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diff --git a/78610-0.txt b/78610-0.txt new file mode 100644 index 0000000..31e0022 --- /dev/null +++ b/78610-0.txt @@ -0,0 +1,7091 @@ +*** START OF THE PROJECT GUTENBERG EBOOK 78610 *** + +[Illustration: + + SADI CARNOT + + AT THE AGE OF 17. + + (From a Portrait by Bailly, 1813.) +] + + + + + REFLECTIONS + ON THE + MOTIVE POWER OF HEAT. + + + _FROM THE ORIGINAL FRENCH OF_ + N.-L.-S. CARNOT, + _Graduate of the Polytechnic School_. + + + ACCOMPANIED BY + AN ACCOUNT OF CARNOT’S THEORY. + BY SIR WILLIAM THOMSON (LORD KELVIN). + + + EDITED BY + + R. H. THURSTON, M.A., LL.D., DR.ENG’G; + _Director of Sibley College, Cornell University_; + “_Officier de l’Instruction Publique de France_,” + _etc., etc., etc._ + +[Illustration: Classical laurel wreath with ribbon banner bearing Greek +text] + + _SECOND, REVISED, EDITION_. + FIRST THOUSAND. + + + NEW YORK: + JOHN WILEY & SONS. + LONDON: CHAPMAN & HALL, LIMITED. + 1897. + + + Copyright, 1890, + ROBERT H. THURSTON. + + + ROBERT DRUMMOND, ELECTROTYPER AND PRINTER, NEW YORK. + + DEDICATED + + TO + + =Sadi Carnot,= + + PRESIDENT OF THE FRENCH REPUBLIC, + + THAT DISTINGUISHED MEMBER OF THE PROFESSION OF ENGINEERING WHOSE WHOLE + LIFE HAS BEEN AN HONOR TO HIS PROFESSION AND TO HIS COUNTRY; + + AND WHO, ELEVATED TO THE HIGHEST OFFICE WITHIN THE GIFT OF THE + + FRENCH NATION, + + HAS PROVEN BY THE QUIET DIGNITY AND THE EFFICIENCY WITH WHICH HE HAS + PERFORMED HIS AUGUST DUTIES THAT HE IS A WORTHY MEMBER OF A NOBLE + FAMILY, ALREADY RENDERED FAMOUS BY AN EARLIER SADI CARNOT, NOW IMMORTAL + IN THE ANNALS OF SCIENCE, AND IS HIMSELF DESERVING OF ENROLMENT IN A + LIST OF GREAT MEN WHICH INCLUDES THAT OTHER DISTINGUISHED ENGINEER, OUR + OWN FIRST PRESIDENT, + + GEORGE WASHINGTON. + + + + + CONTENTS. + + + I. + PAGE + THE WORK OF N.-L.-SADI CARNOT. _By the Editor_, 1 + + II. + THE LIFE OF N.-L.-SADI CARNOT. _By Mons. H. Carnot_, 20 + + III. + REFLECTIONS ON THE MOTIVE POWER OF HEAT AND ON MACHINES FITTED TO + DEVELOP THAT POWER. _By Mons. N.-L.-Sadi Carnot_, 37 + + IV. + ACCOUNT OF CARNOT’S THEORY. _By Sir William Thomson_ (_Lord + Kelvin_), 127 + + APPENDIX. + A. EXTRACTS FROM UNPUBLISHED WRITINGS OF CARNOT, 205 + B. CARNOT’S FOOT-NOTES, 237 + C. NOTE BY THE EDITOR, 261 + + + + + PUBLISHERS’ NOTE. + + +The _raison d’être_ of the following translation of the famous work of +Carnot is not the usual one, either with the Publishers or the +Editor—expectation of gain in either purse or fame. Neither could +reasonably be anticipated from the reproduction of the work of an author +of more than a half-century ago, in a field then unrecognized, and +to-day familiar to but few; and especially when, as is in this case the +fact, the work itself has been long out of date as a scientific +authority, even had it ever held such a position. It could not be +presumed that a very large proportion of even the men of science of the +English-speaking world would be sufficiently familiar with the subject, +or interested in its origin, to purchase such a relic of a primitive +period as is this little book. Nor could the translation of the work, or +the gathering together by the Editor of related matter, be supposed +likely to be productive of any form of compensation. The hook is +published as matter of limited but most intense scientific interest, and +on that score only. + +It has seemed to the Editor and to the Publishers that the product of +the wonderful genius of Carnot,—the great foundation-stone of one of the +most marvellous and important of modern sciences, the first statement of +the grand though simple laws of Thermodynamics,—as illustrated in this +one little treatise, should be made accessible to all who desire to +study the work in English, and preserved, so far as its publication in +this form could accomplish it, as a permanent memorial, in a foreign +tongue, of such grand truths, and of such a great genius as was their +discoverer. It is with this purpose that Publishers and Editor have +cooperated in this project. + +The book consists, as will be seen on inspection, of the translation of +Carnot’s _Réflexions sur la Puissance Motrice du Feu_, preceded by a +notice written by the Editor calling attention to its remarkable +features, and its extraordinary character as the product of a most +remarkable genius; and by a biographical sketch of the great author, +written by his brother, Mons. Hyppolyte Carnot, which sketch we find in +the French copy of the work as published by Gauthier-Villars, the latest +reproduction of the book in the original tongue. To the main portion of +the book, Carnot’s _Réflexions_, is appended the celebrated paper of Sir +William Thomson, his “Account of Carnot’s Theory,” in which that great +physicist first points out to the world the treasure so long concealed, +unnoticed, among the scientific literature, already mainly antiquated, +of the first quarter of the nineteenth century. The distinguished writer +of this paper has kindly interested himself in the scheme of the Editor, +and has consented to its insertion as a natural and desirable commentary +upon the older work, and especially as exhibiting the relations of the +fundamental principles discovered and enunciated by Carnot to the modern +view of the nature of thermodynamic phenomena—relations evidently +understood by that writer, but not by the leaders of scientific thought +of his time, and therefore ignored by him in the construction of his new +science. + +The Appendix contains a number of Carnot’s own notes, too long to be +inserted in the body of the paper in its present form, and which have +therefore been removed to their present location simply as a matter of +convenience in bookmaking. + +The dedication of the work to the grandnephew of the author, who by a +singular coincidence happens to-day to occupy the highest position that +any citizen can aspire to reach in that now prosperous Republic, will be +recognized as in all respects appropriate by every reader of the work of +the earlier Sadi Carnot who is familiar with the character, the history, +the attainments, the achievements, of the later Sadi Carnot in so many +and widely diverse fields. The Carnot talent and the Carnot character +are equally observable in both men, widely as they are separated in time +and in the nature of their professional labors. Both are great +representatives of a noble family, whose honor and fame they have both +splendidly upheld. + +The Publishers offer this little book to its readers as a small, yet in +one sense not unimportant, contribution to the great cause of modern +science, as a relic, a memorial, a corner-stone. + + + + + NOTE BY THE EDITOR. + + +“_Je me suis proposé de grands desseins dans ce petit ouvrage_,” as +Bernardin de Saint-Pierre says in the preface to his pathetic story of +_Paul et Virginie_. I have sought to present to the great +English-speaking world the work of a genius hitherto only known to a few +men of science, and not well known, even among the people of France, for +whose credit he has done so much. In placing before the readers of this +translation his book—small of size but great in matter as it is—I feel +that I have accomplished an easy task, but one of real importance. I +have been asked, as Corresponding Member for the United States of the +Société des Ingénieurs Civils de France, to communicate to my colleagues +scientific and professional memoirs and whatever may be of interest to +them—“_en un mot, que nous resserrions les liens qui font des ingénieurs +en général une seule famille_.” That were a pleasant task; but a grander +and a more agreeable one still is that of bringing “nearer in heart and +thought” the members of that still larger community, the men of science +of the world, and of weaving still more firmly and closely those bonds +of kindly thought and feeling which are growing continually more +numerous and stronger as the nations are brought to see that humanity is +larger and more important than political divisions, and that the labors +of educated men and of the guiding minds in the great industries are +constantly doing more to promote a true brotherhood of mankind than ever +have, or ever can, the greatest statesmen. + +When the wonderful intellectual accomplishments of men like the elder +Sadi Carnot become known and appreciated by the world, much more will +have been accomplished in this direction. It is perhaps from this point +of view that the importance of such work will be most fully recognized. +When the little treatise which is here for the first time published in +English becomes familiar to those for whom it is intended, it will be, +to many at least, a matter of surprise no less than pleasure to discover +that France has produced a writer on this now familiar subject whose +inspiration anticipated many of the principles that those founders of +the modern science, Rankine and Clausius, worked out through the tedious +and difficult methods of the higher mathematics, and which were hailed +by their contemporaries as marvellous discoveries. + + + + + NOTE TO SECOND EDITION. + + +The present edition of this little work is improved by the removal of a +few errata observed in the first issue, and by the addition of a recent +and excellent portrait of Lord Kelvin, as a frontispiece to his +era-making paper, at page 127. This picture, taken within the last year, +is thought by the friends of its distinguished subject to be one of the +best yet produced. That it is satisfactory to him and his friends is +indicated by the fact that the original of this reproduction was +presented to the writer by Lady Kelvin, in 1895, immediately after it +was taken, and the autograph supplied by her distinguished husband. The +Editor takes this occasion to acknowledge cordially the letters of +appreciation and commendation received from those who have agreed with +M. Haton de la Goupillière that the translation of Carnot and its +publication in this manner, with the famous paper of Lord Kelvin, will +be considered as worthy of approval by English-speaking readers as well +as “appreciated by the whole French nation.” + + + + + I. + THE WORK OF SADI CARNOT. + + + BY THE EDITOR. + + +Nicolas-Léonard-Sadi Carnot was, perhaps, the greatest genius, in the +department of physical science at least, that this century has +produced. By this I mean that he possessed in highest degree that +combination of the imaginative faculty with intellectual acuteness, +great logical power and capacity for learning, classifying and +organizing in their proper relations, all the facts, phenomena, and +laws of natural science which distinguishes the real genius from other +men and even from the simply talented man. Only now and then, in the +centuries, does such a genius come into view. Euclid was such in +mathematics; Newton was such in mechanics; Bacon and Compte were such +in logic and philosophy; Lavoisier and Davy were such in chemistry; +and Fourier, Thomson, Maxwell, and Clausius were such in mathematical +physics. Among engineers, we have the examples of Watt as inventor and +philosopher, Rankine as his mathematical complement, developing the +theory of that art of which Watt illustrated the practical side; we +have Hirn as engineer-experimentalist, and philosopher, as well; +Corliss as inventor and constructor; and a dozen creators of the +machinery of the textile manufactures, in which, in the adjustment of +cam-work, the highest genius of the mechanic appears. + +But Carnot exhibited that most marked characteristic of real genius, the +power of applying such qualities as I have just enumerated to great +purposes and with great result while still a youth. Genius is not +dependent, as is talent, upon the ripening and the growth of years for +its prescience; it is ready at the earliest maturity, and sometimes +earlier, to exhibit its marvellous works; as, for example, note Hamilton +the mathematician and Mill the logician; the one becoming master of a +dozen languages when hardly more than as many years of age, reading +Newton’s Principia at sixteen and conceiving that wonderful system, +quaternions, at eighteen; the other competent to begin the study of +Greek at three, learning Latin at seven and reading Plato before he was +eight. Carnot had done his grandest work of the century in his province +of thought, and had passed into the Unseen, at thirty-six; his one +little volume, which has made him immortal, was written when he was but +twenty-three or twenty-four. It is unnecessary, here, to enter into the +particulars of his life; that has been given us in ample detail in the +admirable sketch by his brother which is here republished. It will be +quite sufficient to indicate, in a few words, what were the conditions +amid which he lived and the relation of his work to that great science +of which it was the first exposition. + +At the time of Carnot, the opinion of the scientific world was divided, +as it had been for centuries, on the question of the true nature of heat +and light, and as it still is, to a certain extent, regarding +electricity. On the one hand it was held by the best-known physicists +that heat is a substance which pervades all bodies in greater or less +amount, and that heating and cooling are simply the absorption and the +rejection of this “imponderable substance” by the body affected; while, +on the other hand, it was asserted by a small but increasing number that +heat is a “mode of motion,” a form of energy, not only imponderable, but +actually immaterial; a quality of bodies, not a substance, and that it +is identical, in its nature, with other forms of recognizable energy, +as, for example, mechanical energy. A quarter of a century before Carnot +wrote, the experiments of Rumford and of Davy had been crucial in the +settlement of the question and in the proof of the correctness of the +second of the two opposing parties; but their work had not become so +generally known or so fully accepted as to be acknowledged as +representative of the right views of the subject. The prevalent opinion, +following Newton, was favorable to the first hypothesis; and it was in +deference to this opinion that Carnot based his work on an inaccurate +hypothesis; though, fortunately, the fact did not seriously militate +against its value or his credit and fame. + +“With true philosophical caution, he avoids committing himself to this +hypothesis; though he makes it the foundation of his attempt to discover +how work is produced from heat.”[1] + +The results of Carnot’s reasoning are, fortunately, mainly independent +of any hypothesis as to the nature of heat or the method or mechanism of +development and transfer or transformation of its energy. Carnot was in +error in assuming no loss of heat in a completed cycle and in thus +ignoring the permanent transformation of a definite proportion into +mechanical energy; but his proposition that efficiency increases with +increase of temperature-range is still correct; as is his assertion of +its independence of the nature of the working substance. + +Carnot’s “_Réflexions sur la Puissance Motrice du Feu_,” published in +1824, escaped notice at the time, was only now and then slightly +referred to later, until Clapeyron seized upon its salient ideas and +illustrated them by the use of the Watt diagram of energy, and might, +perhaps, have still remained unknown to the world except for the fact +that Sir William Thomson, that greatest of modern mathematical +physicists, fortunately, when still a youth and at the commencement of +his own great work, discovered it, revealed its extraordinary merit, +and, readjusting Carnot’s principles in accordance with the modern views +of heat-energy, gave it the place that it is so well entitled to in the +list of the era-making books of the age. But it still remained +inaccessible to all who could not find the original paper until, only a +few years since, it was reprinted by Gauthier-Villars, the great +publishing house of Paris, accompanied by a biographical sketch by the +younger brother, which it has been thought wise to reproduce with the +translation of Carnot’s book. In making the translation, also, this +later text has been followed; and now, for the first time, so far as is +known to the writer, the work of Carnot is made accessible to the reader +in English. + +The original manuscript of Carnot has been deposited by his brother in +the archives of the French Academy of Sciences, and thus insured +perpetual care. The work of Carnot includes not only the treatise which +it is the principal object of this translation to give to our readers, +but also a considerable amount of hitherto unpublished matter which has +been printed by his brother, with the new edition of the book, as +illustrative of the breadth and acuteness of the mind of the Founder of +the Science of Thermodynamics. + +These previously unpublished materials consist of memoranda relating to +the specific heats of substances, their variations, and various other +facts and data, and principles as well; some of which are now recognized +as essential elements of the new science, even of its fundamental part. +The book is particularly rich in what have been generally supposed to be +the discoveries of later writers, and in enunciations of principles now +recognized as those forming the base and the supporting framework of +that latest of the sciences. As stated by Tait, in his history of +Thermodynamics, the “two grand things” which Carnot originated and +introduced were his idea of a “cycle” and the notion of its +“reversibility,” when perfect. “Without this work of Carnot, the modern +theory of energy, and especially that branch of it which is at present +by far the most important in practice, the dynamical theory of heat, +could not have attained its now enormous development.” These +conceptions, original with our author, have been, in the hands of his +successors, Clausius and other Continental writers, particularly, most +fruitful of interesting and important results; and Clapeyron’s happy +thought of so employing the Watt diagram of energy as to render them +easy of comprehension has proved a valuable aid in this direction. + +The exact experimental data needed for numerical computations in +application of Carnot’s principles were inaccessible at the date of his +writing; they were supplied, later, by Mayer, by Colding, by Joule, and +by later investigators. Even the idea of equivalence, according to +Hypolyte Carnot, was not originally familiar to the author of this +remarkable work; but was gradually developed and defined as he +progressed with his philosophy. It is sufficiently distinctly enunciated +in his later writings. He then showed a familiarity with those notions +which have been ascribed generally to Mayer and which made the latter +famous, and with those ideas which are now usually attributed to Joule +with similar result. He seems actually to have planned the very kind of +research which Joule finally carried out. All these advanced views must, +of course, have been developed by Carnot before 1832, the date of his +illness and death, and ten or fifteen years earlier than they were made +public by those who have since been commonly considered their +discoverers. These until lately unpublished notes of Carnot contain +equally well-constructed arguments in favor of the now accepted theory +of heat as energy. While submitting to the authority of the greatest +physicists of his time, and so far as to make their view the basis of +his work, to a certain extent, he nevertheless adhered privately to the +true idea. His idea of the equivalence of heat and other forms of energy +was as distinct and exact as was his notion of the nature of that +phenomenon. He states it with perfect accuracy. + +In making his measures of heat-energy, he assumes as a unit a measure +not now common, but one which may be easily and conveniently reduced to +the now general system of measurement. He takes the amount of power +required to exert an energy equal to that needed to raise one cubic +meter of water through a height of one meter, as his unit; this is 1000 +kilogrammeters, taken as his unit of motive power; while he says that +this is the equivalent of 2.7 of his units of heat; which latter +quantity would be destroyed in its production of this amount of power, +or rather work. His unit of heat is thus seen to be 1000 ÷ 2.7, or 370 +kilogrammeters. This is almost identical with the figure obtained by +Mayer, more than ten years later, and from presumably the same +approximate physical data, the best then available, in the absence of a +Regnault to determine the exact values. Mayer obtained 365, a number +which the later work of Regnault enabled us to prove to be 15 per cent. +too low, a conclusion verified experimentally by the labors of Joule and +his successors. Carnot was thus _a_ discoverer of the equivalence of the +units of heat and work, as well as the revealer of the principles which +have come to be known by his name. Had he lived a little longer, there +can be little doubt that he would have established the facts, as well as +the principles, by convincing proof. His early death frustrated his +designs, and deprived the world of one of its noblest intellects, just +when it was beginning its marvellous career. + +The following sentence from Carnot illustrates in brief his wonderful +prescience; one can hardly believe it possible that it should have been +written in the first quarter of the nineteenth century: “_On peut donc +poser en thèse générale que la puissance motrice est en quantité +invariable dans la Nature; qu’elle n’est jamais, à proprement parler, ni +produite, ni détruite. A la vérité, elle change de forme, c’est a dire +qu’elle produit tantôt un genre de mouvement, tantôt un autre; mais elle +n’est jamais anéantie._” It is this man who has probably inaugurated the +development of the modern science of thermodynamics and the whole range +of sciences dependent upon it, and who has thus made it possible to +construct a science of the energetics of the universe, and to read the +mysteries of every physical phenomenon of nature; it is this man who has +done more than any contemporary in his field, and who thus displayed a +more brilliant genius than any man of science of the nineteenth century: +yet not even his name appears in the biographical dictionaries; and in +the Encyclopædia Britannica it is only to be found incidentally in the +article on Thermodynamics. + +Throughout his little book, we find numerous proofs of his clearness of +view and of the wonderful powers of mind possessed by him. He opens his +treatise by asserting that “_C’est à la chaleur que doivent être +attribués les grands mouvements qui frappent nos regards sur la terre; +c’est à elle que sont dues les agitations de l’atmosphère, l’ascension +des nuages, la chute des pluies et des autres météores, les courants +d’eau qui sillonnent la surface du globe et dont l’homme est parvenue à +employer pour son usage une faible partie; enfin les tremblements de +terre, les éruptions volcaniques reconnaissent aussi pour cause la +chaleur._” + +Carnot was the first to declare that the maximum of work done by heat, +in any given case of application of the heat-energy, is determined +solely by the range of temperature through which it fell in the +operation, and is entirely independent of the nature of the working +substance chosen as the medium of transfer of energy and the vehicle of +the heat. His assumption of the materiality of heat led, logically, to +the conclusion that the same quantity of heat was finally stored in the +refrigerator as had, initially, left the furnace, and that the effect +produced was a consequence of a fall of temperature analogous to a fall +of water; but, aside from this error—which he himself was evidently +inclined to regard as such,—his process and argument are perfectly +correct.[2] + +Throughout his whole work are distributed condensed assertions of +principles now well recognized and fully established, which indicate +that he not only had anticipated later writers in their establishment, +but that he fully understood their real importance in a theory of +heat-energy and of heat-engines. In fact, he often italicizes them, +placing them as independent paragraphs to more thoroughly impress the +reader with their fundamental importance. Thus he says: “_Partout où il +existe une différence de température, il peut y avoir production de +puissance motrice_;” and again, this extraordinary anticipation of +modern science: “_le maximum de puissance résultant de l’emploi de la +vapeur est aussi le maximum de puissance motrice réalisable par quelque +moyen que ce soit_.” + +“_La puissance motrice de la chaleur est indépendante des agents mis en +œuvre pour la réaliser; sa quantité est fixée uniquement par les +températures des corps entre lesquels se fait, en dernier résultat, le +transport du calorique._” + +“_Lorsqu’un gaz passe, sans changer de température, d’un volume et d’une +pression déterminés à une autre pression également déterminée, la +quantité de calorique absorbée ou abandonnée est toujours la même, +quelle que soit la nature du gaz choisi comme sujet d’expérience._” + +Perhaps as remarkable a discovery as any one of the preceding (and one +which, like those, has been rediscovered and confirmed by later +physicists; one which was the subject of dispute between Clausius, who +proved its truth by the later methods which are now the source of his +fame, and the physicists of his earlier days, who had obtained +inaccurate measures of the specific heats of the gases;—values which +were finally corrected by Regnault, thus proving Carnot and Clausius to +be right—is thus stated by Carnot, and is italicized in his manuscript +and book: + +“_La différence entre la chaleur spécifique sous pression constante et +la chaleur spécifique sous volume constant est la même pour tous les +gaz._” + +He bases his conclusion upon the simplest of thermodynamic +considerations. He says that the increase of volumes with the same +differences of temperature are the same, according to Gay-Lussac and +Dalton; and that, therefore, according to the laws of thermodynamics as +he has demonstrated them, the heat absorbed with equal augmentations of +volume being the same, the two specific heats are constant, and their +difference as well. As will be seen on referring to the text, he bases +upon this principle a determination of the specific heats of constant +volume, taking as his values of the determined specific heats of +constant pressure those of Delaroche and Bérard, making the constant +difference 0.300, that of air at constant pressure being taken as the +standard and as unity. The establishment of this point, in the face of +the opposition, and apparently of the facts, of the best physicists of +his time, was one of those circumstances which did so much to win for +Clausius his great fame. How much greater credit, then, should be given +Carnot, who not only anticipated the later physicists in this matter, +but who must have enunciated his principle under far more serious +discouragements and uncertainty! + +It must be remembered, when reading Carnot, that all the “constants of +nature” were, in his time, very inaccurately ascertained. It is only +since the time of Regnault’s grand work that it has been the rule that +such determinations have been published only when very exactly +determined. No change has been attempted in Carnot’s figures, in any +respect; as it would be far less satisfactory to read a paraphrased +work, and the exact figures are now easily accessible to every one, and +his computations may all be made, if desired, on the basis of modern +data. Sir William Thomson has already performed this task in the paper +appended. + +Throughout the whole of this treatise, small as it is, we find +distributed a singular number of these anticipations of modern +thermodynamic principles. Studying the relation of heat-energy to work +done, he concludes: + +“_La chute du calorique produit plus de puissance motrice dans les +degrés inférieurs que dans les degrés supérieurs._” + +We to-day admit that, since the one degree at a low temperature, and the +corresponding quantity of heat, are larger fractions of the total +temperature, and the total heat stored in the substance, than the one +degree at a higher point on the scale of absolute temperature, this +principle of Carnot has become obvious. + +In the enunciation of the essential principles of efficiency of the +heat-engine, we find the proofs of this same wonderful prescience. He +asserts that, for best effect: “(1) The temperature of the working fluid +must be raised to the highest degree possible, in order to secure a +commensurate range of temperature; (2) The cooling must be carried to +the lowest point on the scale that may be found practicable; (3) The +passage of the fluid from the upper to the lower limit of temperature +must be produced by expansion;” i.e., “it is necessary that the cooling +of the gas shall occur spontaneously by its rarefaction;” which is +simply his method of stating the now universally understood principle +that, for highest efficiency, the expansion must be adiabatic, from a +maximum to a minimum temperature. He goes on to explain these +principles, and then says that the advantage of high-pressure engines +lies “_essentiellement dans la faculté de rendre utile une plus grande +chute de calorique_.” This principle, as a practical system of +operation, had already, as he tells us, been enunciated by M. Clement, +and had been practised, as we well know, since the days of its +originator, Watt; but Carnot saw clearly the thermodynamic principle +which underlies it, and as clearly states it, for the first time. + +He sees clearly, too, the reasons for the attempts of Hornblower and of +Woolf, premature as they proved and as he also sees, in the introduction +of the compound engine, and even suggests that this idea might be still +further developed by the use of a triple-expansion engine, a type which +is to-day just coming into use, more than a half-century after Carnot’s +date. He recognizes the advantages of the compound engine in better +distribution of pressures and in distribution of the work of expansion, +but does not, of course, perceive the then undiscovered limitation of +the efficiency of the simple engine, due to “cylinder condensation,” +which has finally led, perhaps more than any other circumstance, to its +displacement so largely by the multi-cylinder machine. No one has more +exactly and plainly stated the respective advantages to be claimed for +air and the gases, used as working fluids in heat-engines, than does +Carnot; nor does any one to-day better recognize the difficulties which +lie in the path to success in that direction, in the necessity of +finding a means of handling them at high temperatures and of securing +high mean pressures. + +His closing paragraph shows his extraordinary foresight, and the +precision with which that wonderful intellect detected the practical +elements of the problem which the engineer, from the days of Savery, of +Newcomen, and of Watt has been called upon to study, and the importance +of the work, which he began, in the development of a theory of the +action, or of the operation, of the heat-engines, which should give +effective assistance in the development of their improved forms: + +“_On ne doit pas se flatter de mettre jamais à profit, dans la pratique, +toute la puissance des combustibles. Les tentatives que l’on ferait pour +approcher ce résultat seraient même plus nuisibles qu’utiles, si elles +faisaient négliger d’autres considérations importantes. L’économie du +combustible n’est qu’une des conditions à remplir par les machines à +feu; dans beaucoup de circonstances, elle n’est que secondaire: elle +doit souvent céder le pas à la sûreté, à la solidité, à la durée de la +machine, au peu de place qu’il faut lui faire occuper, au peu de frais +de son établissement, etc. Savoir apprécier, dans chaque cas, à leur +juste valeur, les considérations de convenance et d’économie qui peuvent +se présenter; savoir discerner les plus importantes de celles qui sont +seulement accessoires, les balancer toutes convenablement entre elles, +afin de parvenir, par les moyens les plus faciles, au meilleur résultat: +tel doit être le principal talent de l’homme appelé à diriger, à +co-ordonner entre eux les travaux de ses semblables, à les faire +concourir vers un but utile de quelque genre qu’il soit._” + +Such was the work and such the character of this wonderful man. Those +whose desire to follow more closely and to witness the process of +development of the work of which this initial paper of Carnot was the +introductory, should study the contribution of Sir William Thomson to +this development, as published in 1849,—a paper which constitutes that +physicist the virtual discoverer of Carnot and the godfather of the man +and his thoughts. This paper constitutes the final chapter of this +little book. + +From that time the additional progress so rapidly made in the new +science was as inevitable as the development of a gold-field, once the +precious metal has been found in paying quantities in the hitherto +unvisited cañons and gorges of a distant and unexplored mountain-range. +But great as is the work since done, and great as have been the +discoveries and the discoverers of later years, none claims our +gratitude and compels our respect in greater degree than does the +original discoverer— + + SADI CARNOT. + + + + + II. + LIFE OF SADI CARNOT. + + BY M. H. CARNOT. + + +As the life of Sadi Carnot was not marked by any notable event, his +biography would have occupied only a few lines; but a scientific work by +him, after remaining long in obscurity, brought again to light many +years after his death, has caused his name to be placed among those of +great inventors. In regard to his person, his mind, his character, +nothing whatever has been known. Since there remains a witness of his +private life—the sole witness, has he not a duty to fulfil? Ought he not +to satisfy the natural and legitimate interest which attaches to any man +whose work has deserved a portion of glory? + +Nicolas-Léonard-Sadi Carnot was born June 1, 1796, in the smaller +Luxembourg. This was that part of the palace where our father then dwelt +as a member of the Directory. Our father had a predilection for the name +of Sadi, which recalled to his mind ideas of wisdom and poetry. His +firstborn had borne this name, and despite the fate of this poor child, +who lived but a few months, he called the second also Sadi, in memory of +the celebrated Persian poet and moralist. + +Scarcely a year had passed when the proscription, which included the +Director, obliged him to give up his life, or at least his liberty, to +the conspirators of fructidor. Our mother carried her son far from the +palace in which violation of law had just triumphed. She fled to St. +Omer, with her family, while her husband was exiled to Switzerland, then +to Germany. + +Our mother often said to me, “Thy brother was born in the midst of the +cares and agitations of grandeur, thou in the calm of an obscure +retreat. Your constitutions show this difference of origin.” + +My brother in fact was of delicate constitution. He increased his +strength later, by means of varied and judicious bodily exercises. He +was of medium size, endowed with extreme sensibility and at the same +time with extreme energy, more than reserved, almost rude, but +singularly courageous on occasion. When he felt himself to be contending +against injustice, nothing could restrain him. The following is an +anecdote in illustration. + +The Directory had given place to the Consulate. Carnot, after two years +of exile, returned to his country and was appointed Minister of War. +Bonaparte at the same time was still in favor with the republicans. He +remembered that Carnot had assisted him in the beginning of his military +career, and he resumed the intimate relation which had existed between +them during the Directory. When the minister went to Malmaison to work +with the First Consul, he often took with him his son, then about four +years old, to stay with Madame Bonaparte, who was greatly attached to +him. + +She was one day with some other ladies in a small boat on a pond, the +ladies rowing the boat themselves, when Bonaparte, unexpectedly +appearing, amused himself by picking up stones and throwing them near +the boat, spattering water on the fresh toilets of the rowers. The +ladies dared not manifest their displeasure, but the little Sadi, after +having looked on at the affair for some time, suddenly placed himself +boldly before the conqueror of Marengo, and threatening him with his +fist, he cried “Beast of a First Consul, will you stop tormenting those +ladies!” + +Bonaparte, at this unexpected attack, stopped and looked in astonishment +at the child. Then he was seized with a fit of laughter in which all the +spectators of the scene joined. + +At another time, when the minister, wishing to return to Paris, sought +his son, who had been left with Madame Bonaparte, it was discovered that +he had run away. They found him a long way off, in a mill, the mechanism +of which he was trying to understand. This desire had been in the +child’s mind for days, and the honest miller, not knowing who he was, +was kindly answering all his questions. Curiosity, especially in regard +to mechanics and physics, was one of the essential traits of Sadi’s +mind. + +On account of this disposition so early manifested, Carnot did not +hesitate to give a scientific direction to the studies of his son. He +was able to undertake this task himself when the monarchical tendencies +of the new government had determined him to retire. For a few months +only Sadi followed the course of M. Bourdon at the Charlemagne Lycée to +prepare himself for the Polytechnic School. + +The pupil made rapid progress. He was just sixteen years old when he was +admitted to the school, the twenty-fourth on the list. This was in 1812. +The following year he left it, first in artillery. But he was considered +too young for the school of Metz, and he continued his studies at Paris +for a year. To this circumstance is due the fact that he took part in +March, 1814, in the military exploits of Vincennes, and not of the butte +Chaumont, as almost all the historians of the siege of Paris declared. +M. Chasles, one of Sadi’s school-fellows, took pains to rectify this +error at a séance of the Institute in 1869. + +If the pupils of the Polytechnic School did not earlier enter into the +campaign, it was not because they had not asked to do so. I find in my +brother’s papers the copy of an address to the Emperor, signed by them +December 29, 1813: + +“SIRE: The country needs all its defenders. The pupils of the +Polytechnic School, faithful to their motto, ask to be permitted to +hasten to the frontiers to share the glory of the brave men who are +consecrating themselves to the safety of France. The battalion, proud of +having contributed to the defeat of the enemy, will return to the school +to cultivate the sciences and prepare for new services.” + +General Carnot was at Anvers, which he had just been defending against +the confederate English, Prussians, and Swedes, where the French flag +yet floated, when he wrote to his son, April 12, 1814: + +“MY DEAR SADI: I have learned with extreme pleasure that the battalion +of the Polytechnic School has distinguished itself, and that you have +performed your first military exploits with honor. When I am recalled, I +shall be very glad if the Minister of War will give you permission to +come to me. You will become acquainted with a fine country and a +beautiful city, where I have had the satisfaction of remaining in peace +while disaster has overwhelmed so many other places.” + +Peace being restored, Sadi rejoined his father at Anvers and returned +with him into France. + +In the month of October he left the Polytechnic School, ranking sixth on +the list of young men destined to service in the engineer corps, and +went to Metz as a cadet sub-lieutenant at the school. Many scientific +papers that he wrote there were a decided success. One is particularly +referred to as very clever, a memoir on the instrument called the +_theodolite_ which is used in astronomy and geodesy. + +I obtain these details from M. Ollivier, who was of the same rank as +Sadi and who, later, was one of the founders of the _École Centrale_. +Among his other comrades besides M. Chasles, the learned geometrician +just now referred to, was Gen. Duvivier, lamented victim of the +insurrection of June 1848. I ought also to mention M. Robelin, Sadi’s +most intimate friend, who came to help me nurse him during his last +illness, and who published a notice concerning him in the _Revue +encyclopédique_, t. lv. + +The events of 1815 brought General Carnot back into politics during the +“_Cent Jours_” which ended in a fresh catastrophe. + +This gave Sadi a glimpse of human nature of which he could not speak +without disgust. His little sub-lieutenant’s room was visited by certain +superior officers who did not disdain to mount to the third floor to pay +their respects to the son of the new minister. + +Waterloo put an end to their attentions. The Bourbons re-established on +the throne, Carnot