Heat

HEATBy Sib WILLIAM THOMSON.

Entered according to Act of Parliament of Canada, in the year 1880, by william Thomson, Knt., Glasgow, Scotland, in the office ef the Minister of Agriculture. HEAT is a property of matter which first became known to us by one of six very distinct senses.

§ 1. Sense of Heat— The sense of touch, as commonly meant, has two distinct objects—force and heat. If a person stretches out his hand till it meets anything solid, or holds it out while something solid is placed upon it, he experiences a sensation of force. He perceives resistance to the previous motion of his hand in one case, in the other case the necessity of resisting to prevent his hand from being forced downwards; the immediate object of this perception in each case is force. But there is another very distinct sensation, that of heat or cold, which he may or may not perceive in either of those cases, and which he may also perceive, still by what is commonly called the sense of touch, in other cases even when no sense of force is also experienced. Thus, in the first case, if the solid be a fixed can of warm water, or of iced water, the person perceives a sense of heat or of cold; and, supposing him to have performed the operation with his eyes shut, his mind is informed by the double sense of touch that his hand has met with a hot fixed body or a cold fixed body: in the other case he may perceive that a hot heavy solid, or a cold heavy solid, has been laid upon his hand. But if he dips his hand gently into a can of water, or if he holds it towards a fire, or if he exposes it to a gentle current of air, or waves it about through the air, he perceives heat or cold without any accompanying sense of force.

The earliest scientific thoughts respecting these sensations of heat and cold must have led to the true conclusion that there is some property of external matter on which the sensations depend, and a little advance into the natural philosophy of the subject has suggested and proved that this property is also possessed by the living body, and that the sensatiou of heat or cold in the hand, in the instances referred to above, depends on the change produced in the hand in respect to this property by a change of circumstances which preceded the sensation. We now call heat the property of matter concerned in these sensations, and temperature a certain variable quality of matter varying according to its temporary condition in respect to heat.

In the strictest modern scientific language (compare § 3 below) the word heat is used to denote something communicable from one body or piece of matter to another, and temperature a definite variable quality of matter, varying generally in any particular piece of matter when heat is communicated to it or taken from it, varying also as we shall see (§§ 8 and 9 below; also Thermodynamics) in consequence of operations which can take place within the body itself, or which may be performed upon it from without, but which cannot be described as communication of heat to it or drawing off of heat from it.

§ 2. Latent Heat.— There are exceptional cases in which temperature does not vary in a mass of matter when heat is communicated to it from, or taken from it to, external matter. For instance, when the body is ice at the melting point, heat communicated to it does not raise its temperature; or if the body be water at the freezing point with ever so small a piece of ice in it (see Matter, Properties of), heat taken from it does not cause its temperature to fall; or if the whole mass considered be ice and water well mixed, heat may be either communicated to it or taken from it without altering its temperature; or if the body be water at the boiling point in the open air, heat very slowly communicated to it in however great quantities does not raise its temperature sensibly, but causes it to disappear by evaporation from its surface; or if the body be steam in a cylinder with a little water in the bottom and with a frictionless piston above it for roof (fig. 1), under atmospheric pressure, heat taken from it very slowly does not cool it until the whole steam has become condensed into water, and heat communicated to it very slowly does not warm it until the whole water has become evaporated into steam; or if the body be ice (or frozen water), in place of the liquid water of the last case, and if the pressure on the upper side of the piston, instead of atmospheric pressure of about 1033 grammes per square centimetre (14 ’ 7 lb per square inch), be anything less than ∙ 3 1 θth of a gramme per square centimetre, the same statement will still apply with “ice” substituted for water. Black’s celebrated doctrine of latent heat is merely the declaration of a class of phenomena of which the preceding illustrations sufficiently indicate the character. Modern mysticism has been much exercised in respect to the terms sensible heat and latent heat, whether in decrying them, or in continuing to use them, but with aggravating haziness, instead of the clear wrongness of the old doctrine. It has become of late years somewhat the fashion to decry the designation of latent heat, because it had been very often stated in language involving the assumption of the materiality of heat. Now that we know heat to be a mode of motion, and not a material substance, the old “impressive, clear, and wrong” statements regarding latent heat, evolution and absorption of heat by compression, specific heats of bodies and quantities of heat possessed by them, are summarily discarded. But they have not yet been generally enough followed by equally clear and concise statements of what we now know to be the truth. A combination of impressions surviving from the old erroneous notions regarding the nature of heat with imperfectly developed apprehension of the new theory has somewhat liberally perplexed the modern student of thermodynamics with questions unanswerable by theory or experiment, and propositions which escape the merit of being false by having no assignable meaning. There is no occasion to give up either “sensible heat” or “latent heat”; and there is a positive need to retain the term latent heat, because if it were given up a term would be needed to replace it, and it seems impossible to invent a better. Heat given to a substance and warming it is said to be sensible in the substance. Heat given to a substance and not warming it is said to become latent. These designations express with perfect clearness the relation of certain material phenomena to our sensory perception of them. Thus when heat given to a quantity of water warms it, the heat becomes sensible to a hand held in the water. When a basin of warm water and a basin of water and ice are placed side by side, a hand dipped first in one and then in the other perceives the heat. If now the warm water be poured into the basin of ice and water, and stirred for a few seconds of time (unless there is enough of warm water to melt all the ice), the hand perceives no warmth; on the contrary, it perceives that the temperature is the same as it was in the basin of ice and water at the beginning. Thus the heat which was sensible in the basin of warm water has ceased to be sensible in the water that was in that basin, and has not become sensible in the other. It is therefore well said to have become latent.

Calorimetry.

§ 3. Calorimetry by Latent Heat.— The doctrine of latent heat leads us very smoothly to a most important measurement in thermal science, the measurement of quantities of this wonderful property of matter which we call heat; and this without our knowing anything of what the nature of heat is, —whether it be a subtle elastic fluid, or a state of motion, or possibly some modification of matter related to action of force. Without, in the first place, admitting into our minds any definite idea as to the nature of heat, we may agree to measure quantities of heat by quantities of ice melted into water without change of temperature. Thus if a kilogramme of ice is melted by a large quantity of water at a lukewarm temperature, or by a comparatively small quantity of very hot water, the same quantity of heat has certainly gone from the warm water to the ice in each case, supposing that the result in each case is the ice and warm water left all in a state of ice-cold water. The measurement of quantities of heat, whether thus by the melting of ice, or by any other means, received the name of “Calorimetry,” when the essence of heat was supposed to be a fluid, and this fluid called caloric. The name calorimetry is still by general consent retained to designate measurement of quantities of heat, as distinguished from thermometry, or the measurement of temperature ( § 10-67). As long as the truth or falsity of the materialistic hypothesis seemed an open question, the word caloric was held to imply the materiality of heat. Thus Davy, after discussing some of the fundamental dogmas of the “Calorists,” as he called them, and describing his own experiments, which proved beyond all doubt the falsity of their fundamental hypothesis that heat is matter, varied the statement of his conclusion by saying, “or caloric does not exist.” While accepting Davy’s conclusion, however, we need not accept this way of stating it; and as most of our best modern writers still use the word calorimetry, and as French writers have, in comparatively recent times, introduced the word “calorie” to designate a unit quantity of heat, it is decidedly convenient still to retain the name caloric to denote definitely the measurable essence of heat. This is convenient scientifically as tending to give precision to language and ideas respecting the two classes of measurement, calorimetry and thermometry; and it has the advantage of leaving the more popular word heat available for that somewhat lax general usage, from which we cannot altogether displace it, in which it may sometimes mean high temperature, as when we speak of great heat, or summer heat, or blood heat, sometimes a measurable quantity of heat, as in the term latent heat, and sometimes a branch of study or science dealing with the transference of heat by conduction and radiation, as in the title of Fourier’s great work Théorie analytique de la Chaleur, or the whole province of science concerned with heat, including calorimetry and thermometry, and conduction and radiation of heat, and generation of heat, and dynamical relations of heat, as in English titles of separate books such as Dixon’s, Balfour Stewart’s, and Maxwell’s, or of chapters or divisions of larger treatises, such as even the present article.

f 4. Calorimetry by Meltiny of Ice.— Calorimetry was first practised by means of the melting of ice as explained above, and the first thermal unit, or unit quantity of heat, or “calorie,” although not then called calorie, was the quantity of heat required to melt unit weight of ice. This, for example, is the unit on which Fourier founds his reckoning illustratively when he explains the fundamental principles of his theory of the conduction of heat. Ice seems to have been first used for calorimetry by Wilcke, a Swede. For the systematic application of this method for the measurement of quantities of heat in various physical inquiries Laplace and Lavoisier constructed an instrument, the first to which the name of calorimeter was applied, and described it in the memoirs of the French Academy of Sciences for 1780. 1 Though in the hands of Laplace and Lavoisier it gave good results, it had a great inconvenience, which with less careful and less scientific experimenters might lead to great inaccuracies, on account of the water adhering by capillary attraction to the broken ice, instead of draining away from it completely and showing exactly how much ice had been melted. To avoid this evil Sir John Herschel suggested that, instead of draining away the water from the ice, the water and ice should all be kept together, and the whole bulk measured. The diminution of bulk of the whole thus gives an accurate measurement of the quantity of ice melted, because ice melting into water comes to occupy just 91'675 per cent, of its original volume. This suggestion is admirably carried out by Bunsen 2 in his ice-calorimeter, an instrument possessing also other novel features of remarkable beauty and scientific interest. It is particularly valuable for the measurement of small quantities of heat. Its inventor, for example, by means of it succeeded in making satisfactory determinations of the specific heats of some of those rarer metals, such as indium, of which only a few grammee have been obtained.

§ 5. Calorimetry by the Evaporation of Water.— By another application of Black’s doctrine of latent heat, the evaporation of water may be used for calorimetry with great advantage in many scientific investigations. It is used generally in engineering practice, particularly for testing the heating power of different qualities of coal and the economy of various forms of furnaces. The thermal unit, which presents itself naturally in this system, is the quantity of heat required to evaporate unit weight of water when the pressure of the atmosphere as measured by the barometer is of some conventional standard amount, such as that called one atmosphere, or one atmo, being that for which the barometer, with its mercury column at zero centigrade (or the temperature at which ice melts), stands at 76 centimetres in the latitude of Paris, 48° 50', 3 or at

76 × (1 + ∙00531 sin 2 48° 50 , )

1 + ‘00531 sin e l in any latitude Z. This thermal unit (see Thermodynamics) is, according to Regnault’s observations, equal to 6’8 times the ice-calorimetric unit.

§ 6. Thermometric Calorimetry.— The most prevalent mode of calorimetry in scientific investigation has been hitherto, however, neither that by the melting of ice, nor that by evaporation of water, nor indeed anything founded on the doctrine of latent heat at all. It has been founded on the elevation of temperature produced in water by the communication to it of the heat to be measured; and, for the sake of distinction from calorimetry by latent heat or otherwise, it maybe called thermometric calorimetry. We can only consider it now in anticipation, as we have not yet reached the foundation of any thermometric scale; but even now we can see that, if in any way we fix upon any two particular determinate temperatures, the quantity of water warmed from the lower to the higher of them by the heat to be measured is a perfectly definite measure for the quantity of this heat. The two temperatures chosen for thermometric calorimetry are those marked 0° and 1° on the centigrade scale. The first of these we can understand at present, being the temperature at which ice melts under ordinary atmospheric pressure. The second is fully defined in §§ 35, 37, 51, 67. The quantity of heat required to raise unit mass of water (1 kilogramme, or 1 gramme, or 1 milligramme, or 1 lb, as the case may be) from zero to 1° C. is called the thermal unit centigrade, and sometimes, especially by French writers, the “calorie.”

§ 7. Comparison of Calorimetric Units.— Observations by Prevostaye and Desains, and by Régnault, on the latent heat of fusion of ice, show it to be 79’25 thermal units centigrade, a result differing but little from Black’s original determination, which made it 142 thermal units Fahr.,— this being equal to 78’9 thermal units centigrade. Thus if one kilogramme of ice be put into 79-∣ kilogrammes of water at 1° C., and left till the whole is melted (the process may be accelerated by not too violent stirring, § 9), the result will be 801 kilogrammes of water at 0° C.

Regnault’s experiments on the latent heat of steam show that the quantity of heat required to convert into steam unit 1 The instrument itself is preserved in the Obnsm·a&n're des Artset Métiers in Paris. It is described and explained in Maxwell’s Theory of Heat, chap. iii.

2 Pogg. Ann., Sept. 1870, and Phil. Mag., 871; Maxwell’s Theory of Heat, p. 61.

3 This is chosen because all the most accurate experimental determinations depending on a conventional standard for atmospheric pressure, such as measurements of thermal expansions and specific heats of gases, of latent heat of melting solids in terms of a calorimetric unit depending on the centigrade thermometric scale, of latent heats of vapours, and thermal expansions of mercury and glass, and comparisons of mercury and air thermometers, are those of Regnanlt, and were made in Paris and calculated and given to the world according to an arbitrary standard atmosphere corresponding to 76 centimetres of mercury there.

mass of water at the boiling temperature, uuder standard atmospheric pressure (§ 5), is 536^ thermal units centigrade. This number, which is no doubt very accurate, differs but little from Watt’s final result, 960 thermal units Fahr, (equal to 533 ’ 3 thermal units centigrade), obtained by him, in a repetition in 1781 of experiments which he had commenced in 1765 at the invitation of Black, whose pupil he was.

§ 8. Preliminary regarding the Nature of Heat.—Dynamical Calarimetry.— From the dawn of science till the close of last century two rival hypotheses had been entertained regarding the nature of heat, each with more or less of plausibility, but neither on any sure experimental basis: — one that heat consisted of a subtle elastic fluid permeating through the pores or interstices among the particles of matter, like water in a sponge; the other that it was an intestine commotion among the particles or molecules of matter. In the year 1799 Davy, in his first published work entitled An Essay an Heat, Light, and Combinations of Light f conclusively overthrew the former of these hypotheses, and gave good reason for accepting as true the latter, by his celebrated experiment of converting ice into water by rubbing two pieces of ice together, without communicating any heat from surrounding matter. A few years earlier Rumford had been led to the same conclusion, and had given very convincing evidence of it in his observation of the great amount of heat produced in the process of boring cannon in the military arsenal at Munich, and the experimental investigation on the excitation of heat by friction with which he followed up that observation. He had not, however, given a perfect logical demonstration of his conclusion, nor even quite a complete experimental basis on which it could be established with absolute certainty. According to the materialistic doctrine it would have been held that the heat excited by the friction was not generated, 3, but was produced, squeezed out, or let flow out like honey from a broken honeycomb, from those parts of the solid which were cut or broken into small fragments, or rubbed to powder in the frictional process. If this were true, the very small fragments or powder would contain much less heat in them than an equal mass of continuous solid of the same substance as theirs. But unhappily the caloristic doctrine, besides its fundamental hypothesis, which we now know to be wrong, had given an absurd and illogical test for quantity of heat in a body, of which a not altogether innocuous influence still survives in our modern name “specific heat and Rumford actually, in trying to disprove the materialistic doctrine, was baffled by this sophism. That is to say, he measured the specific heat or “capacity for heat” of the powder, and he found that the powder took as much heat to warm it to a certain degree as did an equal mass of the continuous solid, and from this he concluded that the powder did not contain less heat than the continuous solid at the same temperature. This conclusion is so obviously unwarranted by the premises that it is difficult to imagine how Rumford could have for a moment put forward the “capacity for heat” experiment as proving it, or could have rested in the conclusion without a real proof, or at least the suggestion of a real proof. All that Rumford’s argument proved was that the fundamental hypothesis of the “calorists” and their other altogether gratuitous doctrine of equality of “specific heat” as a test for equality of whole quantities of heat in matter could not be both true; and any one not inclined to give up the materialistic hypothesis might have cheerfully abandoned the minor doctrine, and remained unmoved by Rumford’s argument. If Rumford had but melted a quantity of the powder (or dissolved it in an acid), and compared the heat which it took with that taken by an equal weight of the continuous solid, he would have had no difficulty in proving that the enormous quantity of heat which he had found to be excited by the friction had not been squeezed, or rubbed, or pounded, out of the solid matter, but was really brought into existence, and therefore could not be a material substance. He might even, without experiment, have pointed out that, if the materialistic doctrine were true, it would follow that sufficiently long-continued pounding of any solid substance by pestle and mortar, whether by hand or by aid of machinery, would convert it into a marvellous powder possessing one or other of two properties about equally marvellous. Either the smallest quantity of it thrown into an acid would constitute a freezing mixture of unlimited intensity,—the longer it had been pounded, the more intense would be its frigorifie effect on being dissolved, —or the powder would be incapable of being warmed by friction, because it had already parted with all the heat which friction could rub out of it. The real effect of Rumford’s argument seems to have been to salve the intellectual consciences of those who were not inclined to give up the materialistic doctrine, and to save them from the trouble of reading through Rumford’s paper and thinking for themselves, by which they would have seen that his philosophy was better than his logic, and would inevitably have been forced to agree with him in his conclusion. It is remarkable that Davy’s logic, too, .was at fault, and on just the same point as Rumford’s, but with even more transparently logical fallaciousness, because his argument is put in a more definitely logical form.

“Let heat be considered as matter, and let it be granted that the temperature of bodies cannot be increased unless their capacities are diminished 'from some cause, or heat added to them from some bodies in contact”!!

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“Experiment II.— I procured two parallelepipedons of ice, of the temperature of 29°, 6 inches long, 2 wide, and § of an inch thick; they were fastened by wires to two bars of iron. By a peculiar mechanism their surfaces were placed in contact and kept in a continued and violent friction for some minutes. They were almost entirely converted into water, which water was collected and its temperature ascertained to be 35° after remaining in an atmosphere of a lower temperature for some minutes. The fusion took place only at the plane of contact of the two pieces of ice, and no bodies were in friction but ice. From this experiment it is evident that ice by friction is converted into water, and, according to the supposition, its capacity is diminished; but it is a well-known fact that the capacity of water for heat is much greater than that of ice, and ice must have an absolute quantity of heat added to it before it can be converted into water. Friction consequently does not diminish the capacities of bodies for heat.”—Davy’s Essay on Heat, Light, and Combinations of Light, pp. 10-12.

[Delete from “and, according to the supposition,” to “greater than that of ice” inclusive; and delete the lame and impotent conclusion stated in the last eleven words. The residue constitutes an unanswerable demonstration of Davy’s negative proposition that heat is not matter.]

§ 9. Joulds Dynamical Equivalent of Heat.— It is remarkable That, while Davy’s experiment alone sufficed to overthrow the hypothesis that heat is matter, and Rumford’s, with the addition of just a little consideration of its relations to possibilities or probabilities of inevitable alternatives, did the same, fifty years passed before the scientific world became converted to their conclusion,—a remarkable instance of the tremendous efficiency of bad logic in confounding public opinion and obstructing true philosophic thought.

Joule’s great experiments from 1840 to 1849, creating new provinces of science in the thermodynamics of electricity, and magnetism, and electro-chemistry, recalled attention to Davy and Rumford’s doctrine regarding the nature of heat, and supplied several fresh proofs, each like Davy’s absolutely in itself complete and cogent, that heat is not a material substance, and each advancing with exact dynamical measurement on the way pointed out by Rumford in his measurement of the quantity of heat generated in a certain time by the action of two horses not urged to overwork themselves. The full conversion of the scientific world to the kinetic theory of heat took place about the middle of this century, and was no doubt an immediate consequence of Joule’s work, although Rumford and Davy’s demonstrative experiments, and the ingenious and penetrating speculations of Mohr, and Séguin, and Mayer, and the experimental thermodynamic measurements of Colding, all no doubt contributed to the result. Each of the several subjects of thermodynamic measurement undertaken by Joule gave him a means of estimating the quantity of work required to generate a certain quantity of heat; but after several years of trials be was led to prefer to all others the direct method of simply stirring a quantity of water by a paddle, and measuring the quantity of heat produced by a measured quantity of work; and this method he has accordingly used in all his experiments for the purpose of determining the “dynamical equivalent of heat” from the year 1845 to the present time. By this he found his final result of 1849, which was 772 Manchester foot-pounds for the quantity of work required to warm by 1° Fahr., at any temperature between 55° and 61° Fahr., 1 lb of water weighed in vacuum. In 1870 he commenced work for a fresh determination of the dynamical equivalent of heat at the request of the British Association, and the result was communicated to the Royal Society about the end of 1877, with the following preface :—

“The committee of the British Association on standards of electrical resistance having judged it desirable that a fresh determination of the mechanical equivalent of heat should be made by observing the thermal effects due to the transmission of electrical currents through resistances measured by the unit they had issued, I undertook experiments with that view, resulting in a larger figure (782 - 5, Brit. Assoc. Report, Dundee, 1867, p. 522) than that which I had obtained by the friction of fluids (772’6, Phil. Trans., 1850, p. 82).

“The only way to account for this discrepancy was to admit the existence of error either in my thermal experiments or in the unit of resistance. A committee, consisting of Sir William Thomson, Professor P. G. Tait, Professor Clerk Maxwell, Professor B. Stewart, and myself, were appointed at the meeting of the British Association in 1870; and with the funds thus placed at my disposal I was charged with the present investigation, for the purpose of giving greater accuracy to the results of the direct method.”

