latter as in the former, as the specific heat is greater; that is to say, the increment of total heat, $475^\backslash circ$, for each degree of sensible temperature of gaseous steam, is greater than the increment, $305^\backslash circ$, for saturated steam; and therefore the difference, $1\; -\; 475\; =\; -525^\backslash circ$, being the reduction of latent heat for each degree of temperature for gaseous steam, is less than the difference, $1\; -\; 305\; =\; 695^\backslash circ$, for saturated steam.

The actual expenditure of heat in generating gaseous steam from water of higher temperature than $32^\backslash circ$ Fahr., may be found by substituting the temperature for $32$ in the last formula; as follows, for the common temperature $62^\backslash circ$ Fahr.:—

For a temperature of $100^\backslash circ$ Fahr. the formula is—

For a temperature of $212^\backslash circ$, with the specific heat of water $212.9^\backslash circ$, the formula is—

The appended table of the properties of saturated steam has been calculated by means of the formulas in this article. The first column contains the ascending total pressures in pounds per square inch. The second column, of temperatures, was calculated from the pressures by the formula—

the third column, of total heat, was calculated by the formula $H\; =\; 1081.4\; +\; 305t$; the fourth column, of latent heat, by the formula $L\; =\; 1115.2\; -\; 708t$; the fifth column, of density, by the formula $D\; =\; 941\; \backslash log.\; p\; -\; 2.519$; the sixth column, of the volume of 1 lb. of steam, by the formula $V\; =\; 2.519\; -\; 941\; \backslash log.\; p$; and the seventh column, of relative volume, by the formula $n\; =\; 4.3135\; -\; (941\; \backslash times\; \backslash log.\; p)$.

Mr William Fairbairn and Mr Thomas Tate read a paper at the meeting of the British Association for the advancement of science in September 1859, containing some approximate results of experiments undertaken by them, for the purpose of determining the law of the density of steam and other condensable vapours, and of testing the deductions from the dynamic theory of heat of Professor Rankine and others, with respect to that subject. The new feature of the experiments consists in the use of the "saturation-guage," to aid in determining with precision the point of saturation of steam. Suppose two globes, called A and B, to be connected by a syphon tube, containing mercury, and the whole to be immersed in a bath, in which they can be raised to any required temperature. If an absolute vacuum be created in each globe, and 20 grains of water be introduced into A, and 30 or 40 grains into B; then, if the temperature be slowly and uniformly raised around the globes, the water within them will be gradually evaporated, and filled with steam of a density increasing with the temperature, until the whole of the water in A is converted into steam, when instantly the line of the mercury, heretofore undisturbed, will rise towards the globe A. The disturbance of level thus occasioned constitutes the saturation-test; it is caused by the change of the steam in A, from the condition of saturation to that of superheating, whilst that in B, which contains a surplus of water, must necessarily continue in the state of saturation, and there is no longer an equilibrium of pressure. The volume

of the steam in A is known, and the temperature in the bath at the instant of the disturbance of the mercury is that of saturation. The following is a selection of temperatures and relative volumes of saturated steam approximately reduced from the experiments, with the corresponding volumes reckoned from the foregoing table of the properties of saturated steam:—

The nearness of the experimental results to those theoretically established is very remarkable. The differences, as Professor Rankine suggests, may arise from a difference in the value assumed for the volume of water.

The reader is referred to the end of the article STEAM-ENGINE for a list of references to works on steam. The first chapter of this article, it should be stated, is abridged from the article on STEAM in the previous edition of the Encyclopædia Britannica. (D. K. O.)