was proscribed and Sadi sent successively into many +trying places to pursue his vocation of engineer, to count bricks, to +repair walls, and to draw plans destined to be hidden in portfolios. He +performed these duties conscientiously and without hope of recompense, +for his name, which not long before had brought him so many flatteries, +was henceforth the cause of his advancement being long delayed. + +In 1818 there came an unlooked-for royal ordinance, authorizing the +officers of all branches of the service to present themselves at the +examinations for the new corps of the staff. Sadi was well aware that +favor had much more to do with this matter than ability, but he was +weary of garrison life. The stay in small fortresses to which the nature +of his work confined him did not offer sufficient resources to his love +of study. Then he hoped, and his hope was realized, that a request for a +furlough would be obtained without difficulty, and would insure him the +leisure that he sought. In spite of the friendly opposition of some +chiefs of the engineer corps, testifying to a sincere regret at the +removal from their register of a name which had gained honor among them, +Sadi came to Paris to take the examination, and was appointed lieutenant +on the staff, January 20, 1819. + +He hastened to obtain his furlough, and availed himself of it to lead, +in Paris and in the country round about Paris, a studious life +interrupted but once, in 1821, by a journey to Germany to visit our +father in his exile at Magdeburg. We had then the pleasure of passing +some weeks all three together. + +When, two years later, death took from us this revered father and I +returned alone to France, I found Sadi devoting himself to his +scientific studies, which he alternated with the culture of the arts. In +this way also, his tastes had marked out for him an original direction, +for no one was more opposed than he to the traditional and the +conventional. On his music-desk were seen only the compositions of Lully +that he had studied, and the concerti of Viotti which he executed. On +his table were seen only Pascal, Molière, or La Fontaine, and he knew +his favorite books almost by heart. I call this direction original, +because it was anterior to the artistic and literary movement which +preceded the revolution of 1830. As to the sympathy of Sadi for the +author of the _Provinciales_, it was due not only to the respect of the +young mathematician for one of the masters of science, but his devoutly +religious mind regarded with horror hypocrisy and hypocrites. + +Appreciating the useful and the beautiful, Sadi frequented the museum of +the Louvre and the Italian Theatre, as well as the Jardin des Plantes +and the Conservatoire des Arts et Metiers. Music was almost a passion +with him. He probably inherited this from our mother, who was an +excellent pianist, to whom Dalayrac and especially Monsigny, her +compatriot, had given instruction. Not content with being able to play +well on the violin, Sadi carried to great length his theoretical +studies. + +His insatiable intellect, moreover, would not allow him to remain a +stranger to any branch of knowledge. He diligently followed the course +of the College of France and of the Sorbonne, of the École des Mines, of +the Museum, and of the Bibliothèque. He visited the workshops with eager +interest, and made himself familiar with the processes of manufacture; +mathematical sciences, natural history, industrial art, political +economy,—all these he cultivated with equal ardor. I have seen him not +only practise as an amusement, but search theoretically into, +gymnastics, fencing, swimming, dancing, and even skating. In even these +things Sadi acquired a superiority which astonished specialists when by +chance he forgot himself enough to speak of them, for the satisfaction +of his own mind was the only aim that he sought. + +He had such a repugnance to bringing himself forward that, in his +intimate conversations with a few friends, he kept them ignorant of the +treasures of science which he had accumulated. They never knew of more +than a small part of them. How was it that he determined to formulate +his ideas about the motive power of heat, and especially to publish +them? I still ask myself this question,—I, who lived with him in the +little apartment where our father was confined in the Rue du Parc-Royal +while the police of the first Restoration were threatening him. Anxious +to be perfectly clear, Sadi made me read some passages of his manuscript +in order to convince himself that it would be understood by persons +occupied with other studies. + +Perhaps a solitary life in small garrisons, in the work-room and in the +chemical laboratory, had increased his natural reserve. In small +companies, however, he was not at all taciturn. He took part voluntarily +in the gayest plays, abandoning himself to lively chat. “The time passed +in laughing is well spent,” he once wrote. His language was at such +times full of wit, keen without malice, original without eccentricity, +sometimes paradoxical, but without other pretension than that of an +innocent activity of intelligence. He had a very warm heart under a cold +manner. He was obliging and devoted, sincere and true in his dealings. + +Towards the end of 1826, a new royal ordinance having obliged the staff +lieutenants to return to the ranks, Sadi asked and obtained a return to +the engineer corps, in which he received the following year, as his rank +of seniority, the grade of captain. + +Military service, however, weighed upon him. Jealous of his liberty, in +1828, he laid aside his uniform that he might be free to come and go at +will. He took advantage of his leisure to make journeys and to visit our +principal centres of industry. + +He frequently visited M. Clement Desormes, professor at the +_Conservatoire des Arts et Metiers_, who had made great advances in +applied chemistry. M. Desormes willingly took counsel with him. He was a +native of Bourgogne, our family country, which circumstance, I believe, +brought them together. + +It was before this period (in 1824) that Sadi had published his +_Réflexions sur la puissance motrice du feu_. He had seen how little +progress had been made in the theory of machines in which this power was +employed. He had ascertained that the improvements made in their +arrangement were effected tentatively, and almost by chance. He +comprehended that in order to raise this important art above empiricism, +and to give it the rank of a science, it was necessary to study the +phenomena of the production of motion by heat, from the most general +point of view, independently of any mechanism, of any special agent; and +such had been the thought of his life. + +Did he foresee that this small brochure would become the foundation of a +new science? He must have attached much importance to it to publish it, +and bring himself out of his voluntary obscurity. + +In fact (as his working notes prove), he perceived the existing relation +between heat and mechanical work; and after having established the +principle to which savants have given his name, he devoted himself to +the researches which should enable him to establish with certainty the +second principle, that of equivalence, which he already clearly divined. +Thermodynamics was established from that time. + +But these researches were rudely interrupted by a great event—the +Revolution of July, 1830. + +Sadi welcomed it enthusiastically—not, however, it is evident, as a +personal advantage. + +Several old members of the Convention were still living, even of those +who had become celebrated; no favor of the new government was accorded +them. To the son of Philippe-Egalité was ascribed a saying which, if it +was untrue, at least agreed well with the sentiment of his position: “I +can do nothing for the members of the Convention themselves,” he said, +“but for their families whatever they will.” + +However it may be, some of those about him vaguely questioned my brother +as to his desires in case one of us should be called to the Chamber of +Peers, of which Carnot had been a member in 1815. We had on this +occasion a brief conference. Unknown to us both, this distinction could +be offered only to a title in some sort hereditary. We could not accept +it without forsaking the principles of Carnot, who had combated the +heredity of the peerage. The paternal opinion therefore came to second +our distaste for the proposition, and dictated our reply. + +Sadi frequented the popular reunions at this period without forsaking +his _rôle_ of a simple observer. + +Nevertheless he was, when occasion demanded it, a man of prompt and +energetic action. One incident will suffice to prove this, and to show +the _sang-froid_ which characterized him. + +On the day of the funeral of Gen. Lamarque, Sadi was walking +thoughtfully in the vicinity of the insurrection. A horseman preceding a +company, and who was evidently intoxicated, passed along the street on +the gallop, brandishing his sabre and striking down the passers-by. Sadi +darted forward, cleverly avoided the weapon of the soldier, seized him +by the leg, threw him to the earth and laid him in the gutter, then +continued on his way to escape from the cheers of the crowd, amazed at +this daring deed. + +Before 1830, Sadi had formed part of a _Réunion polytechnique +industrielle_, made up of old pupils of the school, with a plan of study +in common. After 1830, he was a member of the _Association +polytechnique_, consisting also of graduates, the object being the +popular propagation of useful knowledge. The president of this +association was M. de Choiseul-Praslin; the vice-presidents, MM. de +Tracy, Auguste Comte, etc. + +The hopes of the democracy meanwhile seeming to be in abeyance, Sadi +devoted himself anew to study, and pursued his scientific labors with +all the greater energy, as he brought to bear upon them the political +ardor now so completely repressed. He undertook profound researches on +the physical properties of gases and vapors, and especially on their +elastic tensions. Unfortunately, the tables which he prepared from his +comparative experiments were not completed; but happily the excellent +works of Victor Regnault, so remarkable for their accuracy, have +supplied to science, in this respect, the blanks of which Sadi Carnot +was conscious. + +His excessive application affected his health towards the end of June, +1832. Feeling temporarily better, he wrote gayly to one of his friends +who had written several letters to him: “My delay this time is not +without excuse. I have been sick for a long time, and in a very +wearisome way. I have had an inflammation of the lungs, followed by +scarlet-fever. (Perhaps you know what this horrible disease is.) I had +to remain twelve days in bed, without sleep or food, without any +occupation, amusing myself with leeches, with drinks, with baths, and +other toys out of the same shop. This little diversion is not yet ended, +for I am still very feeble.” + +This letter was written at the end of July. + +There was a relapse, then brain fever; then finally, hardly recovered +from so many violent illnesses which had weakened him morally and +physically, Sadi was carried off in a few hours, August 24, 1832, by an +attack of cholera. Towards the last, and as if from a dark presentiment, +he had given much attention to the prevailing epidemic, following its +course with the attention and penetration that he gave to everything. + +Sadi Carnot died in the vigor of life, in the brightness of a career +that he bade fair to run with glory, leaving memory of profound esteem +and affection in the hearts of many friends. + +His copy-books, filled with memoranda, attest the activity of his mind, +the variety of his knowledge, his love of humanity, his clear sentiments +of justice and of liberty. We can follow therein the traces of all his +various studies. But the only work that he actually completed is this +which is here published. It will suffice to preserve his name from +oblivion. + +His moral character has other claims on our recognition. Our only +ambition here is to present a sketch of it. But, much better than +through the perusal of these few pages, Sadi Carnot can be appreciated +by reading the thoughts scattered through his memoranda, which are to be +carefully collected. There are many practical rules of conduct which he +records for himself; many observations that he desires to fix in his +memory; sometimes an impression that has just come to him, grave or gay; +sometimes too, though rarely, a trace of ill-humor directed against men +or society. He never thought that these notes, the outpouring of his +mind, would be read by other eyes than his own, or that they would some +day be used to judge him. I find in them, for my part, touching +analogies with the thoughts of my father, although the father and son +had, unfortunately, lived almost always apart, by force of +circumstances.[3] + + + + + III. + REFLECTIONS ON THE MOTIVE POWER OF HEAT, AND ON MACHINES FITTED TO + DEVELOP THAT POWER.[4] + + BY S. CARNOT. + + +Every one knows that heat can produce motion. That it possesses vast +motive power no one can doubt, in these days when the steam-engine is +everywhere so well known. + +To heat also are due the vast movements which take place on the earth. +It causes the agitations of the atmosphere, the ascension of clouds, the +fall of rain and of meteors, the currents of water which channel the +surface of the globe, and of which man has thus far employed but a small +portion. Even earthquakes and volcanic eruptions are the result of heat. + +From this immense reservoir we may draw the moving force necessary for +our purposes. Nature, in providing us with combustibles on all sides, +has given us the power to produce, at all times and in all places, heat +and the impelling power which is the result of it. To develop this +power, to appropriate it to our uses, is the object of heat-engines. + +The study of these engines is of the greatest interest, their importance +is enormous, their use is continually increasing, and they seem destined +to produce a great revolution in the civilized world. + +Already the steam-engine works our mines, impels our ships, excavates +our ports and our rivers, forges iron, fashions wood, grinds grains, +spins and weaves our cloths, transports the heaviest burdens, etc. It +appears that it must some day serve as a universal motor, and be +substituted for animal power, waterfalls, and air currents. + +Over the first of these motors it has the advantage of economy, over the +two others the inestimable advantage that it can be used at all times +and places without interruption. + +If, some day, the steam-engine shall be so perfected that it can be set +up and supplied with fuel at small cost, it will combine all desirable +qualities, and will afford to the industrial arts a range the extent of +which can scarcely be predicted. It is not merely that a powerful and +convenient motor that can be procured and carried anywhere is +substituted for the motors already in use, but that it causes rapid +extension in the arts in which it is applied, and can even create +entirely new arts. + +The most signal service that the steam-engine has rendered to England is +undoubtedly the revival of the working of the coal-mines, which had +declined, and threatened to cease entirely, in consequence of the +continually increasing difficulty of drainage, and of raising the +coal.[5] We should rank second the benefit to iron manufacture, both by +the abundant supply of coal substituted for wood just when the latter +had begun to grow scarce, and by the powerful machines of all kinds, the +use of which the introduction of the steam-engine has permitted or +facilitated. + +Iron and heat are, as we know, the supporters, the bases, of the +mechanic arts. It is doubtful if there be in England a single industrial +establishment of which the existence does not depend on the use of these +agents, and which does not freely employ them. To take away to-day from +England her steam-engines would be to take away at the same time her +coal and iron. It would be to dry up all her sources of wealth, to ruin +all on which her prosperity depends, in short, to annihilate that +colossal power. The destruction of her navy, which she considers her +strongest defence, would perhaps be less fatal. + +The safe and rapid navigation by steamships may be regarded as an +entirely new art due to the steam-engine. Already this art has permitted +the establishment of prompt and regular communications across the arms +of the sea, and on the great rivers of the old and new continents. It +has made it possible to traverse savage regions where before we could +scarcely penetrate. It has enabled us to carry the fruits of +civilization over portions of the globe where they would else have been +wanting for years. Steam navigation brings nearer together the most +distant nations. It tends to unite the nations of the earth as +inhabitants of one country. In fact, to lessen the time, the fatigues, +the uncertainties, and the dangers of travel—is not this the same as +greatly to shorten distances?[6] + +The discovery of the steam-engine owed its birth, like most human +inventions, to rude attempts which have been attributed to different +persons, while the real author is not certainly known. It is, however, +less in the first attempts that the principal discovery consists, than +in the successive improvements which have brought steam-engines to the +condition in which we find them to-day. There is almost as great a +distance between the first apparatus in which the expansive force of +steam was displayed and the existing machine, as between the first raft +that man ever made and the modern vessel. + +If the honor of a discovery belongs to the nation in which it has +acquired its growth and all its developments, this honor cannot be here +refused to England. Savery, Newcomen, Smeaton, the famous Watt, Woolf, +Trevithick, and some other English engineers, are the veritable creators +of the steam-engine. It has acquired at their hands all its successive +degrees of improvement. Finally, it is natural that an invention should +have its birth and especially be developed, be perfected, in that place +where its want is most strongly felt. + +Notwithstanding the work of all kinds done by steam-engines, +notwithstanding the satisfactory condition to which they have been +brought to-day, their theory is very little understood, and the attempts +to improve them are still directed almost by chance. + +The question has often been raised whether the motive power of heat[7] +is unbounded, whether the possible improvements in steam-engines have an +assignable limit,—a limit which the nature of things will not allow to +be passed by any means whatever; or whether, on the contrary, these +improvements may be carried on indefinitely. We have long sought, and +are seeking to-day, to ascertain whether there are in existence agents +preferable to the vapor of water for developing the motive power of +heat; whether atmospheric air, for example, would not present in this +respect great advantages. We propose now to submit these questions to a +deliberate examination. + +The phenomenon of the production of motion by heat has not been +considered from a sufficiently general point of view. We have considered +it only in machines the nature and mode of action of which have not +allowed us to take in the whole extent of application of which it is +susceptible. In such machines the phenomenon is, in a way, incomplete. +It becomes difficult to recognize its principles and study its laws. + +In order to consider in the most general way the principle of the +production of motion by heat, it must be considered independently of any +mechanism or any particular agent. It is necessary to establish +principles applicable not only to steam-engines[8] but to all imaginable +heat-engines, whatever the working substance and whatever the method by +which it is operated. + +Machines which do not receive their motion from heat, those which have +for a motor the force of men or of animals, a waterfall, an air-current, +etc., can be studied even to their smallest details by the mechanical +theory. All cases are foreseen, all imaginable movements are referred to +these general principles, firmly established, and applicable under all +circumstances. This is the character of a complete theory. A similar +theory is evidently needed for heat-engines. We shall have it only when +the laws of Physics shall be extended enough, generalized enough, to +make known beforehand all the effects of heat acting in a determined +manner on any body. + +We will suppose in what follows at least a superficial knowledge of the +different parts which compose an ordinary steam-engine; and we consider +it unnecessary to explain what are the furnace, boiler, steam-cylinder, +piston, condenser, etc. + +The production of motion in steam-engines is always accompanied by a +circumstance on which we should fix our attention. This circumstance is +the re-establishing of equilibrium in the caloric; that is, its passage +from a body in which the temperature is more or less elevated, to +another in which it is lower. What happens in fact in a steam-engine +actually in motion? The caloric developed in the furnace by the effect +of the combustion traverses the walls of the boiler, produces steam, and +in some way incorporates itself with it. The latter carrying it away, +takes it first into the cylinder, where it performs some function, and +from thence into the condenser, where it is liquefied by contact with +the cold water which it encounters there. Then, as a final result, the +cold water of the condenser takes possession of the caloric developed by +the combustion. It is heated by the intervention of the steam as if it +had been placed directly over the furnace. The steam is here only a +means of transporting the caloric. It fills the same office as in the +heating of baths by steam, except that in this case its motion is +rendered useful. + +We easily recognize in the operations that we have just described the +re-establishment of equilibrium in the caloric, its passage from a more +or less heated body to a cooler one. The first of these bodies, in this +case, is the heated air of the furnace; the second is the condensing +water. The re-establishment of equilibrium of the caloric takes place +between them, if not completely, at least partially, for on the one hand +the heated air, after having performed its function, having passed round +the boiler, goes out through the chimney with a temperature much below +that which it had acquired as the effect of combustion; and on the other +hand, the water of the condenser, after having liquefied the steam, +leaves the machine with a temperature higher than that with which it +entered. + +The production of motive power is then due in steam-engines not to an +actual consumption of caloric, but _to its transportation from a warm +body to a cold body_, that is, to its re-establishment of equilibrium—an +equilibrium considered as destroyed by any cause whatever, by chemical +action such as combustion, or by any other. We shall see shortly that +this principle is applicable to any machine set in motion by heat. + +According to this principle, the production of heat alone is not +sufficient to give birth to the impelling power: it is necessary that +there should also be cold; without it, the heat would be useless. And in +fact, if we should find about us only bodies as hot as our furnaces, how +can we condense steam? What should we do with it if once produced? We +should not presume that we might discharge it into the atmosphere, as is +done in some engines;[9] the atmosphere would not receive it. It does +receive it under the actual condition of things, only because it fulfils +the office of a vast condenser, because it is at a lower temperature; +otherwise it would soon become fully charged, or rather would be already +saturated.[10] + +Wherever there exists a difference of temperature, wherever it has been +possible for the equilibrium of the caloric to be re-established, it is +possible to have also the production of impelling power. Steam is a +means of realizing this power, but it is not the only one. All +substances in nature can be employed for this purpose, all are +susceptible of changes of volume, of successive contractions and +dilatations, through the alternation of heat and cold. All are capable +of overcoming in their changes of volume certain resistances, and of +thus developing the impelling power. A solid body—a metallic bar for +example—alternately heated and cooled increases and diminishes in +length, and can move bodies fastened to its ends. A liquid alternately +heated and cooled increases and diminishes in volume, and can overcome +obstacles of greater or less size, opposed to its dilatation. An +aeriform fluid is susceptible of considerable change of volume by +variations of temperature. If it is enclosed in an expansible space, +such as a cylinder provided with a piston, it will produce movements of +great extent. Vapors of all substances capable of passing into a gaseous +condition, as of alcohol, of mercury, of sulphur, etc., may fulfil the +same office as vapor of water. The latter, alternately heated and +cooled, would produce motive power in the shape of permanent gases, that +is, without ever returning to a liquid state. Most of these substances +have been proposed, many even have been tried, although up to this time +perhaps without remarkable success. + +We have shown that in steam-engines the motive power is due to a +re-establishment of equilibrium in the caloric; this takes place not +only for steam-engines, but also for every heat-engine—that is, for +every machine of which caloric is the motor. Heat can evidently be a +cause of motion only by virtue of the changes of volume or of form which +it produces in bodies. + +These changes are not caused by uniform temperature, but rather by +alternations of heat and cold. Now to heat any substance whatever +requires a body warmer than the one to be heated; to cool it requires a +cooler body. We supply caloric to the first of these bodies that we may +transmit it to the second by means of the intermediary substance. This +is to re-establish, or at least to endeavor to re-establish, the +equilibrium of the caloric. + +It is natural to ask here this curious and important question: Is the +motive power of heat invariable in quantity, or does it vary with the +agent employed to realize it as the intermediary substance, selected as +the subject of action of the heat? + +It is clear that this question can be asked only in regard to a given +quantity of caloric,[11] the difference of the temperatures also being +given. We take, for example, one body _A_ kept at a temperature of 100° +and another body _B_ kept at a temperature of 0°, and ask what quantity +of motive power can be produced by the passage of a given portion of +caloric (for example, as much as is necessary to melt a kilogram of ice) +from the first of these bodies to the second. We inquire whether this +quantity of motive power is necessarily limited, whether it varies with +the substance employed to realize it, whether the vapor of water offers +in this respect more or less advantage than the vapor of alcohol, of +mercury, a permanent gas, or any other substance. We will try to answer +these questions, availing ourselves of ideas already established. + +We have already remarked upon this self-evident fact, or fact which at +least appears evident as soon as we reflect on the changes of volume +occasioned by heat: _wherever there exists a difference of temperature, +motive power can be produced_. Reciprocally, wherever we can consume +this power, it is possible to produce a difference of temperature, it is +possible to occasion destruction of equilibrium in the caloric. Are not +percussion and the friction of bodies actually means of raising their +temperature, of making it reach spontaneously a higher degree than that +of the surrounding bodies, and consequently of producing a destruction +of equilibrium in the caloric, where equilibrium previously existed? It +is a fact proved by experience, that the temperature of gaseous fluids +is raised by compression and lowered by rarefaction. This is a sure +method of changing the temperature of bodies, and destroying the +equilibrium of the caloric as many times as may be desired with the same +substance. The vapor of water employed in an inverse manner to that in +which it is used in steam-engines can also be regarded as a means of +destroying the equilibrium of the caloric. To be convinced of this we +need but to observe closely the manner in which motive power is +developed by the action of heat on vapor of water. Imagine two bodies +_A_ and _B_, kept each at a constant temperature, that of _A_ being +higher than that of _B_. These two bodies, to which we can give or from +which we can remove the heat without causing their temperatures to vary, +exercise the functions of two unlimited reservoirs of caloric. We will +call the first the furnace and the second the refrigerator. + +If we wish to produce motive power by carrying a certain quantity of +heat from the body _A_ to the body _B_ we shall proceed as follows: + +(1) To borrow caloric from the body _A_ to make steam with it—that is, +to make this body fulfil the function of a furnace, or rather of the +metal composing the boiler in ordinary engines—we here assume that the +steam is produced at the same temperature as the body _A_. + +(2) The steam having been received in a space capable of expansion, such +as a cylinder furnished with a piston, to increase the volume of this +space, and consequently also that of the steam. Thus rarefied, the +temperature will fall spontaneously, as occurs with all elastic fluids; +admit that the rarefaction may be continued to the point where the +temperature becomes precisely that of the body _B_. + +(3) To condense the steam by putting it in contact with the body _B_, +and at the same time exerting on it a constant pressure until it is +entirely liquefied. The body _B_ fills here the place of the +injection-water in ordinary engines, with this difference, that it +condenses the vapor without mingling with it, and without changing its +own temperature.[12] + +The operations which we have just described might have been performed in +an inverse direction and order. There is nothing to prevent forming +vapor with the caloric of the body _B_, and at the temperature of that +body, compressing it in such a way as to make it acquire the temperature +of the body _A_, finally condensing it by contact with this latter body, +and continuing the compression to complete liquefaction. + +By our first operations there would have been at the same time +production of motive power and transfer of caloric from the body _A_ to +the body _B_. By the inverse operations there is at the same time +expenditure of motive power and return of caloric from the body _B_ to +the body _A_. But if we have acted in each case on the same quantity of +vapor, if there is produced no loss either of motive power or caloric, +the quantity of motive power produced in the first place will be equal +to that which would have been expended in the second, and the quantity +of caloric passed in the first case from the body _A_ to the body _B_ +would be equal to the quantity which passes back again in the second +from the body _B_ to the body _A_; so that an indefinite number of +alternative operations of this sort could be carried on without in the +end having either produced motive power or transferred caloric from one +body to the other. + +Now if there existed any means of using heat preferable to those which +we have employed, that is, if it were possible by any method whatever to +make the caloric produce a quantity of motive power greater than we have +made it produce by our first series of operations, it would suffice to +divert a portion of this power in order by the method just indicated to +make the caloric of the body _B_ return to the body _A_ from the +refrigerator to the furnace, to restore the initial conditions, and thus +to be ready to commence again an operation precisely similar to the +former, and so on: this would be not only perpetual motion, but an +unlimited creation of motive power without consumption either of caloric +or of any other agent whatever. Such a creation is entirely contrary to +ideas now accepted, to the laws of mechanics and of sound physics. It is +inadmissible.[13] We should then conclude that _the maximum of motive +power resulting from the employment of steam is also the maximum of +motive power realizable by any means whatever_. We will soon give a +second more rigorous demonstration of this theory. This should be +considered only as an approximation. (See page 59.) + +We have a right to ask, in regard to the proposition just enunciated, +the following questions: What is the sense of the word _maximum_ here? +By what sign can it be known that this maximum is attained? By what sign +can it be known whether the steam is employed to greatest possible +advantage in the production of motive power? + +Since every re-establishment of equilibrium in the caloric may be the +cause of the production of motive power, every re-establishment of +equilibrium which shall be accomplished without production of this power +should be considered as an actual loss. Now, very little reflection +would show that all change of temperature which is not due to a change +of volume of the bodies can be only a useless re-establishment of +equilibrium in the caloric.[14] The necessary condition of the maximum +is, then, _that in the bodies employed to realize the motive power of +heat there should not occur any change of temperature which may not be +due to a change of volume_. Reciprocally, every time that this condition +is fulfilled the maximum will be attained. This principle should never +be lost sight of in the construction of heat-engines; it is its +fundamental basis. If it cannot be strictly observed, it should at least +be departed from as little as possible. + +Every change of temperature which is not due to a change of volume or to +chemical action (an action that we provisionally suppose not to occur +here) is necessarily due to the direct passage of the caloric from a +more or less heated body to a colder body. This passage occurs mainly by +the contact of bodies of different temperatures; hence such contact +should be avoided as much as possible. It cannot probably be avoided +entirely, but it should at least be so managed that the bodies brought +in contact with each other differ as little as possible in temperature. +When we just now supposed, in our demonstration, the caloric of the body +_A_ employed to form steam, this steam was considered as generated at +the temperature of the body _A_; thus the contact took place only +between bodies of equal temperatures; the change of temperature +occurring afterwards in the steam was due to dilatation, consequently to +a change of volume. Finally, condensation took place also without +contact of bodies of different temperatures. It occurred while exerting +a constant pressure on the steam brought in contact with the body _B_ of +the same temperature as itself. The conditions for a maximum are thus +found to be fulfilled. In reality the operation cannot proceed exactly +as we have assumed. To determine the passage of caloric from one body to +another, it is necessary that there should be an excess of temperature +in the first, but this excess may be supposed as slight as we please. We +can regard it as insensible in theory, without thereby destroying the +exactness of the arguments. + +A more substantial objection may be made to our demonstration, thus: +When we borrow caloric from the body _A_ to produce steam, and when this +steam is afterwards condensed by its contact with the body _B_, the +water used to form it, and which we considered at first as being of the +temperature of the body _A_, is found at the close of the operation at +the temperature of the body _B_. It has become cool. If we wish to begin +again an operation similar to the first, if we wish to develop a new +quantity of motive power with the same instrument, with the same steam, +it is necessary first to re-establish the original condition—to restore +the water to the original temperature. This can undoubtedly be done by +at once putting it again in contact with the body _A_; but there is then +contact between bodies of different temperatures, and loss of motive +power.[15] It would be impossible to execute the inverse operation, that +is, to return to the body _A_ the caloric employed to raise the +temperature of the liquid. + +This difficulty may be removed by supposing the difference of +temperature between the body _A_ and the body _B_ indefinitely small. +The quantity of heat necessary to raise the liquid to its former +temperature will be also indefinitely small and unimportant relatively +to that which is necessary to produce steam—a quantity always limited. + +The proposition found elsewhere demonstrated for the case in which the +difference between the temperatures of the two bodies is indefinitely +small, may be easily extended to the general case. In fact, if it +operated to produce motive power by the passage of caloric from the body +_A_ to the body _Z_, the temperature of this latter body being very +different from that of the former, we should imagine a series of bodies +_B_, _C_, _D_ ... of temperatures intermediate between those of the +bodies _A_, _Z_, and selected so that the differences from _A_ to _B_, +from _B_ to _C_, etc., may all be indefinitely small. The caloric coming +from _A_ would not arrive at _Z_ till after it had passed through the +bodies _B_, _C_, _D_, etc., and after having developed in each of these +stages maximum motive power. The inverse operations would here be +entirely possible, and the reasoning of page 52 would be strictly +applicable. + +According to established principles at the present time, we can compare +with sufficient accuracy the motive power of heat to that of a +waterfall. Each has a maximum that we cannot exceed, whatever may be, on +the one hand, the machine which is acted upon by the water, and +whatever, on the other hand, the substance acted upon by the heat. The +motive power of a waterfall depends on its height and on the quantity of +the liquid; the motive power of heat depends also on the quantity of +caloric used, and on what may be termed, on what in fact we will call, +the _height of its fall_,[16] that is to say, the difference of +temperature of the bodies between which the exchange of caloric is made. +In the waterfall the motive power is exactly proportional to the +difference of level between the higher and lower reservoirs. In the fall +of caloric the motive power undoubtedly increases with the difference of +temperature between the warm and the cold bodies; but we do not know +whether it is proportional to this difference. We do not know, for +example, whether the fall of caloric from 100 to 50 degrees furnishes +more or less motive power than the fall of this same caloric from 50 to +zero. It is a question which we propose to examine hereafter. + +We shall give here a second demonstration of the fundamental proposition +enunciated on page 56, and present this proposition under a more general +form than the one already given. + +When a gaseous fluid is rapidly compressed its temperature rises. It +falls, on the contrary, when it is rapidly dilated. This is one of the +facts best demonstrated by experiment. We will take it for the basis of +our demonstration.