The result of this final investigation of Joule’s is 772 ’ 43 Manchester foot-pounds for the quantity of heat required to warm from 60° to 61° Fahr, a pound of water weighed in vacuum, which is about ∙ g ⅛th per cent, greater than the final result of 1849 expressed in the same terms. According to Regnault’s measurements of the thermal capacity of water at different temperatures from 0’ to 230° C., it must be about Ό8 per cent, greater at 60° Fahr, than at 32°. According to this, Joule’s thermodynamic result would be 771'81 Manchester foot-pounds, for the work required to warm a pound of water from 32° to 33° Fahr., or 1389 ’ 26 to warm a pound of water from 0° to 1° C. Reducing 1389 ’ 26 feet to metres, we have 423 - 437 metres. At Paris the force of gravity is about τ ⅛θ per cent, less than in Manchester. Hence for about the middle of France and the southern latitudes of Germany, Joule’s result, according to the ordinary reckoning of French and German engineers, may be stated as 423 - 5 kilogramme-metres for the amount of work required to warm 1 kilogramme of water from 0° to 1° C. The force of gravity at Manchester is 981’34 dynes (centimetres per second per second). Multiplying 423 ’ 437 by this, we find accordingly 41,553,000 centimetre-dynes, or “ergs,” for the amount of work in C.G.S. measure required to warm 1 gramme of water from 0° to 1° C.

Thermometry.

§ 10. Preliminary for Thermometry.—Sense of Heat (resumed from § 1).—The sense of heat and cold is not simply dependent on the temperature of the body touched. If a person takes a piece of iron, or a stone, or a piece of wood, or a ball of worsted, or a quantity of finely carded cotton-wool, or of eider down, in his hand, or touches an iron column, or a stone wall, or a wooden beam, or a mass of wool or of down, he will perceive the iron cold, the stone cold, but less cold than the iron, the wood but slightly cold—much less cold than the stone, the wool or down decidedly warm.

We now know that if all the bodies before being touched were near one another in similar exposure, they must have been at the same temperature, and from the iron and stone being felt oold we know that this mean temperature is lower than the temperature of the hand. Each of the bodies touched must at the first instant have taken some heat from the hand, and therefore, if the perception were quick enough, all at the very instant of being touched would have seemed cold to the sense. The iron by its high thermal conductivity (§§ 76, 78, 80) keeps drawing off heat from the hand and lowering its temperature, till after many seconds of time an approximately permanent temperature is reached, which may be considerably lower than the temperature of the hand before·; contact, but somewhat higher than the previous tem-' perature of the iron, because of the internal furnace (see Thermodynamics) generating heat in the hand. A similar result, but in less time and with less ultimate lowering of temperature of the hand, takes place when stone is touched. When wood is touched its comparatively small conductivity (§ 76) allows its surface to be warmed again after the first few seconds, sometimes to a higher temperature than that of the hand before contact; and thus, if the sensation could be perfectly remembered, it would be perceived that the wood was first felt to be cold, and afterwards to be warm. This latter warmth is rendered very perceptible by first holding the hand in contact with a piece of wood, as for instance a mahogany table, for a considerable time, half a minute or more, and then suddenly removing it; a sense of cold is immediately perceived in consequence of the exposure of the hand to the air. The foot is similarly sensitive. If, after holding a bare foot for some time in the air, it be placed on a varnished wooden floor, the floor is perceived to be cold, and if, after standing some time with it pressed to the floor, the foot be suddenly lifted, the air now seems cold by contrast. If a person walks with bare feet on a wooden floor, a continued sense of cold is experienced; and if, immediately after doing so, he sits down, and holds his feet in the air, the air seems to be warm by contrast. The same sensations are perceived even on a carpeted floor, but much less markedly than on a plain wooden floor, and much less markedly on a plain wooden floor than on a varnished wooden floor, and much less markedly on a varnished wooden floor than on a stone floor. In the case of touching soft wool, or finely carded cotton-wool, or eider down, the first instantaneous sensation of cold is scarcely if at all perceived, and that which first provokes consciousness is the subsequent heating; and it is very startling to find a body which we know to be ice-cold on a frosty day feeling positively warm to the first consciously perceptible sensation after it is touched. In this case the small thermal conductivity c? great thermal resistance of the substance is such that heat is carried off by it from the hand slower than it was carried off by radiation and aerial convection (§§ 70, 71) before contact; and thus, after the first momentary cooling of the hand by the initial cataract of heat from it to the cold body touched, in a small fraction of a second of time a higher temperature is attained by the hand than it had before contact.

§ 11. Sense of Temperature.— The sense of heat is in reality a somewhat delicate thermal test when properly used. Even an unskilled hand alternately dipped into two basins of water will, as we have found by experiment, detect a difference of temperature of less than a quarter of a degree centigrade; and there can be no doubt that bath and hospital attendants, and persons occupied with hot liquors in various manufactures, such as dyeing, can detect much smaller differences of temperature than that, and, what is still more remarkable, can remember permanently sensations of absolute temperature sufficiently to tell within less than a degree centigrade that the temperature of a bath, or a poultice, or dyeing liquor is “blood heat,” or “fever heat,” or some other definite temperature to which they have been accustomed.

§ 12. Thermometry by Sense of Heat—with arbitrary Centigrade Scale deduced from Mixtures of Hot and Cold Water.— Without knowing anything of the nature of heat we might found a complete system of thermometry on the mixing of hot and cold water with no other thermoscope (§ 13) than our sense of heat, if we had but two definite constant temperatures of reference. These in practical thermometry are supplied by the melting-point of ice and the temperature of steam from water boiling in air at a definite pressure (the “atmo” or standard atmosphere, § 5). Thus, suppose perfectly abundant supplies of iced water and of water at the boiling temperature to be available, and suppose it to be desired to measure the temperature of a river, or lake, or sea. Take measured quantities of the boiling and of the ice-cold water, and mix them by trial until, tested by the hand, the mixture is found to have the same temperature as that of the mass of water of which the temperature is to be determined. Suppose, for example, the mixture giving the required temperature to consist of 86 ’ 6 parts by weight of ice-cold water, and 13 ’ 4 parts by weight of boiling water; the required temperature is 13 ’ 4 on a perfectly definite scale of thermometry in which the temperature of ice-cold water is called zero, that of boiling water 100, and other temperatures are reckoned according to the law of proportion of mixtures of water in the manner indicated by the example, and defined generally in § 31 below. For temperatures within the range of sensibility of the hand this method would give more accurate results than many common thermometers sold by instrument makers for ordinary popular purposes. It may be relied upon for absolute accuracy within ⅛ths of a degree centigrade, provided the mixing of hot and cold water is performed with sufficiently large quantities of water, and with all proper precautions to obtain in that part of the process all the accuracy obtainable by the living thermoscope.

We shall see (§ 25) that with the most accurate mercury or air thermometers, made for scientific investigation and carefully tested, absolute determinations can scarcely be depended upon within 1 ⅛th of a degree centigrade. The method of mixtures with only the sensory thermoscope is not limited to the range of temperature directly perceptible with unimpaired sensibility; but when the temperature tc be tested is beyond this range an indirect method must be followed, as thus :—

A large quantity of water too warm for the hand is to be tested. Mix it with say twice its weight of ice-cold water, this giving a convenient temperature for the hand; then find by trial what proportions of ice-cold and boiling water give a mixture of the same temperature as tested by the hand; suppose these proportions to be 26 ’ 2 of boiling water and 73 - 8 of ice-cold water. The temperature of the mixture is by definition 26 ’ 2, and on the same principle the required temperature is three times this, or 78 ’ 6.

This system of thermometry is, however, strictly limited to the range between the. freezing and boiling pointsol water, for we do not at present consider the possibilities (see articles Steam; Matter, Properties of; Thermody namics) of obtaining and using thermometrically quantities of water below the freezing point and above the boiling point. It is described here, not only because it is very instructive in respect to the principles of thermometry, but because it is in point of fact the thermometric method used through a large range of processes not only in the arts but in scientific investigation. In many cases the hand is a more convenient and easy test than a common mercury thermo meter, and it has just about the same sensibility, the commonest thermometers in popular use being in fact scarcely to be read to a quarter of a degree centigrade. Ir respect to accuracy a common cheap thermometer, though perhaps a degree or two wrong in its absolute indications, may still be used as an accurate indicator of equality of temperatures just as is the hand in the method of mixtures.

In many cases the hand is more convenient than the thermometer, in other cases the thermometer is more convenient than the hand, but in many cases the thermometer is applicable when the hand is not. When the quantities of water tested are abundant, the hand is always the quicker test, but there must be abundance of water to allow it to be satisfactorily and accurately applicable.

Thermoscopes Differential and Intrinsic. Differential Thermoscopes essentially continuous.—Intrinsic Thermoscopes discontinuous and continuous.—Single and Multiple Intrinsic Thermoscopes {discontinuous').— Continuous Intrinsic Thermoscopes.

§ 13. A thermoscope is an indicator of temperature. A differential thermoscope is a thermoscope which shows difference, or tests equality, of simultaneous temperatures in two places. Its action is essentially continuous, depending on difference of temperature between the two places, and showing zero continuously when the temperatures of the two places are varied, provided they are kept exactly equal. Every kind of differential thermoscope, and of continuous intrinsic thermoscope, must be founded on some property of matter, continuously varying with the temperature, as density of a fluid under constant pressure, pressure of a fluid in constant volume, volume of the liquid part of a whole mass of liquid and solid kept in constant volume, steam-pressure of a solid or liquid, shape or density of an elastic solid under constant stress, stress of an elastic solid in a constant state of strain, viscosity of a fluid, electric current in a circuit of two metals with their junctions at unequal temperatures, electric resistance of a conductor, magnetic moment of a steel or loadstone magnet.

Examples :—(1) Leslie’s differential air thermometer; (2) steampressure differential thermometers 39-44 below); (3) Joule’s hydraulic and pneumatic differential thermoscopes {Memoirs Chem. Soc., vol. iii. p. 201; Broc. LU. and Phil. Soc. Manchester, vol. iii. p. 73; Ibid., vol. vii. p. 35); (4) viscosity differential thermoscope {Proceedings R.S.E., April 5,1880); (5) thermo-electric differential thermometer; (6) Siemens electric resistance differential thermometer; (7) thermo-magnetic differential thermometer (see Proceedings R.S.É. for April 5, 1880).

§ 14. Intrinsic Thermoscopes.— An intrinsic thermoscope is an instrument capable of indicating one definite temperature or several definite temperatures, or all temperatures within the range of the instrument, whatever it may be—the temperature or temperatures indicated being intrinsically determined by the constitution of the instrument and indicated by some recognizable feature of the instrument which changes discontinuously or continuously, as the case may be, and which is always the same when the instrument is brought back again and again to the same temperature, whatever changes it may have experienced in the intervals. Discontinuous intrinsic thermoscopes show only a limited number of temperatures. A continuous intrinsic thermoscope shows any temperature whatever throughout the range of efficiency of the instrument, ideally any temperature whatever, though in practice every thermoscope is limited, some with both inferior and superior limit, as the mercury thermometer by the freezing of mercury at about - 39° C., and the bursting pressure of mercury-steam a little above + 360° C.; others with only a superior limit, as metallic thermoscopes, whether thermo-elastic, or thermo-electric, or electric-resistance, or thermo-magnetic, by the melting of their substances at very high temperatures, or, in the case of the thermo-magnetic instrument, by the total or partial loss of its magnetism at some temperature much below the melting point of its substance. A continuous intrinsic thermoscope, when applied to a body whose temperature is changing, shows continuously every variation of temperature within its range of efficiency.

§ 15. Discontinuous Intrinsic Thermoscopes.— A single intrinsic thermoscope is a thermoscope which shows whether the temperature of the body to which it is applied is higher or lower than some one definito temperature depending on the intrinsic quality of the instrument.

Examples:—(1) a piece of ice, or of wax, or of fusible metal; (2) an apparatus for boiling water or other liquid under a perfectly constant pressure; (3) an apparatus for boiling water under the natural atmospheric pressure, and a barometer to measure exactly what the pressure is at the time.

A multiple intrinsic thermoscope might be made by preparing a graduated series of metallic alloys, numbering them in order of their melting points, and arranging them together conveniently for use. The temperature might be reckoned numerically, according to the number of the alloys that melt, when the whole series is exposed to the temperature to be tested. This discontinuous numerical reckoning of temperature is perfectly analogous to the Birmingham reckoning of wires and sheet metals by numbered gauges. Ideally it may be made infinitely nearly continuous by making a series of alloys with fine enough gradation of composition, but the method is in its essence discontinuous. It is useful for many special applications in science and in the arts, as for instance in that very fundamental one (§ 12) of giving one of the fixed points in the ordinary thermometric scale, the “freezing point also in a form of safety valve for boilers or hot-water pipes, in which a plug fixed by solder is released by the melting of the solder when the temperature reaches a certain limit; also an exceedingly useful guard against overheating in the flue of a stove, by which a stopper is allowed to fall by the melting of a leaden support, and stop the draught, before the temperature reaches the highest limit judged permissible.

§ 16. Continuous Intrinsic Thermoscop)es.— Continuity of indication requires, as said above (§ 13), choice of some property or properties of matter varying continuously with temperature, such as those enumerated in § 13. A continuous intrinsic thermoscope must have a feature, depending on the chosen property of matter, which shall vary with perfect continuity when the temperature is gradually changed, and shall always be the same when the instrument is brought to the same temperature again and again, whatever variation of temperature it may have experienced in the intervals. The accuracy of an intrinsic thermometer, whether discontinuous or continuous, depends upon permanence of quality of the material and of the mechanical constitution of the instrument, according to which the recognized feature shall always be very accurately the same for the same temperature. The sensibility or delicacy of a continuous intrinsic thermometer depends upon the recognizability of change in its indicating feature with very small change of temperature.

§ 17. The property of matter chosen as the foundation of almost all ordinary continuous intrinsic thermoscopes in common use is interdependence of the density, the temperature, and the pressure of a fluid. The only other thermc-scopes which can be said to be in common use at all are “metallic thermometers” (see Thermometer; Matter, Properties of); these depend upon the change of shape of a rigid elastic solid under a stated stress, or on the change of shape of a compound solid, composed of two elastic solids of different substances melted or soldered together. For the present we confine our attention to the former and much larger class of instruments. The general type of all those instruments, except the steam-pressure thermometer (§§ 39-46 below), is a glass measure, measuring the bulk of a fluid. To give the requisite practical sensibility to the measurement, the glass, except for the case of the constant-pressure gas thermometer (§§ 64-67 below) and of the steam-pressure thermometer, is made of a shape which may be generally described as a bottle with a long narrow neck. The body of the bottle, which may either be spherical or of an elongated form, is called the bulb, and the neck is called the tube or stem (stem we shall most frequently call it, to obviate ambiguities without circumlocutions). The thermometric fluid may be all liquid, as mercury, or oil, or alcohol, or ether, or glycerine and water; or it may be all gas, as common air, or hydrogen, or carbonic acid; or it may be partly liquid and partly steam (steam being a name which we shall invariably use to designate the less dense portion of a fluid substance at one temperature and pressure throughout, and in equilibrium, with two parts of different densities). This last case is different from the two preceding, in respect to the character of the thermometric indication: the whole volume of the thermometric substance may be changed from that of all liquid to that of all steam without changing the temperature or the pressure, and the pressure cannot be changed without changing the temperature, provided the substance is kept in the double condition of part liquid and part steam; in other words, in this case the pressure depends upon the temperature alone and is independent of the volume. In the steam-pressure thermometer, therefore, there is no delicate measuring of volume of the thermometric substance, and the vessel containing it is not in the shape of bulb and stem; but the instrument consists essentially of a means of measuring the pressure of the thermometric substance, with a test that it is really in the twofold condition of part liquid and part steam, whether by seeing it through a glass containing-vessel, or by a proper hydraulic appliance for ascertaining that the pressure is not altered by rarefaction or condensation when the temperature is kept constant. Realized thermometers of this species, quite convenient for many practical purposes, with steam of sulphurous acid, of water, and of mercury, to serve for different ranges of temperature, from below - 30° C. to above + 520° C., are described in §§ 39-44 below.

§ 18. In respect to general convenience for large varieties of uses, whether for scientific investigation, or for the arts, or for ordinary life, liquid thermometers are generally and with good reason preferred; but the general preference of either mercury or spirits of wine for the liquid, which is so much the rule, is not (§20 below) so clearly reasonable. For ordinary uses in which the thermometer has to be moved about and placed in various positions, gas thermometers are much less convenient, because they require essentially an accurate measurement of pressqre, and generally for this purpose a column of liquid. But when the thermometer is to be kept always in one position, as for instance when it is devoted to testing the temperature of the air indoors or out of doors, Amonton’s air or gas thermometer is really as convenient and as easily read as any liquid thermometer can be: but even it, simple as it is, involves a triple division of the hermetically sealed space, with three different conditions of occupation, —one part occupied by the thermometric substance, another by the pressure-measuring vapourless liquid, and the third vacuous: and it is by so much the less simple than the liquid thermometer that in the liquid thermometer the enclosed space is divided into only two parts, one occupied by the thermometric liquid, and the other by its steam, with or without some admixture of common air. For accuracy the air or gas thermometer is superior, we might almost say incomparably superior, to the mercury thermometer, and, though in a much less degree, still decidedly superior to even the most accurate liquid thermometer, on account of the imperfect constancy of the glass containing-vessel.

§ 19. If we were quite sure of the bulk measurement given by the glass bulb and tube, liquid thermometers would be quite as accurate as gas thermometers. For there is no difficulty in giving any required degree of sensibility to the instrument by making the bulb large enough; and the quality of the liquid itself, hermetically sealed in glass, may be regarded as being as constant as anything we know of in the material world. But, alas for thermometry, the glass measure is not constant! In fact, glass is a substance of very imperfect elasticity (Elasticity, § 4, vol. vii. p. 797); and it is found that the bulb of a thermometer is not always of the same volume at the same temperature, but that, on the contrary, it experiences uncertain changes exceedingly embarrassing in thermometry. In the course of a few months after a thermometer is filled and sealed, the bulb generally shrinks by some uncertain amount of from t° τ ΰύ^ -ö °f its bulk, sometimes even in the course of years to almost τo 1 σo . This ha3 been discovered by a gradual rising of the freezing point, in new mercury thermometers, generally as much as from ∣ 0 to ∣ o C., sometimes to as much as 1°, which corresponds to a shrinkage of yo'σoiι> as the bulk-expansion of mercury is when its temperature is raised from 0°to 100° C. (Table II. below) ∙g⅜. τ , or Ό1815, of its bulk at 0°. After a few months or a few years this progressive shrinkage ceases to be sensible; but if the thermometer at any time is exposed to the temperature of boiling water or any higher temperature, an abrupt sub-permanent enlargement of the bulb is produced, and the freezing point, if tested for by placing the thermometer in ice and water, is found to be lowered; then again for weeks or months or years there is a gradual shrinkage, as shown by a gradual rising of the freezing point when the thermometer is tested again and again by placing it in ice and water. A very delicate mercury thermometer, which has been kept for years at ordinary atmospheric pressures when out of use, and never when in experimental use exposed to any temperature higher than about 30° C., or much lower than the freezing point, becomes very constant, and probably may never show any change of as much as τ ⅛ of a degree C. in its freezing point or in its indication at any other absolutely definite temperature, within some such range as from - 20° or — 10° C. to + 30° or + 40° C. But the abrupt and irregular changes, produced by exposing the thermometer to temperatures much above or much below some such limited range as that, constitute a very serious difficulty in the way of accurate thermometry by the mercury-in-glass thermometer.

Although the greatest care has been bestowed by Régnault, Joule, and all other accurate thermometric-experimenters to avoid error from this cause, we have still but little definite information as to its natural history in thermometers of different qualities of glass, different shapes of bulb, and constructed differently in respect to processes of glass-blowing, boiling the mercury, and sealing the stem. We do not even know whether the excess of the atmospheric pressure outside the bulb over the pressure due to mercury and Torricellian vacuum inside is influential sensibly, or to any considerable degree, in producing the gradual initial shrinkage. If it were so we might expect that the effect of heating the thermometer up to 100° or more at any time would be rather to produce an accelerated shrinkage for the time than what it is found to be, which is a return towards the original larger volume, followed by gradual shrinkage from day to day and week to week afterwards, A careful comparison between two thermometers constructed similarly in all respects, except sealing one of them with Torricellian vacuum and the other with air above the mercury, would be an important contribution to knowledge of this subject, interesting, not only in respect to thermometry, but also to that very fundamental question of physical science, the imperfect elasticity· of solids (see Elasticity, § 4).

§ 20. The error of a thermometer due to irregular shrinkages and enlargements of the bulb is clearly the less the greater is the expansion of the thermometric fluid with the given change of temperature. By the investigation of § 30 we can calculate exactly how much the error is for any stated amount of abnormal change of bulk in the bulb. But it is enough at present to remark that for different liquids in the same or in similar bulbs the errors are very nearly in the inverse proportions of the expansions of the liquids. Now (Table III. below) in being warmed from 0° to 1° C. alcohol expands 6 times as much as mercury, methyl butyrate 7 times, and sulphuric ether 8⅜ times. Hence if irregular changes of bulk of the bulb leave, as they probably do in practice, an uncertainty of ⅛ths of a degree in respect to absolute temperature by the best possible mercury-in-glass thermometers used freely at all temperatures from the lowest up to 100° C., the uncertainty from this cause will be reduced to τi ⅛th of a degree by using alcohol, or -⅛th by using methyl butyrate instead of mercury; it may therefore, in a glass thermometer of alcohol or of the butyrate, be considered as practically annulled (§ 19) after a few weeks or months have passed, and the first main shrinkage is over.

§ 21. An alcohol-in-glass thermometer is easily made strong enough to bear a temperature of 100° C., as this gives by the pressure of the vapour an internal bursting pressure of not quite an atmosphere and a quarter in excess of the atmospheric pressure outside. The boiling point of methyl butyrate (Table III. below) is 102° C.; a thermometer of it may therefore be used for temperatures considerably above 100° C., but how much above we cannot tell, as we have not experiments as yet on the pressure of its vapour at temperatures above its boiling point. The pressure of vapour of sulphuric ether (Table VI. below) is too great to allow a thermometer of this liquid sealed in glas3 to be used much above - 60° or 70° C., but for low temperatures it makes a very valuable thermometer. It was used in 1850 by W. Thomson in finding by experiment the lowering of the freezing point of water, predicted theoretically (see Thermodynamics) by J. Thomson in 1849 (7Vαns. P. S. Ef, and gave a sensibility of 128 divisions to 1° C. Glass thermometers with ether, or chloroform (whose expansion is about 4 per cent, greater than that of ether), were used by Joule and W. Thomson in experiments on changes of temperature experienced by bodies moving through air, in one of which the sensibility was as great as 330 scale divisions to the 1° C. All these liquids and many others in the modern chemist’s repertory of oils and ethers and alcohols, besides the superior sensibility which they give by their greater expansions, have a great advantage over mercury for some thermometric purposes in their smaller specific gravity. This allows the bulb to be larger, with less liability to break or to give disturbed readings through distortion by the weight of the contained liquid.

§ 22. Liquids which wet the glass have another great advantage over mercury in their smaller capillary attraction (see Capillary Action, vol. v. p. 56; and Matter, Properties of) and in the constancy of their 180° angle of contact with the glass, instead of the much greater absolute intensity of capillary attraction in the mercury, and its very variable angle of contact, averaging about 45° when the mercury is rising, and about as much as 90° when it is falling. On account of these variations the bulb of the mercury thermometer is subjected to abrupt variations of pressure when the mercury is rising or falling. The greatest and least pressures due to this cause are experienced when the angle of contact is respectively least and greatest, and differ by the pressure due to a vertical column of mercury equal in height to the difference of depressions of mercury in a capillary tube of the same bore as the thermometer stem when the angle of contact is changing from one to the other of the supposed extreme values. Hence the mercury in a thermometer rises and falls by jerks very noticeable in a delicate thermometer when looked at with a lens of moderate magnifying power, or even with the naked eye. Dr Joule informs us that this defect is much greater in some thermometers than in others, and that he believes it is greatly owing to the tube being left unsealed for too long a time after the introduction of the mercury (by which it is to be presumed something of a film of oxide of mercury is left on the glass to reappear on the surface of the mercury when it sinks as it cools after the sealing of the end). In Joule’s own thermometers not the smallest indication has ever been detected of what he calls “this untoward phenomenon, which is calculated to drive an observer mad, if he discovers it towards the close of a series of careful experiments.” Their admirable quality in this respect is no doubt due to the great care taken by the maker, Mr Dancer, under Joule’s own instructions, to have the mercury and the interior of the bulb and tube thoroughly clean, and to guard it from exposure to any “matter in its wrong place” until completion of the sealing. But no amount of care could possibly produce a mercury thermometer of moderate dimensions moving otherwise than by jerks of ever so many divisions, if its stem were of fine enough bore to give anything approaching to two or three hundred divisions to the centigrade degree.

§ 23. One chief objection to the use of alcohol or other volatile liquid for the thermometric substance in ordinary glass thermometers is the liability to distillation of some of the liquid into the stem and head reservoir, unless the glass above the level of the liquid be kept at least as warm as the liquid. On this account a spirit thermometer is not suitable for being plunged into a space w armer than the surrounding atmosphere with the stem simply left to take the temperature to which it comes in the circumstances. But whether for elaborate experimental use, or for the most ordinary thermometric purposes, there is little difficulty in arranging to keep the part of the stem which is above the liquid surface somewhat warmer than at the liquid surface, and this suffices absolutely to prevent the evil of distillation. The only other objection of any grave validity

against the use of highly expansive liquids instead of mercury is the difficulty of allowing for the expansion of the liquid in the stem, if it is not at the same temperature as the bulb. With the same difference of temperatures in different parts of the instrument, the error on this account is clearly in simple proportion to the expansibility of the liquid; and therefore, the residual error due to want of perfect accuracy in the data for the allowance will, generally speaking, be greater with the more expansive than with the less expansive liquid. But in every case in which the bulb and stem can all conveniently be kept at one temperature, a thermometer having for its thermometric substance some highly expansive oil or alcohol or ether, or other so-called organic liquid of perfectly permanent chemical constitution, cannot but be much more accurate and sensitive than the mercury thermometer, which has hitherto been used almost exclusively in thermometric work of the highest rank. We shall see (§§ 62 and 64-68) that the ultimate standard for thermometry, according to the absolute thermodynamic scale (§ 34) r is practically attained by the use of hydrogen or nitrogen gas as the thermometric substance, but that for ordinary use a gas thermometer can scarcely be made as convenient as one in which the thermometric substance is a liquid. For practical thermometry of the most accurate kind it seems that the best plan will be to use as ordinary working standard thermometers highly sensitive thermometers constructed of some chosen “organic” liquid, and graduated according to the absolute thermodynamic scale, by aid of the thermodynamically corrected air thermometer (§ 62) used as ultimate standard of reference. The great convenience of the mercury thermometer in respect to freedom from liability to distillation and smallness of error on account of difference of temperature between the bulb and stem renders it the most convenient for a large variety of scientific and practical purposes in which the most minute accuracy or the most extreme sensibility is not required.

§ 24. Without any thermodynamic reason for preferring air to mercury as thermometric fluid, Regnault preferred it for two very good reasons. (1) Its expansion is 20 times that of mercury and 160 times the cubical expansion of glass, and therefore with air the error due to irregularity in the expansion of the glass is 20 times smaller than with mercury, and small enough to produce no practical defalcation from absolute accuracy in thermometry, as he found by elaborate and varied trials. So far as this is concerned, some highly expansive organic liquids would answer nearly as well as air for thermometric fluid, and would have the advantage of giving a thermometer much more easily used. (2) For an ultimate standard of reference air has the advantage over organic liquids generally, that different samples of it taken at different times, or in different parts of the world, and purified of water and carbonic acid by well-known and easily practised processes, are sufficiently uniform to give thermometric results between which the accordance is practically perfect, provided the thermometric plan according to which the different samples are used is the same, or as approximately the same as is easily secured in practice. Two plans for the thermometric use of air naturally present themselves :—(I.) augmentation of volume of air kept in constant pressure; and (II.) augmentation of pressure of air kept in constant volume. Regnault tried both plans, but found that he could only arrange his apparatus to give good results by the second, and on it therefore he founded what he called his “normal air thermometer.” For the sake of perfect definiteness he chose, as the densityof the air in his normal thermometer, the densitywhich air has when at the temperature of melting ice and under the pressure of one atmo. He adopted the centigrade scale in respect to the marking of the freezing and boiling points by 0° and 100°; and the principle which he assumed for the reckoning of other temperatures was to call equal those differences of temperature for which differences of pressure of the air in his normal thermometer are equal. Thus he was led to a definition of temperature expressed by the following formula :—

t= ιoo y ~ π - , (1),

Pιoo~∏

where ∏ denotes the pressure one atmo, and p> and p 100 the pressures of the air of the normal thermometer at the temperatures denoted by t and by 100 respectively, the latter being the temperature of steam issuing from water boiling under the pressure of one atmo. By the most accurate observations which he could make Regnault found for his “normal air” j p 100 = 1’3665 × ∏. Hence his thermometric formula becomes i=100-J⅛5- =272-85/^-1) .... (2).

•3665 ×∏ ∖∏ ∕

§ 25. Regnault compared with his normal air thermometer thermometers on the same plan of constant volume, but with air at other than the normal density of 1 atmo, and with other gases than air; also air and gas thermometers on the plan of constant apparent volume as measured in a glass bulb and stem; also a thermometer founded simply on the dilatation of mercury; also thermometers of mercury in different kinds of glass, each graduated on the glass stem with divisions corresponding to exactly equal volumes of the bore; also overflowing thermometers (thermomètres à déversement), in which a bulb with a short piece of fine stem was perfectly filled with mercury at 0° and the quantity of mercury expelled by the high temperature to be measured was weighed, instead of being volumetrically measured by divisions of a long stem as in the ordinary thermometer.

The whole of this thermometric investigation is full of scientific interest, and abounds with results of great practical value in respect even of the minutest details of Regnault’s work. It will be found fully described in the first of his three volumes, entitled Delation des Experiences entreprises par ordre de Monsieur le Ministre des Travaux Publics et sur la proposition de la Commission centrale des Machines à Vapeur pour Déterminer les Principales Lois et Données Numériques qui entrent dans le Calcul des Machines a Vapeur, which were published at Paris in 1847, 1862, and 1870. Here we can but state some of t·he most important of the general conclusions:—

(1) The air thermometers with pressure at 0° of from 44 to 149 centimetres of mercury agreed perfectly with the normal air thermometer calculated according to the same formula (2), and nearly the same numerical coefficient 272 - 85. A slightly larger value 272 - 98 (or ’ 0036632~ 1 ) gave the best agreement for the 44 c.m. pressure, and the somewhat smaller value 272,*7 (or O03667 -1 ) for the pressure 149 c.m.

(2) The hydrogen gas thermometer, with pressure one atmo at 0°, and with its indications calculated according tc formula (2) but with a different numerical coefficient, agreed perfectly with the normal air thermometer from 0° to 325°.

(3) The carbonic acid gas thermometer with pressure 46 c.m. at 0°, and its indications calculated with the coefficient 27l ’ 59 (O03682~ 1 ), agreed perfectly with the normal air thermometer from 0° to 308°.

(4) The carbonic acid gas thermometer with pressure, at 0°, 74 centimetres (or nearly 1 atmo), calculated with the coefficient 270'64 (O03695~ 1 ) to make it agree with the normal air thermometer at 100°, gave numbers somewhat too large for all temperatures from 200° to 323°. The difference seemed to rise to a maximum at about 180°, when it was about ⅜°, and to diminish so as to be only about τ ⅛° at the highest temperatures of the comparison. Two sulphurous acid gas thermometers, with pressures 59 c.m. and 75 c.m. at 0°, calculated with coefficients 263 , 6 (- 003794 -1 ) and 261’4 (O03825 -1 ) respectively to make them agree at 100° with the normal air thermometer, each gave numbers too small for the higher temperatures by differences increasing gradually from ∣° at 140° to 3° at 320°.

(5) Air and gas thermometers calculated according to differences of pressure of the gas kept at the same apparent volume (that is to say, with the bounding mercury column at a constant mark on the glass stem of the thermometer) give numbers too small at the higher temperatures by differences gradually increasing up to 2∣° at 350° in the case of Choisi le Roi crystal, a hard glass without lead, and to as much as 3∣° in the case of ordinary glass.

In connexion with these observations Regnault remarks that the greatest cause of uncertainty in his air thermometry is the allowance for expansion of the glass. It was only by most carefully made special experiments on each particular bulb and tube, to determine its expansion throughout the range for which it was to be used, that he succeeded in obtaining the great accuracy which we find in his results, according to which the probable error, whether by his normal air thermometer, or by other air or gas thermometers of those stated above to agree with it perfectly, was not more than from ’ 1 to ’ 15 of a degree for any temperature up to 350°.

(6) The mercury-in-glass thermometers which Regnault generally used for comparison with his normal air thermometer were overflowing thermometers, because he found that with such he could more easily obtain the very minute accuracy at which he aimed than with the ordinary volumetric thermometers; but the formula by which he calculated temperature from the overflowing thermometer was adapted to give exactly the same result as would have been obtained by the ordinary thermometer with divisions on the stem corresponding to equal volumes of the bore. It must be remembered, however, that this perfect agreement between the volumetric and overflowing thermometers would not be found unless the expansion of the bulb and tube were uniform and isotropic throughout.

(7) The general results of Regnault’s comparisons of mercury thermometers with his normal air thermometer were given by himself in a diagram of curves from which the accompanying is copied on a reduced scale (fig. 3). It shows that at a temperature of 320° the independent mercury thermometer stands at 329 - 8°, the thermometer of mercury in Choisi le Roi crystal at 327 ’ 25°, and the thermometer of mercury in ordinary glass at 321’8°; and that the independent mercury thermometer and the mercury in Choisi le Roi crystal stand 10° higher than the normal air thermometer at the temperatures by it of 323° and 345° respectively.

§ 26. The curve for the independent mercury thermometer is merely Regnault’s graphic representation of his experiments on the absolute expansion of mercury (Relation des Experiences, vol. i. p. 328). It shows that the addition of bulk given to the same mass of mercury under constant pressure by elevation of temperature iβ for the same difference of temperatures as indicated by his normal air thermometer regularly greater and greater the higher the temperature.

§ 27. It is interesting to see by the diagram that at the high temperatures all the mercury thermometers keep nearer to the air thermometer than does the independent mercury thermometer, and that tha mercury in ordinary soft glass keeps much nearer to the air thermometer than does the mercury in the hard Choisi le Roi glass. We infer that, still reckoning temperature by the air thermometer, we have regular augmentation of expansion at the high temperatures in all the different glasses, each greater than the augmentation of expansion of mercury, and that this augmentation is greater in the soft ordinary glass than in the hard Choisi le Roi glass, being in the ordinary glass great enough to overcompensate in the resulting thermometric indication the augmenting expansion of the mercury from 100° to 245°; while above 245° in the ordinary glass thermometer, and at all temperatures above 100° in the Choisi le Roi thermometer, the compensation is only partial. Between 0° and 100° the independent mercury thermometer stands regularly lower than the air thermometer by as great a difference as ’ 35° at 50°, where it is a maximum. The curves for the mercury-in-glass thermometers are not shown between 0° and 100°, but it is clear from the diagram that the Choisi le Roi thermometer must, like the independent mercury thermometer, stand lower than the air thermometer, but by a smaller difference, probably only about ’ 2° at 50°; and the ordinary glass thermometer higher than the air thermometer from 0° to 100° by a difference which may be ’ 2° or ∙3° at 50°. This last inference from the diagram is confirmed by Regnault’s table of results facing page 227 of his first volume.

§ 28. In the best modern thermometers the graduations are actually engraved on the glass; but in most popular thermometers, and in many for scientific investigation, they are on an attached scale of wood, or ivory, or brass, or paper. Some of the best popular thermometers are the German bath thermometers, in which the graduation is on a paper scale guarded by being enclosed in a wide glass tube hermetically sealed round the stem and over the bulb of the glass which contains the mercury, in the manner sho.wn in fig. 4.

The graduation is clearer and more easily read in this kind of thermometer than in any other. The complete protection of the paper scale against damp and damage afforded by its hermetically sealed glass envelope gives a perennially enduring quality to this form of thermometer, such as is possessed by no others except those graduated on the glass; and the lightness of the paper renders its proper attachment to the inner stem, by gum or otherwise, thoroughly trustworthy, when once.well done by the maker of the instrument. For scientific purposes the paper scale was too cheap, and common, and good, to satisfy the ideas of those instrument makers who in Germany and France substituted the heavy graduated slab of opal glass for the paper, while still adhering to the bath thermometer pattern in hermetically enclosing this scale in an outer containing glass tube,—very unnecessarily, as the glass scale, unlike the paper scale, does not require any such protection.

This is now, however, a thing of the past. At the present time all high-class scientific thermometers are graduated on the glass of the stem without any attached scale of other material. Except in respect to ease of reading the indications this simplest form is, both for popular and for scientific purposes, superior even to the German bath thermometer with hermetically sealed paper scale; and this will be the form intended when we speak of a mercury thermometer, or a spirit thermometer, or a liquid thermometer, without any special qualification.

§ 29. Properties of Matter concerned in Liquid Thermometers.— The indications of the liquid thermometer depend not only upon the expansion of the liquid with heat; they are seriously modified by the expansion experienced also by the containing solid. The instrument in fact consists of a glass measure measuring the bulk of a liquid. If the bulk of the hollow space in the glass and the bulk of the liquid expand by the same amount, the apparent bulk of the liquid as thus measured will remain unchanged. Now, supposing the glass to be perfectly homogeneous and isotropic (see art. Elasticity, §§ 38, 39, and chap. i. of Mathematical Theory), and the bulb to be free from internal strain, the glass will, when warmed uniformly, expand equally in all directions, and the volume of the hollow space will be altered in the same ratio as the volume of the glass itself. Hence the indications of the thermometer depend on a difference between the expansion of the glas3 anl the expansion of the liquid.

§ 30. To define exactly the indications of a thermometer founded on the expansion of a fluid, let the volume of the bore of the stem between two consecutive divisions be called for brevity a degree measure. The degree measure is habitually made as nearly as possible equal throughout the scale in the best mercury-in-glass thermometers; and, as we shall see (§ 62), it ought to be so in an air thermometer to give indications agreeing with the absolute thermodynamic scale nearly enough for the most accurate practical thermometry. But in practical spirit-thermometers the divisions are made to correspond as nearly as may be to degrees of a standard mercury or air thermometer, and the degree measures are therefore (Table II. below) larger and larger from the lower to the upper end of the scale. For the purpose, however, of comparing the thermometric performances of different liquids, we shall suppose the degree measure to be of equal volume throughout the scale in each case.

Let N be the number of degree measures contained in the volume of the bulb and stem up to the point marked zero on the scale; and let Dt denote the volume, at any temperature t, of the degree measure reckoned in absolute units of volume. The volume of the bulb and stem up to zero will be ND i . On the supposition of perfect isotropy and freedom from strain in the glass, N will be independent of the temperature and D i ∕D 0 will be the ratio of the volume of any portion of the glass at temperature t to its volume at the temperature called zero, if D o denote the volume of the degree measure when the glass is at this zero temperature. Let now L t and L o denote the volumes of the whole liquid in a thermometer at the two temperatures t and 0; we have L 0 = ND 0 . And if s be the number of scale divisions marking the place of the liquid surface in the thermometer tube, we have L { ·=(N + s)D ( . Hence L i ∕L 0 = (l+s∕N)D i ∕D 0 . Hence s=N (jξyχy - 1) . Hence, if Ei denote augmentation of bulk of the liquid, and E t augmentation of bulk of each degree-division of the stem, when temperature is raised from 0 to t, each reckoned in terms of the bulk at zero temperature, we have 5 =n(L±⅞-1V

U+e; ) " ι+E ’ i

This is the formula for the ordinary liquid thermometer. It is also applicable to the constant pressure air thermometer, in which, with proper instrumental means to keep the pressure constant, air is allowed to expand or contract with elevation or depression of temperature, and its volume is measured in a properly shaped glass measuring vessel. We may arbitrarily determine to take s as the numeric for the temperature which is indicated by any one particular thermometer of this kind, for instance, a methyl butyrate thermometer, or an alcohol thermometer, or a mercury or an air thermometer. But if s = i for any one individual thermometer, it cannot be exactly so for any other. In the first advances towards accurate thermometry it was taken so for the mercury-in-glass thermometer, and by general consent it was continued so until it was found (§ 25) that different mercury-in-glass thermometers, each made with absolute accuracy, differ largely in their reckonings of temperature.

§31. Numerical Thermometry.— In § 12 above, a perfectly definite and very simple basis for numerical thermometry was described, not as having been adopted in practice, but as an illustration of a very general principle upon which reckoning of temperature may be done in numbers. The principle is this. Two definite temperatures depending on properties of some particular substance or substances are first fixed upon and marked by two arbitrary numbers,— as, for instance, the temperature of melting ice marked zero, and the temperature of steam issuing from boiling water under atmospheric pressure of exactly one atmo, marked 100. Then any intermediate temperature t is obtained by taking t parts of water at 100° and (100 - t) parts at 0° and mixing them together. As said in § 12 this method is limited to temperatures at which liquid water can be obtained, and therefore practically it is only applicable between the melting point of ice and the boiling point of wateι i , under ordinary atmospheric pressure.

§ 32. Any other liquid of permanent chemical constitution might be used instead of water as the thermometric substance in thermometry founded on mixtures; so even might a powdered solid. Oil if used instead of water would have the advantage of being available for higher temperatures; but want of perfect definiteness and constancy of chemical constitution is a fatal disqualification for it as the fundamental thermometric substance for thermometry by mixtures. Liquid mercury might be used with the advantage of being available for both higher and lower temperatures than water, through a much wider whole range indeed than either water or oil. For use as thermometric substance for the method of mixtures both water and mercury, in the conditions of approximate purity in which they are easily obtained in abundance, have a paramount advantage over all other liquids, in near enough approximation to perfect definiteness and constancy of constitution to give practically perfect thermometric results.

§ 33. In §§ 12, 24, 25, 30, 31, 32, several distinct definitions of numerical reckoning of temperature have been given. In each of these the differences of temperature which are to be called equal are defined specially, and this is the essence of the thermometric scale in each case (the marking of 0° and 100° for the “freezing” and “boiling” points being common to all as a matter of practical usage and not an essential of the thermometric principle in any case). Thus in §§ 12, 31, and 32 differences of temperature are called equal which are produced by the communication of equal quantities of heat to a given quantity of the particular thermometric substance chosen—water, for example, or mercury; in other words (§ 68 below), this thermometric system is chosen so as to make the specific heat of a particular thermometric substance the same for all temperatures. Again in § 24 differences of temperature are called equal for which the differences of pressure are equal in air of the particular density which air has if its pressure is one atmo when its temperature is “freezing.” This is Regnault’s “normal thermometry.” In § 25 (1), (2), (3), (4), other reckonings of temperature differing essentially from this, though, as Regnault’s experiments proved, by but very small differences, are given simply by the substitution of air of other than Regnault’s normal density, and of other gases than air, for the air of Regnault’s normal thermometer. In § 25 (5) a thermometry founded on a complex coefficiency of change of pressure and volume of a gas and change of volume of some one particular glass vessel is defined and compared with Regnault’s normal thermometry; and in § 25 (6) and (7) the same is done for the ordinary mercury-in-glass thermometer, which depends on a coefficiency of glass and mercury leading to the reckoning of temperature defined in § 30. Again in § 26 and § 24 (I.) is indicated a system of thermometry founded on the absolute dilatation of some fluid, such as mercury or alcohol or butyrate of oxide of methyl or other permanent liquid or air, at some constant .pressure, such as one atmo, with equal differences of temperature defined as those which give equal dilatations of the particular substance chosen as the thermometric fluid. .