[17] + +If, when the temperature of a gas has been raised by compression, we +wish to reduce it to its former temperature without subjecting its +volume to new changes, some of its caloric must be removed. This caloric +might have been removed in proportion as pressure was applied, so that +the temperature of the gas would remain constant. Similarly, if the gas +is rarefied we can avoid lowering the temperature by supplying it with a +certain quantity of caloric. Let us call the caloric employed at such +times, when no change of temperature occurs, _caloric due to change of +volume_. This denomination does not indicate that the caloric appertains +to the volume: it does not appertain to it any more than to pressure, +and might as well be called _caloric due to the change of pressure_. We +do not know what laws it follows relative to the variations of volume: +it is possible that its quantity changes either with the nature of the +gas, its density, or its temperature. Experiment has taught us nothing +on this subject. It has only shown us that this caloric is developed in +greater or less quantity by the compression of the elastic fluids. + +[Illustration: + + FIG. 1. +] + +This preliminary idea being established, let us imagine an elastic +fluid, atmospheric air for example, shut up in a cylindrical vessel, +_abcd_ (Fig. 1), provided with a movable diaphragm or piston, _cd_. Let +there be also two bodies, _A_ and _B_, kept each at a constant +temperature, that of _A_ being higher than that of _B_. Let us picture +to ourselves now the series of operations which are to be described: + +(1) Contact of the body _A_ with the air enclosed in the space _abcd_ or +with the wall of this space—a wall that we will suppose to transmit the +caloric readily. The air becomes by such contact of the same temperature +as the body _A_; _cd_ is the actual position of the piston. + +(2) The piston gradually rises and takes the position _ef_. The body _A_ +is all the time in contact with the air, which is thus kept at a +constant temperature during the rarefaction. The body _A_ furnishes the +caloric necessary to keep the temperature constant. + +(3) The body _A_ is removed, and the air is then no longer in contact +with any body capable of furnishing it with caloric. The piston +meanwhile continues to move, and passes from the position _ef_ to the +position _gh_. The air is rarefied without receiving caloric, and its +temperature falls. Let us imagine that it falls thus till it becomes +equal to that of the body _B_; at this instant the piston stops, +remaining at the position _gh_. + +(4) The air is placed in contact with the body _B_; it is compressed by +the return of the piston as it is moved from the position _gh_ to the +position _cd_. This air remains, however, at a constant temperature +because of its contact with the body _B_, to which it yields its +caloric. + +(5) The body _B_ is removed, and the compression of the air is +continued, which being then isolated, its temperature rises. The +compression is continued till the air acquires the temperature of the +body _A_. The piston passes during this time from the position _cd_ to +the position _ik_. + +(6) The air is again placed in contact with the body _A_. The piston +returns from the position _ik_ to the position _ef_; the temperature +remains unchanged. + +(7) The step described under number 3 is renewed, then successively the +steps 4, 5, 6, 3, 4, 5, 6, 3, 4, 5; and so on. + +In these various operations the piston is subject to an effort of +greater or less magnitude, exerted by the air enclosed in the cylinder; +the elastic force of this air varies as much by reason of the changes in +volume as of changes of temperature. But it should be remarked that with +equal volumes, that is, for the similar positions of the piston, the +temperature is higher during the movements of dilatation than during the +movements of compression. During the former the elastic force of the air +is found to be greater, and consequently the quantity of motive power +produced by the movements of dilatation is more considerable than that +consumed to produce the movements of compression. Thus we should obtain +an excess of motive power—an excess which we could employ for any +purpose whatever. The air, then, has served as a heat-engine; we have, +in fact, employed it in the most advantageous manner possible, for no +useless re-establishment of equilibrium has been effected in the +caloric. + +All the above-described operations may be executed in an inverse sense +and order. Let us imagine that, after the sixth period, that is to say +the piston having arrived at the position _ef_, we cause it to return to +the position _ik_, and that at the same time we keep the air in contact +with the body _A_. The caloric furnished by this body during the sixth +period would return to its source, that is, to the body _A_, and the +conditions would then become precisely the same as they were at the end +of the fifth period. If now we take away the body _A_, and if we cause +the piston to move from _ef_ to _cd_, the temperature of the air will +diminish as many degrees as it increased during the fifth period, and +will become that of the body _B_. We may evidently continue a series of +operations the inverse of those already described. It is only necessary +under the same circumstances to execute for each period a movement of +dilatation instead of a movement of compression, and reciprocally. + +The result of these first operations has been the production of a +certain quantity of motive power and the removal of caloric from the +body _A_ to the body _B_. The result of the inverse operations is the +consumption of the motive power produced and the return of the caloric +from the body _B_ to the body _A_; so that these two series of +operations annul each other, after a fashion, one neutralizing the +other. + +The impossibility of making the caloric produce a greater quantity of +motive power than that which we obtained from it by our first series of +operations, is now easily proved. It is demonstrated by reasoning very +similar to that employed at page 56; the reasoning will here be even +more exact. The air which we have used to develop the motive power is +restored at the end of each cycle of operations exactly to the state in +which it was at first found, while, as we have already remarked, this +would not be precisely the case with the vapor of water.[18] + +We have chosen atmospheric air as the instrument which should develop +the motive power of heat, but it is evident that the reasoning would +have been the same for all other gaseous substances, and even for all +other bodies susceptible of change of temperature through successive +contractions and dilatations, which comprehends all natural substances, +or at least all those which are adapted to realize the motive power of +heat. Thus we are led to establish this general proposition: + +_The motive power of heat is independent of the agents employed to +realize it; its quantity is fixed solely by the temperatures of the +bodies between which is effected, finally, the transfer of the caloric._ + +We must understand here that each of the methods of developing motive +power attains the perfection of which it is susceptible. This condition +is found to be fulfilled if, as we remarked above, there is produced in +the body no other change of temperature than that due to change of +volume, or, what is the same thing in other words, if there is no +contact between bodies of sensibly different temperatures. + +Different methods of realizing motive power may be taken, as in the +employment of different substances, or in the use of the same substance +in two different states—for example, of a gas at two different +densities. + +This leads us naturally to those interesting researches on the aeriform +fluids—researches which lead us also to new results in regard to the +motive power of heat, and give us the means of verifying, in some +particular cases, the fundamental proposition above stated.[19] + +We readily see that our demonstration would have been simplified by +supposing the temperatures of the bodies _A_ and _B_ to differ very +little. Then the movements of the piston being slight during the periods +3 and 5, these periods might have been suppressed without influencing +sensibly the production of motive power. A very little change of volume +should suffice in fact to produce a very slight change of temperature, +and this slight change of volume may be neglected in presence of that of +the periods 4 and 6, of which the extent is unlimited. + +If we suppress periods 3 and 5, in the series of operations above +described, it is reduced to the following: + +(1) Contact of the gas confined in _abcd_ (Fig. 2) with the body _A_, +passage of the piston from _cd_ to _ef_. + +[Illustration: + + FIG. 2. FIG. 3. +] + +(2) Removal of the body _A_, contact of the gas confined in _abef_ with +the body _B_, return of the piston from _ef_ to _cd_. + +(3) Removal of the body _B_, contact of the gas with the body _A_, +passage of the piston from _cd_ to _ef_, that is, repetition of the +first period, and so on. + +The motive power resulting from the _ensemble_ of operations 1 and 2 +will evidently be the difference between that which is produced by the +expansion of the gas while it is at the temperature of the body _A_, and +that which is consumed to compress this gas while it is at the +temperature of the body _B_. + +Let us suppose that operations 1 and 2 be performed on two gases of +different chemical natures but under the same pressure—under atmospheric +pressure, for example. These two gases will behave exactly alike under +the same circumstances, that is, their expansive forces, originally +equal, will remain always equal, whatever may be the variations of +volume and of temperature, provided these variations are the same in +both. This results obviously from the laws of Mariotte and MM. +Gay-Lussac and Dalton—laws common to all elastic fluids, and in virtue +of which the same relations exist for all these fluids between the +volume, the expansive force, and the temperature. + +Since two different gases at the same temperature and under the same +pressure should behave alike under the same circumstances, if we +subjected them both to the operations above described, they should give +rise to equal quantities of motive power. + +Now this implies, according to the fundamental proposition that we have +established, the employment of two equal quantities of caloric; that is, +it implies that the quantity of caloric transferred from the body _A_ to +the body _B_ is the same, whichever gas is used. + +The quantity of caloric transferred from the body _A_ to the body _B_ is +evidently that which is absorbed by the gas in its expansion of volume, +or that which this gas relinquishes during compression. We are led, +then, to establish the following proposition: + +_When a gas passes without change of temperature from one definite +volume and pressure to another volume and another pressure equally +definite, the quantity of caloric absorbed or relinquished is always the +same, whatever may be the nature of the gas chosen as the subject of the +experiment._ + +Take, for example, 1 litre of air at the temperature of 100° and under +the pressure of one atmosphere. If we double the volume of this air and +wish to maintain it at the temperature of 100°, a certain quantity of +heat must be supplied to it. Now this quantity will be precisely the +same if, instead of operating on the air, we operate upon carbonic-acid +gas, upon nitrogen, upon hydrogen, upon vapor of water or of alcohol, +that is, if we double the volume of 1 litre of these gases taken at the +temperature of 100° and under atmospheric pressure. + +It will be the same thing in the inverse sense if, instead of doubling +the volume of gas, we reduce it one half by compression. The quantity of +heat that the elastic fluids set free or absorb in their changes of +volume has never been measured by any direct experiment, and doubtless +such an experiment would be very difficult, but there exists a datum +which is very nearly its equivalent. This has been furnished by the +theory of sound. It deserves much confidence because of the exactness of +the conditions which have led to its establishment. It consists in this: + +Atmospheric air should rise one degree Centigrade when by sudden +compression it experiences a reduction of volume of ¹⁄₁₁₆.[20] + +Experiments on the velocity of sound having been made in air under the +pressure of 760 millimetres of mercury and at the temperature of 6°, it +is only to these two circumstances that our datum has reference. We +will, however, for greater facility, refer it to the temperature 0°, +which is nearly the same. + +Air compressed ¹⁄₁₁₆, and thus heated one degree, differs from air +heated directly one degree only in its density. The primitive volume +being supposed to be _V_, the compression of ¹⁄₁₁₆ reduces it to _V_ − +¹⁄₁₁₆ _V_. + +Direct heating under constant pressure should, according to the rule of +M. Gay-Lussac, increase the volume of air ¹⁄₂₆₇ above what it would be +at 0°: so the air is, on the one hand, reduced to the volume _V_ − ¹⁄₁₁₆ +_V_; on the other, it is increased to _V_ + ¹⁄₂₆₇ _V_. + +The difference between the quantities of heat which the air possesses in +both cases is evidently the quantity employed to raise it directly one +degree; so then the quantity of heat that the air would absorb in +passing from the volume _V_ − ¹⁄₁₁₆ _V_ to the volume _V_ + ¹⁄₂₆₇ _V_ is +equal to that which is required to raise it one degree. + +Let us suppose now that, instead of heating one degree the air subjected +to a constant pressure and able to dilate freely, we inclose it within +an invariable space, and that in this condition we cause it to rise one +degree in temperature. The air thus heated one degree will differ from +the air compressed ¹⁄₁₁₆ only by its ¹⁄₁₁₆ greater volume. So then the +quantity of heat that the air would set free by a reduction of volume of +¹⁄₁₁₆ is equal to that which would be required to raise it one degree +Centigrade under constant volume. As the differences between the volumes +_V_ − ¹⁄₁₁₆ _V_, _V_, and _V_ + ¹⁄₂₆₇ _V_ are small relatively to the +volumes themselves, we may regard the quantities of heat absorbed by the +air in passing from the first of these volumes to the second, and from +the first to the third, as sensibly proportional to the changes of +volume. We are then led to the establishment of the following relation: + +The quantity of heat necessary to raise one degree air under constant +pressure is to the quantity of heat necessary to raise one degree the +same air under constant volume, in the ratio of the numbers + + ¹⁄₁₁₆ + ¹⁄₂₆₇ to ¹⁄₁₁₆; + +or, multiplying both by 116 × 267, in the ratio of the numbers 267 + 116 +to 267. + +This, then, is the ratio which exists between the capacity of air for +heat under constant pressure and its capacity under constant volume. If +the first of these two capacities is expressed by unity, the other will +be expressed by the number (267)/(267 + 116), or very nearly 0.700; +their difference, 1 − 0.700 or 0.300, will evidently express the +quantity of heat which will produce the increase of volume in the air +when it is heated one degree under constant pressure. + +According to the law of MM. Gay-Lussac and Dalton, this increase of +volume would be the same for all other gases; according to the theory +demonstrated on page 87, the heat absorbed by these equal increases of +volume is the same for all the elastic fluids, which leads to the +establishment of the following proposition: + +_The difference between specific heat under constant pressure and +specific heat under constant volume is the same for all gases._ + +It should be remarked here that all the gases are considered as taken +under the same pressure, atmospheric pressure for example, and that the +specific heats are also measured with reference to the volumes. + +It is a very easy matter now for us to prepare a table of the specific +heat of gases under constant volume, from the knowledge of their +specific heats under constant pressure. Here is the table: + + TABLE OF THE SPECIFIC HEAT OF GASES. + ───────────────────────┬───────────────────────┬─────────────────────── + NAMES OF GASES. │ Specific Heat under │Specific Heat at Const. + │ Const. Press. │ Vol. + ───────────────────────┼───────────────────────┼─────────────────────── + Atmospheric Air, │ 1.000 │ 0.700 + Hydrogen Gas, │ 0.903 │ 0.603 + Carbonic Acid, │ 1.258 │ 0.958 + Oxygen, │ 0.976 │ 0.676 + Nitrogen, │ 1.000 │ 0.700 + Protoxide of Nitrogen, │ 1.350 │ 1.050 + Olefiant Gas, │ 1.553 │ 1.253 + Oxide of Carbon, │ 1.034 │ 0.734 + ───────────────────────┴───────────────────────┴─────────────────────── + +The first column is the result of the direct experiments of MM. +Delaroche and Bérard on the specific heat of the gas under atmospheric +pressure, and the second column is composed of the numbers of the first +diminished by 0.300. + +The numbers of the first column and those of the second are here +referred to the same unit, to the specific heat of atmospheric air under +constant pressure. + +The difference between each number of the first column and the +corresponding number of the second being constant, the relation between +these numbers should be variable. Thus the relation between the specific +heat of gases under constant pressure and the specific heat at constant +volume, varies in different gases. + +We have seen that air when it is subjected to a sudden compression of +¹⁄₁₁₆ of its volume rises one degree in temperature. The other gases +through a similar compression should also rise in temperature. They +should rise, but not equally, in inverse ratio with their specific heat +at constant volume. In fact, the reduction of volume being by hypothesis +always the same, the quantity of heat due to this reduction should +likewise be always the same, and consequently should produce an +elevation of temperature dependent only on the specific heat acquired by +the gas after its compression, and evidently in inverse ratio with this +specific heat. Thus we can easily form the table of the elevations of +temperature of the different gases for a compression of ¹⁄₁₁₆. + + TABLE OF THE ELEVATION OF TEMPERATURE<BR>OF + _Gases through the Effect of Compression_. + ──────────────────────┬──────────────────────────────────────────────── + NAMES OF GASES. │ Elevation of Temperature for a Reduction of + │ Volume of ¹⁄₁₁₆. + ──────────────────────┼──────────────────────────────────────────────── + │ ° + Atmospheric Air, │ 1.000 + Hydrogen Gas, │ 1.160 + Carbonic Acid, │ 0.730 + Oxygen, │ 1.035 + Nitrogen, │ 1.000 + Protoxide of Nitrogen,│ 0.667 + Olefiant Gas, │ 0.558 + Carbonic Oxide, │ 0.955 + ──────────────────────┴──────────────────────────────────────────────── + +A second compression of ¹⁄₁₁₆ (of the altered volume), as we shall +presently see, would also raise the temperature of these gases nearly as +much as the first; but it would not be the same with a third, a fourth, +a hundredth such compression. The capacity of gases for heat changes +with their volume. It is not unlikely that it changes also with the +temperature. + +We shall now deduce from the general proposition stated on page 68 a +second theory, which will serve as a corollary to that just +demonstrated. + +Let us suppose that the gas enclosed in the cylindrical space _abcd_ +(Fig. 2) be transported into the space _a′b′c′d′_ (Fig. 3) of equal +height, but of different base and wider. This gas would increase in +volume, would diminish in density and in elastic force, in the inverse +ratio of the two volumes _abcd_, _a′b′c′d′_. As to the total pressure +exerted in each piston _cd_, _c′d′_, it would be the same from all +quarters, for the surface of these pistons is in direct ratio to the +volumes. + +Let us suppose that we perform on the gas inclosed in _a′b′c′d′_ the +operations described on page 70, and which were taken as having been +performed upon the gas inclosed in _abcd_; that is, let us suppose that +we have given to the piston _c′d′_ motions equal to those of the piston +_cd_, that we have made it occupy successively the positions _c′d′_ +corresponding to _cd_, and _e′f′_ corresponding to _ef_, and that at the +same time we have subjected the gas by means of the two bodies _A_ and +_B_ to the same variations of temperature as when it was inclosed in +_abcd_. The total effort exercised on the piston would be found to be, +in the two cases, always the same at the corresponding instants. This +results solely from the law of Mariotte.[21] In fact, the densities of +the two gases maintaining always the same ratio for similar positions of +the pistons, and the temperatures being always equal in both, the total +pressures exercised on the pistons will always maintain the same ratio +to each other. If this ratio is, at any instant whatever, unity, the +pressures will always be equal. + +As, furthermore, the movements of the two pistons have equal extent, the +motive power produced by each will evidently be the same; whence we +should conclude, according to the proposition on page 68, that the +quantities of heat consumed by each are the same, that is, that there +passes from the body _A_ to the body _B_ the same quantity of heat in +both cases. + +The heat abstracted from the body _A_ and communicated to the body _B_, +is simply the heat absorbed during the rarefaction of the gas, and +afterwards liberated by its compression. We are therefore led to +establish the following theorem: + +_When an elastic fluid passes without change of temperature from the +volume U to the volume V, and when a similar ponderable quantity of the +same gas passes at the same temperature from the volume U′ to the volume +V′, if the ratio of U′ to V′ is found to be the same as the ratio of U +to V, the quantities of heat absorbed or disengaged in the two cases +will be equal._ + +This theorem might also be expressed as follows: + +_When a gas varies in volume without change of temperature, the +quantities of heat absorbed or liberated by this gas are in arithmetical +progression, if the increments or the decrements of volume are found to +be in geometrical progression._ + +When a litre of air maintained at a temperature of ten degrees is +compressed, and when it is reduced to one half a litre, a certain +quantity of heat is set free. This quantity will be found always the +same if the volume is further reduced from a half litre to a quarter +litre, from a quarter litre to an eighth, and so on. + +If, instead of compressing the air, we carry it successively to two +litres, four litres, eight litres, etc., it will be necessary to supply +to it always equal quantities of heat in order to maintain a constant +temperature. + +This readily accounts for the high temperature attained by air when +rapidly compressed. We know that this temperature inflames tinder and +even makes air luminous. If, for a moment, we suppose the specific heat +of air to be constant, in spite of the changes of volume and +temperature, the temperature will increase in arithmetical progression +for reduction of volume in geometrical progression. + +Starting from this datum, and admitting that one degree of elevation in +the temperature corresponds to a compression of ¹⁄₁₁₆, we shall readily +come to the conclusion that air reduced to ¹⁄₁₄ of its primitive volume +should rise in temperature about 300 degrees, which is sufficient to +inflame tinder.[22] + +The elevation of temperature ought, evidently, to be still more +considerable if the capacity of the air for heat becomes less as its +volume diminishes. Now this is probable, and it also seems to follow +from the experiments of MM. Delaroche and Bérard on the specific heat of +air taken at different densities. (See the Mémoire in the _Annales de +Chimie_, t. lxxxv. pp. 72, 224.) + +The two theorems explained on pp. 72 and 81 suffice for the comparison +of the quantities of heat absorbed or set free in the changes of volume +of elastic fluids, whatever may be the density and the chemical nature +of these fluids, provided always that they be taken and maintained at a +certain invariable temperature. But these theories furnish no means of +comparing the quantities of heat liberated or absorbed by elastic fluids +which change in volume at different temperatures. Thus we are ignorant +what relation exists between the heat relinquished by a litre of air +reduced one half, the temperature being kept at zero, and the heat +relinquished by the same litre of air reduced one half, the temperature +being kept at 100°. The knowledge of this relation is closely connected +with that of the specific heat of gases at various temperatures, and to +some other data that Physics as yet does not supply. + +The second of our theorems offers us a means of determining according to +what law the specific heat of gases varies with their density. + +Let us suppose that the operations described on p. 70, instead of being +performed with two bodies, _A_, _B_, of temperatures differing +indefinitely small, were carried on with two bodies whose temperatures +differ by a finite quantity—one degree, for example. In a complete +circle of operations the body _A_ furnishes to the elastic fluid a +certain quantity of heat, which may be divided into two portions: (1) +That which is necessary to maintain the temperature of the fluid +constant during dilatation; (2) that which is necessary to restore the +temperature of the fluid from that of the body _B_ to that of the body +_A_, when, after having brought back this fluid to its primitive volume, +we place it again in contact with the body _A_. Let us call the first of +these quantities _a_ and the second _b_. The total caloric furnished by +the body A will be expressed by _a_ + _b_. + +The caloric transmitted by the fluid to the body _B_ may also be divided +into two parts: one, _b′_, due to the cooling of the gas by the body +_B_; the other, _a′_, which the gas abandons as a result of its +reduction of volume. The sum of these two quantities is _a′_ + _b′_; it +should be equal to _a_ + _b_, for, after a complete cycle of operations, +the gas is brought back exactly to its primitive state. It has been +obliged to give up all the caloric which has first been furnished to it. +We have then + + _a_ + _b_ = _a′_ + _b′_; + +or rather, + + _a_ − _a′_ = _b′_ − _b_. + +Now, according to the theorem given on page 81, the quantities _a_ and +_a′_ are independent of the density of the gas, provided always that the +ponderable quantity remains the same and that the variations of volume +be proportional to the original volume. The difference _a_ − _a′_ should +fulfil the same conditions, and consequently also the difference _b′_ − +_b_, which is equal to it. But _b′_ is the caloric necessary to raise +the gas enclosed in _abcd_ (Fig. 2) one degree; _b′_ is the caloric +surrendered by the gas when, enclosed in _abef_, it is cooled one +degree. These quantities may serve as a measure for specific heats. We +are then led to the establishment of the following proposition: + +_The change in the specific heat of a gas caused by change of volume +depends entirely on the ratio between the original volume and the +altered volume._ That is, the difference of the specific heats does not +depend on the absolute magnitude of the volumes, but only on their +ratio. + +This proposition might also be differently expressed, thus: + +_When a gas increases in volume in geometrical progression, its specific +heat increases in arithmetical progression._ + +Thus, _a_ being the specific heat of air taken at a given density, and +_a_ + _h_ the specific heat for a density one half less, it will be, for +a density equal to one quarter, _a_ + 2_h_; for a density equal to one +eighth, _a_ + 3_h_; and so on. + +The specific heats are here taken with reference to weight. They are +supposed to be taken at an invariable volume, but, as we shall see, they +would follow the same law if they were taken under constant pressure. + +To what cause is the difference between specific heats at constant +volume and at constant pressure really due? To the caloric required to +produce in the second case increase of volume. Now, according to the law +of Mariotte, increase of volume of a gas should be, for a given change +of temperature, a determined fraction of the original volume, a fraction +independent of pressure. According to the theorem expressed on page 76, +if the ratio between the primitive volume and the altered volume is +given, that determines the heat necessary to produce increase of volume. +It depends solely on this ratio and on the weight of the gas. We must +then conclude that: + +_The difference between specific heat at constant pressure and specific +heat at constant volume is always the same, whatever may be the density +of the gas, provided the weight remains the same._ + +These specific heats both increase accordingly as the density of the gas +diminishes, but their difference does not vary.[23] + +Since the difference between the two capacities for heat is constant, if +one increases in arithmetical progression the other should follow a +similar progression: thus one law is applicable to specific heats at +constant pressure. + +We have tacitly assumed the increase of specific heat with that of +volume. This increase is indicated by the experiments of MM. Delaroche +and Bérard: in fact these physicists have found 0.967 for the specific +heat of air under the pressure of 1 metre of mercury (see Mémoire +already cited), taking for the unit the specific heat of the same weight +of air under the pressure of 0^m.760. + +According to the law that specific heats follow with relation to +pressures, it is only necessary to have observed them in two particular +cases to deduce them in all possible cases: it is thus that, making use +of the experimental result of MM. Delaroche and Bérard which has just +been given, we have prepared the following table of the specific heat of +air under different pressures: + + SPECIFIC HEAT OF AIR. + ────────────────────────┬──────────────────────────────────────────── + Pressure in Atmospheres.│Specific Heat, that of Air under Atmospheric + │ Pressure being 1. + ────────────────────────┼──────────────────────────────────────────── + ¹⁄₁₀₂₄ │ 1.840 + ¹⁄₅₁₂ │ 1.756 + ¹⁄₂₅₆ │ 1.672 + ¹⁄₁₂₈ │ 1.588 + ¹⁄₆₄ │ 1.504 + ¹⁄₃₂ │ 1.420 + ¹⁄₁₆ │ 1.336 + ⅛ │ 1.252 + ¼ │ 1.165 + ½ │ 1.084 + 1 │ 1.000 + 2 │ 0.916 + 4 │ 0.832 + 8 │ 0.748 + 16 │ 0.664 + 32 │ 0.580 + 64 │ 0.496 + 128 │ 0.412 + 256 │ 0.328 + 512 │ 0.244 + 1024 │ 0.160 + ────────────────────────┴──────────────────────────────────────────── + +The first column is, as we see, a geometrical progression, and the +second an arithmetical progression. + +We have carried out the table to the extremes of compression and +rarefaction. It may be believed that air would be liquefied before +acquiring a density 1024 times its normal density, that is, before +becoming more dense than water. The specific heat would become zero and +even negative on extending the table beyond the last term. We think, +furthermore, that the figures of the second column here decrease too +rapidly. The experiments which serve as a basis for our calculation have +been made within too contracted limits for us to expect great exactness +in the figures which we have obtained, especially in the outside +numbers. + +Since we know, on the one hand, the law according to which heat is +disengaged in the compression of gases, and on the other, the law +according to which specific heat varies with volume, it will be easy for +us to calculate the increase of temperature of a gas that has been +compressed without being allowed to lose heat. In fact, the compression +may be considered as composed of two successive operations: (1) +compression at a constant temperature; (2) restoration of the caloric +emitted. The temperature will rise through the second operation in +inverse ratio with the specific heat acquired by the gas after the +reduction of volume,—specific heat that we are able to calculate by +means of the law demonstrated above. The heat set free by compression, +according to the theorem of page 81, ought to be represented by an +expression of the form + + _s_ = _A_ + _B_ log _v_, + +_s_ being this heat, _v_ the volume of the gas after compression, _A_ +and _B_ arbitrary constants dependent on the primitive volume of the +gas, on its pressure, and on the units chosen. + +The specific heat varying with the volume according to the law just +demonstrated, should be represented by an expression of the form + + _z_ = _A′_ + _B′_ log _v_, + +_A′_ and _B′_ being the different arbitrary constants of _A_ and _B_. + +The increase of temperature acquired by the gas, as the effect of +compression, is proportional to the ratio (_s_)/(_z_) or to the relation +(_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_). It can be represented by this +ratio itself; thus, calling it _t_, we shall have + + _t_ = (_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_). + +If the original volume of the gas is 1, and the original temperature +zero, we shall have at the same time _t_ = 0, log _v_ = 0, whence _A_ = +0; _t_ will then express not only the increase of temperature, but the +temperature itself above the thermometric zero. + +We need not consider the formula that we have just given as applicable +to very great changes in the volume of gases. We have regarded the +elevation of temperature as being in inverse ratio to the specific heat; +which tacitly supposes the specific heat to be constant at all +temperatures. Great changes of volume lead to great changes of +temperature in the gas, and nothing proves the constancy of specific +heat at different temperatures, especially at temperatures widely +separated. This constancy is only an hypothesis admitted for gases by +analogy, to a certain extent verified for solid bodies and liquids +throughout a part of the thermometric scale, but of which the +experiments of MM. Dulong and Petit have shown the inaccuracy when it is +desirable to extend it to temperatures far above 100°.[24] + +According to a law of MM. Clement and Desormes, a law established by +direct experiment, the vapor of water, under whatever pressure it may be +formed, contains always, at equal weights, the same quantity of heat; +which leads to the assertion that steam, compressed or expanded +mechanically without loss of heat, will always be found in a saturated +state if it was so produced in the first place. The vapor of water so +made may then be regarded as a permanent gas, and should observe all the +laws of one. Consequently the formula + + _t_ = (_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_) + +should be applicable to it, and be found to accord with the table of +tensions derived from the direct experiments of M. Dalton. + +We may be assured, in fact, that our formula, with a convenient +determination of arbitrary constants, represents very closely the +results of experiment. The slight irregularities which we find therein +do not exceed what we might reasonably attribute to errors of +observation.[25] + +We will return, however, to our principal subject, from which we have +wandered too far—the motive power of heat. + +We have shown that the quantity of motive power developed by the +transfer of caloric from one body to another depends essentially upon +the temperature of the two bodies, but we have not shown the relation +between these temperatures and the quantities of motive power produced. +It would at first seem natural enough to suppose that for equal +differences of temperature the quantities of motive power produced are +equal; that is, for example, the passage of a given quantity of caloric +from a body, _A_, maintained at 100°, to a body, _B_, maintained at 50°, +should give rise to a quantity of motive power equal to that which would +be developed by the transfer of the same caloric from a body, _B_, at +50°, to a body, _C_, at zero. Such a law would doubtless be very +remarkable, but we do not see sufficient reason for admitting it _à +priori_. We will investigate its reality by exact reasoning. + +Let us imagine that the operations described on p. 70 be conducted +successively on two quantities of atmospheric air equal in weight and +volume, but taken at different temperatures. Let us suppose, further, +the differences of temperature between the bodies _A_ and _B_ equal, so +these bodies would have for example, in one of these cases, the +temperatures 100° and 100° − _h_ (_h_ being indefinitely small), and in +the other 1° and 1° − _h_. The quantity of motive power produced is, in +each case, the difference between that which the gas supplies by its +dilatation and that which must be expended to restore its primitive +volume. Now this difference is the same in both cases, as any one can +prove by simple reasoning, which it seems unnecessary to give here in +detail; hence the motive power produced is the same. + +Let us now compare the quantities of heat employed in the two cases. In +the first, the quantity of heat employed is that which the body _A_ +furnishes to the air to maintain it at the temperature of 100° during +its expansion. In the second, it is the quantity of heat which this same +body should furnish to it, to keep its temperature at one degree during +an exactly similar change of volume. If these two quantities of heat +were equal, there would evidently result the law that we have already +assumed. But nothing proves that it is so, and we shall find that these +quantities are not equal. + +The air that we shall first consider as occupying the space _abcd_ (Fig. +2), and having 1 degree of temperature, can be made to occupy the space +_abef_, and to acquire the temperature of 100 degrees by two different +means: + +(1) We may heat it without changing its volume, then expand it, keeping +its temperature constant. + +(2) We may begin by expanding it, maintaining the temperature constant, +then heat it, when it has acquired its greater volume. + +Let _a_ and _b_ be the quantities of heat employed successively in the +first of the two operations, and let _b′_ and _a′_ be the quantities of +heat employed successively in the second. As the final result of these +two operations is the same, the quantities of heat employed in both +should be equal. We have then + + _a_ + _b_ = _a′_ + _b′_, + +whence + + _a′_ − _a_ = _b_ − _b′_. + +_a′_ is the quantity of heat required to cause the gas to rise from 1° +to 100° when it occupies the space _abef_. + +_a_ is the quantity of heat required to cause the gas to rise from 1° to +100° when it occupies the space _abcd_. + +The density of the air is less in the first than in the second case, and +according to the experiments of MM. Delaroche and Bérard, already cited +on page 87, its capacity for heat should be a little greater. + +The quantity _a′_ being found to be greater than the quantity _a_, _b_ +should be greater than _b′_. Consequently, generalizing the proposition, +we should say: + +_The quantity of heat due to the change of volume of a gas is greater as +the temperature is higher._ + +Thus, for example, more caloric is necessary to maintain at 100° the +temperature of a certain quantity of air the volume of which is doubled, +than to maintain at 1° the temperature of this same air during a +dilatation exactly equal. + +These unequal quantities of heat would produce, however, as we have +seen, equal quantities of motive power for equal fall of caloric taken +at different heights on the thermometric scale; whence we draw the +following conclusion: + +_The fall of caloric produces more motive power at inferior than at +superior temperatures._ + +Thus a given quantity of heat will develop more motive power in passing +from a body kept at 1 degree to another maintained at zero, than if +these two bodies were at the temperature of 101° and 100°. + +The difference, however, should be very slight. It would be nothing if +the capacity of the air for heat remained constant, in spite of changes +of density. According to the experiments of MM. Delaroche and Bérard, +this capacity varies little—so little even, that the differences noticed +might strictly have been attributed to errors of observation or to some +circumstances of which we have failed to take account. + +We are not prepared to determine precisely, with no more experimental +data than we now possess, the law according to which the motive power of +heat varies at different points on the thermometric scale. This law is +intimately connected with that of the variations of the specific heat of +gases at different temperatures—a law which experiment has not yet made +known to us with sufficient exactness.