§ 34. Each of all these different definitions of temperature is founded on some particular property of a particular substance. A thermometer graduated to fulfil one of the definitions for one particular substance would not agree with another thermometer graduated according to the same definition for another substance, or according to some of the other definitions. A much more satisfactory foundation for thermometry is afforded by thermodynamic science, which (see Thermodynamics)' gives us a definition of temperature depending on certain thermodynamic properties of matter in such a manner that if a thermometer is graduated according to it from observation of one class of thermal effects in one particular substance, it will agree with a thermometer graduated according to the same thermodynamic law from the same class of effects in any other substance, or from the same or from some other class of effects in another substance. Thus we have what is called the absolute thermodynamic scale. This scale is now in modern thermal science, the ultimate scale of reference for all thermometers of whatever kind (§ 67). It is defined in §§ 35 and 37 after the following preliminary. A piece of matter which we shall call the “thermometric body” or “thermometric substance” must be given, and at each instant it must be throughout at one temperature, whatever operations we perform upon it. For simplicity we shall suppose it to be of one substance throughout. It may be all solid, or it may be partly solid and the remainder gaseous (as the contents of a wholly frozen cryophorus or any other form of closed vessel full of ice and vapour of water, but with no ah’); or it may at one particular temperature in the course of its use be partly solid and partly liquid and partly gaseous (as the contents of a partially frozen cryophorus); or it may be partially liquid and partially gaseous (⅛s the contents of an unfrozen cryophorus or of a “philosopher’s hammer”); or it may be all liquid; or it may be all gas; or it may be all fluid at a temperature above the Andrews “critical temperature.” If it be all solid it may be under any homogeneous stress (Elasticity, Mathematical Theory, part i., chap, i.); but in any case we suppose for simplicity the stress to be homogeneous throughout, and therefore if the thermometric body be partly solid and partly fluid, the stress in the solid as well as in the fluid must be uniform pressure in all directions. To avoid excluding the case of all solid from our statements, we shall use generally the word stress, which will mean normal pressure reckoned in number of units of force per unit of area in every case in which the whole or any part of the thermometric body is fluid, and will denote this or any other possible stress when the thermometric body is all solid.

§ 35. (1) Alter the bulk or shape of the thermometric substance till it becomes warmer to any desired degree.

(2) Keeping it now at this higher temperature, alter bulk or shape farther, and generate the heat which the substance takes to keep its temperature constant, by stirring water, or a portion of the substance itself, if it is partly fluid, and measure the quantity of work spent in this stirring.

(3) Bring it back towards its original bulk and shape till it becomes cooled to its original temperature. (4) Keeping it at this temperature, reduce it to its original bulk and shape, carrying off, by a large quantity of water, the heat which it must part with to prevent it from becoming warmed. Find by a special experiment how much work must be done to give an equal amount of heat to an equal amount of water by stirring. Then the ratio of the first measured quantity of work to the second is the ratio of the higher temperature to the lower on the absolute thermodynamic scale.

§ 36. The following is equivalent to § 35, and is more convenient for analytical use. It is derived from § 35 by supposing the first and third operations to be so small that the ratio defined as the ratio of the two temperatures is infinitely nearly unity, and conversely § 35—our first form of definition of absolute temperature—may be derived from the second, which is to be now given, by passing through a finite range of temperature by successive infinitesimal steps, and applying the second definition to each step.

§ 37. Let the thermometric body be infinitesimally warmed by stirring a portion or the whole of itself if it be partially or wholly fluid, or by stirring a quantity of fluid in space around it if it be all solid; and during the process let the stress upon the body be kept unchanged. The body expanding or contracting or changing its shape with the heat, as the case may be, does work upon the surrounding material by which its stress is maintained.

Find the ratio of the amount of work thus done to the amount of work spent in the stirring. For brevity we shall call this the work-ratio. Again, let the stress be infinitesimally increased, the thermometric body being now for the time enclosed in an impermeable envelope so that it may neither gain nor lose caloric. It will rise (or fall) in temperature in virtue of the augmentation of stress. The ratio of this infinitesimal elevation of temperature to the whole absolute temperature is equal to the work-ratio multiplied into the ratio of the infinitesimal augmentation of stress to the whole stress.

§ 38. To show how’ our definition of absolute temperature is to be applied in practice take the following examples. Example 1.—Any case in which·the thermometric substance is part in one condition and the remainder in another of different density, as part solid and part vapour, or part solid and part liquid, or part liquid and part steam. In this last case, as explained above (§ 34), we suppose the stress to be uniform pressure in all directions.

Let/? be its amount, and let t be the absolute temperature corresponding to this pressure. Let σ be the ratio of the density of the rarer to that of the denser portion, ρ the density of the rarer portion, and J κ the quantity of work required to generate the heat taken to convert unit mass of the substance from the lower to the higher condition (∕c the “latent heat” of transition from the lower to the higher condition per unit mass of the substance, and J the dynamical equivalent of the thermal unit in which κ is measured). The work done by the substance in passing from the denser to the rarer condition per unit volume of the latter is - σ), and the amount of work required to generate the heat taken in doing so is Jpκ. Hence the work-ratio of our second definition is

(1). Jp/c

Let now the pressure be increased by an infinitely small quantity dp, and, the substance being still in the two conditions but of uniform temperature throughout, let dt be the corresponding rise in temperature. We have by the definition (§ 37)

dt = p(I - <r) dp = 1 - t J pK p iρK

Hence = (2).

i dp J P κ

Hence by integration

∕2> Ί

( 1 ,-⅜ I

J OK

p i- (3).

∕^ (l-<r)⅞

or i 5 P κ

§ 39. Fig. 6 represents a thermometer constructed to show absolute temperature on the plan of example 1, § 38, realized for the case of water and vapour of water as thermometric substance. The containing vessel consists of a tube with cylindric bulb like an ordinary thermometer; but, unlike an ordinary thermometer, the tube is bent in the manner shown in the drawing. The tube may be of from 1 to 2 or 3 millims. bore, and the cylindrical part of the bulb of about ten times as much. The length of the cylindrical part of the bulb may be rather more than τ ⅜θ∙ of the length of the straight part of the tube. The contents, water and vapour of water, are to be put in and the glass hermetically sealed to enclose them, with the utmost precautions to obtain pure water as thoroughly freed from air as possible, after better than the best manner of instrument makers in making cryophoruses and water hammers. The quantity of water left in at the sealing must be enough to fill the cylindrical part of the bulb and the horizontal branch of the tube. When in use the straight part of the tube must be vertical with its closed end up, and the part of it occupied by the manometric water-column must be kept at a nearly enough definite temperature by a sur rounding glass jacket-tube of iced water. This glass jacket-tube is wide enough to allow little lumps of ice to be dropped into it from its upper end, which is open. By aid of an india-rubber tube connected with its lower end, and a little movable cistern, as shown in the drawing, the level of the water in the jacket is kept from a few inches above to a quarter of an inch below that of the interior manometric column. Thus, by dropping in lumps of ice so as always to keep some unmelted ice floating in the water of the jacket, it is easy to keep the temperature of the top of the manometric watercolumn exactly at the freezing temperature. As we shall see presently, the manometric water below its free surface may be at any temperature from freezing to 10° C. above freezing without more than τ 1 θ per cent, of hydrostatic error. The temperature in the vapour-space above the liquid column may be either freezing or anything higher. It ought not to be lower than freezing, because, if it were so, vapour would condense as hoar frost on the glass, and evaporation from the top of the liquid column would either cryophoruswise (see Liquid and Thermodynamics) freeze the liquid there, or would cool it below the freezing point.

§ 40. The chief object of keeping the top of the manometric column exactly at the freezing point is to render perfectly definite and constant the steam pressure in the space above it.

A second object of considerable importance when the bore of the tube is so small as one millimetre is to give constancy to the capillary tension of the surface of the water. The elevation by capillary attraction of ice-cold water in a tube of oue millimetre bore is about 7 millims. The constancy of temperature provided by the surrounding iced water will be more than sufficient to prevent any perceptible error due to inequality of this effect. To avoid error from capillary attraction the bore of the tube ought to be very uniform, if it is so small as one millimetre. If it be three millimetres or more, a very rough approach to uniformity would suffice.

A third object of the iced-water jacket, and one of much more importance than the second, is to give accuracy to the hydrostatic measurement by keeping the density of the water throughout the long vertical branch definite and constant. But the density of water at the freezing point is only 1 ⅛ per cent, less than the maximum density, and is the same as the density at 8° C.; and therefore when 5 1 σ per cent, is an admissible error on our thermometric pressure, the density will be nearly enough constant with any temperature from 0° to 10° C. throughout the column. But on account of the first object mentioned above the very top of the water column must be kept with exceeding exactness at the freezing temperature.

§ 41. In this instrument the “thermometric substance” (§ 34) is the water and vapour of water in the bulb, or more properly speaking the portions of water and vapour of water infinitely near their separating interface. The rest of the water is merely a means of measuring hydrostatically the fluid pressure at the interface. When the temperature is so high as to make the pressure too great to be conveniently measured by a water column, the hydrostatic measurement may be done, as shown in the annexed drawing (fig. 7), by a mercury column in a glas3 tube, surrounded by a glass water jacket not shown in the drawing, to keep it very accurately at some definite temperature so that the density of the mercury may be accurately known.

The simple form of steam thermometer represented with figured dimensions in fig. 6 will be very convenient for practical use for temperatures from freezing to 60°. Through this range the pressure of water-steam, reckoned in terms of the balancing column of water of maximum density, increases (Table V.) from 6∣ to 202 - 3 centimetres; and for this therefore a tube of a little more than 2 metres will suffice. From 60° to 140° the pressure of steam now reckoned in terms of the length of a balancing column of mercury at 0° increases from 14 - 88 to 271 , 8 centimetres; and for this a tube of 280 centimetres may be provided. For higher temperatures a longer column, or several columns, as in the multiple manometer, or an accurate air pressure-gauge, or some other means, such as a very accurate instrument constructed on the principle of Bourdon’s metallic pressure-gauge, may be employed, so as to allow us still to use water and vapour of water as thermometric substance.

§ 42. At 230° C., the superior limit of Regnault’s high-pressure steam experiments, the pressure is 27 53 atmos, but there is no need for limiting our steam thermometer to this temperature and pressure. Suitable means can easily be found for measuring with all needful accuracy much higher pressures than 27 atmos. But at so high a temperature as 140°, vapour of mercury measured by a water column, as shown in the diagram (fig. 8), becomes available for purposes for which one millimetre to the degree is a sufficient sensibility. The mercury-steampressure thermometer, with pressure measured by watercolumn, of dimensions shown in the drawing, serves from 140’ to 280° C., and will have very ample sensibility through the upper half of its scale. At 280° its sensibility will be about 4∣ centimetres to the degree! For temperatures above 280° sufficient sensibility for most purposes is obtained by substituting mercury for water in that simplest form of steam thermometer shown in fig. 6, in which the pressure of the steam is measured by a column of the liquid itself kept at a definite temperature. When the liquid is mercury there is no virtue in the particular temperature 0 j C., and a stream of water as nearly as may be of atmospheric temperature will be the easiest as well as the most accurate way of keeping the mercury at a definite temperature. As the pressure of mercury steam is at all ordinary atmospheric temperatures quite imperceptible to the hydrostatic test when mercury itself is the balancing liquid, that which was the chief reason for fixing the temperature at the interface between liquid and vapour at the top of the pressure-measuring column when the balancing liquid was water (§ 40) has no weight in the present case; but, on the other hand, a much more precise definiteness than the ten degrees latitude allowed in the former case for the temperature of the main length of the manometric column is now necessary. In fact, a change of temperature of 2 ’ 2° in mercury at any atmospheric temperature produces about the same proportionate change of density as is produced in water by a change of temperature from 0° to 10°, that is to say, about 2⅛∙ per cent.; but there is no difficulty in keeping, by means of a water jacket, the mercury column constant to some definite temperature within a vastly smaller margin % of error than 2 ’ 2°, especially if we choose for the definite temperature something near the atmospheric temperature at the time, or the temperature of whatever abundant water supply may be available. If the vertical tube for the pressure-measuring mercury column be 830 centimetres long, the simple mercury-steam thermometer may be used up to 520° C., the highest temperature reached by Regnault in his experiments (Table V. below) on mercury-steam. By using an iron bulb and tube for the part of the thermometer exposed to the high temperature, and for the lower part of the measuring column to within a few metres of its top, with glass for the upper part to allow the mercury to be seen, a mercury-steam-pressure thermometer can with great ease be made which shall be applicable for temperatures giving pressures up to as many atmospheres as can be measured by the vertical height available. The apparatus may of course be simplified by dispensing with the Torricellian vacuum at the upper end of the tube, and opening the tube to the atmosphere, when the steam-pressure to be measured is so great that a rough and easy barometer observation gives with sufficient accuracy the air-pressure at the top of the measuring column. The easiest, and not necessarily in practice the least accurate, way of measuring very high pressures of mercury-steam will be by enclosing some air above the cool pressure-measuring column of mercury, and so making it into a compressed-air pressuregauge, it being understood that the law of compression of the air under the pressures for which it is to be used in the gauge is known by accurate independent experiments such as those of Regnault on the compressibility of air and other gases.

§ 43. The water-steam thermometer may be used, but somewhat precariously, for temperatures below the freezing point, because water, especially when enclosed and protected as the portion of it in the bulb of our thermometer is, may be cooled many degrees below its freezing point without becoming frozen: but, not to speak of the uncertainty or instability of this peculiar condition of water, the instrument would be unsatisfactory on account of insufficient thermometric sensibility for temperatures more than two or three degrees below the freezing point. Hence, to make a steam thermometer for such temperatures some other substance than water should be taken, and none seems better adapted for the purpose than sulphurous acid, which, in the apparatus represented with figured dimensions in the accompanying diagram (fig. 9), makes an admirably convenient and sensitive thermometer for temperatures from + 20° to something far below — 30°, as we see from the results of Regnault’s measurements (Table VI. below).

§ 44. To sum up, we have in §§ 39 . . 43 a complete series of steam-pressure thermometers, of sulphurous acid, of water, and of mercury, adapted to give absolutely definite and highly sensitive thermometric indications throughout the wide range from something much below - 30’ to considerably above 520° of the centigrade scale. The graduation of the scales of these thermometers to show absolute temperature is to be made by calculation from formula (3) of § 38, when the requisite experimental data, that is to say, the values of σ and pκ for different values of p throughout the range for which each substance is to be used as thermometric fluid are available. Hitherto these requisites have not been given by direct experiment for any one of the three substances with sufficient accuracy for our thermometric purpose through any range whatever. Water, naturally, is the one for which the nearest approach to the requisite information has been obtained. Fur it Regnault’s experiments have given, no doubt with great accuracy, the values of p (the steam pressure) and of κ (the latent heat of steam per unit mass) for all temperatures reckoned by his normal air thermometer, which we now regard merely as an arbitrary scale of temperature, through the range from -30° to +239°. If he, or any other experimenter, had given us with similar accuracy through the same range the values of p (the density of steam) and <τ (the ratio of the density of steam to the density of water in contact with it), for temperatures reckoned on the same arbitrary scale, we should have all the data from experiment required for the graduation of our water-steam thermometer to absolute thermodynamic scale. For it is to be remarked that all reckoning of temperature is eliminated from the second member of formula (3), and that in our use of it Regnault’s normal thermometer has merely been referred to for the values of pκ and of 1 — σ, which correspond to stated values of p, The arbitrary constant of integration, t 0 , is truly arbitrary. It will be convenient to give it such a value that the difference of values of t between the freezing point of water and the temperature for which p is equal to one atmo shall be 100, as this makes it agree with the centigrade scale in respect to the difference between the numbers measuring the temperatures which on the centigrade scale are marked 0° and 100°. We shall see (§ 56 below) that indirectly by means of experiments on hydrogen gas this assignation of the arbitrary constant of integration would give 273 for the absolute temperature 0° C., and 373 for that of 100° C. Meantime, as said above, w e have not the complete data from direct experiments even on water-steam for graduating the water-steam thermometer; but on the other hand we have, from experiments on air and on hydrogen and other gases, data which allow us to graduate indirectly any continuous intrinsic thermoscope (§19 above) according to the absolute scale; and we shall see that by thus indirectly graduating the water-steam thermometer, we learn the density of steam at different temperatures more accurately than it has hitherto been made known by any direct experiments on water-steam itself.

§ 45. Merely viewed as a continuous intrinsic thermoscope, the steam thermometer, in one or other of the forms described above to suit different parts of the entire range from the lowest temperatures to temperatures somewhat above 520°, is no doubt superior in the conditions for accuracy specified in § 16 to every other thermoscope of any of the different kinds hitherto in use; and it may be trusted more surely for accuracy than any other as a thermometric standard when once it has been graduated according to the absolute scale, whether by practical experiments on steam, or indirectly by experiments on air or other gases. In fact, the use of steam-pressure measured in definite units of pressure, as a thermoscopic effect, in the steam thermometer is simply a continuous extension to every temperature of the principle already practically adopted for fixing the temperature which is called 100° on the centigrade scale; and it stands on precisely the same theoretical footing as an air thermometer, or a mercury-in-glass thermometer, or an alcohol thermometer, or a methyl butyrate thermometer, in respect to the graduation of its scale according to absolute temperature. Any one intrinsic thermoscope may be so graduated ideally by thermodynamic experiments on the substance itself without the aid of any other thermometer or any other thermometric substance; but the steam-pressure thermometer has the great practical advantage over all others, except the air thermometer, that these experiments are easily realizable with great accuracy instead of being, though ideally possible, hardly to be considered possible as a practical means of attaining to thermodynamic thermometry. In fact, for water-steam it is only the most easily obtained of experimental data, the measurement of the density of the steam at different pressures, that has not already been actually obtained by direct experiment. Whether or not, when this lacuna has been filled up by direct experiments, the data from water-steam alone may yield more accurate thermodynamic thermometry than we have at present from the hydrogen or nitrogen gas thermometer (§§ 64 . . 69 below), we are unable at present to judge. But when once we have the means, directly from itself, or indirectly from comparison with hydrogen or nitrogen or air thermometers, of graduating once for all a sulphurous acid steam thermometer, water-steam thermometer, or mercury-steam thermometer, that is to say, when once we have a table of the absolute thermodynamic temperatures corresponding to the different steam pressures of the substances sulphurous acid, water, and mercury, we have a much more accurate and more easily reproducible standard than either the air or gas thermometer of any form, or the mercury thermometer, or any liquid thermometer can give.

In fact, the series of steam thermometers for the whole range from the lowest temperatures can be reproduced with the greatest ease in any part of the world by a person commencing with no other material than a piece of sulphur and air to burn it in, some pure water, and some pure mercury, and with no other apparatus than can be made by a moderately skilled glass-blower, and with no other standard of physical measurement of any kind than an accurate linear measure. He may assume the force of gravity to be that calculated for his latitude, with the ordinary rough allowance for his elevation above the sea, and his omission to measure with higher accuracy the actual force of gravity in his locality can lead him into no thermometric error which is not incomparably less than the inevitable errors in the reproduction and use of the air thermometer, or of mercury or other liquid thermometers. In temperatures above the highest for which mercury-steam pressure is not too great to be practically available, nothing hitherto invented but Deville’s air thermometer with hard porcelain bulb suited to resist the high temperature is available for accurate thermometry.

§ 46. We have given the steam thermometer as our first example of thermodynamic thermometry because intelligence in thermodynamics has been hitherto much retarded, and the student unnecessarily perplexed, and a mere quicksand has been given as a foundation for thermometry, by building from the beginning on an ideal substance called perfect gas, with none of its properties realized rigorously by any real substance, and with some of them (see Matter, Liquid, Steam) unknown, and utterly unassignable, even by guess. But after having been moved by this reason to give the steam-pressure thermometer as our first theoretical example, we have been led into the preceding carefully detailed examination of its practical qualities, and we have thus become convinced that though ldtherto usedin scientific investigations only for fixing the “boiling point,” and (through an inevitable natural selection) by practical engineers for knowing the temperatures of their boilers by the pressures indicated by the Bourdon gauge, it is destined to be of great service both in the strictest scientific thermometry and as a practical thermometer for a great variety of useful applications.

§ 47. Example 2 (including example 1).—Any case in which the stress is uniform pressure in all directions.

Let p> and v denote the pressure and volume. The condition of the substance (single, double, or triple, as the case may be) is determinate when p and v are given, and it will therefore be spoken of shortly as the condition (p, ν'). Let e be the energy which must be communicated to the substance to bring it from any conveniently defined zero condition (y> 0 , r 0 ) to any condition whatever (p, ν'). Remark that e is a function of the two independent variables p, v to be found by experiment, and that the finding of it by experiment is a perfectly determinate practical problem, which can be carried out without the aid of any thermαscope, and without any consideration whatever relating to temperature. We shall see in fact that accurate practical solutions of it for many different substances have been obtained by experiment (see Thermodynamics). The absolute temperature t is also a function of p and v to be also determined by experiment, according to the equivalent definitions of §§ 35 and 37. Let heat be communicated to the substance so as to cause its volume to increase by dv, the pressure being kept constant. The energy of the body will be augmented by — . dv. dv

At the same time the body in expanding and pressing out the matter around it does work to the extent of p. dv (4).