[26] + +We will endeavor now to estimate exactly the motive power of heat, and +in order to verify our fundamental proposition, in order to determine +whether the agent used to realize the motive power is really unimportant +relatively to the quantity of this power, we will select several of them +successively: atmospheric air, vapor of water, vapor of alcohol. + +Let us suppose that we take first atmospheric air. The operation will +proceed according to the method indicated on page 70. We will make the +following hypotheses: The air is taken under atmospheric pressure. The +temperature of the body _A_ is ¹⁄₁₀₀₀ of a degree above zero, that of +the body _B_ is zero. The difference is, as we see, very slight—a +necessary condition here. + +The increase of volume given to the air in our operation will be ¹⁄₁₁₆ + +¹⁄₂₆₇ of the primitive volume; this is a very slight increase, +absolutely speaking, but great relatively to the difference of +temperature between the bodies _A_ and _B_. + +The motive power developed by the whole of the two operations described +(page 70) will be very nearly proportional to the increase of volume and +to the difference between the two pressures exercised by the air, when +it is found at the temperatures 0°.001 and zero. + +This difference is, according to the law of M. Gay-Lussac, ¹⁄₂₆₇₀₀₀ of +the elastic force of the gas, or very nearly ¹⁄₂₆₇₀₀₀ of the atmospheric +pressure. + +The atmospheric pressure balances at 10.40 metres head of water; +¹⁄₂₆₇₀₀₀ of this pressure equals ¹⁄₂₆₇₀₀₀ × 10^m.40 of head of water. + +As to the increase of volume, it is, by supposition, ¹⁄₁₁₆ + ¹⁄₂₆₇ of +the original volume, that is, of the volume occupied by one kilogram of +air at zero, a volume equal to 0^{mc}.77, allowing for the specific +weight of the air. So then the product, + + (¹⁄₁₁₆ + ¹⁄₂₆₇) × 0.77 × ¹⁄₂₆₇₀₀₀ × 10.40, + +will express the motive power developed. This power is estimated here in +cubic metres of water raised one metre. + +If we carry out the indicated multiplications, we find the value of the +product to be 0.000000372. + +Let us endeavor now to estimate the quantity of heat employed to give +this result; that is, the quantity of heat passed from the body _A_ to +the body _B_. + +The body _A_ furnishes: + +(1) The heat required to carry the temperature of one kilogram of air +from zero to 0°.001; + +(2) The quantity necessary to maintain at this temperature the +temperature of the air when it experiences a dilatation of + + ¹⁄₁₁₆ + ¹⁄₂₆₇. + +The first of these quantities of heat being very small in comparison +with the second, we may disregard it. The second is, according to the +reasoning on page 74, equal to that which would be necessary to increase +one degree the temperature of one kilogram of air subjected to +atmospheric pressure. + +According to the experiments of MM. Delaroche and Bérard on the specific +heat of gases, that of air is, for equal weights, 0.267 that of water. +If, then, we take for the unit of heat the quantity necessary to raise 1 +kilogram of water 1 degree, that which will be required to raise 1 +kilogram of air 1 degree would have for its value 0.267. Thus the +quantity of heat furnished by the body _A_ is + + 0.267 units. + +This is the heat capable of producing 0.000000372 units of motive power +by its fall from 0°.001 to zero. + +For a fall a thousand times greater, for a fall of one degree, the +motive power will be very nearly a thousand times the former, or + + 0.000372. + +If, now, instead of 0.267 units of heat we employ 1000 units, the motive +power produced will be expressed by the proportion + + (0.267)/(0.000372) = (1000)/(x), whence x = (372)/(267) = 1.395. + +Thus 1000 units of heat passing from a body maintained at the +temperature of 1 degree to another body maintained at zero would +produce, in acting upon the air, + + 1.395 units of motive power. + +We will now compare this result with that furnished by the action of +heat on the vapor of water. + +[Illustration: + + FIG. 4. +] + +Let us suppose one kilogram of liquid water enclosed in the cylindrical +vessel _abcd_ (Fig. 4), between the bottom _ab_ and the piston _cd_. Let +us suppose, also, the two bodies _A_, _B_ maintained each at a constant +temperature, that of _A_ being a very little above that of _B_. Let us +imagine now the following operations: + +(1) Contact of the water with the body _A_, movement of the piston from +the position _cd_ to the position _ef_, formation of steam at the +temperature of the body _A_ to fill the vacuum produced by the extension +of volume. We will suppose the space _abef_ large enough to contain all +the water in a state of vapor. + +(2) Removal of the body _A_, contact of the vapor with the body _B_, +precipitation of a part of this vapor, diminution of its elastic force, +return of the piston from _ef_ to _ab_, liquefaction of the rest of the +vapor through the effect of the pressure combined with the contact of +the body _B_. + +(3) Removal of the body _B_, fresh contact of the water with the body +_A_, return of the water to the temperature of this body, renewal of the +former period, and so on. + +The quantity of motive power developed in a complete cycle of operations +is measured by the product of the volume of the vapor multiplied by the +difference between the tensions that it possesses at the temperature of +the body _A_ and at that of the body _B_. As to the heat employed, that +is to say, transported from the body _A_ to the body _B_, it is +evidently that which was necessary to turn the water into vapor, +disregarding always the small quantity required to restore the +temperature of the liquid water from that of _B_ to that of _A_. + +Suppose the temperature of the body _A_ 100 degrees, and that of the +body _B_ 99 degrees: the difference of the tensions will be, according +to the table of M. Dalton, 26 millimetres of mercury or 0^m.36 head of +water. + +The volume of the vapor is 1700 times that of the water. If we operate +on one kilogram, that will be 1700 litres, or 1^{mc}.700. + +Thus the value of the motive power developed is the product + + 1.700 × 0.36 = 0.611 units, + +of the kind of which we have previously made use. + +The quantity of heat employed is the quantity required to turn into +vapor water already heated to 100°. This quantity is found by +experiment. We have found it equal to 550°, or, to speak more exactly, +to 550 of our units of heat. + +Thus 0.611 units of motive power result from the employment of 550 units +of heat. The quantity of motive power resulting from 1000 units of heat +will be given by the proportion + + ⁵⁵⁰⁄₀.611 = 1000/_x_, whence _x_ = ⁶¹¹⁄₅₅₀ = 1.112. + +Thus 1000 units of heat transported from one body kept at 100 degrees to +another kept at 99 degrees will produce, acting upon vapor of water, +1.112 units of motive power. + +The number 1.112 differs by about ¼ from the number 1.395 previously +found for the value of the motive power developed by 1000 units of heat +acting upon the air; but it should be observed that in this case the +temperatures of the bodies _A_ and _B_ were 1 degree and zero, while +here they are 100 degrees and 99 degrees. The difference is much the +same; but it is not found at the same height in the thermometric scale. +To make an exact comparison, it would have been necessary to estimate +the motive power developed by the steam formed at 1 degree and condensed +at zero. It would also have been necessary to know the quantity of heat +contained in the steam formed at one degree. + +The law of MM. Clement and Desormes referred to on page 92 gives us this +datum. The constituent heat of vapor of water being always the same at +any temperature at which vaporization takes place, if 550 degrees of +heat are required to vaporize water already brought up to 100 degrees, +550 + 100 or 650 will be required to vaporize the same weight of water +taken at zero. + +Making use of this datum and reasoning exactly as we did for water at +100 degrees, we find, as is easily seen, + + 1.290 + +for the motive power developed by 1000 units of heat acting upon the +vapor of water between one degree and zero. This number approximates +more closely than the first to + + 1.395. + +It differs from it only ¹⁄₁₃, an error which does not exceed probable +limits, considering the great number of data of different sorts of which +we have been obliged to make use in order to arrive at this +approximation. Thus is our fundamental law verified in a special +case.[27] + +We will examine another case in which vapor of alcohol is acted upon by +heat. The reasoning is precisely the same as for the vapor of water. The +data alone are changed. Pure alcohol boils under ordinary pressure at +78°.7 Centigrade. One kilogram absorbs, according to MM. Delaroche and +Bérard, 207 units of heat in undergoing transformation into vapor at +this same temperature, 78°.7. + +The tension of the vapor of alcohol at one degree below the +boiling-point is found to be diminished ¹⁄₂₅. It is ¹⁄₂₅ less than the +atmospheric pressure; at least, this is the result of the experiment of +M. Bétancour reported in the second part of _l’Architecture hydraulique_ +of M. Prony, pp. 180, 195.[28] + +If we use these data, we find that, in acting upon one kilogram of +alcohol at the temperatures of 78°.7 and 77°.7, the motive power +developed will be 0.251 units. + +This results from the employment of 207 units of heat. For 1000 units +the proportion must be + + (207)/(0.254) = (1000)/(_x_), whence _x_ = 1.230. + +This number is a little more than the 1.112 resulting from the use of +the vapor of water at the temperatures 100° and 99°; but if we suppose +the vapor of water used at the temperatures 78° and 77°, we find, +according to the law of MM. Clement and Desorme, 1.212 for the motive +power due to 1000 units of heat. This latter number approaches, as we +see, very nearly to 1.230. There is a difference of only ¹⁄₅₀. + +We should have liked to be able to make other approximations of this +sort—to be able to calculate, for example, the motive power developed by +the action of heat on solids and liquids, by the congelation of water, +and so on; but Physics as yet refuses us the necessary data.[29] + +The fundamental law that we propose to confirm seems to us to require, +however, in order to be placed beyond doubt, new verifications. It is +based upon the theory of heat as it is understood to-day, and it should +be said that this foundation does not appear to be of unquestionable +solidity. New experiments alone can decide the question. Meanwhile we +can apply the theoretical ideas expressed above, regarding them as +exact, to the examination of the different methods proposed up to date, +for the realization of the motive power of heat. + +It has sometimes been proposed to develop motive power by the action of +heat on solid bodies. The mode of procedure which naturally first occurs +to the mind is to fasten immovably a solid body—a metallic bar, for +example—by one of its extremities; to attach the other extremity to a +movable part of the machine; then, by successive heating and cooling, to +cause the length of the bar to vary, and so to produce motion. Let us +try to decide whether this method of developing motive power can be +advantageous. We have shown that the condition of the most effective +employment of heat in the production of motion is, that all changes of +temperature occurring in the bodies should be due to changes of volume. +The nearer we come to fulfilling this condition the more fully will the +heat be utilized. Now, working in the manner just described, we are very +far from fulfilling this condition: change of temperature is not due +here to change of volume; all the changes are due to contact of bodies +differently heated—to the contact of the metallic bar, either with the +body charged with furnishing heat to it, or with the body charged with +carrying it off. + +The only means of fulfilling the prescribed condition would be to act +upon the solid body exactly as we did on the air in the operations +described on page 92. But for this we must be able to produce, by a +single change of volume of the solid body, considerable changes of +temperature, that is, if we should want to utilize considerable falls of +caloric. Now this appears impracticable. In short, many considerations +lead to the conclusion that the changes produced in the temperature of +solid or liquid bodies through the effect of compression and rarefaction +would be but slight. + +(1) We often observe in machines (particularly in steam-engines) solid +pieces which endure considerable strain in one way or another, and +although these efforts may be sometimes as great as the nature of the +substances employed permits, the variations of temperature are scarcely +perceptible. + +(2) In the action of striking medals, in that of the rolling-mill, of +the draw-plate, the metals undergo the greatest compression to which we +can submit them, employing the hardest and strongest tools. Nevertheless +the elevation of temperature is not great. If it were, the pieces of +steel used in these operations would soon lose their temper. + +(3) We know that it would be necessary to exert on solids and liquids a +very great strain in order to produce in them a reduction of volume +comparable to that which they experience in cooling (cooling from 100° +to zero, for example). Now the cooling requires a greater abstraction of +caloric than would simple reduction of volume. If this reduction were +produced by mechanical means, the heat set free would not then be able +to make the temperature of the body vary as many degrees as the cooling +makes it vary. It would, however, necessitate the employment of a force +undoubtedly very considerable. + +Since solid bodies are susceptible of little change of temperature +through changes of volume, and since the condition of the most effective +employment of heat for the development of motive power is precisely that +all change of temperature should be due to a change of volume, solid +bodies appear but ill fitted to realize this power. + +The same remarks apply to liquids. The same reasons may be given for +rejecting them.[30] + +We are not speaking now of practical difficulties. They will be +numberless. The motion produced by the dilatation and compression of +solid or liquid bodies would only be very slight. In order to give them +sufficient amplitude we should be forced to make use of complicated +mechanisms. It would be necessary to employ materials of the greatest +strength to transmit enormous pressure; finally, the successive +operations would be executed very slowly compared to those of the +ordinary steam-engine, so that apparatus of large dimensions and heavy +cost would produce but very ordinary results. + +The elastic fluids, gases or vapors, are the means really adapted to the +development of the motive power of heat. They combine all the conditions +necessary to fulfil this office. They are easy to compress; they can be +almost infinitely expanded; variations of volume occasion in them great +changes of temperature; and, lastly, they are very mobile, easy to heat +and to cool, easy to transport from one place to another, which enables +them to produce rapidly the desired effects. We can easily conceive a +multitude of machines fitted to develop the motive power of heat through +the use of elastic fluids; but in whatever way we look at it, we should +not lose sight of the following principles: + +(1) The temperature of the fluid should be made as high as possible, in +order to obtain a great fall of caloric, and consequently a large +production of motive power. + +(2) For the same reason the cooling should be carried as far as +possible. + +(3) It should be so arranged that the passage of the elastic fluid from +the highest to the lowest temperature should be due to increase of +volume; that is, it should be so arranged that the cooling of the gas +should occur spontaneously as the effect of rarefaction. The limits of +the temperature to which it is possible to bring the fluid primarily, +are simply the limits of the temperature obtainable by combustion; they +are very high. + +The limits of cooling are found in the temperature of the coldest body +of which we can easily and freely make use; this body is usually the +water of the locality. + +As to the third condition, it involves difficulties in the realization +of the motive power of heat when the attempt is made to take advantage +of great differences of temperature, to utilize great falls of heat. In +short, it is necessary then that the gas, by reason of its rarefaction, +should pass from a very high temperature to a very low one, which +requires a great change of volume and of density, which requires also +that the gas be first taken under a very heavy pressure, or that it +acquire by its dilatation an enormous volume—conditions both difficult +to fulfil. The first necessitates the employment of very strong vessels +to contain the gas at a very high temperature and under very heavy +pressure. The second necessitates the use of vessels of large +dimensions. These are, in a word, the principal obstacles which prevent +the utilization in steam-engines of a great part of the motive power of +the heat. We are obliged to limit ourselves to the use of a slight fall +of caloric, while the combustion of the coal furnishes the means of +procuring a very great one. + +It is seldom that in steam-engines the elastic fluid is produced under a +higher pressure than six atmospheres—a pressure corresponding to about +160° Centigrade, and it is seldom that condensation takes place at a +temperature much under 40°. The fall of caloric from 160° to 40° is +120°, while by combustion we can procure a fall of 1000° to 2000°. + +In order to comprehend this more clearly, let us recall what we have +termed the fall of caloric. This is the passage of the heat from one +body, _A_, having an elevated temperature, to another, _B_, where it is +lower. We say that the fall of the caloric is 100° or 1000° when the +difference of temperature between the bodies _A_ and _B_ is 100° or +1000°. + +In a steam-engine which works under a pressure of six atmospheres the +temperature of the boiler is 160°. This is the body _A_. It is kept, by +contact with the furnace, at the constant temperature of 160°, and +continually furnishes the heat necessary for the formation of steam. The +condenser is the body _B_. By means of a current of cold water it is +kept at a nearly constant temperature of 40°. It absorbs continually the +caloric brought from the body _A_ by the steam. The difference of +temperature between these two bodies is 160° − 40°, or 120°. Hence we +say that the fall of caloric is here 120°. + +Coal being capable of producing, by its combustion, a temperature higher +than 1000°, and the cold water, which is generally used in our climate, +being at about 10°, we can easily procure a fall of caloric of 1000°, +and of this only 120° are utilized by steam-engines. Even these 120° are +not wholly utilized. There is always considerable loss due to useless +re-establishments of equilibrium in the caloric. + +[Illustration: + + Fig. 5. +] + +It is easy to see the advantages possessed by high-pressure machines +over those of lower pressure. _This superiority lies essentially in the +power of utilizing a greater fall of caloric._ The steam produced under +a higher pressure is found also at a higher temperature, and as, +further, the temperature of condensation remains always about the same, +it is evident that the fall of caloric is more considerable. But to +obtain from high-pressure engines really advantageous results, it is +necessary that the fall of caloric should be most profitably utilized. +It is not enough that the steam be produced at a high temperature: it is +also necessary that by the expansion of its volume its temperature +should become sufficiently low. A good steam-engine, therefore, should +not only employ steam under heavy pressure, _but under successive and +very variable pressures, differing greatly from one another, and +progressively decreasing_.[31] + +In order to understand in some sort _à posteriori_ the advantages of +high-pressure engines, let us suppose steam to be formed under +atmospheric pressure and introduced into the cylindrical vessel _abcd_ +(Fig. 5), under the piston _cd_, which at first touches the bottom _ab_. +The steam, after having moved the piston from _ab_ to _cd_, will +continue finally to produce its results in a manner with which we will +not concern ourselves. + +Let us suppose that the piston having moved to _cd_ is forced downward +to _ef_, without the steam being allowed to escape, or any portion of +its caloric to be lost. It will be driven back into the space _abef_, +and will increase at the same time in density, elastic force, and +temperature. If the steam, instead of being produced under atmospheric +pressure, had been produced just when it was being forced back into +_abef_, and so that after its introduction into the cylinder it had made +the piston move from _ab_ to _ef_, and had moved it simply by its +extension of volume, from _ef_ to _cd_, the motive power produced would +have been more considerable than in the first case. In fact, the +movement of the piston, while equal in extent, would have taken place +under the action of a greater pressure, though variable, and though +progressively decreasing. + +The steam, however, would have required for its formation exactly the +same quantity of caloric, only the caloric would have been employed at a +higher temperature. + +It is considerations of this nature which have led to the making of +double-cylinder engines—engines invented by Mr. Hornblower, improved by +Mr. Woolf, and which, as regards economy of the combustible, are +considered the best. They consist of a small cylinder, which at each +pulsation is filled more or less (often entirely) with steam, and of a +second cylinder having usually a capacity quadruple that of the first, +and which receives no steam except that which has already operated in +the first cylinder. Thus the steam when it ceases to act has at least +quadrupled in volume. From the second cylinder it is carried directly +into the condenser, but it is conceivable that it might be carried into +a third cylinder quadruple the second, and in which its volume would +have become sixteen times the original volume. The principal obstacle to +the use of a third cylinder of this sort is the capacity which it would +be necessary to give it, and the large dimensions which the openings for +the passage of the steam must have. We will say no more on this subject, +as we do not propose here to enter into the details of construction of +steam-engines. These details call for a work devoted specially to them, +and which does not yet exist, at least in France.[32] + +If the expansion of the steam is mainly limited by the dimensions of the +vessels in which the dilatation must take place, the degree of +condensation at which it is possible to use it at first is limited only +by the resistance of the vessels in which it is produced, that is, of +the boilers. + +In this respect we have by no means attained the best possible results. +The arrangement of the boilers generally in use is entirely faulty, +although the tension of the steam rarely exceeds from four to six +atmospheres. They often burst and cause severe accidents. It will +undoubtedly be possible to avoid such accidents, and meantime to raise +the steam to much greater pressures than is usually done. + +Besides the high-pressure double-cylinder engines of which we have +spoken, there are also high-pressure engines of one cylinder. The +greater part of these latter have been constructed by two ingenious +English engineers, Messrs. Trevithick and Vivian. They employ the steam +under a very high pressure, sometimes eight to ten atmospheres, but they +have no condenser. The steam, after it has been introduced into the +cylinder, undergoes therein a certain increase of volume, but preserves +always a pressure higher than atmospheric. When it has fulfilled its +office it is thrown out into the atmosphere. It is evident that this +mode of working is fully equivalent, in respect to the motive power +produced, to condensing the steam at 100°, and that a portion of the +useful effect is lost. But the engines working thus dispense with +condenser and air-pump. They are less costly than the others, less +complicated, occupy less space, and can be used in places where there is +not sufficient water for condensation. In such places they are of +inestimable advantage, since no others could take their place. These +engines are principally employed in England to move coal-wagons on +railroads laid either in the interior of mines or outside of them. + +We have, further, only a few remarks to make upon the use of permanent +gases and other vapors than that of water in the development of the +motive power of heat. + +Various attempts have been made to produce motive power by the action of +heat on atmospheric air. This gas presents, as compared with vapor of +water, both advantages and disadvantages, which we will proceed to +examine. + +(1) It presents, as compared with vapor of water, a notable advantage in +that, having for equal volume a much less capacity for heat, it would +cool more rapidly by an equal increase of volume. (This fact is proved +by what has already been stated.) Now we have seen how important it is +to produce by change of volume the greatest possible changes of +temperature. + +(2) Vapors of water can be formed only through the intervention of a +boiler, while atmospheric air could be heated directly by combustion +carried on within its own mass. Considerable loss could thus be +prevented, not only in the quantity of heat, but also in its +temperature. This advantage belongs exclusively to atmospheric air. +Other gases do not possess it. They would be even more difficult to heat +than vapor of water. + +(3) In order to give to air great increase of volume, and by that +expansion to produce a great change of temperature, it must first be +taken under a sufficiently high pressure; then it must be compressed +with a pump or by some other means before heating it. This operation +would require a special apparatus, an apparatus not found in +steam-engines. In the latter, water is in a liquid state when injected +into the boiler, and to introduce it requires but a small pump. + +(4) The condensing of the vapor by contact with the refrigerant body is +much more prompt and much easier than is the cooling of air. There +might, of course, be the expedient of throwing the latter out into the +atmosphere, and there would be also the advantage of avoiding the use of +a refrigerant, which is not always available, but it would be requisite +that the increase of the volume of the air should not reduce its +pressure below that of the atmosphere. + +(5) One of the gravest inconveniences of steam is that it cannot be used +at high temperatures without necessitating the use of vessels of +extraordinary strength. It is not so with air for which there exists no +necessary relation between the elastic force and the temperature. Air, +then, would seem more suitable than steam to realize the motive power of +falls of caloric from high temperatures. Perhaps in low temperatures +steam may be more convenient. We might conceive even the possibility of +making the same heat act successively upon air and vapor of water. It +would be only necessary that the air should have, after its use, an +elevated temperature, and instead of throwing it out immediately into +the atmosphere, to make it envelop a steam-boiler, as if it issued +directly from a furnace. + +The use of atmospheric air for the development of the motive power of +heat presents in practice very great, but perhaps not insurmountable, +difficulties. If we should succeed in overcoming them, it would +doubtless offer a notable advantage over vapor of water.[33] + +As to the other permanent gases, they should be absolutely rejected. +They have all the inconveniences of atmospheric air, with none of its +advantages. The same can be said of other vapors than that of water, as +compared with the latter. + +If we could find an abundant liquid body which would vaporize at a +higher temperature than water, of which the vapor would have, for the +same volume, a less specific heat, which would not attack the metals +employed in the construction of machines, it would undoubtedly merit the +preference. But nature provides no such body. + +The use of the vapor of alcohol has been proposed. Machines have even +been constructed for the purpose of using it, by avoiding the mixture of +its vapor with the water of condensation, that is, by applying the cold +body externally instead of introducing it into the machine. It has been +thought that a remarkable advantage might be secured by using the vapor +of alcohol in that it possesses a stronger tension than the vapor of +water at the same temperature. We can see in this only a fresh obstacle +to be overcome. The principal defect of the vapor of water is its +excessive tension at an elevated temperature; now this defect exists +still more strongly in the vapor of alcohol. As to the relative +advantage in a greater production of motive power,—an advantage +attributed to it,—we know by the principles above demonstrated that it +is imaginary. + +It is thus upon the use of atmospheric air and vapor of water that +subsequent attempts to perfect heat-engines should be based. It is to +utilize by means of these agents the greatest possible falls of caloric +that all efforts should be directed. + +Finally, we will show how far we are from having realized, by any means +at present known, all the motive power of combustibles. + +One kilogram of carbon burnt in the calorimeter furnishes a quantity of +heat capable of raising one degree Centigrade about 7000 kilograms of +water, that is, it furnishes 7000 units of heat according to the +definition of these units given on page 100. + +The greatest fall of caloric attainable is measured by the difference +between the temperature produced by combustion and that of the +refrigerant bodies. It is difficult to perceive any other limits to the +temperature of combustion than those in which the combination between +oxygen and the combustible may take place. Let us assume, however, that +1000° may be this limit, and we shall certainly be below the truth. As +to the temperature of the refrigerant, let us suppose it 0°. We +estimated approximately (page 104) the quantity of motive power that +1000 units of heat develop between 100° and 99°. We found it to be 1.112 +units of power, each equal to 1 metre of water raised to a height of 1 +metre. + +If the motive power were proportional to the fall of caloric, if it were +the same for each thermometric degree, nothing would be easier than to +estimate it from 1000° to 0°. Its value would be + + 1.112 × 1000 = 1112. + +But as this law is only approximate, and as possibly it deviates much +from the truth at high temperatures, we can only make a very rough +estimate. We will suppose the number 1112 reduced one half, that is, to +560. + +Since a kilogram of carbon produces 7000 units of heat, and since the +number 560 is relatively 1000 units, it must be multiplied by 7, which +gives + + 7 × 560 = 3920. + +This is the motive power of 1 kilogram of carbon. + +In order to compare this theoretical result with that of experiment, let +us ascertain how much motive power a kilogram of carbon actually +develops in the best-known steam-engines. + +The engines which, up to this time, have shown the best results are the +large double-cylinder engines used in the drainage of the tin and copper +mines of Cornwall. The best results that have been obtained with them +are as follows: + +65 millions of lbs. of water have been raised one English foot by the +bushel of coal burned (the bushel weighing 88 lbs.). This is equivalent +to raising, by a kilogram of coal, 195 cubic metres of water to a height +of 1 metre, producing thereby 195 units of motive power per kilogram of +coal burned. + +195 units are only the twentieth of 3920, the theoretical maximum; +consequently ¹⁄₂₀ only of the motive power of the combustible has been +utilized. + +We have, nevertheless, selected our example from among the best +steam-engines known. + +Most engines are greatly inferior to these. The old engine of Chaillot, +for example, raised twenty cubic metres of water thirty-three metres, +for thirty kilograms of coal consumed, which amounts to twenty-two units +of motive power per kilogram,—a result nine times less than that given +above, and one hundred and eighty times less than the theoretical +maximum. + +We should not expect ever to utilize in practice all the motive power of +combustibles. The attempts made to attain this result would be far more +hurtful than useful if they caused other important considerations to be +neglected. The economy of the combustible is only one of the conditions +to be fulfilled in heat-engines. In many cases it is only secondary. It +should often give precedence to safety, to strength, to the durability +of the engine, to the small space which it must occupy, to small cost of +installation, etc. To know how to appreciate in each case, at their true +value, the considerations of convenience and economy which may present +themselves; to know how to discern the more important of those which are +only accessories; to balance them properly against each other, in order +to attain the best results by the simplest means: such should be the +leading characteristics of the man called to direct, to co-ordinate +among themselves the labors of his comrades, to make them co-operate +towards one useful end, of whatsoever sort it may be. + +[Illustration: + + (_To face p. 127._) +] + + + + + IV.[34] + CARNOT’S THEORY OF THE MOTIVE POWER OF HEAT.[35] +WITH NUMERICAL RESULTS DEDUCED FROM REGNAULT’S EXPERIMENTS ON STEAM.[36] + + BY SIR WILLIAM THOMSON [LORD KELVIN]. + + +1. The presence of heat may be recognized in every natural object; and +there is scarcely an operation in nature which is not more or less +affected by its all-pervading influence. An evolution and subsequent +absorption of heat generally give rise to a variety of effects; among +which may be enumerated, chemical combinations or decompositions; the +fusion of solid substances; the vaporization of solids or liquids; +alterations in the dimensions of bodies, or in the statical pressure by +which their dimensions may be modified; mechanical resistance overcome; +electrical currents generated. In many of the actual phenomena of nature +several or all of these effects are produced together; and their +complication will, if we attempt to trace the agency of heat in +producing any individual effect, give rise to much perplexity. It will, +therefore, be desirable, in laying the foundation of a physical theory +of any of the effects of heat, to discover or to imagine phenomena free +from all such complication, and depending on a definite thermal agency; +in which the relation between the cause and effect, traced through the +medium of certain simple operations, may be clearly appreciated. Thus it +is that Carnot, in accordance with the strictest principles of +philosophy, enters upon the investigation of the theory of the motive +power of heat. + +2. The sole effect to be contemplated in investigating the motive power +of heat is _resistance overcome_, or, as it is frequently called, “_work +performed_,” or “_mechanical effect_.” The questions to be resolved by a +complete theory of the subject are the following: + +(1) What is the precise nature of the thermal agency by means of which +_mechanical effect_ is to be produced, without effects of any other +kind? + +(2) How may the amount of this thermal agency necessary for performing a +given quantity of work be estimated? + +3. In the following paper I shall commence by giving a short abstract of +the reasoning by which Carnot is led to an answer to the first of these +questions; I shall then explain the investigation by which, in +accordance with his theory, the experimental elements necessary for +answering the second question are indicated; and, in conclusion, I shall +state the _data_ supplied by Regnault’s recent observations on steam, +and apply them to obtain, as approximately as the present state of +experimental science enables us to do, a complete solution of the +question. + +I. On the nature of Thermal agency, considered as a motive power. + +4. There are [at present known] two, and only two, distinct ways in +which mechanical effect can be obtained from heat. One of these is by +means of the alterations of volume, which bodies may experience through +the action of heat; the other is through the medium of electric agency. +Seebeck’s discovery of thermo-electric currents enables us at present to +conceive of an electro-magnetic engine supplied from a thermal origin, +being used as a motive power; but this discovery was not made until +1821, and the subject of thermo-electricity can only have been generally +known in a few isolated facts, with reference to the electrical effects +of heat upon certain crystals, at the time when Carnot wrote. He makes +no allusion to it, but confines himself to the method for rendering +thermal agency available as a source of mechanical effect, by means of +the expansions and contractions of bodies. + +5. A body expanding or contracting under the action of force may, in +general, either produce mechanical effect by overcoming resistance, or +receive mechanical effect by yielding to the action of force. The amount +of mechanical effect thus developed will depend not only on the +calorific agency concerned, but also on the alteration in the physical +condition of the body. Hence, after allowing the volume and temperature +of the body to change, we must restore it to its original temperature +and volume; and then we may estimate the aggregate amount of mechanical +effect developed as due solely to the thermal origin. + +6. Now the ordinarily-received, and almost universally-acknowledged, +principles with reference to “quantities of caloric” and “latent heat” +lead us to conceive that, at the end of a cycle of operations, when a +body is left in precisely its primitive physical condition, if it has +absorbed any heat during one part of the operations, it must have given +out again exactly the same amount during the remainder of the cycle. The +truth of this principle is considered as axiomatic by Carnot, who admits +it as the foundation of his theory; and expresses himself in the +following terms regarding it, in a note on one of the passages of his +treatise:[37] + +“In our demonstrations we tacitly assume that after a body has +experienced a certain number of transformations, if it be brought +identically to its primitive physical state as to density, temperature, +and molecular constitution, it must contain the same quantity of heat as +that which it initially possessed; or, in other words, we suppose that +the quantities of heat lost by the body under one set of operations are +precisely compensated by those which are absorbed in the others. This +fact has never been doubted; it has at first been admitted without +reflection, and afterwards verified, in many cases, by calorimetrical +experiments. To deny it would be to overturn the whole theory of heat, +in which it is the fundamental principle. It must be admitted, however, +that the chief foundations on which the theory of heat rests, would +require a most attentive examination. Several experimental facts appear +nearly inexplicable in the actual state of this theory.” + +7. Since the time when Carnot thus expressed himself, the necessity of a +most careful examination of the entire experimental basis of the theory +of heat has become more and more urgent. Especially all those +assumptions depending on the idea that heat is a _substance_, invariable +in quantity; not convertible into any other element, and incapable of +being _generated_ by any physical agency; in fact the acknowledged +principles of latent heat,—would require to be tested by a most +searching investigation before they ought to be admitted, as they +usually have been, by almost every one who has been engaged on the +subject, whether in combining the results of experimental research, or +in general theoretical investigations. + +8. The extremely important discoveries recently made by Mr. Joule of +Manchester, that heat is evolved in every part of a closed electric +conductor, moving in the neighborhood of a magnet,[38] and that heat is +_generated_ by the friction of fluids in motion, seem to overturn the +opinion commonly held that heat cannot be _generated_, but only produced +from a source, where it has previously existed either in a sensible or +in a latent condition. + +In the present state of science, however, no operation is known by which +heat can be absorbed into a body without either elevating its +temperature or becoming latent, and producing some alteration in its +physical condition; and the fundamental axiom adopted by Carnot may be +considered as still the most probable basis for an investigation of the +motive power of heat; although this, and with it every other branch of +the theory of heat, may ultimately require to be reconstructed upon +another foundation, when our experimental data are more complete. On +this understanding, and to avoid a repetition of doubts, I shall refer +to Carnot’s fundamental principle, in all that follows, as if its truth +were thoroughly established. + +9. We are now led to the conclusion that the origin of motive power, +developed by the alternate expansions and contractions of a body, must +be found in the agency of heat entering the body and leaving it; since +there cannot, at the end of a complete cycle, when the body is restored +to its primitive physical condition, have been any absolute absorption +of heat, and consequently no conversion of heat, or caloric, into +mechanical effect; and it remains for us to trace the precise nature of +the circumstances under which heat must enter the body, and afterwards +leave it, so that mechanical effect may be produced. As an example, we +may consider that machine for obtaining motive power from heat with +which we are most familiar—the steam-engine. + +10. Here, we observe, that heat enters the machine from the furnace, +through the sides of the boiler, and that heat is continually abstracted +by the water employed for keeping the condenser cool. According to +Carnot’s fundamental principle, the quantity of heat thus discharged, +during a complete revolution (or double stroke) of the engine, must be +precisely equal to that which enters the water of the boiler;[39] +provided the total mass of water and steam be invariable, and be +restored to its primitive physical condition (which will be the case +rigorously, if the condenser be kept cool by the external application of +cold water instead of by injection, as is more usual in practice), and +if the condensed water be restored to the boiler at the end of each +complete revolution. Thus we perceive that a certain quantity of heat is +_let down_ from a hot body, the metal of the boiler, to another body at +a lower temperature, the metal of the condenser; and that there results +from this transference of heat a certain development of mechanical +effect. + +11. If we examine any other case in which mechanical effect is obtained +from a thermal origin, by means of the alternate expansions and +contractions of any substance whatever, instead of the water of a +steam-engine, we find that a similar transference of heat is effected, +and we may therefore answer the first question proposed, in the +following manner: + +_The thermal agency by which mechanical effect may be obtained is the +transference of heat from one body to another at a lower temperature._ + +11. On the measurement of Thermal Agency, considered with reference to +its equivalent of mechanical effect. + +12. A _perfect_ thermodynamic engine of any kind is a machine by means +of which the greatest possible amount of mechanical effect can be +obtained from a given thermal agency; and, therefore, if in any manner +we can construct or imagine a perfect engine which may be applied for +the transference of a given quantity of heat from a body at any given +temperature to another body at a lower given temperature, and if we can +evaluate the mechanical effect thus obtained, we shall be able to answer +the question at present under consideration, and so to complete the +theory of the motive power of heat. But whatever kind of engine we may +consider with this view, it will be necessary for us to prove that it is +a perfect engine; since the transference of the heat from one body to +the other may be wholly, or partially, effected by conduction through a +solid,[40] without the development of mechanical effect; and, +consequently, engines may be constructed in which the whole or any +portion of the thermal agency is wasted. Hence it is of primary +importance to discover the criterion of a perfect engine. This has been +done by Carnot, who proves the following proposition: + +13. _A perfect thermodynamic engine is such that, whatever amount of +mechanical effect it can derive from a certain thermal agency, if an +equal amount be spent in working it backwards, an equal reverse thermal +effect will be produced._[41] + +14. This proposition will be made clearer by the applications of it +which are given later (§ 29), in the cases of the air-engine and the +steam-engine, than it could be by any general explanation; and it will +also appear, from the nature of the operations described in those cases, +and the principles of Carnot’s reasoning, that a perfect engine may be +constructed with any substance of an indestructible texture as the +alternately expanding and contracting medium. Thus we might conceive +thermodynamic engines founded upon the expansions and contractions of a +perfectly elastic solid, or of a liquid; or upon the alterations of +volume experienced by substances in passing from the liquid to the solid +state,[42] each of which being perfect, would produce the same amount of +mechanical effect from a given thermal agency; but there are two cases +which Carnot has selected as most worthy of minute attention, because of +their peculiar appropriateness for illustrating the general principles +of his theory, no less than on account of their very great practical +importance: the steam-engine, in which the substance employed as the +transferring medium is water, alternately in the liquid state and in the +state of vapor; and the air-engine, in which the transference is +effected by means of the alternate expansions and contractions of a +medium always in the gaseous state. The details of an actually +practicable engine of either kind are not contemplated by Carnot in his +general theoretical reasonings, but he confines himself to the ideal +construction, in the simplest possible way in each case, of an engine in +which the economy is perfect. He thus determines the degree of +perfectibility which cannot be surpassed; and by describing a +conceivable method of attaining to this perfection by an air-engine or a +steam-engine, he points out the proper objects to be kept in view in the +practical construction and working of such machines. I now proceed to +give an outline of these investigations. + + + CARNOT’S THEORY OF THE STEAM-ENGINE. + +15. Let _CDF_{2}E_{2}_ be a cylinder, of which the curved surface is +perfectly impermeable to heat, with a piston also impermeable to heat, +fitted in it; while the fixed bottom _CD_, itself with no capacity for +heat, is possessed of perfect conducting power. Let _K_ be an +impermeable stand, such that when the cylinder is placed upon it the +contents below the piston can neither gain nor lose heat. Let _A_ and +_B_ be two bodies permanently retained at constant temperatures, _S°_ +and _T°_, respectively, of which the former is higher than the latter. +Let the cylinder, placed on the impermeable stand, _K_, be partially +filled with water, at the temperature _S_, of the body _A_, and (there +being no air below it) let the piston be placed in a position _EF_, near +the surface of the water. The pressure of the vapor above the water will +tend to push up the piston, and must be resisted by a force applied to +the piston,[43] till the commencement of the operations, which are +conducted in the following manner: + +[Illustration] + +(1) The cylinder being placed on the body _A_, so that the water and +vapor may be retained at the temperature _S_, _let the piston rise any +convenient height EE_{1}, to a position E_{1}F_{1}, performing work by +the pressure of the vapor below it during its ascent_. + + [During this operation a certain quantity, _H_, of heat, the amount of + latent heat in the fresh vapor which is formed, is abstracted from the + body _A_.] + +(2) The cylinder being removed, and placed on the impermeable stand _K, +let the piston rise gradually, till, when it reaches a position +E_{2}F_{2}, the temperature of the water and vapor is T, the same as +that of the body B_. + + [During this operation the fresh vapor continually formed requires + heat to become latent; and, therefore, as the contents of the cylinder + are protected from any accession of heat, their temperature sinks.] + +(3) The cylinder being removed from _K_, and placed on _B, let the +piston be pushed down, till, when it reaches the position E_{3}F_{3}, +the quantity of heat evolved and abstracted by B amounts to that which, +during the first operation, was taken from A_. + +[Note of Nov. 5, 1881. The specification of this operation, with a view +to the return to the primitive condition, intended as the conclusion to +the four operations, is the only item in which Carnot’s temporary and +provisional assumption of the materiality of heat has effect. To exclude +this hypothesis, Prof. James Thomson has suggested the following +corrected specification for the third operation: _Let the piston be +pushed down, till it reaches a position E_{3}F_{3}, determined so as to +fulfil the condition, that at the end of the fourth operation the +primitive temperature S shall be reached_:[44]] + + [During this operation the temperature of the contents of the cylinder + is retained constantly at _T°_, and all the latent heat of the vapor + which is condensed into water at the same temperature is given out to + _B_.] + +(4) The cylinder being removed from _B_, and placed on the impermeable +stand, _let the piston be pushed down from E_{3}F_{3} to its original +position EF_. + + [During this operation, the impermeable stand preventing any loss of + heat, the temperature of the water and air must rise continually, till + (since the quantity of heat evolved during the third operation was + precisely equal to that which was previously absorbed) at the + conclusion it reaches its primitive value, _S_, in virtue of Carnot’s + fundamental axiom.] + + [Note of Nov. 5, 1881. With Prof. James Thomson’s correction of + operation (3), the words in virtue of “Carnot’s Fundamental Axiom” + must be replaced by “the condition fulfilled by operation (3),” in the + description of the results of operation (4).] + +16. At the conclusion of this cycle of operations[45] the total thermal +agency has been the _letting down_ of _H_ units of heat from the body +_A_, at the temperature _S_, to _B_, at the lower temperature _T_; and +the aggregate of the mechanical effect has been a certain amount of +_work produced_, since during the ascent of the piston in the first and +second operations, the temperature of the water and vapor, and therefore +the pressure of the vapor on the piston, was on the whole higher than +during the descent, in the third and fourth operations. It remains for +us actually to evaluate this aggregate amount of work performed; and for +this purpose the following graphical method of representing the +mechanical effect developed in the several operations, taken from Mons. +Clapeyron’s paper, is extremely convenient. + +17. Let _OX_ and _OY_ be two lines at right angles to one another. Along +_OX_ measure off distances _ON_{1}_, _N_{1}N_{2}_, _N_{2}N_{3}_, +_N_{3}O_, respectively proportional to the spaces described by the +piston during the four successive operations described above; and, with +reference to these four operations respectively, let the following +constructions be made: + +(1) Along _OY_ measure a length _OA_, to represent the pressure of the +saturated vapor at the temperature _S_; and draw _AA_{1}_ parallel to +_OX_, and let it meet an ordinate through _N_{1}_, in _A_{1}_. + +(2) Draw a curve _A_{1}PA_ such that, if _ON_ represent, at any instant +during the second operation, the distance of the piston from its +primitive position, _NP_ shall represent the pressure of the vapor at +the same instant. + +(3) Through _A__{2} draw _A_{2}A_{3}_ parallel to _OX_, and let it meet +an ordinate through _N_{3}_ in _A_{3}_. + +(4) Draw the curve _A_{3}A_ such that the abscissa and ordinate of any +point in it may represent respectively the distances of the piston from +its primitive position, and the pressure of the vapor, at each instant +during the fourth operation. The last point of this curve must, +according to Carnot’s fundamental principle, coincide with _A_, since +the piston is, at the end of the cycle of operations, again in its +primitive position, and the pressure of the vapor is the same as it was +at the beginning. + +[Illustration] + +18. Let us now suppose that the lengths, _ON_{1}_, _N_{1}N_{2}_, +_N_{2}N_{3}_, and _N_{3}O_, _represent numerically_ the volumes of the +spaces moved through by the piston during the successive operations. +It follows that the mechanical effect obtained during the first +operation will be _numerically represented_ by the area +_AA_{1}N_{1}O_; that is, the number of superficial units in this area +will be equal to the number of “foot-pounds” of work performed by the +ascending piston during the first operation. The work performed by the +piston during the second operation will be similarly represented by +the area _A_{1}A_{2}N_{2}N_{1}_. Again, during the third operation a +certain amount of work is spent on the piston, which will be +represented by the area _A_{2}A_{3}N_{3}N_{2}_; and lastly, during the +fourth operation, work is spent in pushing the piston to an amount +represented by the area _A_{3}AON_{3}_. + +19. Hence the mechanical effect (represented by the area +_OAA_{1}A_{2}N_{2}_) which was obtained during the first and second +operations, exceeds the work (represented by _N_{2}A_{2}A_{3}AO_) spent +during the third and fourth, by an amount represented by the area of the +quadrilateral figure _AA_{1}A_{2}A_{3}_; and, consequently, it only +remains for us to evaluate this area, that we may determine the total +mechanical effect gained in a complete cycle of operations. Now, from +experimental data, at present nearly complete, as will be explained +below, we may determine the length of the line _AA_{1}_ for the given +temperature _S_, and a given absorption _H_, of heat, during the first +operation; and the length of _A_{2}A_{3}_ for the given lower +temperature _T_, and the evolution of the same quantity of heat during +the fourth operation: and the curves _A_{1}PA_{2}_, _A_{3}P′A_ may be +drawn as graphical representations of actual observations. The figure +being thus constructed, its area may be measured, and we are, therefore, +in possession of a graphical method of determining the amount of +mechanical effect to be obtained from any given thermal agency. As, +however, it is merely the area of the figure which it is required to +determine, it will not be necessary to be able to describe each of the +curves _A_{1}PA_{2}_, _A_{3}P′A_, but it will be sufficient to know the +difference of the abscissas corresponding to any equal ordinates in the +two; and the following analytical method of completing the problem is +the most convenient for leading to the actual numerical results. + +20. Draw any line _PP′_ parallel to _OX_, meeting the curvilinear sides +of the quadrilateral in _P_ and _P′_. Let ξ denote the length of this +line, and _p_ its distance from _OX_. The area of the figure, according +to the integral calculus, will be denoted by the expression + + ∫_{_p_{3}_} ^{_p_{1}_} ξ_dp_, + +where _p_{1}_ and _p_{3}_ (the limits of integration indicated according +to Fourier’s notation) denote the lines _OA_ and _N_{3}A_{3}_, which +represent respectively the pressures during the first and third +operations. Now, by referring to the construction described above, we +see that ξ is the difference of the volumes below the piston at +corresponding instants of the second and fourth operations, or instants +at which the saturated steam and the water in the cylinder have the same +pressure _p_, and consequently the same temperature, which we may denote +by _t_. Again, throughout the second operation the entire contents of +the cylinder possess a greater amount of heat by _H_ units than during +the fourth; and, therefore, at any instant of the second operation there +is as much more steam as contains _H_ units of latent heat than at the +corresponding instant of the fourth operation. Hence if _k_ denote the +latent heat in a unit of saturated steam at the temperature _t_, the +volume of the steam at the two corresponding instants must differ by +(_H_)/(_k_). Now, if σ denote the ratio of the density of the steam to +that of the water, the volume (_H_)/(_k_) of steam will be formed from +the volume σ (_H_)/(_k_) of water; and consequently we have, for the +difference of volumes of the entire contents at the corresponding +instants, + + ξ = (1 - σ)(_H_)/(_k_). + +Hence the expression for the area of the quadrilateral figure becomes + + ∫^{_p_{1}_}_{_p_{3}_}(1 - σ)(_H_)/(_k_)_dp_. + +Now, σ, _k_, and _p_, being quantities which depend upon the +temperature, may be considered as functions of _t_; and it will be +convenient to modify the integral so as to make _t_ the independent +variable. The limits will be from _t_ = _T_ to _t_ = _S_, and, if we +denote by _M_ the value of the integral, we have the expression + + _M_ = _H_ ∫_{_T_}^{_S_}(1 - σ)((_dp_/_dt_)/_k_)_dt_. (1) + +for the total amount of mechanical effect gained by the operations +described above. + +21. If the interval of temperatures be extremely small,—so small that (1 +− σ)(_dp_)/(_dt_/_k_) will not sensibly vary for values of _t_ between +_T_ and _S_,—the preceding expression becomes simply + + _Μ_ = (1 - σ)(_dp_)/(_dt_)/(_k_). _Η_(_S_ - _Τ_). (2) + +This might, of course, have been obtained at once by supposing the +breadth of the quadrilateral figure _AA_{1}A_{2}A_ to be extremely small +compared with its length, and then taking for its area, as an +approximate value, the product of the breadth into the line _AA_{1}_, or +the line _A_{3}A_{2}_, or any line of intermediate magnitude. + +The expression (2) is rigorously correct for any interval _S_ − _T_, if +the mean value of (1 − σ)((_dp_/_dt_)/_k_) for that interval be employed +as the coefficient of _H_(_S_ − _T_). + + + CARNOT’S THEORY OF THE AIR-ENGINE. + +22. In the ideal air-engine imagined by Carnot four operations performed +upon a mass of air or gas enclosed in a closed vessel of variable volume +constitute a complete cycle, at the end of which the medium is left in +its primitive physical condition; the construction being the same as +that which was described above for the steam-engine, a body _A_, +permanently retained at the temperature _S_, and _B_ at the temperature +_T_; an impermeable stand _K_; and a cylinder and piston, which in this +case contains a mass of air at the temperature _S_, instead of water in +the liquid state, at the beginning and end of a cycle of operations. The +four successive operations are conducted in the following manner: + +(1) The cylinder is laid on the body _A_, so that the air in it is kept +at the temperature _S_; and the piston is allowed to rise, performing +work. + +(2) The cylinder is placed on the impermeable stand _K_, so that its +contents can neither gain nor lose heat, and the piston is allowed to +rise farther, still performing work, till the temperature of the air +sinks to _T_. + +(3) The cylinder is placed on _B_, so that the air is retained at the +temperature _T_, and the piston is pushed down till the air gives out to +the body _B_ as much heat as it had taken in from _A_, during the first +operation. + + [Note of Nov. 5, 1881. To eliminate the assumption of the materiality + of heat, make Professor James Thomson’s correction here also; as above + in § 15; or take Maxwell’s rearrangement of the cycle described in the + foot-note to § 15, p. 144.] + +(4) The cylinder is placed on _K_, so that no more heat can be taken in +or given out, and the piston is pushed down to its primitive position. + +23. _At the end of the fourth operation the temperature must have +reached its primitive value S, in virtue of_ CARNOT’S _axiom_. + +24. Here, again, as in the former case, we observe that work is +performed by the piston during the first two operations; and during the +third and fourth work is spent upon it, but to a less amount, since the +pressure is on the whole less during the third and fourth operations +than during the first and second, on account of the temperature being +lower. Thus, at the end of a complete cycle of operations, mechanical +effect has been obtained; and the thermal agency from which it is drawn +is the taking of a certain quantity of heat from _A_, and _letting it +down_, through the medium of the engine, to the body _B_ at a lower +temperature. + +25. To estimate the actual amount of effect thus obtained, it will be +convenient to consider the alterations of volume of the mass of air in +the several operations as extremely small. We may afterwards pass by the +integral calculus, or, practically, by summation to determine the +mechanical effect whatever be the amplitudes of the different motions of +the piston. + +26. Let _dq_ be the quantity of heat absorbed during the first +operation, which is evolved again during the third; and let _dv_ be the +corresponding augmentation of volume which takes place while the +temperature remains constant, as it does during the first operation.[46] +The diminution of volume in the third operation must be also equal to +_dv_, or only differ from it by an infinitely small quantity of the +second order. During the second operation we may suppose the volume to +be increased by an infinitely small quantity φ; which will occasion a +diminution of pressure and a diminution of temperature, denoted +respectively by ω and τ. During the fourth operation there will be a +diminution of volume and an increase of pressure and temperature, which +can only differ, by infinitely small quantities of the second order, +from the changes in the other direction, which took place in the second +operation, and they also may, therefore, be denoted by φ, ω, and τ, +respectively. The alteration of pressure during the first and third +operations may at once be determined by means of Mariotte’s law, since +in them the temperature remains constant. Thus, if, at the commencement +of the cycle, the volume and pressure be _v_ and _p_, they will have +become _v_ + _dv_ and _pv_/(_v_ + _dv_) at the end of the first +operation. Hence the diminution of pressure during the first operation +is _p_ − _pv_/(_v_ + _dv_) or _pdv_/(_v_ + _dv_) and therefore, if we +neglect infinitely small quantities of the second order, we have +_pdv_/_v_ for the diminution of pressure during the first operation; +which to the same degree of approximation, will be equal to the increase +of pressure during the third. If _t_ + τ and _t_ be taken to denote the +superior and inferior limits of temperature, we shall thus have for the +volume, the temperature, and the pressure at the commencements of the +four successive operations, and at the end of the cycle, the following +values respectively: + + (1) _v_, _t_ + τ, _p_; + (2) _v_ + _dv_, _t_ + τ, _p_(1 − (_dv_)/(_v_)); + (3) _v_ + _dv_ + φ, _t_, _p_(1 − (_dv_)/(_v_)) − ω; + (4) _v_ + φ, _t_, _p_ − ω; + (5) _v_, _t_ + τ, _p_. + +Taking the mean of the pressures at the beginning and end of each +operation, we find + + (1) _p_(1 − ½(_dv_)/(_v_)), + + (2) _p_(1 − (_dv_)/(_v_)) − ½ω, + + (3) _p_(1 − ½(_dv_)/(_v_))) − ω, + + (4) _p_ − ½ω, + +which, as we are neglecting infinitely small quantities of the second +order, will be the expressions for the mean pressures during the four +successive operations. Now, the mechanical effect gained or spent, +during any of the operations, will be found by multiplying the mean +pressure by the increase or diminution of volume which takes place; and +we thus find + + (1) _p_(1 − ½(_dv_)/(_v_))_dv_, + + (2) {_p_(1 − (_dv_)/(_v_)) − ½ω}φ, + + (3) {_p_(1 − ½(_dv_)/(_v_)) − ω}_dv_, + + (4) (_p_ − ½ω)φ. + +for the amounts gained during the first and second, and spent during the +third and fourth operations; and hence, by addition and subtraction, we +find + + ω_dv_ − _p_φ(_dv_)/(_v_), or (_v_ω − _p_φ)(_dv_)/(_v_), + +for the aggregate amount of mechanical effect gained during the cycle of +operations. It only remains for us to express this result in terms of +_dq_ and τ, on which the given thermal agency depends. For this purpose +we remark that φ and ω are alterations of volume and pressure which take +place along with a change of temperature τ, and hence, by the laws of +compressibility and expansion, we may establish a relation[47] between +them in the following manner: + +Let _p_{0}_ be the pressure of the mass of air when reduced to the +temperature zero, and confined in a volume _v_{0}_; then, whatever be +_v_{0}_, the product _p_{0}v_{0}_ will, by the law of compressibility, +remain constant; and, if the temperature be elevated from 0 to _t_ + τ, +and the gas be allowed to expand freely without any change of pressure, +its volume will be increased in the ratio of 1 to 1 + _E_(_t_ + τ), +where _E_ is very nearly equal to .00366 (the Centigrade scale of the +air-thermometer being referred to), whatever be the gas employed, +according to the researches of Regnault and of Magnus on the expansion +of gases by heat. If, now, the volume be altered arbitrarily with the +temperature continually at _t_ + τ, the product of the pressure and +volume will remain constant; and therefore we have + + _pv_ = _p_{0}v_{0}_{1 + _E_(_t_ + τ)}. + +Similarly, + + (_p_ − ω)(_v_ + φ) = _p_{0}v_{0}_{1 + _Et_}. + +Hence, by subtraction, we have + + _v_ω − _p_φ + ωφ = _p_{0}v_{0}E_τ, + +or, neglecting the product ωφ, + + _v_ω − _p_φ = _p_{0}v_{0}E_τ. + +Hence the preceding expression for mechanical effect, gained in the +cycle of operations, becomes + + _p_{0}v_{0}_. _E_τ . _dv_/_v_. + +Or, as we may otherwise express it, + + (_Ep_{0}v_{0}_)/(_vdq_/_dv_). _dq_. τ. + +Hence, if we denote by _M_ the mechanical effect due to _H_ units of +heat descending through the same interval τ, which might be obtained by +repeating the cycle of operations described above, (_H_)/(_dq_) times, +we have + + _M_ = (_Ep_{0}v_{0}_)/(_vdq_/_dv_). _H_τ. (3) + +27. If the _amplitudes_ of the operations had been finite, so as to give +rise to an absorption of _H_ units of heat during the first operation, +and a lowering of temperature from _S_ to _T_ during the second, the +amount of work obtained would have been found to be expressed by means +of a double definite integral thus:[48] + + _M_ = ∫_{0}^{_H_} _dq_ ∫_{_T_}^{_S_} _dt_. (_Ep_{0}v_{0}_)/(_vdq_/_dv_), ⎫ + or ⎬. (4) + _M_ = _Ep_{0}v_{0}_ ∫_{0}^{_H_} ∫_{_T_}^{_S_} (1)/(_v_) (_dv_)/(_dq_). _dtdq_; ⎭ + +this second form being sometimes more convenient. + +28. The preceding investigations, being founded on the approximate laws +of compressibility and expansion (known as the law of Mariotte and +Boyle, and the law of Dalton and Gay-Lussac), would require some slight +modifications to adapt them to cases in which the gaseous medium +employed is such as to present sensible deviations from those laws. +Regnault’s very accurate experiments show that the deviations are +insensible, or very nearly so, for the ordinary gases at ordinary +pressures; although they may be considerable for a medium, such as +sulphurous acid, or carbonic acid under high pressure, which approaches +the physical condition of a vapor at saturation; and therefore, in +general, and especially in practical applications to real air-engines, +it will be unnecessary to make any modification in the expressions. In +cases where it may be necessary, there is no difficulty in making the +modifications, when the requisite data are supplied by experiment. + +29.[49] Either the steam-engine or the air-engine, according to the +arrangements described above, gives all the mechanical effect that can +possibly be obtained from the thermal agency employed. For it is clear +that in either case the operations may be performed in the reverse +order, with every thermal and mechanical effect reversed. Thus, in the +steam-engine, we may commence by placing the cylinder on the impermeable +stand, allow the piston to rise, performing work, to the position +_E_{3}F_{3}_; we may then place it on the body _B_, and allow it to +rise, performing work, till it reaches _E_{2}F_{2}_ after that the +cylinder may be placed again on the impermeable stand, and the piston +may be pushed down to _E_{1}F_{1}_; and, lastly, the cylinder being +removed to the body _A_, the piston may be pushed down to its primitive +position. In this inverse cycle of operations a certain amount of work +has been spent, precisely equal, as we readily see, to the amount of +mechanical effect gained in the direct cycle described above; and heat +has been abstracted from _B_, and deposited in the body _A_, at a higher +temperature, to an amount precisely equal to that which in the direct +style was _let down_ from _A_ to _B_. Hence it is impossible to have an +engine which will derive more mechanical effect from the same thermal +agency than is obtained by the arrangement described above; since, if +there could be such an engine, it might be employed to perform, as a +part of its whole work, the inverse cycle of operations, upon an engine +of the kind we have considered, and thus to continually restore the heat +from _B_ to _A_, which has descended from _A_ to _B_ for working itself; +so that we should have a complex engine, giving a residual amount of +mechanical effect without any thermal agency, or alteration of +materials, which is an impossibility in nature. The same reasoning is +applicable to the air-engine; and we conclude, generally, that any two +engines, constructed on the principles laid down above, whether +steam-engines with different liquids, an air-engine and a steam-engine, +or two air-engines with different gases, must derive the same amount of +mechanical effect from the same thermal agency. + +30. Hence, by comparing the amounts of mechanical effect obtained by the +steam-engine and the air-engine from the letting down of the _H_ units +of heat from _A_ at the temperature (_t_ + τ) to _B_ at _t_, according +to the expressions (2) and (3), we have + + _M_ = (1 − σ)(_dp_)/(_kdt_). _H_τ = (_Ep_{0}v_{0}_)/(_vdq_/_dv_). _H_τ. (5) + +If we denote the coefficient of _Η_τ in these equal expressions by μ, +which maybe called “Carnot’s coefficient,” we have + + μ = (1 − σ)(_dp_)/(_kdt_) = (_Ep_{0}v_{0}_)/(_vdq_/_dv_), (6) + +and we deduce the following very remarkable conclusions: + +(1) For the saturated vapors of all different liquids, at the same +temperature, the value of (1 − σ)(_dp_/_kdt_) must be the same. + +(2) For any different gaseous masses, at the same temperature, the value +of _Ep_{0}v_{0}_/(_vdq_/_dv_) must be the same. + +(3) The values of these expressions for saturated vapors and for gases, +at the same temperature, must be the same. + +31. No conclusion can be drawn _a priori_ regarding the values of this +coefficient μ for different temperatures, which can only be determined, +or compared, by experiment. The results of a great variety of +experiments, in different branches of physical science (Pneumatics and +Acoustics), cited by Carnot and by Clapeyron, indicate that the values +of μ for low temperatures exceed the values for higher temperatures; a +result amply verified by the continuous series of experiments performed +by Regnault on the saturated vapor of water for all temperatures from 0° +to 230°, which, as we shall see later, give values for μ gradually +diminishing from the inferior limit to the superior limit of +temperature. When, by observation, μ has been determined as a function +of the temperature, the amount of mechanical effect, _M_, deducible from +_H_ units of heat descending from a body at the temperature _S_ to a +body at the temperature _T_, may be calculated from the expression + + _M_ = _H_ ∫_{_S_}^{_T_} μ_dt_, (7) + +which is, in fact, what either of the equations (1) for the +steam-engine, or (4) for the air-engine, becomes, when the notation μ, +for Carnot’s multiplier, is introduced. + +The values of this integral may be practically obtained, in the most +convenient manner, by first determining, from observation, the mean +values of μ for the successive degrees of the thermometric scale, and +then adding the values for all the degrees within the limits of the +extreme temperatures _S_ and _T_.[50] + +32. The complete theoretical investigation of the motive power of heat +is thus reduced to the experimental determination of the coefficient μ; +and may be considered as perfect, when, by any series of experimental +researches whatever, we can find a value of μ for every temperature +within practical limits. The special character of the experimental +researches, whether with reference to gases or with reference to vapors, +necessary and sufficient for this object, is defined and restricted in +the most precise manner, by the expressions (6) for μ, given above. + +33. The object of Regnault’s great work, referred to in the title of +this paper, is the experimental determination of the various physical +elements of the steam-engine; and when it is complete, it will furnish +all the _data_ necessary for the calculation of μ. The valuable +researches already published in a first part of that work make known the +latent heat of a given weight, and the pressure, of saturated steam for +all temperatures between 0° and 230° Cent. of the air-thermometer. +Besides these data, however, the density of saturated vapor must be +known, in order that _k_, the latent heat of a unit of volume, may be +calculated from Regnault’s determination of the latent heat of a given +weight.[51] Between the limits of 0° and 100°, it is probable, from +various experiments which have been made, that the density of vapor +follows very closely the simple laws which are so accurately verified by +the ordinary gases;[52] and thus it may be calculated from Regnault’s +table giving the pressure at any temperature within those limits. +Nothing as yet is known with accuracy as to the density of saturated +steam between 100 and 230°, and we must be contented at present to +estimate it by calculation from Regnault’s table of pressures; although, +when accurate experimental researches on the subject shall have been +made, considerable deviations from the laws of Boyle and Dalton, on +which this calculation is founded, may be discovered. + +34. Such are the experimental data on which the mean values of μ for the +successive degrees of the air-thermometer, from 0 to 230°, at present +laid before the Royal Society, is founded. The unit of length adopted is +the English foot; the unit of weight, the pound; the unit of work, a +“foot-pound;” and the unit of heat that quantity which, when added to a +pound of water at 0°, will produce an elevation of 1° in temperature. +The mean value of μ for any degree is found to a sufficient degree of +approximation by taking, in place of σ, _dp_/_dt_ and _k_; in the +expression + + (1 − σ). (_dp_)/(_kdt_); + +the mean values of those elements; or, what is equivalent to the +corresponding accuracy of approximation, by taking, in place of σ and +_k_ respectively, the mean of the values of those elements for the +limits of temperature, and in place of _dp_/_dt_, the difference of the +values of _p_, at the same limits. + +35. In Regnault’s work (at the end of the eighth memoir), a table of the +pressures of saturated steam for the successive temperatures 0°, 1°, +2°, ... 230°, expressed in millimetres of mercury, is given. On account +of the units adopted in this paper, these pressures must be estimated in +pounds on the square foot, which we may do by multiplying each number of +millimetres by 2.7896, the weight in pounds of a sheet of mercury, one +millimetre thick, and a square foot in area. + +36. The value of _k_, the latent heat of a cubic foot, for any +temperature _t_, is found from λ, the latent heat of a pound of +saturated steam, by the equation + + _k_ = (_p_)/(760). (1 + .00366 × 100)/(1 + .00366 × _t_). × .036869[53] . λ, + +where _p_ denotes the pressure in millimetres, and λ the latent heat of +a pound of saturated steam; the values of λ being calculated by the +empirical formula[54] + + λ = (606.5 + 0.305_t_) − (_t_ + .00002_t_^2 + 0.0000003_t_^3), + +given by Regnault as representing, between the extreme limits of his +observations, the latent heat of a unit weight of saturated steam. + + + EXPLANATION OF TABLE I. + +37. The mean values of μ for the first, for the eleventh, for the +twenty-first, and so on, up to the 231st[55] degree of the +air-thermometer, have been calculated in the manner explained in the +preceding paragraphs. These, and interpolated results, which must agree +with what would have been obtained, by direct calculation from +Regnault’s data, to three significant places of figures (and even for +the temperatures between 0° and 100°, the experimental data do not +justify us in relying on any of the results to a greater degree of +accuracy), are exhibited in Table I. + +_To find the amount of mechanical effect due to a unit of heat, +descending from a body at a temperature S to a body at T, if these +numbers be integers, we have merely to add the values of μ in Table I. +corresponding to the successive numbers._ + + _T_ + 1, _T_ + 2, ... _S_ − 2, _S_ − 1. + + + EXPLANATION OF TABLE II. + +38. The calculation of the mechanical effect, in any case, which might +always be effected in the manner described in § 37 (with the proper +modification for fractions of degrees, when necessary), is much +simplified by the use of Table II., where the first number of Table I., +the sum of the first and second, the sum of the first three, the sum of +the first four, and so on, are successively exhibited. The sums thus +tabulated are the values of the integrals + + ∫_{0}^1 μ_dt_, ∫_{0}^2 μ_dt_, ∫_{0}^3 μ_dt_, ... ∫_{0}^{231} μ_dt_; + +and, if we denote ∫_{0}^t μ_dt_ by the letter _M_, Table II. may be +regarded as a table of the value of _M_. + +_To find the amount of mechanical effect due to a unit of heat +descending from a body at a temperature S to a body at T, if these +numbers be integers, we have merely to subtract the value of M, for the +number T, from the value for the number S, given in Table II._ + + TABLE I.[56] + MEAN VALUES OF Μ FOR THE SUCCESSIVE DEGREES OF THE AIR-THERMOMETER FROM + 0° TO 230°. + ───────────────────────────────────┬─────────────────────────────────── + ° │ μ + ───────────────────────────────────┼─────────────────────────────────── + 1│ 4.960 + 2│ 4.946 + 3│ 4.932 + 4│ 4.918 + 5│ 4.905 + 6│ 4.892 + 7│ 4.878 + 8│ 4.865 + 9│ 4.852 + 10│ 4.839 + 11│ 4.826 + 12│ 4.812 + 13│ 4.799 + 14│ 4.786 + 15│ 4.773 + 16│ 4.760 + 17│ 4.747 + 18│ 4.735 + 19│ 4.722 + 20│ 4.709 + 21│ 4.697 + 22│ 4.684 + 23│ 4.672 + 24│ 4.659 + 25│ 4.646 + 26│ 4.634 + 27│ 4.621 + 28│ 4.609 + 29│ 4.596 + 30│ 4.584 + 31│ 4.572 + 32│ 4.559 + 33│ 4.547 + 34│ 4.535 + 35│ 4.522 + 36│ 4.510 + 37│ 4.498 + 38│ 4.486 + 39│ 4.474 + 40│ 4.462 + 41│ 4.450 + 42│ 4.438 + 43│ 4.426 + 44│ 4.414 + 45│ 4.402 + 46│ 4.390 + 47│ 4.378 + 48│ 4.366 + 49│ 4.355 + 50│ 4.343 + 51│ 4.331 + 52│ 4.319 + 53│ 4.308 + 54│ 4.296 + 55│ 4.285 + 56│ 4.273 + 57│ 4.262 + 58│ 4.250 + 59│ 4.239 + 60│ 4.227 + 61│ 4.216 + 62│ 4.205 + 63│ 4.194 + 64│ 4.183 + 65│ 4.172 + 66│ 4.161 + 67│ 4.150 + 68│ 4.140 + 69│ 4.129 + 70│ 4.119 + 71│ 4.109 + 72│ 4.098 + 73│ 4.088 + 74│ 4.078 + 75│ 4.067 + 76│ 4.057 + 77│ 4.047 + 78│ 4.037 + 79│ 4.028 + 80│ 4.018 + 81│ 4.009 + 82│ 3.999 + 83│ 3.990 + 84│ 3.980 + 85│ 3.971 + 86│ 3.961 + 87│ 3.952 + 88│ 3.943 + 89│ 3.934 + 90│ 3.925 + 91│ 3.916 + 92│ 3.907 + 93│ 3.898 + 94│ 3.889 + 95│ 3.880 + 96│ 3.871 + 97│ 3.863 + 98│ 3.854 + 99│ 3.845 + 100│ 3.837 + 101│ 3.829 + 102│ 3.820 + 103│ 3.812 + 104│ 3.804 + 105│ 3.796 + 106│ 3.788 + 107│ 3.780 + 108│ 3.772 + 109│ 3.764 + 110│ 3.757 + 111│ 3.749 + 112│ 3.741 + 113│ 3.734 + 114│ 3.726 + 115│ 3.719 + 116│ 3.712 + 117│ 3.704 + 118│ 3.697 + 119│ 3.689 + 120│ 3.682 + 121│ 3.675 + 122│ 3.668 + 123│ 3.661 + 124│ 3.654 + 125│ 3.647 + 126│ 3.640 + 127│ 3.633 + 128│ 3.627 + 129│ 3.620 + 130│ 3.614 + 131│ 3.607 + 132│ 3.601 + 133│ 3.594 + 134│ 3.586 + 135│ 3.579 + 136│ 3.573 + 137│ 3.567 + 138│ 3.561 + 139│ 3.555 + 140│ 3.549 + 141│ 3.543 + 142│ 3.537 + 143│ 3.531 + 144│ 3.525 + 145│ 3.519 + 146│ 3.513 + 147│ 3.507 + 148│ 3.501 + 149│ 3.495 + 150│ 3.490 + 151│ 3.484 + 152│ 3.479 + 153│ 3.473 + 154│ 3.468 + 155│ 3.462 + 156│ 3.457 + 157│ 3.451 + 158│ 3.446 + 159│ 3.440 + 160│ 3.435 + 161│ 3.430 + 162│ 3.424 + 163│ 3.419 + 164│ 3.414 + 165│ 3.409 + 166│ 3.404 + 167│ 3.399 + 168│ 3.394 + 169│ 3.389 + 170│ 3.384 + 171│ 3.380 + 172│ 3.375 + 173│ 3.370 + 174│ 3.365 + 175│ 3.361 + 176│ 3.356 + 177│ 3.351 + 178│ 3.346 + 179│ 3.342 + 180│ 3.337 + 181│ 3.332 + 182│ 3.328 + 183│ 3.323 + 184│ 3.318 + 185│ 3.314 + 186│ 3.309 + 187│ 3.304 + 188│ 3.300 + 189│ 3.295 + 190│ 3.291 + 191│ 3.287 + 192│ 3.282 + 193│ 3.278 + 194│ 3.274 + 195│ 3.269 + 196│ 3.265 + 197│ 3.261 + 198│ 3.257 + 199│ 3.253 + 200│ 3.249 + 201│ 3.245 + 202│ 3.241 + 203│ 3.237 + 204│ 3.233 + 205│ 3.229 + 206│ 3.225 + 207│ 3.221 + 208│ 3.217 + 209│ 3.213 + 210│ 3.210 + 211│ 3.206 + 212│ 3.202 + 213│ 3.198 + 214│ 3.195 + 215│ 3.191 + 216│ 3.188 + 217│ 3.184 + 218│ 3.180 + 219│ 3.177 + 220│ 3.173 + 221│ 3.169 + 222│ 3.165 + 223│ 3.162 + 224│ 3.158 + 225│ 3.155 + 226│ 3.151 + 227│ 3.148 + 228│ 3.144 + 229│ 3.141 + 230│ 3.137 + 231│ 3.134 + ───────────────────────────────────┴─────────────────────────────────── + + TABLE II. + MECHANICAL EFFECT IN FOOT-POUNDS DUE TO A THERMIC UNIT CENTIGRADE, + PASSING FROM A BODY, AT ANY TEMPERATURE LESS THAN 230° TO A BODY AT 0°. + ───────────────────────────────────┬─────────────────────────────────── + Superior Limit of Temperature. │ Mechanical Effect. + ───────────────────────────────────┼─────────────────────────────────── + ° │ Ft.-Pounds. + │ + 1│ 4.960 + 2│ 9.906 + 3│ 14.838 + 4│ 19.756 + 5│ 24.661 + 6│ 29.553 + 7│ 34.431 + 8│ 39.296 + 9│ 44.148 + 10│ 48.987 + 11│ 53.813 + 12│ 58.625 + 13│ 63.424 + 14│ 68.210 + 15│ 72.983 + 16│ 77.743 + 17│ 82.490 + 18│ 87.225 + 19│ 91.947 + 20│ 96.656 + 21│ 101.353 + 22│ 106.037 + 23│ 110.709 + 24│ 115.368 + 25│ 120.014 + 26│ 124.648 + 27│ 129.269 + 28│ 133.878 + 29│ 138.474 + 30│ 143.058 + 31│ 147.630 + 32│ 152.189 + 33│ 156.736 + 34│ 161.271 + 35│ 165.793 + 36│ 170.303 + 37│ 174.801 + 38│ 179.287 + 39│ 183.761 + 40│ 188.223 + 41│ 192.673 + 42│ 197.111 + 43│ 201.537 + 44│ 205.951 + 45│ 210.353 + 46│ 214.743 + 47│ 219.121 + 48│ 223.487 + 49│ 227.842 + 50│ 232.185 + 51│ 236.516 + 52│ 240.835 + 53│ 245.143 + 54│ 249.439 + 55│ 253.724 + 56│ 257.997 + 57│ 262.259 + 58│ 266.509 + 59│ 270.748 + 60│ 274.975 + 61│ 279.191 + 62│ 283.396 + 63│ 287.590 + 64│ 291.773 + 65│ 295.945 + 66│ 300.106 + 67│ 304.256 + 68│ 308.396 + 69│ 312.525 + 70│ 316.644 + 71│ 320.752 + 72│ 324.851 + 73│ 328.939 + 74│ 333.017 + 75│ 337.084 + 76│ 341.141 + 77│ 345.188 + 78│ 349.225 + 79│ 353.253 + 80│ 357.271 + 81│ 361.280 + 82│ 365.279 + 83│ 369.269 + 84│ 373.249 + 85│ 377.220 + 86│ 381.181 + 87│ 385.133 + 88│ 389.076 + 89│ 393.010 + 90│ 396.935 + 91│ 400.851 + 92│ 404.758 + 93│ 408.656 + 94│ 412.545 + 95│ 416.425 + 96│ 420.296 + 97│ 424.159 + 98│ 428.013 + 99│ 431.858 + 100│ 435.695 + 101│ 439.524 + 102│ 443.344 + 103│ 447.156 + 104│ 450.960 + 105│ 454.756 + 106│ 458.544 + 107│ 462.324 + 108│ 466.096 + 109│ 469.860 + 110│ 473.617 + 111│ 477.366 + 112│ 481.107 + 113│ 484.841 + 114│ 488.567 + 115│ 492.286 + 116│ 495.998 + 117│ 499.702 + 118│ 503.399 + 119│ 507.088 + 120│ 510.770 + 121│ 514.445 + 122│ 518.113 + 123│ 521.174 + 124│ 525.428 + 125│ 529.075 + 126│ 532.715 + 127│ 536.348 + 128│ 539.975 + 129│ 543.595 + 130│ 547.209 + 131│ 550.816 + 132│ 554.417 + 133│ 558.051 + 134│ 561.597 + 135│ 565.176 + 136│ 568.749 + 137│ 572.316 + 138│ 575.877 + 139│ 579.432 + 140│ 582.981 + 141│ 586.524 + 142│ 590.061 + 143│ 593.592 + 144│ 597.117 + 145│ 600.636 + 146│ 604.099 + 147│ 607.656 + 148│ 611.157 + 149│ 614.652 + 150│ 618.142 + 151│ 621.626 + 152│ 625.105 + 153│ 628.578 + 154│ 632.046 + 155│ 635.508 + 156│ 638.965 + 157│ 642.416 + 158│ 645.862 + 159│ 649.302 + 160│ 652.737 + 161│ 656.167 + 162│ 659.591 + 163│ 663.010 + 164│ 666.424 + 165│ 669.833 + 166│ 673.237 + 167│ 676.636 + 168│ 680.030 + 169│ 683.419 + 170│ 686.803 + 171│ 690.183 + 172│ 693.558 + 173│ 696.928 + 174│ 700.293 + 175│ 703.654 + 176│ 707.010 + 177│ 710.361 + 178│ 713.707 + 179│ 717.049 + 180│ 720.386 + 181│ 723.718 + 182│ 727.046 + 183│ 730.369 + 184│ 733.687 + 185│ 737.001 + 186│ 740.310 + 187│ 743.614 + 188│ 746.914 + 189│ 750.209 + 190│ 753.500 + 191│ 756.787 + 192│ 760.069 + 193│ 763.347 + 194│ 766.621 + 195│ 769.890 + 196│ 773.155 + 197│ 776.416 + 198│ 779.673 + 199│ 782.926 + 200│ 786.175 + 201│ 789.420 + 202│ 792.661 + 203│ 795.898 + 204│ 799.131 + 205│ 802.360 + 206│ 805.585 + 207│ 808.806 + 208│ 812.023 + 209│ 815.236 + 210│ 818.446 + 211│ 821.652 + 212│ 824.854 + 213│ 828.052 + 214│ 831.247 + 215│ 834.438 + 216│ 837.626 + 217│ 840.810 + 218│ 843.990 + 219│ 847.167 + 220│ 850.340 + 221│ 853.509 + 222│ 856.674 + 223│ 859.836 + 224│ 862.994 + 225│ 866.149 + 226│ 869.300 + 227│ 872.448 + 228│ 875.592 + 229│ 878.733 + 230│ 881.870 + 231│ 885.004 + ───────────────────────────────────┴─────────────────────────────────── + + + _Note on the curves described in Clapeyron’s graphical method of + exhibiting Carnot’s Theory of the Steam-Engine._ + +39. At any instant when the temperature of the water and vapor is _t_, +during the fourth operation (see above, § 16, and suppose, for the sake +of simplicity, that at the beginning of the first and at the end of the +fourth operation the piston is absolutely in contact with the surface of +the water), the latent heat of the vapor must be precisely equal to the +amount of heat that would be necessary to raise the temperature of the +whole mass, if in the liquid state, from _t_ to _S_.[57] Hence, if _v′_ +denote the volume of the vapor, _c_ the mean capacity for heat of a +pound of water between the temperatures _S_ and _t_, and _W_ the weight +of the entire mass, in pounds, we have + + _kv′_ = _c_(_S_ − _t_)_W_. + +Again, the circumstances during the second operation are such that the +mass of liquid and vapor possesses _H_ units of heat more than during +the fourth; and consequently, at the instant of the second operation, +when the temperature is _t_, the volume _v_ of the vapor will exceed +_v′_ by an amount of which the latent heat is _H_, so that we have + + _v_ = _v′_ + (_H_)/(_k_). + +40. Now, at any instant, the volume between the piston and its primitive +position is less than the actual volume of vapor by the volume of the +water evaporated. Hence, if _x_ and _x′_ denote the abscissæ of the +curve at the instants of the second and fourth operations respectively, +when the temperature is _t_, we have + + _x_ = _v_ − σ_v_, _x′_ = _v′_ − σ_v′_, + +and, therefore, by the preceding equations, + + _x_ = (1 − σ)/(_k_){_H_ + _c_(_S_ − _t_)_W_}, (_a_) + _x′_ = (1 − σ)/(_k_)_c_(_S_ − _t_)_W_. (_b_) + These equations, along with _y_ = _y′_ = _p_, (_c_) + +enable us to calculate, from the data supplied by Regnault, the abscissa +and ordinate for each of the curves described above (§ 17) corresponding +to any assumed temperature _t_. After the explanations of §§ 33, 34, 35, +36, it is only necessary to add that _c_ is a quantity of which the +value is very nearly unity, and would be exactly so were the capacity of +water for heat the same at every temperature as it is between 0° and 1°; +and that the value of _c_(_S_ − _t_), for any assigned values of _S_ and +_t_, is found, by subtracting the number corresponding to _t_ from the +number corresponding to _s_, in the column headed “_Nombre des unités de +chaleur abandonnées par un kilogramme d’eau en descendant de T° à 0°_,” +of the last table (at the end of the tenth memoir) of Regnault’s work. +By giving _S_ the value 230°, and by substituting successively 220, 210, +200, etc., for _t_, values for _x_, _y_, _x′_, _y′_, have been found, +which are exhibited in the table opposite. + + ─────────────┬─────────────────┬────────────────────────┬───────────── + Temperatures.│ Volumes to be │ Volumes from the │Pressures of + │described by the │ primitive position of │ saturated + │ piston, to │ the piston to those │ steam, in + │ complete the │occupied at instants of │pounds on the + │fourth operation.│ the second operation. │square foot. + _t_ │ _x′_ │ _x_ │_y_ = _y′_ = + │ │ │ _p_ + ─────────────┼─────────────────┼────────────────────────┼───────────── + 0°│ 1269. _W_ │_x′_ + 5.409._H_ │ 12.832 + 10│ 639.6. _W_ │_x′_ + 2.847._H_ │ 25.567 + 20│ 337.3. _W_ │_x′_ + 1.571._H_ │ 48.514 + 30│ 185.5. _W_ │_x′_ + .9062._H_ │ 88.007 + 40│ 105.9. _W_ │_x′_ + .5442._H_ │ 153.167 + 50│ 62.62. _W_ │_x′_ + .3392._H_ │ 256.595 + 60│ 38.19. _W_ │_x′_ + .2188._H_ │ 415.070 + 70│ 21.94. _W_ │_x′_ + .1456._H_ │ 650.240 + 80│ 15.38. _W_ │_x′_ + .09962._H_ │ 989.318 + 90│ 10.09. _W_ │_x′_ + .06994._H_ │ 1465.80 + 100│ 6.744. _W_ │_x′_ + .05026._H_ │ 2120.11 + 110│ 4.578. _W_ │_x′_ + .03688._H_ │ 2999.87 + 120│ 3.141. _W_ │_x′_ + .02758._H_ │ 4160.10 + 130│ 2.176. _W_ │_x′_ + .02098._H_ │ 5663.70 + 140│ 1.519. _W_ │_x′_ + .01625._H_ │ 7581.15 + 150│ 1.058. _W_ │_x′_ + .01271._H_ │ 9990.26 + 160│ 0.7369. _W_ │_x′_ + .01010._H_ │ 12976.2 + 170│ 0.5085. _W_ │_x′_ + .008116._H_ │ 16630.7 + 180│ 0.3454. _W_ │_x′_ + .006592._H_ │ 21051.5 + 190│ 0.2267. _W_ │_x′_ + .005406._H_ │ 26341.5 + 200│ 0.1409. _W_ │_x′_ + .004472._H_ │ 32607.7 + 210│ 0.0784. _W_ │_x′_ + .003729._H_ │ 39960.7 + 220│ 0.3310. _W_ │_x′_ + .003130._H_ │ 48512.4 + 230│ 0 │_x′_ + .002643._H_ │ 58376.6 + ─────────────┴─────────────────┴────────────────────────┴───────────── + + + _Appendix._ + + (Read April 30, 1849.) + +41. In p. 30 some conclusions drawn by Carnot from his general reasoning +were noticed; according to which it appears, that if the value of μ for +any temperature is known, certain information may be derived with +reference to the saturated vapor of any liquid whatever, and, with +reference to any gaseous mass, without the necessity of experimenting +upon the specific medium considered. Nothing in the whole range of +Natural Philosophy is more remarkable than the establishment of general +laws by such a process of reasoning. We have seen, however, that doubt +may exist with reference to the truth of the axiom on which the entire +theory is founded, and it therefore becomes more than a matter of mere +curiosity to put the inferences deduced from it to the test of +experience. The importance of doing so was clearly appreciated by +Carnot; and, with such data as he had from the researches of various +experimenters, he tried his conclusions. Some very remarkable +propositions which he derives from his theory coincide with Dulong and +Petit’s subsequently discovered experimental laws with reference to the +heat developed by the compression of a gas; and the experimental +verification is therefore in this case (so far as its accuracy could be +depended upon) decisive. In other respects, the data from experiment +were insufficient, although, so far as they were available as tests, +they were confirmatory of the theory. + +42. The recent researches of Regnault add immensely to the experimental +data available for this object, by giving us the means of determining +with considerable accuracy the values of μ within a very wide range of +temperature, and so affording a trustworthy standard for the comparison +of isolated results at different temperatures, derived from observations +in various branches of physical science. + +In the first section of this Appendix the theory is tested, and shown to +be confirmed by the comparison of the values of μ found above, with +those obtained by Carnot and Clapeyron from the observations of various +experimenters on air, and the vapors of different liquids. In the second +and third sections some striking confirmations of the theory arising +from observations by Dulong, on the specific heat of gases, and from Mr. +Joule’s experiments on the heat developed by the compression of air, are +pointed out; and in conclusion, the actual methods of obtaining +mechanical effect from heat are briefly examined with reference to their +economy. + + +I. _On the values of μ derived by Carnot and Clapeyron from observations + on Air, and on the Vapors of various liquids._ + +43. In Carnot’s work, pp. 80–82, the mean value of μ between 0° and 1° +is derived from the experiments of Delaroche and Bérard on the specific +heat of gases, by a process approximately equivalent to the calculation +of the value of (_Ep_{0}v_{0}_)/(_vdq_/_dv_) for the temperature ½°. +There are also, in the same work, determinations of the values of μ from +observations on the vapors of alcohol and water; but a table given in M. +Clapeyron’s paper, of the values of μ derived from the data supplied by +various experiments with reference to the vapors of ether, alcohol, +water, and oil of turpentine, at the respective boiling-points of these +liquids, affords us the means of comparison through a more extensive +range of temperature. In the cases of alcohol and water, these results +ought of course to agree with those of Carnot. There are, however, +slight discrepancies which must be owing to the uncertainty of the +experimental data.[58] In the opposite table, Carnot’s results with +reference to air, and Clapeyron’s results with reference to the four +different liquids, are exhibited, and compared with the values of μ +which have been given above (Table I.) for the same temperatures, as +derived from Regnault’s observations on the vapor of water. + + ────────────┬──────────────┬──────────────┬──────────────┬───────────── + │ │ │ Values of μ │ + │ │ │ deduced from │ + Names of the│ │ │ Regnault’s │ + Media. │Temperatures. │ Values of μ. │Observations. │Differences. + ────────────┼──────────────┼──────────────┼──────────────┼───────────── + │ ° │ (Carnot) │ │ + Air │ 0.5│ 4.377│ 4.960│ .383 + Sulphuric │ (Boil. pt.)│ (Clapeyron)│ │ + Ether │ 35.5│ 4.478│ 4.510│ .032 + Alcohol │ 78.8│ 3.963│ 4.030│ .071 + Water │ 100│ 3.658│ 3.837│ .179 + Essence of │ │ │ │ + Turpentine│ 156.8│ 3.530│ 3.449│ −.081 + ────────────┴──────────────┴──────────────┴──────────────┴───────────── + +44. It may be observed that the discrepancies between the results +founded on the experimental data supplied by the different observers +with reference to water at the boiling-point, are greater than those +which are presented between the results deduced from any of the other +liquids, and water at the other temperatures; and we may therefore feel +perfectly confident that the verification is complete to the extent of +accuracy of the observations.[59] The considerable discrepancy presented +by Carnot’s result deduced from experiments on air, is not to be +wondered at when we consider the very uncertain nature of his data. + +45. The fact of the gradual decrease of μ through a very extensive range +of temperature, being indicated both by Regnault’s continuous series of +experiments and by the very varied experiment on different media, and in +different branches of Physical Science, must be considered as a striking +verification of the theory. + + + II. _On the Heat developed by the Compression of Air._ + +46. Let a mass of air, occupying initially a given volume _V_, under a +pressure _P_, at a temperature _t_, be compressed to a less volume _V′_, +and allowed to part with heat until it sinks to its primitive +temperature _t_. The quantity of heat which is evolved may be +determined, according to Carnot’s theory, when the particular value of +μ, corresponding to the temperature _t_, is known. For, by § 30, +equation (6), we have + + _v_(_dq_)/(_dv_) = (_Ep_{0}v_{0}_)/(μ), + +where _dq_ is the quantity of heat absorbed, when the volume is allowed +to increase from _v_ to _v_ + _dv_; or the quantity evolved by the +reverse operation. Hence we deduce + + _dq_ = (_Ep_{0}v_{0}_)/(μ) (_dv_)/(_v_). (8) + +Now, (_Ep_{0}v_{0}_)/(μ) is constant, since the temperature remains +unchanged; and therefore we may at once integrate the second number. By +taking it between the limits _V′_ and _V_, we thus find + + _Q_ = (_Ep_{0}v_{0}_)/(μ) log (_V_)/(_V′_)[60], (9) + +where _Q_ denotes the required amount of heat evolved by the compression +from _V_ to _P′_. This expression may be modified by employing the +equations _PV_ = _P′V′_ = _p_{0}v_{0}_(1 + _Et_); and we thus obtain + + _Q_ = (_EPV_)/(μ(1 + _Et_)) log (_V_)/(_V′_) = (_EP′V′_)/(μ(1 + _Et_)) log (_V_)/(_V′_). (10) + +From this result we draw the following conclusion: + +47. _Equal volumes of all elastic fluids, taken at the same temperature +and pressure, when compressed to smaller equal volumes, disengage equal +quantities of heat._ + +This extremely remarkable theorem of Carnot’s was independently laid +down as a probable experimental law by Dulong, in his “_Recherches sur +la Chaleur Spécifique des Fluides Élastiques_,” and it therefore affords +a most powerful confirmation of the theory.[61] + +48. In some very remarkable researches made by Mr. Joule upon the heat +developed by the compression of air, the quantity of heat produced in +different experiments has been ascertained with reference to the amount +of work spent in the operation. To compare the results which he has +obtained with the indications of theory, let us determine the amount of +work necessary actually to produce the compression considered above. + +49. In the first place, to compress the gas from the volume _v_ + _dv_ +to _v_, the work required is _pdv_, or, since + + _pv_ = _p_{0}v_{0}_(1 + _Et_), + _p_{0}v_{0}_(1 + _Et_)(_dv_)/(_v_). + +Hence, if we denote by _W_ the total amount of work necessary to produce +the compression from _V_ to _V′_, we obtain, by integration, + + _W_ = _p_{0}v_{0}_(1 + _Et_) log (_V_)/(_V′_). + +Comparing this with the expression above, we find + + (_W_)/(_Q_) = (μ(1 + _Et_))/(_E_). (11) + +50. Hence we infer that— + +(1) The amount of work necessary to produce a unit of heat by the +compression of a gas is the same for all gases at the same temperature; + +(2) And that the quantity of heat evolved in all circumstances, when the +temperature of the gas is given, is proportional to the amount of work +spent in the compression. + +51. The expression for the amount of work necessary to produce a unit of +heat is + + μ(1 + _Et_)/(_E_), + +and therefore Regnault’s experiments on steam are available to enable us +to calculate its value for any temperature. By finding the values of μ +at 0°, 10°, 20°, etc., from Table I., and by substituting successively +the values 0, 10, 20, etc., for _t_, the following results have been +obtained: + + TABLE OF THE VALUES OF (μ(1 + _Et_))/(_E_). + ───────────────────────────────────┬─────────────────────────────────── + Work requisite to produce a unit of│ Temperature of the Gas. + Heat by the compression of a Gas. │ + ───────────────────────────────────┼─────────────────────────────────── + Ft.-pounds. │ ° + 1357.1 │ 0 + 1368.7 │ 10 + 1379.0 │ 20 + 1388.0 │ 30 + 1395.7 │ 40 + 1401.8 │ 50 + 1406.7 │ 60 + 1412.0 │ 70 + 1417.6 │ 80 + 1424.0 │ 90 + 1430.6 │ 100 + 1438.2 │ 110 + 1446.4 │ 120 + 1455.8 │ 130 + 1465.3 │ 140 + 1475.8 │ 150 + 1489.2 │ 160 + 1499.0 │ 170 + 1511.3 │ 180 + 1523.5 │ 190 + 1536.5 │ 200 + 1550.2 │ 210 + 1564.0 │ 220 + 1577.8 │ 230 + ───────────────────────────────────┴─────────────────────────────────── + +Mr. Joule’s experiments were all conducted at temperatures from 50° to +about 60° Fahr., or from 10° to 16° Cent.; and consequently, although +some irregular differences in the results, attributable to errors of +observation inseparable from experiments of such a very difficult +nature, are presented, no regular dependence on the temperature is +observable. From three separate series of experiments, Mr. Joule deduces +the following numbers for the work, in foot-pounds, necessary to produce +a thermic unit Fahrenheit by the compression of a gas. + + 820, 814, 760. + +Multiplying these by 1.8, to get the corresponding number for a thermic +unit Centigrade, we + + 1476, 1465, and 1368. + +The largest of these numbers is most nearly conformable with Mr. Joule’s +views of the relation between such experimental “equivalents,” and +others which he obtained in his electro-magnetic researches; but the +smallest agrees almost perfectly with the indications of Carnot’s +theory; from which, as exhibited in the preceding table, we should +expect, from the temperature in Mr. Joule’s experiments, to find a +number between 1369 and 1379 as the result.[62] + + + + + III. _On the Specific Heats of Gases._ + + +52. The following proposition is proved by Carnot as a deduction from +his general theorem regarding the specific heats of gases. + +_The excess of specific heat[63] under a constant pressure above the +specific heat at a constant volume, is the same for all gases at the +same temperature and pressure._ + +53. To prove this proposition, and to determine an expression for the +“excess” mentioned in its enunciation, let us suppose a unit of volume +of a gas to be elevated in temperature by a small amount, τ. The +quantity of heat required to do this will be _A_τ, if _A_ denote the +specific heat at a constant volume. Let us next allow the gas to expand +without going down in temperature, until its pressure becomes reduced to +its primitive value. The expansion which will take place will be +(_E_τ)/(1 + _Et_), if the temperature be denoted by _t_; and hence, by +(8), the quantity of heat that must be supplied, to prevent any lowering +of temperature, will be + + (_Ep_{0}v_{0}_)/(μ) . (_E_τ)/(1 + _Et_), or (_E^2p_)/(μ(1 + _Et_)^2)τ. + +Hence the total quantity added is equal to + + _Α_τ + (_E^2p_)/(μ(1 + _Et_)^2)τ. + +But, since _B_ denotes the specific heat under constant pressure, the +quantity of heat requisite to bring the gas into this state, from its +primitive condition, is equal to _Β_τ, and hence we have + + _B_ = _A_ + (_E^2p_)/(μ(1 + _Et_)^2). (12) + + + IV. _Comparison of the Relative Advantages of the Air-engine and + Steam-engine._ + +54. In the use of water-wheels for motive power, the economy of the +engine depends not only upon the excellence of its adaptation for +actually transmitting any given quantity of water through it, and +producing the equivalent of work, but upon turning to account the entire +available fall; so, as we are taught by Carnot, the object of a +thermodynamic engine is to economize in the best possible way the +transference of all the heat evolved, from bodies at the temperature of +the source, to bodies at the lowest temperature at which the heat can be +discharged. With reference, then, to any engine of the kind, there will +be two points to be considered: + +(1) The extent of the _fall_ utilized. + +(2) The economy of the engine, with the fall which it actually uses. + +55. In the first respect, the air-engine, as Carnot himself points out, +has a vast advantage over the steam-engine; since the temperature of the +hot part of the machine may be made very much higher in the air-engine +than would be possible in the steam-engine, on account of the very high +pressure produced in the boiler, by elevating the temperature of the +water which it contains to any considerable extent above the atmospheric +boiling-point. On this account a “perfect air-engine” would be a much +more valuable instrument than a “perfect steam-engine.”[64] + +Neither steam-engines nor air-engines, however, are nearly perfect; and +we do not know in which of the two kinds of machine the nearest approach +to perfection may be actually attained. The beautiful engine invented by +Mr. Stirling of Galston may be considered as an excellent beginning for +the air-engine;[65] and it is only necessary to compare this with +Newcomen’s steam-engine, and consider what Watt has effected, to give +rise to the most sanguine anticipations of improvement. + + + V. _On the Economy of Actual Steam-engines._ + +56. The steam-engine being universally employed at present as the means +for deriving motive power from heat, it is extremely interesting to +examine, according to Carnot’s theory, the economy actually attained in +its use. In the first place we remark, that out of the entire “fall” +from the temperature of the coals to that of the atmosphere it is only +part—that from the temperature of the boiler to the temperature of the +condenser—that is made available; while the very great fall from the +temperature of the burning coals to that of the boiler, and the +comparatively small fall from the temperature of the condenser to that +of the atmosphere, are entirely lost as far as regards the mechanical +effect which it is desired to obtain. We infer from this, that the +temperature of the boiler ought to be kept as high as, according to the +strength, is consistent with safety, while that of the condenser ought +to be kept as nearly down at the atmospheric temperature as possible. To +take the entire benefit of the actual fall, Carnot showed that the +“principle of expansion” must be pushed to the utmost.[66] + +57. To obtain some notion of the economy which has actually been +obtained, we may take the alleged performances of the best Cornish +engines, and some other interesting practical cases, as examples.[67] + +(1) The engine of _the Fowey Consols mine_ was reported, in 1845, to +have given 125,089,000 foot-pounds of effect, for the consumption of one +bushel or 94 lbs. of coals. Now the average amount evaporated from +Cornish boilers, by one pound of coal, is 8½ lbs. of steam; and hence +for each pound of steam evaporated 156,556 foot-pounds of work are +produced. + +The pressure of the saturated steam in the boiler may be taken as 3½ +atmospheres;[68] and, consequently, the temperature of the water will be +140°. Now (Regnault, end of Mémoire X.) the latent heat of a pound of +saturated steam at 140° is 508, and since, to compensate for each pound +of steam removed from the boiler in the working of the engine, a pound +of water, at the temperature of the condenser, which may be estimated at +30°, is introduced from the hot-well; it follows that 618 units of heat +are introduced to the boiler for each pound of water evaporated. But the +work produced, for each pound of water evaporated, was found above to be +156,556 foot-pounds. Hence ¹⁵⁶⁵⁵⁶⁄₆₁₈, or 253 foot-pounds, is the amount +of work produced for each unit of heat transmitted through the Fowey +Consols engine. Now in Table II. we find 583.0 as the theoretical effect +due to a unit descending from 140° to 0°, and 143 as the effect due to a +unit descending from 30° to 0°. The difference of these numbers, or +440,[69] is the number of foot-pounds of work that a _perfect_ engine +with its boiler at 140° and its condenser at 30° would produce for each +unit of heat transmitted. Hence the Fowey Consols engine, during the +experiments reported on, performed ²⁵³⁄₄₄₀ of its theoretical duty, or +57½ per cent. + +(2) The best duty on record, as performed by an engine at work (not for +merely experimental purposes), is that of Taylor’s engine, at the United +Mines, which in 1840 worked regularly for several months at the rate of +98,000,000 foot-pounds for each bushel of coals burned. This is ⁹⁸⁄₁₂₅, +or .784 of the experimental duty reported in the case of the Fowey +Consols engine. Hence the best useful work on record is at the rate of +198.3 foot-pounds for each unit of heat transmitted, and is +(198.3)/(440) or 45 per cent of the theoretical duty, on the supposition +that the boiler is at 140° and the condenser at 30°. + +(3) French engineers contract (in Lille, in 1847, for example) to make +engines for mill-power which will produce 30,000 metre-pounds or 98,427 +foot-pounds of work for each pound of steam used. If we divide this by +618, we find 159 foot-pounds for the work produced by each unit of heat. +This is 36.1 per cent of 440, the theoretical duty.[70] + +(4) English engineers have contracted to make engines and boilers which +will require only 3⅓ lbs. of the best coal per horse-power per hour. +Hence in such engines each pound of coal ought to produce 565,700 +foot-pounds of work, and if 7 lbs. of water be evaporated by each pound +of coal, there would result 83,814 foot-pounds of work for each pound of +water evaporated. If the pressure in the boiler be 3½ atmospheres +(temperature 140°) the amount of work for each unit of heat will be +found, by dividing this by 618, to be 130.7 foot-pounds, which is +(130.7)/(440) or 29.7 per cent of the theoretical duty.[71] + +(5) The actual average of work performed by good Cornish engines and +boilers is 55,000,000 foot-pounds for each bushel of coal, or less than +half the experimental performance of the Fowey Consols engine, more than +half the actual duty performed by the United Mines engine in 1840; in +fact, about 25 per cent of the theoretical duty. + +(6) The average performances of a number of Lancashire engines and +boilers have been recently found to be such as to require 12 lbs. of +Lancashire coal per horse-power per hour (i.e., for performing 60 × +33,000 foot-pounds), and of a number of Glasgow engines such as to +require 15 lbs. (of a somewhat inferior coal) for the same effect. There +are, however, more than twenty large engines in Glasgow at present[72] +which work with a consumption of only 6½ lbs. of dross, equivalent to 5 +lbs. of the best Scotch or 4 lbs. of the best Welsh coal, per +horse-power per hour. The economy may be estimated from these data, as +in the other cases, on the assumption which, with reference to these, is +the most probable we can make, that the evaporation produced by a pound +of best coal is 7 lbs. of steam. + +58. The following tables afford a synoptic view of the performances and +theoretical duties in the various cases discussed above. + +In Table A the numbers in the second column are found by dividing the +numbers in the first by 8½ in cases (1), (2), and (5), and by 7 in cases +(4), (6), and (7), the estimated numbers of pounds of steam actually +produced in the different boilers by the burning of 1 lb. of coal. + +The numbers in the third column are found from those in the second, by +dividing by 618 in Table A, and 614 in Table B, which are respectively +the quantities of heat required to convert a pound of water taken from +the hot-well at 30°, into saturated steam, in the boiler, at 140° or at +121°. + +With reference to the cases (3), (4), (6), (7), the hypothesis of Table +B is probably in general nearer the truth than that of Table A. In (4), +(6), and (7), especially upon hypothesis B, there is much uncertainty as +to the amount of evaporation that will be actually produced by 1 lb. of +fuel. The assumption on which the numbers in the second column in Table +B are calculated, is, that each pound of coal will send the same number +of units of heat into the boiler, whether hypothesis A or hypothesis B +be followed. Hence, except in the case of the French contract, in which +the _evaporation_, not the fuel, is specified, the numbers in the third +column are the same as those in the third column of Table A. + + TABLE A. + VARIOUS ENGINES IN WHICH THE TEMPERATURE OF THE BOILER IS 140° C. AND + THAT OF THE CONDENSER 30° C. + _Theoretical Duty for each Unit of Heat transmitted, 440[73] + foot-pounds._ + ─────────────────┬─────────────┬──────────────┬─────────────┬────────── + CASES. │Work produced│Work produced │Work produced│Percentage + │for each lb. │ for each lb. │for each unit│ of + │ of coal │ of water │ of heat │theoretical + │ consumed. │ evaporated. │transmitted. │ duty. + ─────────────────┼─────────────┼──────────────┼─────────────┼────────── + │ Ft.-lbs. │ Ft.-lbs. │ Ft.-lbs. │ + (1) Fowey Consols│ │ │ │ + experiment,│ 1,330,734│ 156,556│ 253│ 57.5 + reported in│ │ │ │ + 1845 │ │ │ │ + (2) Taylor’s │ │ │ │ + engine at │ │ │ │ + the United │ 1,042,553│ 122,653│ 198.4│ 45.1 + Mines, │ │ │ │ + working in │ │ │ │ + 1840 │ │ │ │ + (3) French │ │ │ │ + engines, │ │ 98,427│ 159│ 36.1 + according │ │ │ │ + to contract│ │ │ │ + (4) English │ │ │ │ + engines, │ 565,700│ 80,814│ 130.8│ 29.7 + according │ │ │ │ + to contract│ │ │ │ + (5) Average │ │ │ │ + actual │ │ │ │ + performance│ 585,106│ 68,836│ 111.3│ 25.3 + of Cornish │ │ │ │ + engines │ │ │ │ + (6) Common │ │ │ │ + engines, │ │ │ │ + consuming │ │ │ │ + 12 lbs. of │ 165,000│ 23,571│ 38.1│ 8.6 + best coal │ │ │ │ + per │ │ │ │ + horse-power│ │ │ │ + per hour │ │ │ │ + (7) Improved │ │ │ │ + engines │ │ │ │ + with │ │ │ │ + expansion │ │ │ │ + cylinders, │ │ │ │ + consuming │ │ │ │ + an │ 495,000│ 70,710│ 114.4│ 26 + equivalent │ │ │ │ + to 4 lbs. │ │ │ │ + of best │ │ │ │ + coal per │ │ │ │ + horse-power│ │ │ │ + per hour │ │ │ │ + ─────────────────┴─────────────┴──────────────┴─────────────┴────────── + + TABLE B. + VARIOUS ENGINES IN WHICH THE TEMPERATURE OF THE BOILER IS 121° C.[74] + AND THAT OF THE CONDENSER 30° C. + _Theoretical Duty for each Unit of Heat transmitted, 371 foot-pounds._ + ─────────────────┬─────────────┬──────────────┬─────────────┬────────── + CASES. │Work produced│Work produced │Work produced│Percentage + │for each lb. │ for each lb. │for each unit│ of + │ of coal │ of water │ of heat │theoretical + │ consumed. │ evaporated. │transmitted. │ duty. + ─────────────────┼─────────────┼──────────────┼─────────────┼────────── + │ Ft.-lbs. │ Ft.-lbs. │ Ft.-lbs. │ + (3) French │ │ │ │ + engines, │ │ 98,427│ 160.3│ 43.2 + according │ │ │ │ + to contract│ │ │ │ + (4) English │ │ │ │ + engines, │ 565,700│⁶¹⁴⁄₆₁₈×80,814│ 130.8│ 35 + according │ │ │ │ + to contract│ │ │ │ + (6) Common │ │ │ │ + engines, │ │ │ │ + consuming │ │ │ │ + 12 lbs. of │ 165,000│⁶¹⁴⁄₆₁₈×23,571│ 38.1│ 10.3 + coal per │ │ │ │ + horse-power│ │ │ │ + per hour │ │ │ │ + (7) Improved │ │ │ │ + engines │ │ │ │ + with │ │ │ │ + expansion │ │ │ │ + cylinders, │ │ │ │ + consuming │ │ │ │ + an │ 495,000│⁶¹⁴⁄₆₁₈×70,710│ 114.4│ 30.7 + equivalent │ │ │ │ + to 4 lbs. │ │ │ │ + best coal │ │ │ │ + per │ │ │ │ + horse-power│ │ │ │ + per hour │ │ │ │ + ─────────────────┴─────────────┴──────────────┴─────────────┴────────── + + + + + APPENDIX A. + EXTRACTS FROM UNPUBLISHED WRITINGS OF CARNOT. + + + I. NOTES. + +Let us first open at the memoranda relating to his daily occupations: + + +“Plan in the morning the work of the day, and reflect in the evening on +what has been done.” + +“Carry when walking a book, and a note-book to preserve the ideas, and a +piece of bread in order to prolong the walk if need be.” + +“Vary the mental and bodily exercises with dancing, horsemanship, +swimming, fencing with sword and with sabre, shooting with gun and +pistol, skating, the sling, stilts, tennis, bowls; hop on one foot, +cross the arms, jump high and far, turn on one foot propped against the +wall, exercise in shirt in the evening to get up a perspiration before +going to bed; turning, joinery, gardening, reading while walking, +declamation, singing, violin, versification, musical composition; eight +hours of sleep; a walk on awakening, before and after eating; great +sobriety; eat slowly, little, and often; avoid idleness and useless +meditation.” + + +Then come more general precepts: + + +“Adopt good habits when I change my method of life.” + +“Never turn to the past unless to enlighten the future. Regrets are +useless.” + +“Form resolutions in advance in order not to reflect during action. Then +obey thyself blindly.” + +“The promptitude of resolutions most frequently accords with their +justice.” + +“Yield frequently to the first inspiration. Too much meditation on the +same subject ends by suggesting the worst part, or at least causes loss +of precious time.” + +“Suffer slight disagreeables without seeming to perceive them, but +repulse decisively any one who evidently intends to injure or humiliate +you.” + +“One should never feign a character that he has not, or affect a +character that he cannot sustain.” + +“Self-possession without self-sufficiency. Courage without effrontery.” + +“Make intimate acquaintances only with much circumspection; perfect +confidence in those who have been thoroughly tested. Nothing to do with +others.” + +“Question thyself to learn what will please others.” + +“No useless discourse. All conversation which does not serve to +enlighten ourselves or others, to interest the heart or amuse the mind, +is hurtful.” + +“Speak little of what you know, and not at all of what you do not know.” + +“Why not say more frequently, ‘I do not know’?” + +“Speak to every one of that which he knows best. This will put him at +his ease, and be profitable to you.” + +“Abstain from all pleasantry which could wound.” + +“Employ only expressions of the most perfect propriety.” + +“Listen attentively to your interlocutor, and so prepare him to listen +in the same way to your reply, and predispose him in favor of your +arguments.” + +“Show neither passion nor weariness in discussion. + +“Never direct an argument against any one. If you know some particulars +against your adversary, you have a right to make him aware of it to keep +him under control, but proceed with discretion, and do not wound him +before others.” + +“When discussion degenerates into dispute, be silent; this is not to +declare yourself beaten.” + +“How much modesty adds to merit! A man of talent who conceals his +knowledge is like a branch bending under a weight of fruit.” + +“Why try to be witty? I would rather be thought stupid and modest than +witty and pretentious.” + +“Men desire nothing so much as to make themselves envied.” + +“Egotism is the most common and most hated of all vices. Properly +speaking, it is the only one which should be hated.” + +“The pleasures of self-love are the only ones that can really be turned +into ridicule.” + +“I do not know why these two expressions, good sense and common sense, +are confounded. There is nothing less common than good sense.” + +“The strain of suffering causes the mind to decay.” + + +We will quote one of those misanthropic sallies the rarity of which we +are glad to remark: + + +“It must be that all honest people are in the galleys; only knaves are +to be met with elsewhere.” + + +But serenity of mind returns immediately after the above: + + +“I rejoice for all the misfortunes which might have happened to me, and +which I have escaped.” + +“Life is a short enough passage. I am half the journey. I will complete +the remainder as I can.” + +“Hope being the greatest of all blessings, it is necessary, in order to +be happy, to sacrifice the present to the future.” + +“Let us not be exacting; perfection is so rare.” + +“Indulgence! Indulgence!” + +“The more nearly an object approaches perfection, the more we notice its +slightest defects.” + +“To neglect the opportunity of an innocent pleasure is a loss to +ourselves. It is to act like a spendthrift.” + +“_Recherché_ pleasures cause simple pleasures to lose all their +attractions.” + +“It may sometimes be necessary to yield the right, but how is one to +recover it when wanted?” + +“Love is almost the only passion that the good man may avow. It is the +only one which accords with delicacy.” + +“Do nothing that all the world may not know.” + +“The truly wise man is he who loves virtue for its own sake.” + +“We say that man is an egotist, and nevertheless his sweetest pleasures +come to him through others. He only tastes them on condition of sharing +them.” + +“If one could continually satisfy his desires, he would never have time +to desire. Happiness then is necessarily composed of alternatives. It +could not exist at a constant level.” + + +On the subject of nations and conquerors: + + +“To each conqueror can be said, when he has ceased tormenting our poor +globe, ‘Would you not have been able to tilt equally well against a +little globe of pasteboard?’” + +“The laws of war, do they say? As if war were not the destruction of all +laws.” + +“War has been represented as necessary to prevent the too rapid increase +of the population, but war mows down the flower of the young men, while +it spares the men disgraced by nature. Hence it tends to the +degeneration of the species.” + + +Then the writer turns his shafts against medicine: + + +“In some respects medicine is directly opposed to the will of nature, +which tends to perpetuate the strongest and best of the species, and to +abandon the delicate to a thousand forms of destruction. This is what +occurs among animals and savage men. Only the most robust attain the +adult age, and these only reproduce the species. Medicine and the aids +of the social state prolong the lives of feeble creatures whose +posterity is usually equally feeble. Among the Spartans, barbarous +regulations put an end to the existence of malformed infants, that the +strength and beauty of the race might be preserved. Such regulations are +antipathetic to our customs; nevertheless it might be desirable that we +should devote ourselves to the preservation of the human race from the +causes of weakness and degeneracy.” + +“The decadence of the Greeks and Romans without change of race proves +the influence of institutions upon customs.” + + +We will give here a fragment on political economy, to show the variety +contained in the pages on which we draw: + + +“According to the system of modern economists, it would be desirable +that the government should interfere as little as possible in the +commerce and industry of the country. Nevertheless we cannot deny that +in certain circumstances this intervention is very useful.” + +“Taxes are regarded by economists as an evil, but as a necessary evil, +since they provide for public expenses. Consequently, economists think +that if the government possessed sufficient revenues, in domains for +example, the suppression of all taxes would be a desirable measure.” + +“Taxes are a means of influencing production and commerce to give to +them a direction which they would not naturally have taken. Such an +influence may undoubtedly have disagreeable consequences if the taxes +are imposed without discrimination or exclusively for a fiscal purpose, +but it is entirely otherwise if wisdom and tact preside at their +institution.” + +“A tax on the rent of a farm would be much better than a tax on the land +itself. Proprietors then could only avoid taxes by themselves improving +their property. As it is, they merely collect the rents, and usually +employ their surplus in unproductive expenditure, while the proprietary +farmers voluntarily devote theirs to the improvement of the land.” + +“A tax on the farms would then result in the proprietors themselves +working the lands, and this would mean better cultivation, and +improvements which would yield returns indeed, but at too remote a +period for the tenant. It would tend to a division of landed property, +men of small fortune uniting in the purchase with capitalists who seek +only the rent or payment for the land.” + +“Great capitalists could not themselves cultivate vast extents of land, +and not wanting to diminish their revenues by renting them, would be +induced to sell portions suitable for cultivation by their new owners, +and would then carry their money into new industrial and commercial +enterprises.” + +“The competition of the sellers would cause a momentary fall in the +price of the lands, and would enable small farmers to become +land-owners. The number of vast estates often badly managed would then +be diminished, and considerable fortunes, changing hands more easily, +would naturally pass into those which would be most likely to increase +their value.” + +“Proprietors, becoming cultivators to escape the taxes, would settle in +the country, where their presence would disseminate intelligence and +comfort; their revenues, before spent unprofitably, would then pay +expenses and improvements on their property.” + +“The establishment of such a tax would certainly find many opponents +among proprietors, landed non-cultivators who form in fact the +influential _personnel_ in the state, for it is they who usually make +the laws.” + +“Perhaps it would be necessary to weaken their opposition by not +subjecting the actual proprietors to the new tax, which might take +effect only with the next change either by sale or by inheritance. A +restriction of the right of transfer would also facilitate the passage +from one situation to the other. All changes in taxes should, as a +general thing, be made gradually, in order to avoid sudden changes of +fortune.” + +“We may consider the renting of a property for several years as a sale +of the usufruct during the time of the lease. Now nine years’ +possession, for example, is equal to more than a third of the value of +the property, supposing the annual product to be one twentieth of the +capital. It would then be reasonable to apply to this sort of sale the +laws which govern that of landed property, and consequently the mutation +tax. The person who cannot or will not cultivate his soil, instead of +alienating the property itself, binds himself to alienate the usufruct +for a time, and the price is paid at stated intervals instead of all at +once. There is farm rent.” + +“Now it is by a fiction that the purchaser pays the mutation tax. In +fact, it is always the seller who