Hence the whole work required to generate the heat given to it amounts to

(⅛+⅜ < δ >∙

Hence the ratio of (4) to (5), or dj~ (β)∙

is the “work-ratio” of § 37. Hence by the definition

Di dp p dp , f ,,

~ - 7 ⅛ry - de— · · · · (7> · * +j, ⅛ + ∙"

where Di denotes the change of temperature produced by augmenting the pressure by dp, and at the same time preventing the substance from either giving heat to or taking heat from the surrounding matter. To express this last condition analytically, let dv be the augmentation of volume (negative, of course, if dp be positive) which it implies. The work done on the substance by the pressure from without is —pdv, and the energy of the substance is augmented by just this amount, because of the condition to be expressed. Hence

<8): whence de dv =fτ⅛~ dp ^∙ dv + p

But ta , dt , dt , Dt^- r dp + ~dv , dp - r dv and so we have ( dt de ∖ d ±- d f ⅞ (10).

dp de dv +p l

Eliminating ^Dt ∣ dp from this by (7) we find j ( de ∖ dt de dt zιιx

V + dv)dp dp dv ’ ' '

§ 48. This is a linear partial differential equation of the first order for the determination of t, supposing, as we do for the present, that e is a known function of p and v. The following graphical illustration of the well-known analytical process for finding the complete solutions of such equations shows exactly how much towards determination of temperature can be done with no other data from experiment than the values of d f- and — as functions of p and v, dv dp and what additional information is required to fully determine t.

First remark that (11) is the condition that -ibe a factor rendering { p + - ∖ lv + -dp a complete differential of a function V dv J c lp of two independent variables p and v. Let φ be this function,— that is to say, let φ be such that dφ 1 ∕ de ∖ , dφ 1 de ~, τ(y + ⅛)' a,lj ⅛-T⅛ de de , „ or t P + ^d^> dp ' - (12) ‘ dφ dφ dv dp .

Then every solution of the differential equation p + — dp do (1 g)

dv de dp renders φ constant; and conversely, every series of values of p and v which renders φ constant constitutes a solution of (13). Now this differential equation may be solved graphically by taking p and v as rectangular coordinates of a point in a plane, and drawing the whole series of curves which satisfy it as follows. Commence with any point and calculate for its values of p and v the value of the second member of (13). Draw through this point an infinitesimal line in the direction of the tangent to the curve given by the value so found for ⅛ . With the altered values of p and v corresponding dv to the other end of this infinitesimal line, calculate a fresh value of ⅛ , and continue the curve in the slightly altered direction thus dv ’ found, and so on. Take another point anywhere infinitesimally near this curve but not in it, and draw by a similar process the curve through it satisfying the equation. Take a third point infinitely near this second curve, and draw through it a third curve satisfying (13), and so on till the whole area of values p, v, possible for the substance in question, is filled with a series of curves one of which passes through, or infinitely nearly through, every point of the area. Assign arbitrarily a particular value of φ to each of these curves; then graphically find '⅛ and ⅛ for any or every value of dv dp p and v. Then either of the two second forms of equation (12) gives us explicitly a value of t for any values whatever of p and v.

§ 49. The solution for i thus obtained involves the arbitrary assumption of a particular value of ≠ for each one of the series of curves which we have determinately traced. Hence, to render t wholly determinate, something more must be given than c as a function of p and v. Now the only thing that can be given respecting temperature for any particular substance before we have a thermometric scale is the relation subsisting between p and v when the temperature is constant. This relation can, with merely a single-tern perature-thermoscope (§ 15 above), in addition to dynamical instruments, be determined for some one particular temperature; and this, if e be known for every value of p and v, is the only additional knowledge required for the determination of t for every value of p and v. For let p ≈f(v) be the relation between p and υ for some one particular temperature, t 0 . If by this we eliminate p from (12) we find

S=⅛∣> + s] '<*>≈

where — , when p=f(v), becomes a known function of v alone. clv

Hence by integration we find + c] (15), where F denotes a known function and C an arbitrary constant. Now trace the curve p=f(v) cm our diagram. It must generally cut every one of the previously drawn determinate series of curves. Hence equation (15), with two arbitrarily assigned constants t 0 and C, gives determinately the value of <j> for every one of the diagram of curves, and thus φ is determined for every value of p and ν. Either of equations (12) then gives t determinately as a function of p and υ, with only the value t 0 arbitrary. The information from experiment, regarding the properties of the thermometric substance, on which this determination is founded, consists of a knowledge of the relation between p and υ ior any one temperature, and of the value of e - e 0 for all values of p, v, (c 0 denoting the unknown value of e for some particular values 2> o r o)∙ Although, theoretically, this information is attainable by purely dynamical operations and measurements, with no other thermal guidance or test than that afforded by a single-temperature-intrinsic-thermoscope (§15), the whole of it has not in fact been explicitly obtained for any one substance. But less than the whole of it suffices to make a perfect absolute thermometer of any given substance.

§ 50. For this purpose it is not necessary to find t for all values of p and υ: it is enough to know it for all values mutually related in any manner convenient for thermometric practice. For example, if we could find t for every value of v with p constant at some one particular chosen value,—this would give a “constant pressure” absolute thermometer. Or again, if we find t for every value of p with v kept constant,—this would give us a “constant volume” absolute thermometer. Let us now examine into the restricted dynamic and thermoscopic investigations upon any particular substance, which will suffice to allow us to make of it a standard absolute thermometer of one or other of these species.

§ 51. Dynamical and thermoscopic investigation required to graduate, according to the absolute scale, a constant-pressure thermometer of any particular fluid.— Let a large quantity of fluid be given, and let proper mechanical means be taken to cause it to flow slowly and uniformly through a pipe, in one short length of which there is a fixed porous plug. If, as is the case with coιρmon air, nitrogen, oxygen, carbonic acid, and no doubt many other gases, the fluid leaves the plug cooler than it enters it, let there be a paddle in the stream flowing from the plug, and let this paddle be turned so as to stir the fluid and cause the temperature, when the rapids are fairly past and the eddies due to the stirring subsided, to be the same as in the stream flowing towards the plug. When, as in the case of hydrogen and of all ordinary liquids, the fluid flows away from the plug warmer than it entered it, let a uniform stream of water be kept flowing in a separate canal outside the tube round a portion of it in which the internal flow is from the plug, and by this means let the temperature of the internal fluid be brought to equality with that which it had on entering the plug. By a separate thermodynamic experiment find how much work would have to be spent in stirring the external stream of water by a paddle to warm it as much as it is warmed by conduction from the internal fluid across the separating tube. Returning now to the internal fluid flowing towards and from the plug, let p + δp be the pressure in the steady stream approaching the disturbed region, and ∕> the pressure in the steady stream flowing from the disturbed region; and let δw be the quantity of work done by the paddle per unit of mass of the fluid passing by, reckoned positive in the first case, that namely in which the paddle compensates a cooling effect experienced in passing through the porous plug. In the second case - 8w (in this case a positive quantity) must denote the work done by the paddle upon the supposed external stream of water in the separate thermodynamic experiment. It is to be reckoned per unit mass of the internal fluid, irrespectively of the rate of flow of the external water. Let t denote the temperature of the fluid according to the thermodynamic scale, and let 8t denote the infinitely small change of temperature which it must experience to produce an infinitesimal expansion from volume v to volume v + δw under constant pressure. We have

Λ dt - 1 (16).

t dv _ 1 δw 1 + — Γ~ V δp Proof.— Let r + δw, υ, and c + δe, e, be respectively the amounts of the volume and of the energy of the fluid per unit mass, in the tranquil stream before and after passing the disturbed region. The work done by an ideal piston pressing the fluid in towards the disturbed regionis (p + δy>)(r + δv), and the work done by the emergent stream upon an ideal piston moving before it is pv, each reckoned per unit of mass, of the fluid. The whole work done on the fluid per unit mass by these ideal pistons is pδv + vδ^; add to this δw done by the paddle, and we find that, on the whole, an amount of work equal to pδv + vδp + δw is done on the fluid in passing through the disturbed region. Hence e exceeds e + δe by this amount; that is to say, - δe=pδυ+vδρ + δw (17).

Now the paddle and plug together act so as to render the temperatures equal in the tranquil streams at pressures p and∕> + δy>. But if there were change of temperature its analytical expressior would be dt _ dt. ∕, o∖

δt=- r δυ + - r hp (18).

dv dp

Hence δv and δp are so proportioned as to make this vanish. That is to say, we have dt

=

dv and we have - de de t , δe = - r δυ + ~ r δp; dv dp hence (17) divided by δρ becomes dt _ dt d fL. f⅛TJ± = v ⅛L + v+ 5 Ti . . . (20). dv dt dp i dt δp dv dv

Using this in (11) we find

-* ν+ s ^ (21).

df δp> j dv

Dividing (21) by v, and taking the reciprocal of both members, we have the equation (16) which was to be proved.

§ 52. Now if for any particular fluid at some one given pressure p, with infinitesimal excess δp above this pressure for the higher pressure in the thermodynamic experiment, we find neither heating nor cooling effect in passing through the porous plug, the paddle has nothing to do; that is, δitf = 0. If, with always the same pressure ρ, but with different values of v, that is to say, with the fluid given at different temperatures, but with pressures infinitely nearly the same, we always find the same result, 8ιv-Q, it follows from (16) that for this particular fluid at the particular pressure of the experiment, and for all the temperatures of the experiment, we have r = ι (22).

t dν

Hence by integration i = C-y (23).

Hence we infer that with this fluid for thermometric substance, with the particular pressure of the experiment, and throughout the range of temperatures for wdιich experiment has given us 8w = 0, absolute temperature is shown on a scale graduated and numbered in simple proportion to the whole volume of the fluid.

§ 53. If the thermodynamic test repeated for the same fluid at different pressures gives still the same result, we have, for all pressures and temperatures within the range for which the supposed result 8ιv = 0 has been found by the experiment, t=βB)v (24); where f(p) denotes a quantity which depends only on the pressure of the fluid and is independent of its density.

§ 54. Joule and Thomson’s experiments on the thermal effects of fluids in motion showed that for pressures of from one to five or six atmos hydrogen gas, common air, nitrogen, oxygen, and carbonic acid, all somewhat approximately fulfil the condition of passing through the porous plug without change of temperature,—hydrogen much more approximately, carbonic acid much less approximately, than any of the others. Hence we infer that absolute temperature is somewhat approximately proportional to the volume of the fluid, if any one of these gases be used as the thermometric fluid in a constant-pressure thermometer. We shall presently see that the requisite correction of this statement for the case of hydrogen is so small as to be almost within the limits of accuracy of the most accurate thermometric usage.

§ 55. In the case of common air, nitrogen, oxygen, and carbonic acid, the experiments showed a slight cooling effect upon the fluid in passing through the porous plug; in the case of hydrogen, a much smaller heating effect. According to the rigorous dynamical form of our statement of § 51, we have no right to measure these heating and cooling effects on any scale of temperature, as we have not yet formed a thermometric scale. And it is interesting to remark that in point of fact the thermodynamic experiment described in that section involves the use of a differential thermoscope (§ 13) and not of any intrinsic thermoscope at all; and in respect to this requisite it maybe contrasted with the thermodynamic investigation of § 49 previously, which involved the use, not of any continuous thermoscope, but only of a single-temperature intrinsic thermoscope (§ 14). Now, instead of reckoning on any thermometric scale the cooling effect or the heating effect of passage through the plug, we have to measure the quantity of work (δw) required to annul it, in the case of the majority of gases; and in the case of hydrogen, instead of reckoning on any thermometric scale the heating effect, we have to measure - 8w as explained in § 51. The experiments as actually made by Joule and Thomson simply gave the cooling effects and heating effects shown by mercury thermometers in the tranquil stream towards and from the plug; but the very thermometers that were used had been used by Joule in his original experiments determining the dynamical equivalent of heat, and again in his later experiments by which for the first time the specific heat of air at constant pressure was measured with sufficient accuracy for our present purpose. Hence by putting together different experiments which had actually been made with those thermometers of Joule’s, the operation of measuring δzc, at all events for the case of air, was virtually completed. Thus according to our present view the mercury thermometers are merely used as a step in aid of the measurement of 8ιv, and their scales may be utterly arbitrary, provided we know the quantity of work required to raise unit mass of any of the fluids concerned through the particular differences of temperature actually shown by the thermometers in the Joule and Thomson experiment. The best way of doing this of course is to take advantage of the best measurements, that is to say Regnault’s, of the thermal capacity of air at constant pressure, and then to calculate according to Joule’s own measurement the dynamical equivalent of the heat required to warm water through one degree of his own thermometers.

§ 56. Let K be the thermal capacity, pressure constant, of the fluid experimented on, J the dynamical equivalent of the thermal unit, and St the cooling effect (reckoned negative when the effect is rise of temperature), as measured by Joule’s thermometers. We have δw = JKδ( (25).

Hence (16) becomes v - d l≈ (26).

t dv JK δ£

1 + v Sp

The experiment shoλved St to be simply proportional to Sp not merely for an infinitesimal difference of pressures but for pressures up to 5 or 6 atmos. For the case of hydrogen the heating effect observed amounted, per 100 inches of mercury, to ’ 100 of a degree centigrade at temperatures of 4° or 5° centigrade, and to ∙155 of a degree centigrade at temperatures of from 89° to 93° centigrade. The investigation was not carried out in sufficient detail to give any law of variation of this effect with temperature, and it was not even absolutely proved to be greater for the higher than for the lower temperature. In the circumstances we may take the mean of the results for the higher and lower temperatures, say ‘13 per 100 inches of mercury, or Ό39 per atmo. Hence if ∏ denotes the force per unit of area in the pressure called “one atmo,” we have St _ _ Ό39 Sp~ ∏ ’ v - d L 1 (27).

t dv , λoλ JK k 1

1 - ∙039 —

∏t l

Hence dt dv ~t≈υ- ∙0⅞JK7∏ ’

which gives by integration i = C(y-∙039JK∕Π) ^28).

The arbitrary constant C depends on the unit adopted for temperature. Let this be such that the difference of temperature between freezing and boiling is 100 (which will make our arbitrary scale agree with the ordinary centigrade scale in respect to the difference between these two temperatures). Denote now by t 0 the absolute temperature corresponding to 0° C. The absolute temperature corresponding to 100° C. will be ⅛ + 100. Denote also by v 0 and tq 0ft , for the same two temperatures, the bulks of unit mass of hydrogen at any constant pressure within the limits of Joule and Thomson’s experiments, say, from one to five or six atmos. Then by dividing the value of each member of (28) for 0° C. by the difference of its values for 0° and 100°, we find tn _ ⅞~ '039JK∕Π (29)

100“ t ’ joo~⅞

Hence Zo=1 4’ (1_, ° 39JK/n " o) · · · · (30)ί where E denotes the expansion of hydrogen, pressure constant, from 0° to 100° C. in terms of its volume at 0°, that is to say,

Ε=” 10 °-” η (31).

v o

Let V o denote what the volume would be at 0° C. if the pressure were ∏ instead of the actual pressure p. We have io=⅜°(1 ~⅞ l ' 039jκ ∕ πv °) · · · · (32 >∙

Régnault finds (Expériences, vol. ii. p. 122) that the value of K∕∏V 0 for hydrogen agrees within ⅜ per cent, with its value for common air; and for common air he finds K= - 238. Thus with 423‘5 for the value of J in metres (§ 9 above) we find JK = 100’79 metres. And Regnault’s observations on the density of air give for ∏V 0 (or the height of the homogeneous atmosphere at 0° C.) 7990 metres. Hence for common air, and therefore also for hydrogen, JK∕∏V 0 = '0126; and thus (32) becomes i ∙≈⅛(1 ÷⅞ ’) < ss >∙

with c= - Ό0049 for hydrogen. For this gas expanding under constant pressure of one atmo Régnault found (Experiences, vol. i. p. 80) E='36613, which gives ^^-273 ’ 13. Hence (33), with ⅞ = V 0 , gives

Z o = 273'OO (34); that is to say :—

§ 57. We conclude from Regnault’s observations on the expansion of hydrogen from 0° to 100° C. under a constant pressure of one atmo, and from the small heating effect discovered in Joule and Thomson’s experiments on the forcing of hydrogen through a porous plug, that the absolute temperature of melting ice is 273'00°, if the unit or degree of absolute temperature is so chosen as to make the difference one hundred between the temperatures of melting ice and of water with steam at one atmo of pressure.

§ 58. An almost identical number for that most important physical constant, the absolute temperature of melting ice, is obtained from observations on common air, and a not very different number from observations on carbonic acid, the only two gases besides hydrogen for which Régnault ( Expériences, vol. i. p. 90) measured the expansion under constant pressure, and for which Joule and Thomson made their experiment on the thermal effect of passage through a porous plug. For each of these two gases the thermal effect observed was a lowering of temperature, and was found to vary at different temperatures very nearly in the inverse proportion of the square of the temperature C., by mercury thermometer, with 273 added. Hence nearly enough for use in the small term of the denominator of (26) we have, for air and carbonic acid,

^ = A(2 2- 3 F (35), iρ ∖ t Jn v h where i denotes as before absolute temperature, and A the amount of the cooling effect per atmo of difference of pressures, on the two sides of the plug, at the temperature of melting ice. The values of A found for common air and carbonic acid are ’ 275 and 1 ’ 388. Régnault {Experiences, vol. ii. p. 126) finds JK∕ΠV 0 greater for carbonic acid than for common air in the ratio of 1 ’ 39 to 1 on the average of temperatures from 0° to 210°. But he found also that the specific heat of carbonic acid varies greatly with the temperature; and, taking the mean of the values which he finds for it at 0° and 100°, p. 130, as the proper mean for our present purpose, we find for JK∕ΠV 0 , a value D29 times its value for common air. From these experimental results we find by the mathematical process below (§ 61) still the same approximate formula (33), but with c== + ’ 0026 for common air and c = +Ό163 for carbonic acid. At constant pressure of one atmo Regnault’s measurements gaveE = ∙36706 for common air, and E = *3710 for carbonic acid; and dividing 100 by these decimals we find respectively 272'44 and 269 ’ 5. The corrections on these numbers by formula (33) to give the absolute temperature of freezing are accordingly + ’ 70 and +4 ’ 4, and the corresponding estimates for the required absolute temperature are 273 14 and 273 ’ 9. Bringing together the results in the three cases, we see them conveniently in the following table [table]

1 Investigated in § 61 below.

The close agreement of the results from hydrogen and common air is very satisfactory, and it is interesting to see it brought about with so large a correction calculated from the Joule and Thomson effect. It is also interesting to see the sevenfold larger correction of nearly 5° bringing so nearly the same result from the 1 per cent, larger expansion of carbonic acid. The ⅜ per cent, discrepance which remains between the results from carbonic acid and from hydrogen is not satisfactory, and requires explanation, particularly when we remark that, of five measurements by Régnault {Experiences, vol. i. p. 84) of the expansion of carbonic acid under constant pressure of one atmo, all lie within -L per cent, of the mean number ’ 3710 which 57 r he has given, and we have taken, as his result.

Notwithstanding that the Joule and Thomson correction is so much greater for common air than for hydrogen, the result from common air is probably the most trustworthy of the three, because both Regnault’s experiments and Joule and Thomson’s were probably more accurate for air than for either of the other two gases. The true result to one place of decimals may therefore be considered as most probably being 273'1°, but the probability that it is nearer 273 , 1° than 273 0° is scarcely enough to make it worth while to use in any ordinary thermodynamic calculations any other number than 273°, which is exactly that found from hydrogen.

§ 59. The real meaning of our result 273 - 1° for the absolute temperature of melting ice, expressed without any choice of degrees or units for temperature, is that the ratio of the temperature at which vapour of water has a pressure of one atmo to the temperature at which ice melts is 373 1∕273 ’ 1. Still another way of saying the same thing, this time eliminating all numerical reckoning of temperature, is (see Thermodynamics) as follows :—

For every hundred units of heat converted into WORK BY A PERFECT THERMODYNAMIC ENGINE, 373 1 ARE TAKEN FROM THE SOURCE, AND 273 , 1 REJECTED TO THE REFRIGERATOR, IF THE TEMPERATURE OF THE SOURCE BE THAT AT WHICH STEAM OF WATER HAS A PRESSURE OF ONE ATMO, AND THE TEMPERATURE OF THE REFRIGERATOR THAT AT WHICH ICE MELTS.

§ 60. Integration of differential equation (26), § 56, between volume and absolute temperature for a gas, derived from the Joule and Thomson experiment.

Returning to § 56 we may write equation (26) as follows :— ,dυ rιr St

⅞" φ ~ j ⅛ (36) ·

For each of the five gases experimented on, namely, common air, oxygen, nitrogen, carbonic acid, hydrogen, the experiment showed that, for all pressures up to five or six atmos, blfp was sensibly independent of the pressure, but that it varied very considerably with the temperature. Hence, if we put δi∕δw = 0∕∏, 0, which will thus denote the cooling effect per atmo of differential pressure, is a function of the temperature, and is independent of the whole pressure. With this notation (36) becomes t dv JK.

t--ν = —θ. di ∏

This is a linear equation in z with a second member, if for a moment we put z = log t. Integrating it, and replacing t, we find, as the complete integral, 1, ='⅛ + κ∕'⅛i < 37 >∙

§ 61. Expansions of different gases,pressure constant, calculated from the Joule and Thomson experiment. We have from equation (37) 9→ 0 J-⅛∣ 1 JKi 0 1 P t θdl ) nsλ

V, ⅛ i 1 + π , t ,· J ?[· · · (≡ 8 >∙

t o

For each of the gases experimented on, except hydrogen, Ö was found to vary nearly in the inverse ratio of t i . Putting then 0 = A(i o ∕<) 2 , we find

- (⅜) 3 ] ·

Hence, for these gases at pressures from 0 to 5 or 6 atmos, (38) Ιίρρλτϊί pq

~∙=⅛Ψ÷⅛M⅜y∏. . w This shows that the “proper mean cooling effect” (M in the table of § 58) is 5 D÷100 + √(lb0 + J ] Ä ’ = 5 (1 + L3663 + 1 '3663 2 ) a ~ ' 756A ’ which differs so little from 4 ( 1 + V3⅛5-∙) a " 769a ' the arithmetic mean of the cooling effècts at 0° and 100° C., that if we had simply taken the arithmetic mean for each of the other gases, as for want of knowing better we took it for hydrogen, the difference in the result would have been barely perceptible.

§ 62. Modifying (39) or (38) to suit any two temperatures, t, t', we have v-d t-tC JK ∖ ....

—" f(1 +ffi∕”) (40 >'

m denoting the proper mean cooling effect per atmo in the Joule and Thomson experiment (to be reckoned as negative in the case of hydrogen or any other gas, if there is any other, in which the experiment shows a heating effect). This “proper mean’’may be taken as the arithmetic mean of the values for t and f, unless t-f considerably exceeds 100.

To reduce (40) to numbers, let V o be the volume of unit mass of the gas when at the temperature of melting ice, and under one atmo of pressure. Régnault (vol. ii. p. 303) finds that the value of K∕V 0 is within 1 per cent, the same for oxygen, nitrogen, and hydrogen as for common air. He also (vol. ii. pp. 224-226) finds K to be the same for common air at from 1 to 12 atmos; for hydrogen, 1 to 9; for carbonic acid, 1 to 37.

No doubt similar constancy would be found for oxygen and nitrogen. Hence, as above (§ 56), for common air we still have JK∕∏V 0 = ’ 0126, and thus (40) becomes

, , x , . -tr . ( for common air, λ

V-⅛ i (ι + >∙° 1 H Ä- · <«>.

x 7 ( nitrogen )

If in this formula we take t and tf for the temperatures of 100° C. and 0° C., (v-√)∕ v' becomes 273’· y 0r - 3θ62; and we therefore find E= ∙3662^1 + ∑θ ∙0126m) (42), which agrees with (33) above.

§ 63. The values given by Joule and Thomson’s experiment for m are -0O39° for hydrogen, + 0 ’ 208° for air, + 0 ’ 253° for oxygen, and + 0 ’ 249° for nitrogen.

From these and from the previous results for carbonic acid (§ 58) we have the following table for calculating the expansions from 0° C. to 100° C. of the gases named :— [table]

These formulae must be exceedingly near the truth for all pressures from 0 to 6 atmos, because within this range the thermal effects in the Joule and Thomson experiment were very approximately in simple proportion to the differences of pressure on the two sides of the plug. The following table of results calculated from (43) for several pressures of from 0 to 6 atmos is interesting as showing such different expansions for the different cases, determined by thermodynamic theory from Regnault’s measurements of specific heats and Joule and Thomson’s of their particular thermal effect, with absolutely no direct measurement of expansion except the one for common air at one atmo, shown as the third entry of column 5 in the table. The other five entries of column 5 show a fair amount of agreement between our theoretical results and the only direct measurements by Regnault. More of direct measurement, to allow a more extensive comparison, is very desirable. [table]

§ 64. We are now quite prepared to make a practical working thermometer directly adapted to show temperature on the absolute thermodynamic scale through the whole range of temperature, from the lowest attainable by any means to the highest for which glass remains solid. It is to be remarked that our investigation of § 51, and all the deduced formulae and relative calculations, are absolutely independent of the approximate fulfilment of Boyle’s law by the gases to which we have applied them, and are equally applicable without any approach to fulfilment of Boyle’s law; also that the only experimental data on which are founded our special numerical conclusions of §§ 59 to 63 are Regnault’s measurements of specific heats under constant pressures, and Joule and Thomson’s measurements of the thermal effect of forcing through a porous plug. From these experimental data alone we see by formula (38) of § 61 how to graduate a constant-pressure gas thermometer so that it shall show temperature on the absolute thermodynamic scale. Hence, notwithstanding the difficulty (§ 24 above) which Regnault found in the thermometric use of air or other gases on the system of constant pressure, and his practical preference for the constant-volume air thermometer, it becomes of the highest importance to construct a practical constant-pressure gas thermometer. This we believe may be done by avoiding the objectionable expedient adopted by Pouillet and Régnault of allowing a portion (when high temperatures are to be measured the greater portion) of the whole gas to be pressed into a cool volumetric chamber out of the thermometric chamber proper by the expansion of the portion which remains in; and instead fulfilling the condition, stated, but pronounced practically impossible, by Régnault (Experiences, vol. i. pp. 168, 169), that the thermometric gas “shall, like the mercury of a mercury thermometer, be allowed to expand freely at constant pressure in a calibrated reservoir maintained throughout at one temperature,” We have accordingly designed a constant-pressure gas thermometer to fulfil this condition. It is represented in the accompanying drawing (fig. 10), and described in the following section.

§65. The vessel containing the thermometric fluid, which in this case is to be either hydrogen or nitrogen, consists in the main of a glass bulb and tube placed vertically with bulb up and mouth down; but there is to be a secondary tube of much finer bore opening into the, bulb or into the main tube near its top, as may be found most convenient in any particular case. The main tube which, to distinguish it from the secondary tube, will be called the volumetric tube, is to be of large bore, not less than 2 or 3 centimetres, and is to be ground internally to a truly cylindric form. To allow this to be done it must be made of thick, well-annealed glass like that of the French glass-barrelled air-pumps. The secondary tube, which will be called the manometric capillary, is to be of round bore, not very fine, say from half a millimetre to a millimetre diameter. Its lower end is to be connected with a mercury manometer to show if the pressure of the thermometric air is either greater or less than the definite pressure to which it is to be brought every time a thermometric measurement is made by the instrument. The change of volume required to do this for every change of temperature is made and measured by means of a micrometer screw lifting or lowering along solid glass piston, fitting easily in the glass tube, and caulked air-tight by mercury between its lower end and an iron sole-plate by which the mouth of the volumetric tube is closed. To perform this mercury caulking, when the piston is raised and lowered, mercury is allowed to flow in and out through a hole in the iron sole-plate by an iron pipe, connected with two mercury cisterns at two different levels by branches each provided with a stopcock. When the piston is being raised the stopcock of the branch leading to the lower cistern is closed, and the other is opened enough to allow the mercury to flow up after the piston and press gently on its lower side, without entering more than infinitesimally into the space between it and the surrounding glass tube (the condition of the upper bounding surface of the mercury in this respect being easily seen by the observer looking at it through the glass tube). When the piston is being lowered the stopcock in the branch leading from the upper cistern is closed, and the one in the branch leading to the lower cistern is opened enough to let. the mercury go down before the piston, instead of being forced to any sensible distance into the space between it and the surrounding tube, but not enough to allow it to part company with the lower surface of the piston. The manometer is simply a mercury barometer of the form commonly called a siphon barometer, with its lower end not open to the air but connected to the lower end of the manometric capillary. This connexion is made below the level of the mercury in the following manner. The lower end of the capillary widens into a small glass bell or stout tube of glass of about 2 centimetres bore and 2 centimetres depth, with its lip ground flat like the receiver of an air-pump. The lip or upper edge of the open cistern of the barometer (that is to say, the cistern which would be open to the atmosphere were it used as an ordinary barometer) is also ground flat, and the two lips are pressed together with a greased leather washer between them to obviate risk of breaking the glass, and to facilitate the making of the joint mercury tight. To keep this joint perennially good, and to make quite sure that no air shall ever leak in, in case of the interior pressure being at any time less than the external barometric pressure or being arranged to be so always, it is preserved and caulked by an external mercury-jacket not shown in the drawing. The mercury in the thus constituted lower reservoir of the manometer is above the level of the leather joint, and the space in the upper part of the reservoir over the surface of the mercury, up to a little distance into the capillary above, is occupied by a fixed oil or some other practically vapourless liquid. This oil or other liquid is introduced for the purpose of guarding against error in the reckoning of the whole bulk of the thermometric gas, on account of slight irregular changes in the capillary depression of the border of the mercury surface in the reservoir.

§ 66. In the most accurate use of the instrument, the glass and mercury and oil of the manometer are all kept at one definite temperature according to some convenient and perfectly trustworthy intrinsic thermoscope (§§15 and 16 above), by means of thermal appliances not represented in the drawing but easily imagined. This condition being fulfilled, the one desired pressure of the thermometric gas is attained with exceedingly minute accuracy by working the micrometer screw up or down until the oil is brought precisely to a mark upon the manometric capillary.

In fact, if the glass and mercury and oil are all kept rigorously at one constant temperature, the only access for error is through irregular variations in the capillary depressions in the borders of the mercury surfaces. With so large a diameter as the 2 centimetres chosen in the figured dimensions of the drawing, the error from this cause can hardly amount to τ -J- σ ∙ per cent, of the whole pressure, supposing this to be one atmo or thereabouts.

For ordinary uses of our constant-pressure gas thermometer, where the most minute accuracy is not needed, the rule will still be to bring the oil to a fixed mark on the manometric capillary; and no precaution in respect to temperature will be necessary except to secure that it is approximately uniform throughout the mercury and containing glass, from lower to higher level of the mercury. The quantity of oil is so small that, whatever its temperature may be, the bringing of its free surface to a fixed mark on the capillary secures that the mercury surface below the oil in the lower reservoir is very nearly at one constant point relatively to the glass, much more nearly so than it could be made by direct observation of the mercury surface, at all events without optical magnifying power. Now if the mercury surface be at a constant point of the glass, it is easily proved that the difference of pressures between the two mercury surfaces will be constant, notwithstanding considerable variations of the common temperature of the mercury and glass, provided a certain easy condition is fulfilled, through which the effect of the expansion of the glass is compensated by the expansion of the mercury. This condition is that the whole volume of the mercury shall beιr to the volume in the cylindric vertical tube from the upper surface to the level of the lower surface the ratio (λ - ⅜σ)∕(λ - σ), where λ denotes the cubic expansion of the mercury, and σ the cubic expansion of the solid for the same elevation of temperature, it being supposed for simplicity of statement that the tube is truly cylindric from the upper surface to the level of the lower surface, and that the sectional area of the tube is the same at the two mercury surfaces. The cubic expansion of mercury is approximately seven times the cubic expansion of glass. Hence

(λ- ⅛<r)∕(λ- <r) = (7- ⅛)∕6 = 1'111. ,

Hence the whole volume of the mercury is to be about 1Ί11 time3 the volume from its upper surface to the level of the lower surface; that is to say, the volume from the lower surface in the bend to the same level in the vertical branch is to be of the volume in the vertical tube above this surface. A special experiment on each tube is easily made to find the quantity of mercury that must be put in to cause the pressure to be absolutely constant when the surface in the lower reservoir is kept at a fixed point relatively to the glass, and when the temperature is varied through such moderate differences of temperature as are to be found in the use of the instrument at different times and seasons.

A sheet-iron can containing water or oil or fusible metal, with external thermal appliances of gas or charcoal furnace, or low-pressure or high-pressure steam heater, and with proper internal stirrer or stirrers, is fitted round the bulb and manometric tube to produce uniformly throughout the mass of the thermometric gas the temperature to be measured. This part of the apparatus, which will be called for brevity the heater, must not extend so far down the manometric tube that when raised to its highest temperature it can warm the caulking mercury to as high a temperature as 40° C., because at somewhat higher temperatures than this the pressure of vapour of mercury begins to be perceptible (see Table V. below), and would vitiate the thermometric use of the pure hydrogen or nitrogen of our thermometer. To secure sufficient coolness of the mercury it will probably be advisable to have an open glass jacket of cold water (not shown in the drawing) round the volumetric tube, 2 or 3 centimetres below the bottom of the heater, and reaching to about half a centimetre above the highest position of the bottom of the piston.

§ 67. It seems probable that the constant-pressure hydrogen or nitrogen gas thermometer which we have now described may give even more accurate thermometry than Regnault’s constant-volume air thermometers (§ 24 above), and it seems certain that it will be much more easily used in practice.

We have only to remark here further that, if Boyle’s law were rigorously fulfilled, thermometry by the two methods would be identical, provided the scale in each case be graduated or calculated so as to make the numerical reckoning of the temperature agree at two points,—for example, 0° C. and 100° C. The very close agreement which Régnault found among his different gas thermometers and his air thermometers with air of different densities (§ 25 above), and the close approach to rigorous fulfilment of Boyle’s law which he and other experimenters have ascertained to be presented by air and other gases used in his thermometers, through the ranges of density, pressure, and temperature at which they were used in these thermometers, renders it certain that in reality the difference between Regnault’s normal air thermometry and thermometry by our hydrogen gas constant-pressure thermometer must be exceedingly small. It is therefore satisfactory to know that for all practical purposes absolute temperature is to be obtained with very great accuracy from Regnault’s thermometric system by simply adding 273 to his numbers for temperature on the centigrade scale. It is probable that at the temperatures of 250° or 300° C. (or 523 or 573 absolute) the greatest deviation of temperature thus reckoned from correct absolute temperature is not more than half a degree.

§ 68. The thermometric scale being now thoroughly established in theory and practice (§§ 33-69 and §§ 18-30), we are prepared to define, without any ambiguity, the expressions thermal capacity and specific heat with reference to matter at any temperature and in any physical condition.

Definition 1.—The thermal capacity of a body (whether it be a portion of matter homogeneous throughout or of homogeneous substance in two different conditions as liquid and steam, or solid and vapour of solid, or a piece of apparatus consisting of different parts as glass and metals, and containing as the case may be liquids or gases,—subject only to the condition that the whole matter considered is at one temperature) is the quantity of heat required to raise its temperature by one degree on the absolute thermodynamic scale.

Definition 2.—The specific heat of a substance is the thermal capacity of a stated quantity of it. This stated quantity is generally understood to be the unit of mass, unless some other definite quantity is explicitly designated, as for instance the quantity of the substance which occupies unit of volume at some definite pressure and temperature, for instance, one atmo and temperature 273 absolute. It is of no consequence what unit of mass is chosen provided it be the same as that which is used in defining the thermal unit; but, unless the contrary be explicitly stated, we always understand one gramme as the unit of mass and the thermal unit as the quantity of heat required to raise one gramme of water from 273 to 274 (compare § 6 above).

§ 69. There is scarcely any subject upon which more skilled labour in scientific laboratories, chemical and physical, has been spent than the measurement of specific heats, whether of solids, liquids, or gases. An ample and well-arranged table of results is to be found in Clarke’s Constants of Nature, a compilation of numerical results of scientific experiments made in all parts of the world by various observers and experimenters, a most valuable aid to scientific knowledge given to the world as No. 255 of the Smithsonian Miscellaneous Collections. It is most interesting as showing how very differently different substances behave in respect to constancy or variation of specific heat with temperature. Thus it shows that, according to the results of all the experimenters, the specific heats of all the substances experimented on, whether simple or compound, are very nearly constant at all events for ranges between - 10° and 200° or 300° C., except the three elementary substances, boron, carbon, silicon. The specific heats of these three have been found by F. Weber to vary greatly - with temperature. Thus for diamond he finds the specific heat to be 4 at 0° C. and ’ 27 at 20G o C., or nearly threefold of the amount at 0°; at - 50° C. the specific heat is Ό63; and at + 985° it is ’ 459 or about seven and a half times the specific heat at - 50° (a curious practical commentary, we may remark in passing, on the doctrine of the calorists on specific heat referred to in § 8 above). The specific heats of carbon in its other forms of graphite and charcoal through wide ranges of temperature according to the same observer, F. Weber, are particularly interesting and significant. The approximate equality of the product of specific heat into the atomic weight for the simple metals is interesting and important; no less so is the utter want of constancy and uniformity in the corresponding product for other substances, whether simple or compound. If we were to define a metal as a substance for which through the range of temperature from 0° to 250° C. the product of the specific heat into the atomic weight is not less than 5 - 86 and not greater than 6 - 93, we should include every substance commonly called a metal, and no substance not commonly called a metal, except phosphorus, and solid sulphur lately fused,

Some important results of Regnault’s regarding the specific heats of gases under constant pressure have been already quoted in §§ 56, 58, and 62 above. Further information from experiments aided by thermodynamic theory, regarding specific heats of gases and vapours under constant pressure and of gases in constant volume, will be found under the heading Thermodynamics. To this also, and to the articles Matter (Properties of), Liquid, and Steam, the reader is referred for information respecting latent heats of liquefaction of solids and of evaporation of solids and liquids, also respecting the thermal capacity of a portion of homogeneous substance in two different states, such as the water-liquid and water-steam of an ordinary cryophorus or philosopher’s hammer, or of the sulphurous acid liquid and steam of a sulphurous acid steam thermometer (§ 43 above).

Transference of Heat.

S 70. When two contiguous portions of'matter are at different temperatures, heat is transferred from the warmer to the colder. This process is called conduction of heat.

When two bodies at different temperatures are separated by a transparent medium, such as air, or water, or glass, or ice, heat passes from the warmer to the colder irrespectively of the temperature of the intervening medium, except in so far as its transparency may in some slight degree be affected by the temperature. Thus the colder of the two bodies becomes actually heated above the temperature of theinterven-ing medium if the warmer be kept above this temperature, and if heat is not otherwise drawn off from the colder body in greater quantity than the heat entering it from the warmer. This process of transference from one body to another body at a distance through an intervening medium is called radiation of heat. The condition of the intervening matter in virtue of which heat is thus transferred is called light; and radiant heat is light if we could but see it with the eye, and not merely discern with the mind, as we do, that it is perfectly continuous in quality with the species of radiant heat which we see with the material eye through its affecting the retina with the sense of light. Thus a white hot poker in a room perfectly darkened from all other lights is seen as a brilliant white light gradually becoming reddish and less bright, until it absolutely fades from vision in a dull red glow. Long after it has ceased to be visible to the eye, the fact that heat is being transferred from it to colder bodies all round it, or above it or below it, is proved by our sense of heat in a hand or face held near it on any side or above it or below it. By considering the whole phenomenon of the white hot mass, without much of experimental investigation, we judge that there is perfect continuity through the whole process, in the first part of which the radiant heat is visible and in the second part invisible to the human eye: and thorough experimental investigation confirms this conclusion. Thus radiant heat is brought under the undulatory theory of light, which in its turn becomes annexed to heat as a magnificent outlying province of the kinetic theory of heat.

§ 71. In this article we confine ourselves to a practical evaluation of rate of gain or loss of heat across the surface of an isolated solid placed in a medium such as air, and enclosed in a solid surface all at one temperature, as is approximately the case with the air and the floor, walls, and ceiling of an ordinary room. A rough approximation to the law of this action, founded on supposing the rate of motion to be in simple proportion to the excess of the temperature of the isolated solid above the temperature of the surrounding medium and enclosure was used by Fourier in those of his solutions in which surface emissivity or, as he called it, “Conductibilité extérieure,” is concerned. Without adopting any hypothesis, we define thermal emissivity as the quantity of heat per unit of time, per unit of surface, per degree of excess of temperature, which the isolated body loses in virtue of the combined effect of radiation and convection by currents of air. This definition does not involve the hypothesis of simple proportionality; and the surface emissivity is simply to be determined by experiment for any given temperature of the enclosure, and any given temperature of the isolated body. Dulong and Petit made elaborate experiments on this subject, but did not give any results in absolute measure.

So far as we know the first thoroughly trustworthy experiments giving emissivities in absolute measure were made in the laboratory of the natural philosophy class in the university of Glasgow by Mr D. Macfarlane, in a series of experiments on the cooling of a copper ball. The results are given in Table VII. The ball experimented on was 4 centimetres in diameter, and was suspended in the interior of a double-walled tin-plate vessel. The space between the double walls of this vessel was filled with water at the temperature of the air, and the interior surface was coated with lampblack. Two thermo-electric junctions, one at the centre of the ball the other in contact with the exterior surface of the enclosure, in circuit with a sensitive mirror galvanometer, served to measure the difference of temperatures between the centre of the ball and the exterior surface of the enclosure. By this arrangement the exterior junction was kept very uniformly at a temperature of 11’ C., while the other had the varying temperature of the centre of the ball. Two sets of experiments were made. In one the ball had a bright surface, in the other it was coated with soot from the flame of a lamp, and in both the air was kept moist by a saucer of water placed in the interior of the tinplate enclosure. The results are given in terms of the number of units of heat lost per second, per square centimetre of surface of the copper, per degree of difference between the temperatures of the two junctions.

§ 72. Returning to the conduction of heat, we have first to say that the theory of it was discovered by Fourier and given to the world through the French Academy in his Theorie Analytique de la Chaleur, with solutions of problems naturally arising from it, of which it is difficult to say whether their uniquely original quality, or their transcendently intense mathematical interest, or their perennially important instructiveness for physical science, is most to be praised. Here we can but give the very slightest sketch of the elementary law of conduction in an isotropic substance, the mathematical expression for it in terms of orthogonal plane or curved coordinates, and a few of the elementary solutions in Fourier’s theory.

§ 73. Consider a slab of homogeneous solid bounded by two parallel planes. Let the substance be kept at two different temperatures over these parallel planes by suitable sources of heat and cold. For example, let one side be kept cold by a stream of cold water, or by a large quantity of ice and water in contact with it, and the other kept warm by a large quantity of warm water or by steam blown against it. Whatever particular plans of heater and refrigerator be adopted, care must be taken that the temperature be kept uniform over the whole, or over a sufficiently large area of each side of the slab, to render the isothermal surfaces sensibly parallel planes through the whole of the slab intercepted between the two calorimetric areas, and that the temperature at each side is prevented from varying with time. It will be found that heat must continually be applied at one side and removed from the other, to keep the circumstances in the constant condition thus defined. When this constant condition of surface temperature is maintained long enough, the temperature at every point of the slab settles towards a constant limiting value; and when this limiting value has been sensibly reached by every point of the slab, the temperature throughout remains sensibly constant so long as the surface temperatures are kept constant. In this condition of affairs the temperature varies continuously from one side of the slab to the other; and it is constant throughout each interior plane parallel to the sides; in other words, the isothermal surfaces are parallel planes. Let V and V' be the temperatures in two of these isothermals and a the distance between them. The quotient —-— is the average rate of variation of temperature per unit of length between these two isothermals. Let Q be the quantity of heat taken in per unit of time at a certain area A on one side, and emitted at the corresponding area of the other side of the slab, measured by proper calorimetrical appliances to these areas, which we shall call the calorimetric areas of the apparatus. It will generally be found that the value of the quotient (V - V')∕α is not the same for consecutive isothermal surfaces. For metals it is ascertained by experiment that it increases continuously from the cold side to the hot side of the slab. In other words, as we shall see presently, the thermal conductivity of the substance is not generally the same at different temperatures, and for metals it is smaller the higher the temperature.