pays it. The buyer compares the money +that he spends with the advantage that he gains, and this comparison +determines it. If he did not make money out of it he would not buy it. +When the registration tax did not exist, the purchaser had to pay the +same sum for the same purpose, and this sum went into the pocket of the +seller.” + +“Proprietors of lands, then, after all, have to bear the mutation taxes. +All increase of these taxes is a loss for them, and these taxes are +heavier on the small proprietors than on the large, because their +changes are more frequent. The tax on the farms, on the contrary, would +bear more heavily on large estates.” + +“The tax on farms not affecting the owners of timber, would be made up +by a tax on the felling, a very justifiable tax, for standing timber is +landed property. Standing timber is often worth much more than the land +on which it stands.” + + +Finally, we will give some thoughts which reveal the religious +sentiments of Sadi Carnot: + + +“Men attribute to chance those events of the causes of which they are +ignorant. If they succeed in divining these causes, chance disappears. +To say that a thing has happened by chance, is to say that we have not +been able to foresee it. I do not myself believe that any other +acceptation can be given to this word. What to an ignorant man is +chance, cannot be chance to one better instructed.” + +“If human reason is incapable of discovering the mysteries of Divinity, +why has not Divinity made human reason more clear-sighted?” + +“God cannot punish man for not believing when he could so easily have +enlightened and convinced him.” + +“If God is absolutely good, why should He punish the sinner for all +eternity, since He does not lead him to good, or give him an example?” + +“According to the doctrine of the church, God resembles a sphinx +proposing enigmas, and devouring those who cannot guess them.” + +“The church attributes to God all human passions—anger, desire for +vengeance, curiosity, tyranny, partiality, idleness.” + +“If Christianity were pruned of all which is not Christ, this religion +would be the simplest in the world.” + +“What motives have influenced the writers who have rejected all +religious systems? Is it the conviction that the ideas which they oppose +are all injurious to society? Have they not rather included in the same +proscription religion and the abuse of it?” + +“The belief in an _all-powerful_ Being, who loves us and watches over +us, gives to the mind great strength to endure misfortune.” + +“A religion suited to the soul and preached by men worthy of respect +would exercise the most salutary influence upon society and customs.” + + + II. NOTES OF SADI CARNOT ON MATHEMATICS, PHYSICS, AND OTHER SUBJECTS. + +Up to the present time the changes caused in the temperature of bodies +by motion have been very little studied. This class of phenomena merits, +however, the attention of observers. When bodies are in motion, +especially when that motion disappears, or when it produces motive +power, remarkable changes take place in the distribution of heat, and +perhaps in its quantity. + +We will collect a few facts which exhibit this phenomenon most clearly. + +1. _The Collision of Bodies._—We know that in the collision of bodies +there is always expenditure of motive power. Perfectly elastic bodies +only form an exception, and none such are found in nature. + +We also know that always in the collision of bodies there occurs a +change of temperature, an elevation of temperature. We cannot, as did M. +Berthollet, attribute the heat set free in this case to the reduction of +the volume of the body; for when this reduction has reached its limit +the liberation of heat would cease. Now this does not occur. + +It is sufficient that the body change form by percussion, without change +of volume, to produce disengagement of heat. + +If, for example, we take a cube of lead and strike it successively on +each of its faces, there will always be heat liberated, without sensible +diminution in this disengagement, so long as the blows are continued +with equal force. This does not occur when medals are struck. In this +case the metal cannot change form after the first blows of the die, and +the effect of the collision is not conveyed to the medal, but to the +threads of the screw which are strained, and to its supports. + +It would seem, then, that heat set free should be attributed to the +friction of the molecules of the metal, which change place relatively to +each other, that is, the heat is set free just where the moving force is +expended. + +A similar remark will apply in regard to the collision of two bodies of +differing hardness—lead and iron for instance. The first of these metals +becomes very hot, while the second does not vary sensibly in +temperature. But the motive power is almost wholly exhausted in changing +the form of the first of these metals. We may also cite, as a fact of +the same nature, the heat produced by the extension of a metallic rod +just before it breaks. Experiment has proved that, other things being +equal, the greater the elongation before rupture, the more considerable +is the elevation of temperature. + + +(2) [The remainder is blank.] + + +When a hypothesis no longer suffices to explain phenomena, it should be +abandoned. + +This is the case with the hypothesis which regards caloric as matter, as +a subtile fluid. + +The experimental facts tending to destroy this theory are as follows: + +(1) The development of heat by percussion or the friction of bodies +(experiments of Rumford, friction of wheels on their spindles, on the +axles, experiments to be made). Here the elevation of temperature takes +place at the same time in the body rubbing and the body rubbed. +Moreover, they do not change perceptibly in form or nature (to be +proved). Thus heat is produced by motion. If it is matter, it must be +admitted that the matter is created by motion. + +(2) When an air-pump is worked, and at the same time air is admitted +into the receiver, the temperature remains constant in the receiver. It +remains constant on the outside. Consequently, the air compressed by the +pumps must rise in temperature above the air outside, and it is expelled +at a higher temperature. The air enters then at a temperature of 10°, +for instance, and leaves at another, 10° + 90° or 100°, for example. +Thus heat has been created by motion. + +(3) If the air in a reservoir is compressed, and at the same time +allowed to escape through a little opening, there is by the compression +elevation of temperature, by the escape lowering of temperature +(according to Gay-Lussac and Welter). The air then enters at one side at +one temperature and escapes at the other side at a higher temperature, +from which follows the same conclusion as in the preceding case. + +(Experiment to be made: To fit to a high-pressure boiler a cock and a +tube leading to it and emptying into the atmosphere; to open the cock a +little way, and present a thermometer to the outlet of the steam; to see +if it remains at 100° or more; to see if steam is liquefied in the pipe; +to see whether it comes out cloudy or transparent.) + +(4) The elevation of temperature which takes place at the time of the +entrance of the air into the vacuum, an elevation that cannot be +attributed to the compression of the air remaining (air which may be +replaced by steam), can therefore be attributed only to the friction of +the air against the walls of the opening, or against the interior of the +receiver, or against itself. + +(5) M. Gay-Lussac showed (it is said) that if two receivers were put in +communication with each other, the one a vacuum, the other full of air, +the temperature would rise in one as much as it would fall in the other. +If, then, both be compressed one half, the first would return to its +previous temperature and the second to a much higher one. Mixing them, +the whole mass would be heated. + +When the air enters a vacuum, its passage through one small opening and +the motion imparted to it in the interior appear to produce elevation of +temperature. + + +We may be allowed to express here an hypothesis in regard to the nature +of heat. + +At present, light is generally regarded as the result of a vibratory +movement of the ethereal fluid. Light produces heat, or at least +accompanies the radiating heat, and moves with the same velocity as +heat. Radiating heat is then a vibratory movement. It would be +ridiculous to suppose that it is an emission of matter while the light +which accompanies it could be only a movement. + +Could a motion (that of radiating heat) produce matter (caloric)? + +No, undoubtedly; it can only produce a motion. Heat is then the result +of a motion. + +Then it is plain that it could be produced by the consumption of motive +power, and that it could produce this power. + +All the other phenomena—composition and decomposition of bodies, passage +to the gaseous state, specific heat, equilibrium of heat, its more or +less easy transmission, its constancy in experiments with the +calorimeter—could be explained by this hypothesis. But it would be +difficult to explain why, in the development of motive power by heat, a +cold body is necessary; why, in consuming the heat of a warm body, +motion cannot be produced. + + +It appears very difficult to penetrate into the real essence of bodies. +To avoid erroneous reasoning, it would be necessary to investigate +carefully the source of our knowledge in regard to the nature of bodies, +their form, their forces; to see what the primitive notions are, to see +from what impressions they are derived; to see how one is raised +successively to the different degrees of abstraction. + + +Is heat the result of a vibratory motion of molecules? If this is so, +quantity of heat is simply quantity of motive power. As long as motive +power is employed to produce vibratory movements, the quantity of heat +must be unchangeable; which seems to follow from experiments with the +calorimeter; but when it passes into movements of sensible extent, the +quantity of heat can no longer remain constant. + + +Can examples be found of the production of motive power with actual +consumption of heat? It seems that we may find production of heat with +consumption of motive power (re-entrance of the air into a vacuum, for +example). + + +What is the cause of the production of heat in combinations of +substances? What is radiant caloric? + + +Liquefaction of bodies, solidification of liquids, crystallization—are +they not forms of combinations of integrant molecules? + + +Supposing heat due to a vibratory movement, how can the passage from the +solid or the liquid to the gaseous state be explained? + + +When motive power is produced by the passage of heat from the body _A_ +to the body _B_, is the quantity of this heat which arrives at _B_ (if +it is not the same as that which has been taken from _A_, if a portion +has really been consumed to produce motive power) the same whatever may +be the substance employed to realize the motive power? + +Is there any way of using more heat in the production of motive power, +and of causing less to reach the body _B_? Could we even utilize it +entirely, allowing none to go to the body _B_? If this were possible, +motive power could be created without consumption of combustible, and by +mere destruction of the heat of bodies. + + +Is it absolutely certain that steam after having operated an engine and +produced motive power can raise the temperature of the water of +condensation as if it had been conducted directly into it? + + +Reasoning shows us that there cannot be loss of living force, or, which +is the same thing, of motive power, if the bodies act upon each other +without directly touching each other, without actual collision. Now +everything leads us to believe that the molecules of bodies are always +separated from each other by some space, that they are never actually in +contact. If they touched each other, they would remain united, and +consequently change form. + + +If the molecules of bodies are never in close contact with each other +whatever may be the forces which separate or attract them, there can +never be either production or loss of motive power in nature. This power +must be as unchangeable in quantity as matter. Then the direct +re-establishment of equilibrium of the caloric, and its re-establishment +with production of motive power, would be essentially different from +each other. + + +Heat is simply motive power, or rather motion which has changed form. It +is a movement among the particles of bodies. Wherever there is +destruction of motive power there is, at the same time, production of +heat in quantity exactly proportional to the quantity of motive power +destroyed. Reciprocally, wherever there is destruction of heat, there is +production of motive power. + +We can then establish the general proposition that motive power is, in +quantity, invariable in nature; that it is, correctly speaking, never +either produced or destroyed. It is true that it changes form, that is, +it produces sometimes one sort of motion, sometimes another, but it is +never annihilated. + + +According to some ideas that I have formed on the theory of heat, the +production of a unit of motive power necessitates the destruction of +2.70 units of heats. + +A machine which would produce 20 units of motive power per kilogram of +coal ought to destroy (20 × 2.70)/(7000) of the heat developed by the +combustion. (20 × 2.70)/(7000) = (8)/(1000) about; that is, less than +(1)/(100). + +(Each unit of motive power, or dyname, representing the weight of one +cubic metre of water raised to the height of one metre.) + + + _Experiments to be made on Heat and Motive Power._ + +To repeat Rumford’s experiments in the drilling of a metal in water, but +to measure the motive power consumed at the same time as the heat +produced; same experiments on several metals and on wood. + + +To strike a piece of lead in various ways, to measure the motive power +consumed and the heat produced. Same experiments on other metals. + + +To strongly agitate water in a small cask or in a double-acting pump +having a piston pierced with a small opening. + +Experiment of the same sort on the agitation of mercury, alcohol, air +and other gases. To measure the motive power consumed and heat produced. + + +To admit air into a vacuum or into air more or less rarefied; _id._ for +other gases or vapors. To examine the elevation of temperature by means +of the manometer and the thermometer of Bréguet. Estimation of the error +of the thermometer in the time required for the air to vary a certain +number of degrees. These experiments would serve to measure the changes +which take place in the temperature of the gas during its changes of +volume. They would also furnish means of comparing these changes with +the quantities of motive power produced or consumed. + + +Expel the air from a large reservoir in which it is compressed, and +check its velocity in a large pipe in which solid bodies have been +placed; measure the temperature when it has become uniform. See if it is +the same as in the reservoir. Same experiments with other gases and with +vapor formed under different pressures. + + +To repeat Dalton’s experiments and carry them on to pressures of thirty +or forty atmospheres. To measure the constituent heat of the vapor +within these limits. + +_Id._ on the vapor of alcohol, of ether, of essence of turpentine, of +mercury, to prove whether the agent employed makes any difference in the +production of motive power. + +_Id._ on water charged with a deliquescent salt, the calcium chloride, +for instance. + +Is the law of tensions always the same? To measure the specific heat of +vapor. + + + _Experiments to be made on the Tension of Vapors._ + +A graduated capillary tube filled with water, mercury, or with oil and +air. Plunge this tube into a bath of oil, of mercury, or of melted lead. +To measure the temperature by an air-thermometer. + +Same experiments with alcohol, ether, sulphide of carbon, muriatic +ether, essence of turpentine, sulphur, phosphorus. + +Experiments on the tension of steam with a boiler, and a thermometric +tube full of air. A thermometer will be placed in a tube immersed in the +boiler, open outwards and filled with oil or mercury. + + +Experiments by means of a simple capillary tube filled with three +successive parts—first of air, second of mercury, third of water or +other liquid of which the tension can be measured (of alcohol, of ether, +of essence of turpentine, of lavender, of sulphide of carbon, of +muriatic ether, etc.). One end of the tube may be immersed in a bath of +mercury or oil, the temperature of which is to be measured. The column +of mercury can be made long enough to allow of the air being previously +compressed or rarefied. + +[Illustration: + + FIG. 6. +] + +The tube will be bent into a spiral at one end, the straight part being +graduated (thus permitting the tension of mercurial vapor to be +measured). + + +[Illustration: + + FIG. 7. +] + +Experiments on the tension of vapors at low temperature, with a +thermometric tube bent round, and filled partly with mercury, partly +with water or alcohol. The mercury will operate by its weight. The upper +part of the tube will be empty and sealed, or fully open to the +atmosphere. + +The bulb will be immersed in water the temperature of which is to be +measured. If the tube is sealed, the upper part must be cooled. + +The bulb might contain water, ether, or essence of turpentine. + +If the tube is sealed, the tension of mercurial vapor could be measured. + + +Experiments on the constituent heat of vapors by means of a barometric +tube having two enlarged bulbs. One of the bulbs may be immersed in cold +water, and the elevation of temperature of this water will indicate the +constituent heat of the vapor. + +[Illustration: + + FIG. 8. +] + +The other bulb may be warmed either by boiling liquid or by fire. + +Water, alcohol, steam, ether, mercury, acetic acid, sulphide of carbon. + +The operation may be repeated and add the results. + + + _Experiments to be made on Gases and Vapors._ + +To measure the temperature acquired by the air introduced into a vacuum +or space containing previously rarefied air. + +[Illustration: + + FIG. 9. +] + +If the vacuum is made under the glass receiver of an air-pump, and the +cock admitting the outer air be suddenly opened, the introduction of +this air will cause a Bréguet thermometer to rise to 50° or 60°. To +examine the movement of this thermometer when the reintroduction takes +place only by degrees, to compare it with the movement of the manometer. + +Construction of a manometer which may give the pressure almost +instantaneously. + +Imagine a capillary tube bent into a spiral at one end, and having one +extremity closed, the other open. This tube will be perfectly dry and a +small index of mercury may be introduced into it. The diameter of the +tube will be small enough for the air enclosed in it to take almost +instantly the temperature of the glass. We shall try to ascertain the +time necessary for the establishment of this equilibrium of temperature +by placing the tube under the receiver of the air-pump, making a partial +vacuum, and admitting the air. We shall see whether, some seconds after +the introduction, the index perceptibly moves. The index must be of very +light weight to avoid oscillation as much as possible. + +For the same reason, the capillary tube should be also as narrow as +possible. If the straight part of the tube is equal to the bent part and +the index be placed at the beginning of the bent part, for a pressure +equal to atmospheric pressure, it would not be necessary to subject the +instrument to a less pressure than ½ atmosphere. It is between these two +limits that it would serve as a measure. + +It might end in an open enlargement to prevent the projection of the +mercury outside the tube. Disposed in this way, it could be used as a +general measure for pressures between _p_ and (½)_p_; _p_ being anything +whatever. The apparatus will be fastened to a board bearing a graduated +scale placed against the straight tube. The scale will be, for instance, +numbered by fives or tens. A corresponding table denoting pressures +would be required. + +Placing the instrument under the receiver and forming a partial vacuum, +the index will rise into the enlargement. Then, admitting the air by +degrees and very slowly, we may note the correspondence between the +heights of the ordinary mercury manometer and the point which will be +reached by the lower face of the index of the instrument. This will +answer to form a comparative table of the pressures and the numbers of +the scale. The pressures would be represented by their relations to the +observed pressure at the moment of the passage of the index over zero, +for any other fixed number of the scale. + +Thus, for example, suppose that we observed on the manometer 400 or _n_ +millimetres of mercury when the index is on _o_, then _n′_ when the +index is on 1, _n″_ when on 2, and so on. This will give the ratios +_n′_/_n_, _n″_/_n_, ... which must be inscribed in the table. Then _n_ +could be varied at pleasure, and the table could still be used. + +In fact, according to the law of Mariotte, volumes preserving the same +ratios, pressures should also preserve the same ratios to each other. + +Let _p_ be the pressure when the index is on _o_, _v_ the volume of air +at the same moment, _p′_ and _v′_ the same pressures and volume at the +moment when the index is on 1. Whether the air be expelled or admitted +the pressures would be instead of _p_ and _p′_, _q_ and _q′_. But there +would follow + + _p_ : _p′_ :: _v′_ : _v_ and _q_ : _q′_ :: _v′_ : _v_; + then _p_ : _p′_ :: _q_ : _q′_. + +We should moreover work at a uniform temperature and note the +variations. + +If the straight part of the tube were perfectly calibrated, the volumes, +and consequently the pressures, would form a geometrical progression, +when the figures of the scale would be found to be in arithmetical +progression, and a table of logarithms would enable one to be found from +the other. + +In order to increase as required the mass of air enclosed in the tube +the instrument must be placed on its side or flat, in the air-pump +receivers. The mercury index would be placed in the lateral part of the +enlargement of the tube, and the atmospheric air would enter. The +instrument might also be heated in this position. + +Care must be taken to admit only very dry air, which could be obtained +by placing under the receiver calcium chloride or any other substance +which absorbs moisture greedily. + +Instead of bending the tube into a spiral, it might be bent in the +middle in the form of a ᑌ, or it might be better to form three, four or +more parallel branches. Making the tube very long, the index would have +a larger range for the same changes of pressure, and the results +produced could then be measured by a slight variation in density in the +air of the receiver. + + +_Comparison of the Rapidity with which the Air cools in the Receiver and + in the Tube._ + +Let us suppose, what I believe to be very near the truth, that the heat +absorbed is proportional to the surface of the bodies in contact. From +this we can infer without difficulty, that the rapidity of the cooling +of the air in two cylindrical tubes would be inversely as their +diameters. + +If the receiver is considered as a tube of two decimetres in diameter, +and the manometer as a tube of one millimetre diameter, the rapidity of +the cooling of the air would be in the ratio, very nearly, of 1 to 200. + + + _Extent of the Movement of the Index._ + +Suppose the tube turned up on itself five times and having a total +length of 1 metre; a variation of density equal to ⅒ in the air will +give a movement of 1 decimetre; a variation of heat of 1 degree supposed +to be equivalent to a variation of density of ¹⁄₂₆₆ will give ¹⁄₂₆₆ of a +metre, or about 3^{mm}.70, quite an appreciable quantity. As to the time +required to move the mercury index, regard being had to its mass, if we +suppose it 1 centimetre long, and the variation of pressure ¹⁄₁₀₀ of an +atmosphere, it would require about ⅙ of a second to make it pass over +one decimetre. + + _Use of the Instrument in Measuring the Variations of the Tensions of + the Air under a Pneumatic Receiver._ + +At each stroke of the piston which expands the air under the pneumatic +receiver when a vacuum is to be created, a lowering of pressure is +produced, and undoubtedly a change of temperature. It can be determined +approximately, at least, by observing the position of the manometer at +the instant after the dilatation has taken place, and again after a time +long enough for the temperature to have returned to its original point, +that of the surrounding bodies. Comparison of the elastic force in the +two cases will lead to comparison of the temperatures. + +The temperature having returned to its original point, we will give a +second stroke of the piston which will rarefy the air more than the +former, and thus we will make two observations of the manometer, before +and after the return to the former temperature. And so on. + + + + + APPENDIX B. + CARNOT’S FOOT-NOTES. + + +NOTE A.—The objection may perhaps be raised here, that perpetual motion, +demonstrated to be impossible by mechanical action alone, may possibly +not be so if the power either of heat or electricity be exerted; but is +it possible to conceive the phenomena of heat and electricity as due to +anything else than some kind of motion of the body, and as such should +they not be subjected to the general laws of mechanics? Do we not know +besides, _à posteriori_, that all the attempts made to produce perpetual +motion by any means whatever have been fruitless?—that we have never +succeeded in producing a motion veritably perpetual, that is, a motion +which will continue forever without alteration in the bodies set to work +to accomplish it? The electromotor apparatus (the pile of Volta) has +sometimes been regarded as capable of producing perpetual motion; +attempts have been made to realize this idea by constructing dry piles +said to be unchangeable; but however it has been done, the apparatus has +always exhibited sensible deteriorations when its action has been +sustained for a time with any energy. + +The general and philosophic acceptation of the words _perpetual motion_ +should include not only a motion susceptible of indefinitely continuing +itself after a first impulse received, but the action of an apparatus, +of any construction whatever, capable of creating motive power in +unlimited quantity, capable of starting from rest all the bodies of +nature if they should be found in that condition, of overcoming their +inertia; capable, finally, of finding in itself the forces necessary to +move the whole universe, to prolong, to accelerate incessantly, its +motion. Such would be a veritable creation of motive power. If this were +a possibility, it would be useless to seek in currents of air and water +or in combustibles this motive power. We should have at our disposal an +inexhaustible source upon which we could draw at will. + +NOTE B.—The experimental facts which best prove the change of +temperature of gases by compression or dilatation are the following: + +(1) The fall of the thermometer placed under the receiver of a pneumatic +machine in which a vacuum has been produced. This fall is very sensible +on the Bréguet thermometer: it may exceed 40° or 50°. The mist which +forms in this case seems to be due to the condensation of the watery +vapor caused by the cooling of the air. + +(2) The inflammation of German tinder in the so-called pneumatic +tinder-boxes; which are, as we know, little pump-chambers in which the +air is rapidly compressed. + +(3) The fall of a thermometer placed in a space where the air has been +first compressed and then allowed to escape by the opening of a cock. + +(4) The results of experiments on the velocity of sound. M. de Laplace +has shown that, in order to secure results accurately by theory and +computation, it is necessary to assume the heating of the air by sudden +compression. + +The only fact which may be adduced in opposition to the above is an +experiment of MM. Gay-Lussac and Welter, described in the _Annales de +Chimie et de Physique_. A small opening having been made in a large +reservoir of compressed air, and the ball of a thermometer having been +introduced into the current of air which passes out through this +opening, no sensible fall of the temperature denoted by the thermometer +has been observed. + +Two explanations of this fact may be given: (1) The striking of the air +against the walls of the opening by which it escapes may develop heat in +observable quantity. (2) The air which has just touched the bowl of the +thermometer possibly takes again by its collision with this bowl, or +rather by the effect of the _détour_ which it is forced to make by its +rencounter, a density equal to that which it had in the receiver,—much +as the water of a current rises against a fixed obstacle, above its +level. + +The change of temperature occasioned in the gas by the change of volume +may be regarded as one of the most important facts of Physics, because +of the numerous consequences which it entails, and at the same time as +one of the most difficult to illustrate, and to measure by decisive +experiments. It seems to present in some respects singular anomalies. + +Is it not to the cooling of the air by dilatation that the cold of the +higher regions of the atmosphere must be attributed? The reasons given +heretofore as an explanation of this cold are entirely insufficient; it +has been said that the air of the elevated regions receiving little +reflected heat from the earth, and radiating towards celestial space, +would lose caloric, and that this is the cause of its cooling; but this +explanation is refuted by the fact that, at an equal height, cold reigns +with equal and even more intensity on the elevated plains than on the +summit of the mountains, or in those portions of the atmosphere distant +from the sun. + +NOTE C.—We see no reason for admitting, _à priori_, the constancy of the +specific heat of bodies at different temperatures, that is, to admit +that equal quantities of heat will produce equal increments of +temperature, when this body changes neither its state nor its density; +when, for example, it is an elastic fluid enclosed in a fixed space. +Direct experiments on solid and liquid bodies have proved that between +zero and 100°, equal increments in the quantities of heat would produce +nearly equal increments of temperature. But the more recent experiments +of MM. Dulong and Petit (see _Annales de Chimie et de Physique_, +February, March, and April, 1818) have shown that this correspondence no +longer continues at temperatures much above 100°, whether these +temperatures be measured on the mercury thermometer or on the +air-thermometer. + +Not only do the specific heats not remain the same at different +temperatures, but, also, they do not preserve the same ratios among +themselves, so that no thermometric scale could establish the constancy +of all the specific heats. It would have been interesting to prove +whether the same irregularities exist for gaseous substances, but such +experiments presented almost insurmountable difficulties. + +The irregularities of specific heats of solid bodies might have been +attributed, it would seem, to the latent heat employed to produce a +beginning of fusion—a softening which occurs in most bodies long before +complete fusion. We might support this opinion by the following +statement: According to the experiments of MM. Dulong and Petit, the +increase of specific heat with the temperature is more rapid in solids +than in liquids, although the latter possess considerably more +dilatability. The cause of irregularity just referred to, if it is real, +would disappear entirely in gases. + +NOTE D.—In order to determine the arbitrary constants _A_, _B_, _A′_, +_B′_, in accordance with the results in M. Dalton’s table, we must begin +by computing the volume of the vapor as determined by its pressure and +temperature,—a result which is easily accomplished by reference to the +laws of Mariotte and Gay-Lussac, the weight of the vapor being fixed. + +The volume will be given by the equation + + _v_ = _c_ (267 + _t_)/(_p_), + +in which _v_ is this volume, _t_ the temperature, _p_ the pressure, and +_c_ a constant quantity depending on the weight of the vapor and on the +units chosen. We give here the table of the volumes occupied by a gramme +of vapor formed at different temperatures, and consequently under +different pressures. + + ───────────────────────┬───────────────────────┬─────────────────────── + _t_ │ _p_ │ _v_ + or degrees Centigrade. │or tension of the vapor│ or volume of a gramme + │ expressed in │ of vapor expressed in + │millimetres of mercury.│ litres. + ───────────────────────┼───────────────────────┼─────────────────────── + ° │ mm. │ lit. + 0│ 5.060 │ 185.0 + 20│ 17.32 │ 58.2 + 40│ 53.00 │ 20.4 + 60│ 144.6 │ 7.96 + 80│ 352.1 │ 3.47 + 100│ 760.0 │ 1.70 + ───────────────────────┴───────────────────────┴─────────────────────── + +The first two columns of this table are taken from the _Traité de +Physique_ of M. Biot (vol. i., p. 272 and 531). The third is calculated +by means of the above formula, and in accordance with the result of +experiment, indicating that water vaporized under atmospheric pressure +occupies a space 1700 times as great as in the liquid state. + +By using three numbers of the first column and three corresponding +numbers of the third column, we can easily determine the constants of +our equation + + _t_ = (_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_). + +We will not enter into the details of the calculation necessary to +determine these quantities. It is sufficient to say that the following +values, + + _A_ = 2268, _A′_ = 19.64, + _B_ = −1000, _B′_ = 3.30, + +satisfy fairly well the prescribed conditions, so that the equation + + _t_ = (2268 − 1000 log _v_)/(19.64 + 3.30 log _v_) + +expresses very nearly the relation which exists between the volume of +the vapor and its temperature. We may remark here that the quantity _B′_ +is positive and very small, which tends to confirm this proposition—that +the specific heat of an elastic fluid increases with the volume, but +follows a slow progression. + +NOTE E.—Were we to admit the constancy of the specific heat of a gas +when its volume does not change, but when its temperature varies, +analysis would show a relation between the motive power and the +thermometric degree. We will show how this is, and this will also give +us occasion to show how some of the propositions established above +should be expressed in algebraic language. + +Let _r_ be the quantity of motive power produced by the expansion of a +given quantity of air passing from the volume of one litre to the volume +of _v_ litres under constant temperature. If _v_ increases by the +infinitely small quantity _dv_, _r_ will increase by the quantity _dr_, +which, according to the nature of motive power, will be equal to the +increase _dv_ of volume multiplied by the expansive force which the +elastic fluid then possesses; let _p_ be this expansive force. We should +have the equation + + _dr_ = _pdv_. (1) + +Let us suppose the constant temperature under which the dilatation takes +place equal to _t_ degrees Centigrade. If we call _q_ the elastic force +of the air occupying the volume 1 litre at the same temperature _t_, we +shall have, according to the law of Mariotte, + + (_v_)/(1) = (_q_)/(_p_), whence _p_ = (_q_)/(_v_). + +If now _P_ is the elastic force of this same air at the constant volume +1, but at the temperature zero, we shall have, according to the rule of +M. Gay-Lussac, + + _q_ = _P_ + _P_ (_t_)/(267) = (_P_)/(267)(267 + _t_); + +whence + + _q_ = _p_ = (_P_)/(267) (267 + _t_)/(_v_). + +If, to abridge, we call _N_ the quantity (_P_)/(267), the equation would +become + + _p_ = _N_ (_t_ + 267)/(_v_); + +whence we deduce, according to equation (1), + + _dr_ = _N_ (_t_ + 267)/(_v_)_dv_. + +Regarding _t_ as constant, and taking the integral of the two numbers, +we shall have + + _r_ = _N_(_t_ + 267) log _v_ + _C_. + +If we suppose _r_ = 0 when _v_ = 1, we shall have _C_ = 0; whence + + _r_ = _N_(_t_ + 267) log _v_. (2) + +This is the motive power produced by the expansion of the air which, +under the temperature _t_, has passed from the volume 1 to the volume +_v_. If instead of working at the temperature _t_ we work in precisely +the same manner at the temperature _t_ + _dt_, the power developed will +be + + _r_ + δ_r_ = _N_(_t_ + _dt_ + 267) log _v_. + +Subtracting equation (2), we have + + δ_r_ = _N_ log _vdt_. (3) + +Let _e_ be the quantity of heat employed to maintain the temperature of +the gas constant during its dilatation. According to the reasoning of +page 69, δ_r_ will be the power developed by the fall of the quantity +_e_ of heat from the degree _t_ + _td_ to the degree _t_. If we call _u_ +the motive power developed by the fall of unity of heat from the degree +_t_ to the degree zero, as, according to the general principle +established page 68, this quantity _u_ ought to depend solely on _t_, it +could be represented by the function _Ft_, whence _u_ = _Ft_. + +When _t_ is increased it becomes _t_ + _td_, _u_ becomes _u_ + _du_; +whence + + _u_ + _du_ = _F_(_t_ + _dt_). + +Subtracting the preceding equation, we have + + _du_ = _F_(_t_ + _dt_) − _Ft_ = _F′tdt_. + +This is evidently the quantity of motive power produced by the fall of +unity of heat from the temperature _t_ + _dt_ to the temperature _t_. + +If the quantity of heat instead of being a unit had been _e_, its motive +power produced would have had for its value + + _edu_ = _eF′tdt_. (4) + +But _edu_ is the same thing as δ_r_; both are the power developed by the +fall of the quantity _e_ of heat from the temperature _t_ + _dt_ to the +temperature _t_; consequently, + + _edu_ = δ_r_, + +and from equations (3), (4), + + _eF′tdt_ = _N_ log _vdt_; + +or, dividing by _F′tdt_, + + _e_ = (_N_)/(_F′t_) log _v_ = _T_ log _v_. + +Calling _T_ the fraction (_N_)/(_F′t_) which is a function of _t_ only, +the equation + + _e_ = _T_ log _v_ + +is the analytical expression of the law stated pp. 80, 81. It is common +to all gases, since the laws of which we have made use are common to +all. + +If we call _s_ the quantity of heat necessary to change the air that we +have employed from the volume 1 and from the temperature zero to the +volume _v_ and to the temperature _t_, the difference between _s_ and +_e_ will be the quantity of heat required to bring the air at the volume +1 from zero to _t_. This quantity depends on _t_ alone; we will call it +_U_. It will be any function whatever of _t_. We shall have + + _s_ = _e_ + _U_ = _T_ log _v_ + _U_. + +If we differentiate this equation with relation to _t_ alone, and if we +represent it by _T′_ and _U′_, the differential coefficients of _T_ and +_U_, we shall get + + (_ds_)/(_dt_) = _T′_ log _v_ + _U′_; (5) + +_ds_/_dt_ is simply the specific heat of the gas under constant volume, +and our equation (1) is the analytical expression of the law stated on +page 86. + +If we suppose the specific heat constant at all temperatures (hypothesis +discussed above, page 92), the quantity _ds_/_dt_ will be independent of +_t_; and in order to satisfy equation (5) for two particular values of +_v_, it will be necessary that _T′_ and _U′_ be independent of _t_; we +shall then have _T′_ = _C_, a constant quantity. Multiplying _T′_ and +_C_ by _dt_, and taking the integral of both, we find + + _T_ = _Ct_ + _C_{1}_; + +but as _T_ = _N_/_F′t_, we have + + _F′t_ = (_N_)/(_T_) = (_N_)/(_Ct_ + _C_{1}_). + +Multiplying both by _dt_ and integrating, we have + + _Ft_ = (_N_)/(_C_) log (_Ct_ + _C_{1}_) + _C_{2}_; + +or changing arbitrary constants, and remarking further that _Ft_ is 0 +when _t_ = 0°, + + _Ft_ = _A_ log (1 + (_t_)/(_B_)). (6) + +The nature of the function _Ft_ would be thus determined, and we would +thus be able to estimate the motive power developed by any fall of heat. +But this latter conclusion is founded on the hypothesis of the constancy +of the specific heat of a gas which does not change in volume—an +hypothesis which has not yet been sufficiently verified by experiment. +Until there is fresh proof, our equation (6) can be admitted only +throughout a limited portion of the thermometric scale. + +In equation (5), the first member represents, as we have remarked, the +specific heat of the air occupying the volume _v_. Experiment having +shown that this heat varies little in spite of the quite considerable +changes of volume, it is necessary that the coefficient _T′_ of log _v_ +should be a very small quantity. If we consider it nothing, and, after +having multiplied by _dt_ the equation + + _T′_ = 0, + +we take the integral of it, we find + + _T_ = _C_, constant quantity; + +but + + _T_ = _N_/_F′t_, + +whence + + _F′t_ = _N_/_T_ = _N_/_C_ = _A_; + +whence we deduce finally, by a second integration, + + _Ft_ = _At_ + _B_. + +As _Ft_ = 0 when _t_ = 0, _B_ is 0; thus + + _Ft_ = _At_; + +that is, the motive power produced would be found to be exactly +proportional to the fall of the caloric. This is the analytical +translation of what was stated on page 98. + +NOTE F.