§ 74. Circumstances being as described in § 73, the thèrmal conductivity of the substance between the isothermals v and √ is the value of

-⅜ L- (i).

A(ι>-√) k '

It must be remembered that the temperatures v, v' used in this definition are temperatures of the substance itself. Some experimenters have given largely erroneous results through assuming that the temperatures of the two sides of the slab were equal to those of the calorimetric fluids, such aS warm water or steam on one side, and iced water or cold water, with its temperature measured by a thermometer, on the other side. To obtain correct results, the actual temperatures at two points in the conducting body itself must be ascertained by aid of suitable thermometers, or thermometers and differential thermoscopes, applied in such a way as not sensibly to disturb the isothermal surfaces. This, so far as we know, has not been done by any experimenter hitherto in attempting to measure thermal conductivity directly by the method indicated in the definition; and therefore if any results obtained by this method hitherto are trustworthy, it is only in a few cases,—caseβ in which, unless the substance experimented upon has been of such small conducting power, and the stirring of the calorimetric fluids on its two sides so energetic, that we can feel sure that the observed or assumed temperatures of these fluids, or of the portions of them of which the temperatures have been measured by thermometers, have not differed sensibly from the temperatures of the slab at its surfaces in contact with them.

§ 75. What utter confusion has permeated scientific literature, from experiments on thermal conductivity vitiated through non-fulfilment of this condition, is illustrated by results quoted in Everett’s Units and Physical Constants (London, 1879), among which we find ’ 19 for the conductivity of copper according to Péclet, and IT according to Angström (which we now know to be correct). When we look to Péclet’s and Angstrom’s own papers the confusion becomes aggravated. Péclet, in his Hémoire sur la determination des coefficients de conductibilité des métaux par la chaleur ? quotes old experiments of Clément, and others more recent of Thomas and Laurent, regarding which he gives certain details. Taking his information no doubt from Péclet’s paper, Angström gives a statement for the conductivity of copper, according to experimenters who had preceded him, which, with the decimal point shifted two places to the left to reduce to C. G. S., is as follows - [table]

But Angström did not notice that Péclet had stated the thickness of the plate experimented on by Clément to be between 2 and 3 millimetres. Péclet himself in his next sentence seems to have forgotten this when he compares the figure , 23 which he had calculated from Clément’s results, without taking account of the thickness of the plate, with 1 , 22 which he calculates from Thomas and Laurent’s experiments on copper, without stating any thickness for the tube of copper on which (instead of a flat plate) they had •experimented. Thus we have no data for finding what their results really were in either Péclet’s or Angstrom’s paper; but Péclet seems to show enough regarding it to let us now feel perfectly sure that it is only a question of whether it is tens or hundreds of times too small. Omitting it then from the preceding statement, completing the correction by multiplying the Ό023 by 2⅛ (assuming the thickness of the plate to be 2∣ millimetres, as Péclet says it was between 2 and 3) to give Clément’s result, and appending Angstrom’s result, which we now know to be correct, we have the following statement for thermal conductivity of copper in C. G. S. units :—

•0057, according to Clément. •178, „ „ Péclet.

1Ί, „ ,, Angstrom.

§ 76. The comparison of these results is highly instructive. Clement’s result is two hundred times too small, and Péclet’s five. times too small. Clément experimented by exposing one side of a plate of copper of a square metre surface and about two and a half millimetres thick to steam at 100° C., and the other side to water at 28° C. It was assumed that the difference of temperatures between the two sides was 72°. The difference really was about - 3G of a degree, as we know from the quantity of heat actually conducted through it in Clément’s experiment, indicated by the amount of steam which he found to be condensed into water. In fact, the amount of steam condensed did not differ sensibly from what it would have been if the copper plate had been infinitely thin, or its substance of infinite thermal conductivity. It is important in engineering, and in many of the arts and manufactures involving thermal processes, and particularly in that one of them of greater everyday value to the human race than all the others put together, cookery, to know that for copper or iron boilers, or steampipes, or pots or frying-pans, the transmission of heat from radiant burning coal or charcoal, or red or white hot fireclay or other solids, and from hot air in contact with them, on one side, to hot water or steam or oil or melted fat on the other side, or hot liquid or steam on one side and cool air on the other side, is for practical purposes sensibly the same as if the thermal conductivity of the metal were infinite, or its resistance to the transmission of heat nothing. The explanation is obvious to us now with the definite and sure knowledge regarding thermal conductivities of different substances and of matter in different conditions, solid, liquid, and gaseous, gained within the last twenty years. Angström, Forbes, F. Neumann, and Tait have given, each one of them with thoroughly sufficient experimental evidence to leave no room to doubt the substantial accuracy of his results, absolute values for the thermal conductivities of copper and iron. Clausius and Maxwell have given us thermal conductivities of air and other gases, from their splendid development of the kinetic theory, which are undoubtedly trustworthy as somewhat close approximations to the true values, and which it is quite possible are more accurate than we can hope to see obtained from direct measurements of the conduction of heat through gases. J. T. Bottomley has given a trustworthy and somewhat closely accurate direct measurement of the . thermal conductivity of water. From the results of these experimenters’ work, reduced to uniform C. G. S. reckoning in our table of thermal conductivities (Table VIII.), we see that the thermal conductivity of iron is 80 times, and that of copper 500 times that of water. The thermal conductivity of iron is 3500 times, and that of copper is 20,000 times that of air. Hence, although the water or air at the very interface of its contact with the metal is essentially at the same temperature as the metal, there must be great differences of temperature in very thin layers of the fluid close to the interface when there is large flux of heat through the metal, and the temperature of the fluid as measured by any practicable thermometer, or inferred from knowledge of the average temperature of the whole fluid, or from the temperatures of entering and leaving currents of fluid, may differ by scores of degrees from the actual temperature of the solid at the interface. It is remarkable that Péclet, while perceiving that Clément’s result was largely erroneous on this account, and improving the mode of experimenting by introducing a rotating mechanical stirrer to change very rapidly the fluid in contact with the solid, only multiplied Clément’s conductivity by 30 instead of by 200, which would have been necessary to annul the error. Notwithstanding his failure to obtain accurate results for metals, we have ventured to include his results for wood, and solids of lower conductivity than wood, in our table, because we perceive that he was alive to the necessity for very energetic stirring of the liquid, and the mechanical means which he adopted for it, though utterly insufficient for the case of even the least conductive of the metals, were probably not so for wood and solids of lower conductivity than wood; and because it is not probable that the complication of heat generated by the stirring (which Angström suggests as an objection to Péclet’s method) was in any case sufficient to produce a sensible influence upon the experimental results.

§ 77. The first correct determinations of thermal conductivities were given by Forbes in his paper on the temperature of the earth, in the Transactions of the Royal Society of Edinburgh for 1846, as calculated from his observations of underground temperature at three localities in the neighbourhood of Edinburgh—the trap rock of Calton Hill, the sand of the Experimental Garden, and the sandstone of Craigleith Quarry—by an imperfect approximate method indicated by Poisson. A more complete analytical treatment of the observational results, analysed harmonically and interpreted by application of Fourier’s formula (equation (19) of Math. App.) to each term separately by W. Thomson, gave results (quoted in Table VIII. below) for the conductivities, which differed but little from Forbes’s approximate determinations.

§78. It has always seemed to us that the best mode of experimenting on the conductivity of metals must, without doubt, be by an artificial imitation in a metallic bar, of the natural periodic variations of underground temperature, produced by periodically varied thermal appliances at one end of the bar. The effect of loss or gain of heat through the sides (or lateral surface) of the bar (ideally annullable by a coating of ideal varnish impermeable to heat) may be practically annulled by making the period of the variation small enough.

Let k be the thermal conductivity of the substance and c its thermal capacity per unit bulk. Let e be the emissivity (§ 71) of its surface. Let the bar be circular-cylindric, and r the radius of its cross section. At time t, let v be the mean temperature in a cross section at distance x from the end, and √ the surface temperature at the circular boundary of this section,—all temperatures being reckoned as differences from the temperature of the surrounding medium, called zero temporarily for brevity. The heat lost from the circumference of the bar between the cross sections x - ⅛clx and X + ∖ dx in time dt is ev'. 2πrdxdt, and the heat conducted lengthwise, across the cross section x, in the· same time, is πr 2 -. Hence we dx readily find (see Math. App. below) as the ecpiation of conduction of heat along the bar, very approximately if υ ’ differs very little from v (that is to say, if the temperature is very nearly uniform throughout each cross section), dν d h,dv ∖ 2e , z 1 - v c M~dx ∖ k d^)~ ~r V )r

To estimate √, let v" be the temperature at the centre of the cross section x, and let ( BJ- ∖ denote the rate of decrease of tempera- ∖ dr ) ture from within outwards in the substance of the bar close to its surface. We have clearly y '- y < r (z ¾) ι and, because the emission is supplied by conduction from within, , ∕-dv ∖ ' e "- i ∖ ~ l Γr)-

Hence v" — r'<≤ — υ'. k

The value of e for a blackened globe hung within a hollow, with blackened bounding surface, is about τσ ⅛- σ according to Maefarlane’s experiments (Table VII. below), and considerably less for surfaces with any degree of polish. We may therefore take τ √<∏τ as a maximum value for e. The values of k for copper and iron at ordinary temperatures are, in C. G∙. S., approximately ’ 95 and ‘18. Hence, ifr=5 cms. (or the diameter of the bar 10 cms., which is more than it is likely to be in any laboratory experiments), we find

√' —√<'.dL for copper, ^880 11 and

<Γ— for iron.

IGO

Hence the error will be practically nothing if we take √=ι>. Thus, and if we suppose k to be independent of temperature, (1) becomes dυ _k d i v 2e dt c dx i rc 3 ’ or dv ~ d 2 υ , dt κ dx i ° ’ which is Fourier’s equation for the conduction of heat along a bar or the circumference of a thin ring. Its solution to express simple harmonic variations of temperature produced in an infinitely long bar by properly varied thermal appliances at one end is υ = R,e~s χ cos(pt-fx + e) (4), n,B, e being arbitrary constants,—the “speed,” the semi-range, and the epoch for ic=0; and f g constants given by the formulae ∕=a∕ 2H (w2 + A2 ^" 7i Π

∕ 1 )» · · · · (5)∙

<7=V 2ic((,1 + as ) 1 + a j J

For iron and copper the values of c are respectively ’ 95 and ∙845. Hence, with the previously used values of k for these metals, and with 1/4000 for c, we find κ = l'l for copper and κ= '2 for iron; and for either, Λ = l∕1700r nearly enough.

Suppose, for example, r=2 cm., this makes A=l∕3400; and s o uppose the period to be 32 m. (the greatest of those chosen by Angström), this makes n = 2π∕(60 × 32), or roughly n=l∕310 and h ∣ n=l∕17. Now when h ∣ n is small, we have approximately ∕-√2⅞∙('-⅛)∙ '-√ ’ ≡ - (1+ ⅛) i and therefore with the assumed numbers z "√ z 2⅛(1 -⅛) , ωdf -√ z 2L(1 + 3⅛) ’ by which we see that the propagation of the variation of temperature is but little affected by the lateral surface emissivity. Little as this effect is, it is very perfectly eliminated by the relation

Λ=2- (6), κ which we find from (5).

It is convenient to remark that g is the rate of diminution of the Napierian logarithm of the range, and f the rate of retardation of the epoch (reckoned in radians) per centimetre of the bar. Were there no lateral emissivity these would be equal, and the diffusivity might be calculated from each separately. This was done by William Thomson in his analysis of the Edinburgh underground temperature observations. But in the propagation of periodic variation of temperature along a ⅛χr (as of electric potential along the conductor of a submarine cable) lateral emissivity (or imperfect insulation) augments the rate of diminution of the logarithm of the amplitude, and diminishes the rate of retardation of the phase, leaving the product of the two rates unaffected, and. allowing the diffusivity to be calculated from it by (6). This was carried out for copper and iron by Ängström in Sweden, and the results communicated to the Royal Swedish Academy in January 1861. German and English editions of his paper have been published in Poggendorff’s Annalen for 1863, and the Phil. Mag. for 1863 (first half year). The details of the apparatus and of the actual experiments, in which Angström had the assistance of Thalen, are sufficiently described in this paper, 1 and in a subsequent paper (Pogg. Annalen for 1863, p. 428), to allow us to feel perfect confidence in the very approximate accuracy of the results. Hence we have included them in our Table.

§ 79. The question, Does thermal conductivity vary with temperature! was experimentally investigated by Forbes about thirty years ago; and in a first provisional statement of results communicated to the British Association at Belfast in 1852 it was stated that the thermal conductivity of iron is less at high temperatures than at low. Forbes’s investigation was conducted by an elaborate method of experimenting, in which the static temperature of a long bar of metal is observed after the example of the earlier experiments of Despretz, with a most important additional experiment and measurement by means of which the static result is reduced to give conductivity in absolute measure, and not merely as in Despretz’s experiments to give comparisons between the conductivities of different metals. In 1861 and 1865 Forbes published results calculated from his experiments, including the first determination of thermal conductivity of a metal (iron) in absolute measure, and a confirmation of his old result that the conductivity of iron diminishes with rise of temperature. Forbes’s bars have been inherited and further utilized, and bars of copper, lead, and other metals have been made and experimented upon according to'the same method, by his successor in the university of Edinburgh, Professor Tait. The investigation was conducted partly with a view to test whether the electric conductivities and the thermal conductivities of different metals, more or less approximately pure, and of metallic alloys, are in the same order, and, further, if their thermal conductivities are approximately in the same proportion as their electric conductivities. The following results quoted from his paper on “Thermal and Electric Conductivity” (Transactions R.S.H., 1878) are valuable as an important instalment, but expressly only an instalment, towards the answering of this interesting question:—

“Taking the inferior copper (‘ Copper C ’) as unit both for thermal and for electric conductivity, we find the following table of conductivities at ordinary temperatures, with the rough results as to specific gravity and specific heat referred to in § 15 above :— [table]

1 The first paper is marred unhappily by two or three algebraic and arithmetical errors. One algebraic error is very disturbing to a careful reader, and might even to a hasty judgment seem to throw doubt on the validity of the experimental use which is made of the formulae. There is, however, no real foundation for any such doubt. The following little correction suffices to put the matter right. For the general term as printed in Angstrom’s paper read

,-*0i√i . i2iιτt , .. . ο , ∖ e sin (-¾√⅛ + 3 1, with the following values for ι ¾, g' i :— ∕ ∕ ÿ -≡ · A V V K 2 T 2 + 4K⅛ + 2Ki ’ √= ∕ ∕ * 2 . ∏ ii h ’ V V κ 2 τ 2 + 4κ⅜ s 2Kι'’

instead of these formulae without the i, as Angstrom gives them. Here we see that g i g' i — -A.; and it is the product gf { that Angström

* K.T

uses in his experimental application, not the separate values of either g. or gi. Hence no error is introduced by his having overlooked that gi is not equal to g 1 except for i≈=l.

tt The agreement of these numbers is by no means so close as is generally stated; but this is no longer remarkable, for it is well known that the electric conductivity of all pure metals alters very much with the temperature, while we have seen that as regards thermal conductivity there is but slight change with either copper or lead, though there is a large change with iron. This accords with some results of my own on the electric conductivity of iron at high temperatures (Proc. R.S.B., 1872-73, p. 32), and with the results of the repetition of these experiments by a party of my laboratory students.”— Proc. R.S.E., 1875-76, p. 629.

§ 80. The absolute values of Tait’s results for the five metals of the preceding list are given in C. G. S. units in our general table. As to change of conductivity with temperature there is a discrepance between the results of Angstrom’s and Tait’s experiments. Tait finds but little difference in the conductivity of copper through the wide range of temperatures from 0° to 300° C., and that difference an augmentation instead of a diminution at the high temperatures as shown in the following results (where t is measured from 0° C.) d— [table]

On the other hand, Ängström finds for copper from experiments described in his second paper referred to above, at mean temperatures of from 28 0- 8 to 71 0, 5 C., results which reduced to C.G.S. are as follows :— [table]

From the admirable method of experimenting, and the care with which experiments were carried out by himself and Thalen, it is impossible for us to doubt the validity and somewhat closely approximate numerical truth of the result. On account of the discrepance from Tait’s results it is desirable that Angstrom’s method should be carried out for copper through a much wider range of temperatures. This can be done with great ease from the lowest temperature obtainable by freezing mixtures to temperatures up to the melting point of copper, so excellently plastic is Angstrom’s method. Our proposed extension of it is to be carried out by proper thermal appliances to the end of the bar which Angstrom left to itself,—appliances by which in one series of experiments it may be kept constant at - 50° or - 60° C., in others left to itself to take nearly the atmospheric temperature, in others kept at high temperatures limited only by the melting temperature of copper, if the experimenter desires to go so far. We would also suggest that the thermo-electric method first introduced by Wiedemann and Franz in their experiments on the static temperature of bars or wires heated at one end and allowed to lose heat by convection and radiation from their sides, (which was rejected, not, we think, judiciously, by Angström), might be used with advantage instead of the mercury thermometers inserted in holes in the bar in Angstrom’s apparatus; or that, if thermometers are to be used, air thermometers in which the bulb of the thermometer is itself a very small hole in the bar experimented on, and the tube a fine-bore glass tube fitted to this hole, would be much preferable to the mercury thermometers hitherto employed in, we believe, all experiments except those of Wiedemann and Franz, on the conduction of heat along metallic bars.

§ 81. Fourier’s ninth chapter is entitled “De la Diffusion de la Chaleur.” The idea embodied in this title is the spreading of heat in a solid tending to ultimate equalization of temperature throughout it, instead of the transference of heat from one body to another by conduction through the solid considered. Though Fourier makes the special subject of his chapter on “Diffusion” the conduction of heat through an infinite solid, we may conveniently regard as coming under the several designations “Diffusion of Heat” every case of thermal conduction in which the heat conducted across any part of the solid has the effect of warming contiguous parts on one side of it, or of leaving contiguous parts on the other side cooler,—in other words, every case in which the temperature of the body through which the conduction of heat takes place is varying with time, as distinguished from what Fourier calls “Uniform Motion of Heat,” or the class of cases in which the temperature at every point of the body is constant. The experiments of Γ0clet, Despretz, Forbes and Tait, Wiedemann and Franz, were founded on the uniform conduction of heat across slabs or along bores, and their determinations of relative and absolute conductivities were made by comparing or by measuring absolutely quantities of heat that were conducted out of the body tested. On the other hand, it is the diffusion of heat that is used in the determinations of thermal conductivity in absolute measure by Forbes and William Thomson from the periodic variations of underground temperature; in those of Angström, from his experiments on the spreading of periodic variations of temperature through bars of iron and copper, and a series of valuable experiments a year or two later by F. Neumann, applying the same general method to bars of brass, zinc, German silver, and iron; in experiments by F. Neumann on substances of lower conductivity (coal, cast sulphur, ice, snow, frozen earth, gritstone) formed into cubes or globes of 5 or 6 inches diameter, and heated uniformly, and then left to cool in an atmosphere of lower temperature, and from time to time during the cooling explored by thermo-electric junctions imbedded in them to show their internal distribution; in similar experiments on the cooling of globes of 14 cm. diameter of porphyritic trachyte by Ayrton and Perry in Japan; and in Kirchhoff and Hansemann’s recent experiments, to find the thermal conductivity of iron by the not well-chosen method of suddenly cooling one side of a cube of iron of 14 cm., and observing the temperatures by aid of thermo-electric junctions in several points of the line perpendicular to this side through its middle.

§ 82. When the effect of heat conducted across any part of a body in heating the substance on one side or leaving the substance on the other side cooler is to be reckoned, it is convenient to measure the thermal conductivity in terms, not of the ordinary general gramme water-unit of heat, but of a special unit, the quantity required to raise unit bulk of the substance in 1°. In other words, if k be the conductivity in terms of any thermal unit, and c the thermal capacity of unit bulk of the substance, it is k∕c, not merely k, that expresses the quantity of the substance on which the phenomenon chiefly depends. We therefore propose to give to k/c the name of thermal diffusivity (or simply diffusivity when heat is understood to be the subject), while still using the word thermal conductivity to denote the conducting power as defined in § 73, without restriction as to the thermal unit employed. It is interesting and important to remark that “diffusivity” is essentially to be reckoned in units of area per unit of time, that its “dimensions” are L 2 ∕T (see Dimensions). Its regular C.G.S. reckoning is therefore in square centimetres per second. In the article Diffusion the relation between diffusion of heat and diffusion of matter is explained. We have added diffusion of electricity through a submarine cable, which has been shown to follow the same law as the “linear” diffusion of heat, as Fourier calls the diffusion of heat when the isothermal surfaces are parallel planes. The curves of the following diagram and Tables A, B, and C show in a practically useful way the result in the course of the times noted, of from fractions of a second to thousands of millions of years, of linear diffusion of two different qualities in an infinite line from an initial condition in which there is sudden transition from one quality to the other, in the thoroughly practical cases specified in the accompanying explanations.

. Equation of No. 2 is σ - x 2 ∣ 4xt V “2(1 rκ<!)* e proved in Math. App. xii. (27). Curve No. 1 is found by quadrature from (2) thus—

y =const. × fvdx. '

Curve No. 1 shows temperature; or quantity of substance in solution; or potential in the conductor of a submarine cable through which electricity is diffusing. Curve No. 2 shows rate per unit of distance of variation of the temperature, or of the quantity of substance in solution. Vertical ordinates are actual distances through the medium. Horizontal ordinates represent temperature or quantity of diffusing substances in No. 1 curve, and rate of variation of temperature or of diffusing substance or of electric potential in No. 2 curve.

Diffusions.—Table A. [table]

Diffusions {Secular).—Table B. [table]

1 Instructive as to the proportion of carbonic acid in air at different heights, proving its approximate uniformity due to convection, not to diffusion.

2 Instructive as to geological theories respecting terrestrial temperature.

i Instructive as to theories respecting the saltness of the sea.

Table C. [table]

§ 83. The following tables contain useful information regarding various thermal properties of matter. Every known property of a piece of matter, except its gravity and its inertia, varies with variation of temperature. For further information respecting the effect of variation of temperature in causing changes in properties of matter reference is made to the articles Elasticity, Electricity. Magnetism, Light, Matter (Properties of), Ac.

Table I.— Linear Expansions of Solids. [table]

[table] [table]

Table II. 1 — Cubical Expansions of Solids and Liquids. [table] [table]

1 Abridged from Clarke's Constants of Nature. Table ΠI.— Expansions of Liquids.

Γierre, having measured the expansions of many different liquids, embodies the results in the empirical formula δ» = at + δZ 2 + cZ 3 , where δ< represents the expansion of unit volume from 0° to V cent-, and a, b, c have the values given in the Table for the different liquids specified. The substances and formulae in quotation marks are quoted from Dixon’s Heat, Dublin, 1849. [table]

Table IV. — Density of Water. [table]

Table V.— Steam: Pressures (in Centimetres of Mercury). [table] [table]

Table VI.— Steam Pressures. [table]

Table VII.— Emissivity for Heat of Polished and Blackened, Copper Surfaces. [table]

Table VIII.— Thermal Conductivities. [table]

Table IX.— Diffusivities {Thermal, Material, and Electric'). [table] [table]

1 This result has been accidentally printed , 00000116 in some impressions of vol. vii. p. 217.

Table X.— Relative Diffusivities of different Substances through Water’. [table]

Mathematical Appendix.

Let V be the temperature at any point P specified by ξ, η, f according to any system of three sets of plane or curved orthogonal surfaces used for co-ordinates. Let λ<fξ, μd∙η, vdζ be the lengths of the edges of the infinitesimal rectangular parallelepiped having P for its centre, and its sides parts of the six surfaces ξ-⅛<Zξ, ξ + ⅛dξ, V~⅛dη, v + ⅛hη, ζ-⅛lζ, ζ+⅛dζ. .

The rates of variation of temperature per unit of length in the directions corresponding to the variations of ξ, ∙η, ζ are respectively 1 dv 1 dv 1 dν

A dζ ’ μ dη ’ V dζ

Hence the fluxes across three infinitesimal rectangles having their edges parallel to the three pairs of sides of the parallelepiped, and each having its centre at P, are respectively - ⅛ · μvd∙ηdζ, - - ^ v rλdζdξ , - - ^hμdξdη . A dξ μ dp V dς

Hence the excess of the quantities of heat conducted in to the parallelepiped above those conducted out across the three pairs of faces is j d ∕ ^μv dv ∖ + di 1 vK d ∕ k Xμ Jv ∖ ) dξw .,..( 1 ),

( rfξ∖ A dξj d∙η ∖ μ d∙ηj dζ ∖ v dv J )

The effect of this gain of heat is to warm the matter of the parallelepiped at a rate per unit of time equal to the rate of gain of heat divided by 6λμvdξdηdζ, the thermal capacity of the matter.

Hence dv _ 1 ∖ d Cμv dυ ∖ d ( 1 vK dv ∖ d( 1 Kμ dυ ∖∖ . 2 × dt cKμv ∖ dξ ∖ A dξj d-η ∖ μ dη) dζ ∖ v dζ) )

This, for the case of the uniform motion of heat {dvidt = O), was first given by Lamé, to whom the generalized system of curvilinear coordinates for a point is due (“Mémoire sur les Lois de l’Équilibre des Fluides Ethéres,” Journal de VÉcole Polytechnique, vol. iii., cahier xxiii.). He deduced it from Fourier’s equation [(3) below] in terms of plane rectangular coordinates by a laborious transformation. Equation (2) was first given, proved as above, as the direct expression of Fourier’s fundamental law of conduction, by W. Thomson {Cambridge Mathematical Journal, Nov. 1843).

For plane rectangular coordinates we have λ=*∕χ = v=l, ami if we put x, y, z for ξ, η, ζ in this case, (2) becomes dv 1 (d ( 1 dv ∖ , d ∕ 1 dv ∖ , d ∕ j dv ∖ ) , fn which is Fourier’s celebrated fundamental equation. From it we may deduce by transformation the proper forms of the corresponding equation for polar coordinates; but they are more easily got direct from the equation (2) for generalized coordinates. Thus for ordinary polar coordinates r, θ, φ we have, if we take these for ξ, ∙η, £ respectively,

λ = l, μ = r, v=r sinø .

Hence (2) becomes

*>l'(1 ⅛) t Λ⅛ t ⅛⅛) + X⅛*)! (4). dt cr-(dr ∖ dr J sinø dθ γ dθ ∕ sιn 2 0 dφ ∖ dφj ) φ

If k be constant, and we put k ∣ c=κ, this becomes 1 d Λj n ∕ φ V 1 fβV

dt r 3 ∙ ( <Zr∖ dr J sinø (Z0∖ dθ j sin 2 0 dφ 2 ) ' ’ or du_ 1 dPu 1 di. c ftu ∖ 1 d?u ) i dt K ( dr 2 sinø d0∖ in dθ J + sin 2 0 dφ' i ) ( · (θ)∙

where u = vr )

If again we take for the coordinates r, φ, z (polar coordinates in the plane perpendicular to z being denoted by r, φ), we have λ = l, μ = r, y=l, and so find dv 1 (d f l dv ∖ 1 d (k dv ∖ l d ( 1 dv ∖ ) , h ,.

For the case of k constant we may take it outside the brackets in each of these equations, as we have already done in (4); thus (2) becomes (fø_ k f cPv d?v d 2 υ ∖ . dt~ c ∖ dx i + djf + d¾) or, with κ for kjc, the diffusivity (§§ 81, 82), dv _ (d 2 v d s v , d 2 v ∖ , n . dt κ ∖ djP + dy i + d^) (

It is this restricted form which, with the further restriction that c be constant, is .most generally recognized as Fourier’s equation of conduction, and it is for it, with these restrictions, that his brilliant solutions were given. These solutions are available for practical use by limiting the range of temperature within which any one solution is continuously applied to a range of temperature within which thé values of⅛ and c are each nearly enough constant. We may expect 10 or 20 C. on each side of the mean temperature to be practically not too wide a range for any case, judging from copper and iron (§ 80), the only substances for which hitherto we have any information as to variations of both k and c with temperature.

Each of the following expressions I XVII. for v satisfies

(9) or its equivalent (6), as the reader will readily verify for himself. The special condition corresponding to the peculiar character of the particular solution is specially noted in each case.

I. Instantaneous simple point-source; a quantity Q of heat suddenly generated at the point (0, 0, 0) at time i = 0, and left to diffuse through an infinite homogeneous solid.

Every other solution is obtainable from this by summation.

Q e →W ) ’“W < 1 °)∙

where r 2 = x 2 + y 2 + z 2 1

Verify that f· f f vdxdydz = iπ j vr 2 dr=Q∙, and that _Ζ-χ Z-∞ Jo v = 0 when i = 0; unless also x=0, j∕ = 0, z = 0.

Remark that

^=0 when i-Δ (11).

<ft 6κ

II. Constant simple point-source, rate q:

JL . . . (12).

L /-co 8√>(κ i)^ J 4πr

The formula within the brackets shows how this obvious solution is derivable from (11).

III. Continued point-source; rate per unit of time, at time t, an arbitrary function, ff): ∕∙∞ c~ r 2 ∕4κχ

∙√ · ∙. · ∙. < 18, ∙

IV. Time-periodic simple point-source, rate per unit of time at time t, q sin 2?ft:

t∙Γ = q ∕'¾sin2n(iχ) É Ί =≡ -s- 8 in [2ni- (κn)U] (14).

L jo 8π¾κχ)⅛ j i7rr * ∕7η

Verify that v satisfies (6); also that - 4ιrτ 2 — = q sin2?ft where r = 0.

V. Instantaneous spherical surface source; a quantity Q suddenly generated over a spherical surface of radius a, and left to diffuse outwards and inwards:

ζ-(,ra) t ∕4κt (r ÷α) 2 ∕4κ< t,-Q .... (15).

8π≡aF(κ0≡

To prove this most easily, verify that it satisfies (6); and farther verify that

4π ∕" w 2 (Zr=Q; Jo and that v∙=0 when t = 0, unless also r=α.

Remark that (15) becomes identical with (10) when α∙=0; remark farther that (15) is obtainable from (10) by integration over the spherical surface.

VI. Constant spherical surface source; rate per unit of time from the whole surface, q:

Γ^ ∕* 0 ° f-(r - α ) 2 ∕ 4κi —c-(r + α ) 2 ∕ 4κlι ~∣ Ί

v ∖ ,- ’ f m - l L.<ι β ), = qfπr , where rz=-a and = q ∣ 4∙πa, where r<α J

The formula within the brackets shows how this obvious solution is derivable from (13).

VII. Time-periodic spherical surface source; rate per unit of time, at time t, from whole surface, q sin %nt:

Γ ∕* c0 , · „ , j xe -(r-α) 2 ∕4κχ -e -(r+α) 2 ∕4∕cχ~] ∖

< ι dχ s ' n2 "' i - x) —ÄW J

=-∈ - sin[2πi - (κn)U +B], where r>α, -(17),

= ~{∈~(κn )⅛in[27ft-(∕cn)b∙ + D]-e^ n )¼in[2ni + (κn)U+D]) where r<α J where A B, C, D are constants determined by the conditions that ⅞>α-' t 'r<α = 0 > and -4^j(g)^-(g)^J- is in2,⅛ when the two values of r exceed a and fall short of a by infinitely small differences. Verify that v satisfies (6). Also that v is finite when r=0.

VIII. Fourier’s “Linear Motion of Heat”; instantaneous planesource; quantity per unit surface, <r:

σ p-x ι ∣ 4κt v= σ ∖ ⅛- (18).

2π⅛κt) i

Verify that this satisfies (9) for the case of v independent of y and z, and that ∕∙00

∕ vdκ = σ.

Remark that (18) is obtainable from (15) by putting Q∕4πα 2 = <r, and a=-∞ ; or directly from (10) by integration over the plane.

IX. “Linear Motion of Heat”; time-periodic plane-source; rate per unit of area, per unit of time, at time t, σ sin 2nt:

Γ ∕* α0 e -W i *x~] λ

*'L-√ ⅛ sinto < i -× ’ s⅜⅛J

— —- 1 —e"“Ia! sin(2n<-n⅛-i.) where as-0 ) . (19). (2n y p2π

= ——∈+ 7i ^ z sin(2⅛i+ ?iU-4τr) where z<0 1

(2n)i2π i j

Verify that ν satisfies κ e L θ which is what (9) becomes at dx i when ν is independent of y and z ; also that d,v

- 2π = asin 2nt when x is infinitely small positive, and +2π^= σ sin 2nt when x is infinitely small negative.

X. “Linear Motion of Heat”; space-periodic simple harmonic solid source, with plane isothermal surfaces. Initial distribution, τ = V sin ax, when t~0. Solution for any value of ∕

r^ ∕*° 0 „ -,

V v Z dξ βillβiaι '‰∣]° vr “8inαx (20) · Modifying the integral within the brackets to make it appear as an analytical expression belonging to the general theory of images, for the case of a single infinite row of images,· and equating the result to the right-hand, member, ∖ve see that 2τr

I ∕*~λ~ l=αo ∕ 2iτr∖ 2

ξf t ∕ dξ ≡ i 0 n s a(x-ξ) 2 ∈-V + v) z4κ, = e -* a ⅛^ (V), ' 0 i-- ∞ i being any integer.

It is obvious that for a we may substitute ja in the second member, and in the factor α(z - ξ ) under the integral sign of the first member, without altering it elsewhere, j being any integer: thus we have

2ιr T κ tf^ d * ∞ n s>^- i)S = e^ κj2α2i c s * n s j⅛]

1 Q <= ” ~(ξ+ ¥) 2/4κ< where S = ∈ ' « > i≈—co J

XI. “Linear Motion of Heat”; space - periodic arbitrary solid source, isothermals plane. Initial distribution, v=f(x), when Z = 0, /denoting an arbitrary periodic function, period l ; so that/(a: + il) =fx), i being any integer. Two solutions (A), (B).

(A) derived synthetically from (10):

l -2(⅛ i ∕'^ z '< χ - ξis · · · ∙< 23 > '> where 8- ⅞" e (24 ).

(B) derived analytically and synthetically from (20).

Find A o , A 1 , A 2 , ..., B 1 , B 2 , &c., by the harmonic analysis, to satisfy the condition fx = A q , + A 1 cosø + A 2 cos20 + &c. Ί

+ B 1 sinβ + B 2 sin20+ &c. , n ..

, 2πx ι * ’ · (25) ·

where 0 = -y-

Thent>~A 0 + £ €~*' TV t (χ. cos ¾⅛ + B i sin⅛ a ^) . (26).

∕=1 ∖ l l)

XII. Uniform row of simple instantaneous plane sources. Two solutions (A) and (B).

(A) , from XI. (A), (23):

)-i⅛Λ rl,wιβ <≈>∙

The No. 2 diffusion curve of § 82 is the representation of the first term (t=0) of this formula.

(B) , from XI. (B), (25) and (26):

v =^jl + 2 ⅞ x e -√Υ r ∕ i c0 s 2 >^i . . (28).

l I j=ι l )

The comparison of these two solutions is verv interesting nbvsi-

cally, and useful arithmetically. To facilitate the comparison, put l ∣ 2(τtκt)⅛≈q, and z∕2(π√)⅛ =p (29); the two solutions become — =q ∑ e "*O ’ +⅛) 2 =1 + 2 ∑ e~2 r cos½⅛ . (30). σ *=-» j=-co q

The equation between the second and third member, virtually due originally to Fourier, is also an interesting formula of Jacobi’s, Fundamenta Nova Theorioe Functionum Ellipticarum, as was long ago pointed out by Cayley. Each formula is a series which converges for every value of t however small or however great; the first, (27), the more rapidly the less is t; the second, (28), the more rapidly the greater is 't. For the case of t = l" 2 ∣ iκπ, and x= 0 (that is, q — 1 and p = 0), the two series become identical. For the more comprehensive case of p = 0, but q unrestricted, the comparison gives the following very curious arithmetical theorem—

. ⅜e-) 1

ι + 2(∈-^ r '+∈- to ' i ∙+e- a '' a *+Sc.) · · l '

When i≡≡Z 2 ∕4κπ (or g ’ >l) the first solution (27) converges with so great suddenness that three terms suffice for most practical purposes; when i^Z 2 ∕4κπ (or 7<1) the second solution (28) converges with so great suddenness that one term (after the constant first term) suffices for most practical purposes. Thus by using the solution (27) for all values of t from zero to something less than Z 2 ∕4κπ∙, and (28) for all values greater than the greatest for which (27) is used, we have an exceedingly rapid convergence and easy calculation to find v for any values of x and t. These formulae, thus used, have been of great practical value in calculating what is now known as the arrival curve of signals through a submarine cable, and in designing instruments to record it automatically and allow its telegraphic meaning to be read, or without recording it to allow its meaning to be read by watching the motions of a spot of light. It is clear that (27) and (28) express the potential at a point at distance x from one end of a cable of length ∖ l, at time t from an instant when a quantity ⅞<τc of electricity has been suddenly communicated to that end of the cable, both ends being always kej>t insulated after that instant (as is done practically by an exceedingly short contact with one pole of a voltaic battery, the other being kept to earth). The value of v for x = ⅛Z, and for all values of t from 0 to ∞, represents the rise of the potential at the remote end towards the limiting value -, c towards which the potential rises throughout the conductor.

XIII. (X. in three dimensions.) Space triple periodic solid source; in other words 0 = V sin ax sin βy sin yz, when Z = 0. Solution for any value of t— ⅛ 2 +¾ 2 +< 2 ' rfZ ∕*X z ∙X ~ 4∕√∕ t [ = γ Λ ∞ A« ‰ <Wη<lζ sin a (x - f) sin β(y - η) siny (z - ζ) θ j ∣ ∙

= V sin our sin βy sin yz∈ - ' c ^ α2 ^^^ 2 ^*^I' 2 ^ (32);

Remark that, as an analytical expression for the present case of the general theory of triply-multiple images, the triple integral within the brackets may be written 2π 2π 2π v ∕* a f β f y aξuηΙ'ζ l Sinα(j∙- ξ) sinβ(yη) sinγ(z - ζ) —^—5 ,

Jo Jo Jo 8ar∖κlp

= ,= ... (<+v) i ÷(∙.+^) ’ ÷(t+^) s < 33 >∙ where S= ∑ ∑ ∑ e” 4 κ t i = -x> j = -∞ k=-ao i, j, k, being any positive or negative integers.

XIV. and XV. (X. and XI. in three dimensions.; The formulae may be written down by inspection; from I., with X. and XI. for guides. The analytical theorem thus obtained, corresponding to (30), in three dimensions, is interesting to pure mathematicians.

XVI. Harmonic solutions. Any distribution of heat, whether in an infinite or in a bounded solid, which keeps its type unchanged in subsiding towards uniformity, when left without positive or negative sources, except such, essentially negative, as are required to fulfil a proper boundary condition, is called a harmonic distribution, provided the temperature does not increase to infinity in any direction. The boundary condition, if the solid is bounded, is essentially that the rate of emission from the surface at every point of it varies in simple proportion to the temperature, and at such a rate per l 0 of temperature at each part of the surface as the solution requires. X. and XIII. are examples. The general con-" dition for a harmonic solution is ν=Hf)F(x,y,z) . · . '. . . (34); and this tried in (9) gives d 2 F <Z' 2 F d 2 F

1 dff)~ dx i + dy 1 dz i (35).

∕(i) dt κ F «“

The first member being independent of x, y, z, and the second being independent of t, the common value of the two must be independent of x, y, z, t, that is to say, must be an absolute constant. Let it be denoted by - r: we have

∕(<) = Ce-pi (36),

, d 2 F rf 2 F d a F p v n ∕o⅛∖

dx i dy i dz i κ or in terms of polar coordinates, by (6), rf s U 1 di . dU∖ 1 rf2 R P rτ- n Ί

dr i + r 2 sinø d0∖ sm θd0 J r- siii-’Ø d<i> s κ ! (gg)

where U =—

r j

Of this we have a spherical harmonic solution,

U = [A≠.{r) + Bψ i (r)]S i (39), where S; denotes a spherical surface-harmonic of order and <∕>.(r), ψ.(r), two particular solutions of the equation ci 2 u Γ p i(i+l) - ∣ λ , λ

^ + Lf-⅛-J u - o < ω >∙

Then (36) and (35) give finally fl = e ^ pi 4±irLt⅛ir)s i (4i).

This solution is in its generality applicable to an infinite solid occupying all space except a hollow round the origin. The solid may of course be bounded externally also by a finite closed surface. If there be no hollow, A/B must fulfil the condition that [A<∕>i(r) + Bψ,∙(r)]∕r is finite when r=0. If there are two boundaries, concentric spherical surfaces, with their common centre at the origin of coordinates, the boundary condition obviously requires uniform emissivity over each, but not necessarily equal for the two. If the two emissivities be denoted by h and h', and the radii of the surfaces by a (outer) and α ’ (inner), the boundary conditions are ~ = - hv, when r=a d

‰ ) (42)∙

and —=⅛⅛, when r = a' J

From these we may find h and h', so as to let the harmonic character of the solution be fulfilled in the subsidence. Or if h and 7√ be given, we have in (41) two equations which determine the two unknown quantities A/B and p. Eliminating A/B, we thus find a single transcendental equation for ρ, which is proved to have no imaginary or negative roots, and an infinite number of real positive roots, each >-κi(i + l) ∣ r 2 . In the case of i=0, or temperature independent of ≠ and θ, (40) gives u= A cos r H — + B sin r H-∙ κ κ

For this case the transcendental equation for determining values for ρ is very simple, and its roots are calculated numerically with great ease. With the further restriction of no central hollow, w e must have A≈=0, so that u ∣ r may be finite when r=0. This case was fully investigated by Fourier, and very beautifully worked out in his fifth chapter. The more general problem of a solid sphere, with any given initial distribution of temperature, without the restriction of temperature independent of 0 and <ρ, was solved first we believe by Poisson in the 11th chapter of his Théorie Mathématique de la Chaleτιr, in terms of the formulae (36), (38), (40) above.

XVII. The equation of the transference of heat in terms of columnar coordinates, (7) above, affords naturally another beautiful case of harmonic solution. Assume v= e pt Ui ⅛s i Φ c 1 os mz (43 ); we find by (7)

ePu, 1 dni ∕ ρ „ i i ∖ . ,. .. *≈ + 7*+(L b r<∕ i '^ 0 · · ∙(44)∙

The treatment of this equation and its integral (obviously derivable by i differentiations from u 0 , which is a Bessel’s function) for the full solution of the thermal problem is most interesting, and very instructive and suggestive in respect to pure analysis. It was splendidly worked out for the case of m∙=0 and i =0 by Fourier in his 6th chapter, “The Motion of Heat in a Solid Cylinder,” truly a masterpiece of art. When it was printed in 1821, and published after having with the rest of Fourier’s work been buried alive for fourteen years in the archives of the French Academy, and when Bessel found in it so thorough an investigation and so strikingly beautiful an application of the “Besselsche Function,” we can imagine the ordinary feeling towards those “qui ante nos nostra dixerunt” reversed into the pleasure of genuine admiration.