—M. Dalton believed that he had discovered that the vapors of +different liquids at equal thermometric distances from the boiling-point +possess equal tensions; but this law is not precisely exact; it is only +approximate. It is the same with the law of the proportionality of the +latent heat of vapors with their densities (see Extracts from a Mémoire +of M. C. Despretz, _Annales de Chimie et de Physique_, t. xvi. p. 105, +and t. xxiv. p. 323). Questions of this nature are closely connected +with those of the motive power of heat. Quite recently MM. H. Davy and +Faraday, after having conducted a series of elegant experiments on the +liquefaction of gases by means of considerable pressure, have tried to +observe the changes of tension of these liquefied gases on account of +slight changes of temperature. They have in view the application of the +new liquids to the production of motive power (see _Annales de Chimie et +de Physique_, January, 1824, p. 80). + +According to the above-mentioned theory, we can foresee that the use of +these liquids would present no advantages relatively to the economy of +heat. The advantages would be found only in the lower temperature at +which it would be possible to work, and in the sources whence, for this +reason, it would become possible to obtain caloric. + +NOTE G.—This principle, the real foundation of the theory of +steam-engines, was very clearly developed by M. Clement in a memoir +presented to the Academy of Sciences several years ago. This Memoir has +never been printed, and I owe the knowledge of it to the kindness of the +author. Not only is the principle established therein, but it is applied +to the different systems of steam-engines actually in use. The motive +power of each of them is estimated therein by the aid of the law cited +page 92, and compared with the results of experiment. + +The principle in question is so little known or so poorly appreciated, +that recently Mr. Perkins, a celebrated mechanician of London, +constructed a machine in which steam produced under the pressure of 35 +atmospheres—a pressure never before used—is subjected to very little +expansion of volume, as any one with the least knowledge of this machine +can understand. It consists of a single cylinder of very small +dimensions, which at each stroke is entirely filled with steam, formed +under the pressure of 35 atmospheres. The steam produces no effect by +the expansion of its volume, for no space is provided in which the +expansion can take place. It is condensed as soon as it leaves the small +cylinder. It works therefore only under a pressure of 35 atmospheres, +and not, as its useful employment would require, under progressively +decreasing pressures. The machine of Mr. Perkins seems not to realize +the hopes which it at first awakened. It has been asserted that the +economy of coal in this engine was ⁹⁄₁₀ above the best engines of Watt, +and that it possessed still other advantages (see _Annales de Chimie et +de Physique_, April, 1823, p. 429). These assertions have not been +verified. The engine of Mr. Perkins is nevertheless a valuable +invention, in that it has proved the possibility of making use of steam +under much higher pressure than previously, and because, being easily +modified, it may lead to very useful results. + +Watt, to whom we owe almost all the great improvements in steam-engines, +and who brought these engines to a state of perfection difficult even +now to surpass, was also the first who employed steam under +progressively decreasing pressures. In many cases he suppressed the +introduction of the steam into the cylinder at a half, a third, or a +quarter of the stroke. The piston completes its stroke, therefore, under +a constantly diminishing pressure. The first engines working on this +principle date from 1778. Watt conceived the idea of them in 1769, and +took out a patent in 1782. + +We give here the Table appended to Watt’s patent. He supposed the steam +introduced into the cylinder during the first quarter of the stroke of +the piston; then, dividing this stroke into twenty parts, he calculated +the mean pressure as follows: + + Portions of the descent from the top of the Decreasing pressure of the + cylinder. steam, the entire pressure + being 1. + Steam arriving + 0.05 freely from the 1.000 Total pressure. + boiler. + 0.10 „ 1.000 „ + 0.15 „ 1.000 „ + 0.20 „ 1.000 „ + Quarter 0.25 „ 1.000 „ + The steam being cut + off and the + 0.30 descent taking 0.830 + place only by + expansion. + 0.35 „ 0.714 + 0.40 „ 0.625 + 0.45 „ 0.555 + Half 0.50 „ 0.500 Half original + pressure. + 0.55 „ 0.454 + 0.60 „ 0.417 + 0.65 „ 0.385 + 0.70 „ 0.375 + 0.75 „ 0.333 One third. + 0.80 „ 0.312 + 0.85 „ 0.294 + 0.90 „ 0.277 + 0.95 „ 0.262 + Bottom of cylinder 1.00 „ 0.025 Quarter. + Total, 11.583 + + Mean pressure (11.583)/(20) = 0.579. + +On which he remarked, that the mean pressure is more than half the +original pressure; also that in employing a quantity of steam equal to a +quarter, it would produce an effect more than half. + +Watt here supposed that steam follows in its expansion the law of +Mariotte, which should not be considered exact, because, in the first +place, the elastic fluid in dilating falls in temperature, and in the +second place there is nothing to prove that a part of this fluid is not +condensed by its expansion. Watt should also have taken into +consideration the force necessary to expel the steam which remains after +condensation, and which is found in quantity as much greater as the +expansion of the volume has been carried further. Dr. Robinson has +supplemented the work of Watt by a simple formula to calculate the +effect of the expansion of steam, but this formula is found to have the +same faults that we have just noticed. It has nevertheless been useful +to constructors by furnishing them approximate data practically quite +satisfactory. We have considered it useful to recall these facts because +they are little known, especially in France. These engines have been +built after the models of the inventors, but the ideas by which the +inventors were originally influenced have been but little understood. +Ignorance of these ideas has often led to grave errors. Engines +originally well conceived have deteriorated in the hands of unskilful +builders, who, wishing to introduce in them improvements of little +value, have neglected the capital considerations which they did not know +enough to appreciate. + +NOTE H.—The advantage in substituting two cylinders for one is evident. +In a single cylinder the impulsion of the piston would be extremely +variable from the beginning to the end of the stroke. It would be +necessary for all the parts by which the motion is transmitted to be of +sufficient strength to resist the first impulsion, and perfectly fitted +to avoid the abrupt movements which would greatly injure and soon +destroy them. It would be especially on the working beam, on the +supports, on the crank, on the connecting-rod, and on the first +gear-wheels that the unequal effort would be felt, and would produce the +most injurious effects. It would be necessary that the steam-cylinder +should be both sufficiently strong to sustain the highest pressure, and +with a large enough capacity to contain the steam after its expansion of +volume, while in using two successive cylinders it is only necessary to +have the first sufficiently strong and of medium capacity,—which is not +at all difficult,—and to have the second of ample dimensions, with +moderate strength. + +Double-cylinder engines, although founded on correct principles, often +fail to secure the advantages expected from them. This is due +principally to the fact that the dimensions of the different parts of +these engines are difficult to adjust, and that they are rarely found to +be in correct proportion. Good models for the construction of +double-cylinder engines are wanting, while excellent designs exist for +the construction of engines on the plan of Watt. From this arises the +diversity that we see in the results of the former, and the great +uniformity that we have observed in the results of the latter. + +NOTE I.—Among the attempts made to develop the motive power of heat by +means of atmospheric air, we should mention those of MM. Niepce, which +were made in France several years ago, by means of an apparatus called +by the inventors a pyréolophore. The apparatus was made thus: There was +a cylinder furnished with a piston, into which the atmospheric air was +introduced at ordinary density. A very combustible material, reduced to +a condition of extreme tenuity, was thrown into it, remained a moment in +suspension in the air, and then flame was applied. The inflammation +produced very nearly the same effect as if the elastic fluid had been a +mixture of air and combustible gas, of air and carburetted hydrogen gas, +for example. There was a sort of explosion, and a sudden dilatation of +the elastic fluid—a dilatation that was utilized by making it act upon +the piston. The latter may have a motion of any amplitude whatever, and +the motive power is thus realized. The air is next renewed, and the +operation repeated. + +This machine, very ingenious and interesting, especially on account of +the novelty of its principle, fails in an essential point. The material +used as a combustible (it was the dust of Lycopodium, used to produce +flame in our theatres) was so expensive, that all the advantage was lost +through that cause; and unfortunately it was difficult to employ a +combustible of moderate price, since a very finely powdered substance +was required which would burn quickly, spread rapidly, and leave little +or no ash. + +Instead of working as did MM. Niepce, it would seem to us preferable to +compress the air by means of pumps, to make it traverse a perfectly +closed furnace into which the combustible had been introduced in small +portions by a mechanism easy of conception, to make it develop its +action in a cylinder with a piston, or in any other variable space; +finally, to throw it out again into the atmosphere, or even to make it +pass under a steam-boiler in order to utilize the temperature remaining. + +The principal difficulties that we should meet in this mode of operation +would be to enclose the furnace in a sufficiently strong envelope, to +keep the combustion meanwhile in the requisite state, to maintain the +different parts of the apparatus at a moderate temperature, and to +prevent rapid abrasion of the cylinder and of the piston. These +difficulties do not appear to be insurmountable. + +There have been made, it is said, recently in England, successful +attempts to develop motive power through the action of heat on +atmospheric air. We are entirely ignorant in what these attempts have +consisted—if indeed they have really been made. + +NOTE J.—The result given here was furnished by an engine whose large +cylinder was 45 inches in diameter and 7 feet stroke. It is used in one +of the mines of Cornwall called Wheal Abraham. This result should be +considered as somewhat exceptional, for it was only temporary, +continuing but a single month. Thirty millions of lbs. raised one +English foot per bushel of coal of 88 lbs. is generally regarded as an +excellent result for steam-engines. It is sometimes attained by engines +of the Watt type, but very rarely surpassed. This latter product +amounts, in French measures, to 104,000 kilograms raised one metre per +kilogram of coal consumed. + +According to what is generally understood by one horse-power, in +estimating the duty of steam-engines, an engine of ten horse-power +should raise per second 10 × 75 kilograms, or 750 kilograms, to a height +of one metre, or more, per hour; 750 × 3600 = 2,700,000 kilograms to one +metre. If we suppose that each kilogram of coal raised to this height +104,000 kilograms, it will be necessary, in order to ascertain how much +coal is burnt in one hour by our ten-horse-power engine, to divide +2,700,000 by 104,000, which gives ²⁷⁰⁰⁄₁₀₄ = 26 kilograms. Now it is +seldom that a ten-horse-power engine consumes less than 26 kilograms of +coal per hour. + + + + + APPENDIX C. + NOTE BY THE EDITOR. + + +All the preceding data are to-day subject to modification. + +Thus a duty of 150,000,000 ft.-lbs. per 100 lbs. good coal is to-day +attainable, and two thirds that figure is extremely common. With engines +of large size the coal-consumption has fallen to one half, sometimes +even to one fourth, the figure in the text. + +Hot air-engines are superseded by the gas-engine and the oil-vapor +engine; which even threaten, in the opinion of many engineers, to +ultimately displace the steam-engine. + +Compound and other multiple-cylinder engines, with two, three, and even +four cylinders in series, are now always employed where fuel is costly. +The reason of their success is, in part, that given in Note H; but in +only small part. The real cause of their general adoption is the fact +that the internal thermal waste by “cylinder condensation”—which in +simple engines ordinarily amounts, according to size, to from 25 to 50 +per cent, or more, of all heat supplied by the boiler—is reduced nearly +in proportion to the number of steam-cylinders in series. + +For the applied thermodynamics of the steam-engine, following Carnot and +Thomson, see the pages of Rankine and of Clausius of 1850 to 1860, and +especially the treatise of Rankine on the Steam-engine. The editor has +adopted the methods of these great successors of Carnot in his “Manual +of the Steam-engine” (2 vols. 8vo; N. Y., J. Wiley & Sons), which may be +consulted in this connection, and especially for details of the theory +and the structure of this prime mover. + +----- + +Footnote 1: + + Tait: Thermodynamics, p. 13. + +Footnote 2: + + Account of Carnot’s Theory of the Motive Power of Heat; Sir Wm. + Thomson; Trans. Roy. Soc. of Edinburgh, xvi. 1849; and Math. and Phys. + Papers, xli. vol. 1 (Cambridge, 1882), p. 113. In this paper the + corrections due to the introduction of the dynamic theory are first + applied. + +Footnote 3: + + See the Appendix for these memoranda, and for other previously + unpublished matter. + +Footnote 4: + + Sadi Carnot’s _Réflexions sur la puissance motrice du feu_ (Paris, + Bachelier 1824) was long ago completely exhausted. As but a small + number of copies were printed, this remarkable work remained long + unknown to the earlier writers on Thermodynamics. It was therefore for + the benefit of savants unable to study a work out of print, as well as + to render honor to the memory of Sadi Carnot, that the new publishers + of the _Annales Scientifique de l’École Normale supérieure_ (ii. + series, t. 1, 1872) published a new edition, from which this + translation is reproduced. + +Footnote 5: + + It may be said that coal-mining has increased tenfold in England since + the invention of the steam-engine. It is almost equally true in regard + to the mining of copper, tin, and iron. The results produced in a + half-century by the steam-engine in the mines of England are to-day + paralleled in the gold and silver mines of the New World—mines of + which the working declined from day to day, principally on account of + the insufficiency of the motors employed in the draining and the + extraction of the minerals. + +Footnote 6: + + We say, to lessen the dangers of journeys. In fact, although the use + of the steam-engine on ships is attended by some danger which has been + greatly exaggerated, this is more than compensated by the power of + following always an appointed and well-known route, of resisting the + force of the winds which would drive the ship towards the shore, the + shoals, or the rocks. + +Footnote 7: + + We use here the expression motive power to express the useful effect + that a motor is capable of producing. This effect can always be + likened to the elevation of a weight to a certain height. It has, as + we know, as a measure, the product of the weight multiplied by the + height to which it is raised. + +Footnote 8: + + We distinguish here the steam-engine from the heat-engine in general. + The latter may make use of any agent whatever, of the vapor of water + or of any other, to develop the motive power of heat. + +Footnote 9: + + Certain engines at high pressure throw the steam out into the + atmosphere instead of the condenser. They are used specially in places + where it would be difficult to procure a stream of cold water + sufficient to produce condensation. + +Footnote 10: + + The existence of water in the liquid state here necessarily assumed, + since without it the steam-engine could not be fed, supposes the + existence of a pressure capable of preventing this water from + vaporizing, consequently of a pressure equal or superior to the + tension of vapor at that temperature. If such a pressure were not + exerted by the atmospheric air, there would be instantly produced a + quantity of steam sufficient to give rise to that tension, and it + would be necessary always to overcome this pressure in order to throw + out the steam from the engines into the new atmosphere. Now this is + evidently equivalent to overcoming the tension which the steam retains + after its condensation, as effected by ordinary means. + + If a very high temperature existed at the surface of our globe, as it + seems certain that it exists in its interior, all the waters of the + ocean would be in a state of vapor in the atmosphere, and no portion + of it would be found in a liquid state. + +Footnote 11: + + It is considered unnecessary to explain here what is quantity of + caloric or quantity of heat (for we employ these two expressions + indifferently), or to describe how we measure these quantities by the + calorimeter. Nor will we explain what is meant by latent heat, degree + of temperature, specific heat, etc. The reader should be familiarized + with these terms through the study of the elementary treatises of + physics or of chemistry. + +Footnote 12: + + We may perhaps wonder here that the body _B_ being at the same + temperature as the steam is able to condense it. Doubtless this is not + strictly possible, but the slightest difference of temperature will + determine the condensation, which suffices to establish the justice of + our reasoning. It is thus that, in the differential calculus, it is + sufficient that we can conceive the neglected quantities indefinitely + reducible in proportion to the quantities retained in the equations, + to make certain of the exact result. + + The body _B_ condenses the steam without changing its own + temperature—this results from our supposition. We have admitted that + this body may be maintained at a constant temperature. We take away + the caloric as the steam furnishes it. This is the condition in which + the metal of the condenser is found when the liquefaction of the steam + is accomplished by applying cold water externally, as was formerly + done in several engines. Similarly, the water of a reservoir can be + maintained at a constant level if the liquid flows out at one side as + it flows in at the other. + + One could even conceive the bodies _A_ and _B_ maintaining the same + temperature, although they might lose or gain certain quantities of + heat. If, for example, the body _A_ were a mass of steam ready to + become liquid, and the body _B_ a mass of ice ready to melt, these + bodies might, as we know, furnish or receive caloric without + thermometric change. + +Footnote 13: + + Note A, Appendix B. + +Footnote 14: + + We assume here no chemical action between the bodies employed to + realize the motive power of heat. The chemical action which takes + place in the furnace is, in some sort, a preliminary action,—an + operation destined not to produce immediately motive power, but to + destroy the equilibrium of the caloric, to produce a difference of + temperature which may finally give rise to motion. + +Footnote 15: + + This kind of loss is found in all steam-engines. In fact, the water + destined to feed the boiler is always cooler than the water which it + already contains. There occurs between them a useless re-establishment + of equilibrium of caloric. We are easily convinced, _à posteriori_, + that this re-establishment of equilibrium causes a loss of motive + power if we reflect that it would have been possible to previously + heat the feed-water by using it as condensing water in a small + accessory engine, when the steam drawn from the large boiler might + have been used, and where the condensation might be produced at a + temperature intermediate between that of the boiler and that of the + principal condenser. The power produced by the small engine would have + cost no loss of heat, since all that which had been used would have + returned into the boiler with the water of condensation. + +Footnote 16: + + The matter here dealt with being entirely new, we are obliged to + employ expressions not in use as yet, and which perhaps are less clear + than is desirable. + +Footnote 17: + + Note 13, Appendix B. + +Footnote 18: + + We tacitly assume in our demonstration, that when a body has + experienced any changes, and when after a certain number of + transformations it returns to precisely its original state, that is, + to that state considered in respect to density, to temperature, to + mode of aggregation—let us suppose, I say, that this body is found to + contain the same quantity of heat that it contained at first, or else + that the quantities of heat absorbed or set free in these different + transformations are exactly compensated. This fact has never been + called in question. It was first admitted without reflection, and + verified afterwards in many cases by experiments with the calorimeter. + To deny it would be to overthrow the whole theory of heat to which it + serves as a basis. For the rest, we may say in passing, the main + principles on which the theory of heat rests require the most careful + examination. Many experimental facts appear almost inexplicable in the + present state of this theory. + +Footnote 19: + + We will suppose, in what follows, the reader to be _au courant_ with + the later progress of modern Physics in regard to gaseous substances + and heat. + +Footnote 20: + + M. Poisson, to whom this figure is due, has shown that it accords very + well with the result of an experiment of MM. Clement and Desormes on + the return of air into a vacuum, or rather, into air slightly + rarefied. It also accords very nearly with results found by MM. + Gay-Lussac and Welter. (See note, p. 87.) + +Footnote 21: + + The law of Mariotte, which is here made the foundation upon which to + establish our demonstration, is one of the best authenticated physical + laws. It has served as a basis to many theories verified by + experience, and which in turn verify all the laws on which they are + founded. We can cite also, as a valuable verification of Mariotte’s + law and also of that of MM. Gay-Lussac and Dalton, for a great + difference of temperature, the experiments of MM. Dulong and Petit. + (See _Annales de Chimie et de Physique_, Feb. 1818, t. vii. p. 122.) + + The more recent experiments of Davy and Faraday can also be cited. + + The theories that we deduce here would not perhaps be exact if applied + outside of certain limits either of density or temperature. They + should be regarded as true only within the limits in which the laws of + Mariotte and of MM. Gay-Lussac and Dalton are themselves proven. + +Footnote 22: + + When the volume is reduced ¹⁄₁₁₆, that is, when it becomes ¹¹⁵⁄₁₁₆ of + what it was at first, the temperature rises one degree. Another + reduction of ¹⁄₁₁₆ carries the volume to (¹¹⁵⁄₁₁₆)^2, and the + temperature should rise another degree. After _x_ similar reductions + the volume becomes (¹¹⁵⁄₁₁₆)^{_x_}, and the temperature should be + raised _x_ degrees. If we suppose (¹¹⁵⁄₁₁₆)^{_x_} = ¹⁄₁₄, and if we + take the logarithms of both, we find + + _x_ = about 300°. + + If we suppose (¹¹⁵⁄₁₁₆)^{_x_} = ½, we find + + _x_ = 80°; + + which shows that air compressed one half rises 80°. + + All this is subject to the hypothesis that the specific heat of air + does not change, although the volume diminishes. But if, for the + reasons hereafter given (pp. 86, 89), we regard the specific heat of + air compressed one half as reduced in the relation of 700 to 616, the + number 80° must be multiplied by ⁷⁰⁰⁄₆₁₆, which raises it to 90°. + +Footnote 23: + + MM. Gay-Lussac and Welter have found by direct experiments, cited in + the _Mécanique Céleste_ and in the _Annales de Chimie et de Physique_, + July, 1822, p. 267, that the ratio between the specific heat at + constant pressure and the specific heat at constant volume varies very + little with the density of the gas. According to what we have just + seen, the difference should remain constant, and not the ratio. As, + further, the specific heat of gases for a given weight varies very + little with the density, it is evident that the ratio itself + experiences but slight changes. + + The ratio between the specific heat of atmospheric air at constant + pressure and at constant volume is, according to MM. Gay-Lussac and + Welter, 1.3748, a number almost constant for all pressures, and even + for all temperatures. We have come, through other considerations, to + the number (267 + 116)/(267) = 1.44, which differs from the former + (1)/(20), and we have used this number to prepare a table of the + specific heats of gases at constant volume. So we need not regard this + table as very exact, any more than the table given on p. 89. These + tables are mainly intended to demonstrate the laws governing specific + heats of aeriform fluids. + +Footnote 24: + + Note C, Appendix B. + +Footnote 25: + + Note D, Appendix B. + +Footnote 26: + + Note E, Appendix B. + +Footnote 27: + + We find (_Annales de Chimie et de Physique_, July, 1818, p. 294) in a + memoir of M. Petit an estimate of the motive power of heat applied to + air and to vapor of water. This estimate leads us to attribute a great + advantage to atmospheric air, but it is derived by a method of + considering the action of heat which is quite imperfect. + +Footnote 28: + + Note F, Appendix B. + +Footnote 29: + + Those that we need are the expansive force acquired by solids and + liquids by a given increase of temperature, and the quantity of heat + absorbed or relinquished in the changes of volume of these bodies. + +Footnote 30: + + The recent experiments of M. Oerstedt on the compressibility of water + have shown that, for a pressure of five atmospheres, the temperature + of this liquid exhibits no appreciable change. (_See Annales de Chimie + et de Physique_, Feb. 1823, p. 192.) + +Footnote 31: + + Note G, Appendix B. + +Footnote 32: + + We find in the work called _De la Richesse Minérale_, by M. Heron de + Villefosse, vol. iii. p. 50 and following, a good description of the + steam-engines actually in use in mining. In England the steam-engine + has been very fully discussed in the _Encyclopedia Britannica_. Some + of the data here employed are drawn from the latter work. + +Footnote 33: + + Note I, Appendix B. + +Footnote 34: + + From _Transactions of the Edinburgh Royal Society_, xiv. 1849; + _Annales de Chimie_, xxxv. 1852. + +Footnote 35: + + Published in 1824, in a work entitled “_Réflexions sur la Puissance + Motrice du Feu, et sur les Machines Propres à Developer cette + Puissance. Par S. Carnot._” [Note of Nov. 5, 1881. The original work + has now been republished, with a biographical notice, Paris, 1878.] + +Footnote 36: + + An account of the first part of a series of researches undertaken by + Mons. Regnault, by order of the late French Government, for + ascertaining the various physical data of importance in the theory of + the steam-engine, has been recently published (under the title + “_Relation des Expériences_,” etc.) in the _Mémoires de l’Institut_, + of which it constitutes the twenty-first volume (1847). The second + part of these researches has not yet been published. [Note of Nov. 5, + 1881. The continuation of these researches has now been published; + thus we have for the whole series, vol. i. in 1847; vol. ii. in 1862; + and vol. iii. in 1870.] + +Footnote 37: + + Carnot, p. 67. + +Footnote 38: + + The _evolution_ of heat in a fixed conductor, through which a + galvanic current is sent from any source whatever, has long been + known to the scientific world; but it was pointed out by Mr. Joule + that we cannot infer from any previously-published experimental + researches, the actual _generation_ of heat when the current + originates in electro-magnetic induction; since the question occurs, + _is the heat which is evolved in one part of the closed conductor + merely transferred from those parts which are subject to the + inducing influence?_ Mr. Joule, after a most careful experimental + investigation with reference to this question, finds that it must be + answered in the negative. (See a paper “On the Calorific Effects of + Magneto-Electricity, and on the Mechanical Value of Heat; by J. P. + Joule, Esq.” Read before the British Association at Cork in 1843, + and subsequently communicated by the Author to the _Philosophical + Magazine_, vol. xxiii., pp. 263, 347, 435.) + + Before we can finally conclude that heat is absolutely generated in + such operations, it would be necessary to prove that the inducing + magnet does not become lower in temperature, and thus compensate for + the heat evolved in the conductor. I am not aware that any examination + with reference to the truth of this conjecture has been instituted; + but, in the case where the inducing body is a pure electro-magnet + (without any iron), the experiments actually performed by Mr. Joule + render the conclusion probable that the heat evolved in the wire of + the electro-magnet is not affected by the inductive action, otherwise + than through the reflected influence which increases the strength of + its own current. + +Footnote 39: + + So generally is Carnot’s principle tacitly admitted as an axiom, that + its application in this case has never, so far as I am aware, been + questioned by practical engineers. (1849). + +Footnote 40: + + When “thermal agency” is thus spent in conducting heat through a + solid, what becomes of the mechanical effect which it might produce? + Nothing can be lost in the operations of nature—no energy can be + destroyed. What effect, then, is produced in place of the mechanical + effect which is lost? A perfect theory of heat imperatively demands an + answer to this question; yet no answer can be given in the present + state of science. A few years ago, a similar confession must have been + made with reference to the mechanical effect lost in a fluid set in + motion in the interior of a rigid closed vessel, and allowed to come + to rest by its own internal friction; but in this case the foundation + of a solution of the difficulty has been actually found in Mr. Joule’s + discovery of the generation of heat, by the internal friction of a + fluid in motion. Encouraged by this example, we may hope that the very + perplexing question in the theory of heat, by which we are at present + arrested, will before long be cleared up. [Note of Sept., 1881. The + Theory of the Dissipation of Energy completely answers this question + and removes the difficulty.] + + It might appear that the difficulty would be entirely avoided by + abandoning Carnot’s fundamental axiom; a view which is strongly urged + by Mr. Joule (at the conclusion of his paper “On the Changes of + Temperature produced by the Rarefaction and Condensation of Air.” + _Phil. Mag._, May 1845, vol. xxvi.) If we do so, however, we meet with + innumerable other difficulties—insuperable without farther + experimental investigation, and an entire reconstruction of the theory + of heat from its foundation. It is in reality to experiment that we + must look—either for a verification of Carnot’s axiom, and an + explanation of the difficulty we have been considering; or for an + entirely new basis of the Theory of Heat. + +Footnote 41: + + For a demonstration, see § 29. + +Footnote 42: + + A case minutely examined in another paper, to be laid before the + Society at the present meeting. “Theoretical Considerations on the + Effect of Pressure in Lowering the Freezing-point of Water,” by Prof. + James Thomson. + +Footnote 43: + + In all that follows, the pressure of the atmosphere on the upper side + of the piston will be included in the applied forces, which, in the + successive operations described, are sometimes overcome by the upward + motion, and sometimes yielded to in the motion downwards. It will be + unnecessary, in reckoning at the end of a cycle of operations, to take + into account the work thus spent upon the atmosphere, and the + restitution which has been made, since these precisely compensate for + one another. + +Footnote 44: + + [Note of Nov. 5, 1881. Maxwell has simplified the correction by + beginning the cycle with Carnot’s second operation, and completing it + through his third, fourth, and first operations, with his third + operation nearly as follows: + + + _let the piston be pushed down to any position E_{3}F_{3}_; + + then Carnot’s fourth operation altered to the following: + + _let the piston be pushed down from E_{3}F_{3} until the temperature + reaches its primitive value S_; + + and lastly, Carnot’s first operation altered to the following: + + _let the piston rise to its primitive position_.] + + +Footnote 45: + + In Carnot’s work some perplexity is introduced with reference to the + temperature of the water, which, in the operations he describes, is + not brought back exactly to what it was at the commencement; but the + difficulty which arises is explained by the author. No such difficulty + occurs with reference to the cycle of operation described in the text, + for which I am indebted to Mons. Clapeyron. + +Footnote 46: + + Thus, _dq_/_dv_ will be the partial differential coefficient, with + respect to _v_, of that function of _v_ and _t_ which expresses the + quantity of heat that must be added to a mass of air when in a + “standard” state (such as at the temperature zero, and under the + atmospheric pressure), to bring it to the temperature _t_ and the + volume _v_. That there is such a function, of two independent + variables _v_ and _t_, is merely an analytical expression of Carnot’s + fundamental axiom, as applied to a mass of air. The general principle + may be analytically stated in the following terms:—If _Mdv_ denote the + accession of heat received by a mass of any kind, not possessing a + destructible texture, when the volume is increased by _dv_, the + temperature being kept constant, and if _Ndt_ denote the amount of + heat which must be supplied to raise the temperature by _dt_, without + any alteration of volume; then _Mdv_ + _Ndt_ must be the differential + of a function of _v_ and _t_. [Note of Nov. 5, 1881. In the corrected + theory it is (_M_ − _Jp_)_dv_ + _Ndt_, that is a complete + differential, not _Mdv_ + _Ndt_. See _Dynamical Theory of Heat_ (Art. + XLVIII., below), § 20.] + +Footnote 47: + + We might also investigate another relation, to express the fact that + there is no accession or removal of heat during either the second or + the fourth operation; but it will be seen that this will not affect + the result in the text, although it would enable us to determine both + φ and ω in terms of τ. + +Footnote 48: + + This result might have been obtained by applying the usual notation of + the integral calculus to express the area of the curvilinear + quadrilateral, which, according to Clapeyron’s graphical construction, + would be found to represent the entire mechanical effect gained in the + cycle of operations of the air-engine. It is not necessary, however, + to enter into the details of this investigation, as the formula (3), + and the consequences derived from it, include the whole theory of the + air-engine, in the best practical form; and the investigation of it + which I have given in the text will probably give as clear a view of + the reasoning on which it is founded as could be obtained by the + graphical method, which in this case is not so valuable as it is from + its simplicity in the case of the steam-engine. + +Footnote 49: + + This paragraph is the demonstration, referred to above, of the + proposition stated in § 13, as it is readily seen that it is + applicable to any conceivable kind of thermodynamic engine. + +Footnote 50: + + The results of these investigations are exhibited in Tables I and II. + +Footnote 51: + + It is, comparatively speaking, of little consequence to know + accurately the value of σ, for the factor (1 − σ) of the expression + for μ, since it is so small (being less than ¹⁄₁₇₀₀ for all + temperatures between 0° and 100°) that, unless all the data are known + with more accuracy than we can count upon at present, we might neglect + it altogether, and take _dp_/_kdt_ simply, as the expression for μ, + without committing any error of important magnitude. + +Footnote 52: + + This is well established, within the ordinary atmospheric limits, in + Regnault’s Études Météorologiques, in the _Annales de Chimie_, vol. + xv., 1846. + +Footnote 53: + + It appears that the vol. of 1 kilog. must be 1.69076 according to the + data here assumed. + + The density of saturated steam at 100° is taken as ¹⁄₁₆₉₃.5 of that of + water at its maximum. Rankine takes it as ¹⁄₁₆₉₆. + +Footnote 54: + + The part of this expression in the first vinculum (see Regnault, end + of ninth memoir) is what is known as “the total heat” of a pound of + steam, or the amount of heat necessary to convert a pound of water at + 0° into a pound of saturated steam at _t°_; which, according to + “Watt’s law” thus approximately verified, would be constant. The + second part, which would consist of the single term _t_, if the + specific heat of water were constant for all temperatures, is the + number of thermic units necessary to raise the temperature of a pound + of water from 0° to _t°_, and expresses empirically the results of + Regnault’s experiments on the specific heat of water (see end of the + tenth memoir), described in the work already referred to. + +Footnote 55: + + In strictness, the 230th is the last degree for which the experimental + data are complete; but the data for the 231st may readily be assumed + in a sufficiently satisfactory manner. + +Footnote 56: + + The numbers here tabulated may also be regarded as _the actual values + of μ for t_ = ½, _t_ = 1½, _t_ = 2½, _t_ = 3½, etc. + +Footnote 57: + + For at the end of the fourth operation the whole mass is liquid, and + at the temperature _S_. Now, this state might be arrived at by first + compressing the vapor into water at the temperature _t_, and then + raising the temperature of the liquid to _S_; and however this state + may be arrived at, there cannot, on the whole, be any heat added to or + subtracted from the contents of the cylinder, since, during the fourth + operation, there is neither gain nor loss of heat. This reasoning is, + of course, founded on Carnot’s fundamental principle, which is tacitly + assumed in the commonly-received ideas connected with “Watt’s law,” + the “latent heat of steam,” and “the total heat of steam.” + +Footnote 58: + + Thus, from Carnot’s calculations, we find, in the case of alcohol + 4.035, and in the case of water 3.648, instead of 3.963 and 3.658, + which are Clapeyron’s results in the same cases. + +Footnote 59: + + A still closer agreement must be expected when more accurate + experimental data are afforded with reference to the other media. + Mons. Regnault informs me that he is engaged in completing some + researches, from which we may expect, possibly before the end of the + present year, to be furnished with all the data for five or six + different liquids which we possess at present for water. It is + therefore to be hoped that, before long, a most important test of the + validity of Carnot’s theory will be afforded. + +Footnote 60: + + The _Napierian_ logarithm of _V_/_V′_ is here understood. + +Footnote 61: + + Carnot varies the statement of his theorem, and illustrates it in a + passage, pp. 81, 82, of which the following is translation: + + “_When a gas varies in volume without any change of temperature, the + quantities of heat absorbed or evolved by this gas are in arithmetical + progression, if the augmentation or diminutions of volume are in + geometrical progression._ + + “When we compress a litre of air maintained at the temperature 10°, + and reduce it to half a litre, it disengages a certain quantity of + heat. If, again, the volume be reduced from half a litre to a quarter + of a litre, from a quarter to an eighth, and so on the quantities of + heat successively evolved will be the same. + + “If, in place of compressing the air, we allow it to expand to two + litres, four litres, eight litres, etc., it will be necessary to + supply equal quantities of heat to maintain the temperature always at + the same degree.” + +Footnote 62: + + The best figure (1896) is _J_ = 778 ft.-lbs. = 1 B.T.U., or _J_ = + 426.8 kgm. = 1 calorie, and probably with great accuracy. + +Footnote 63: + + Or the capacity of a unit of volume for heat. + +Footnote 64: + + Carnot suggests a combination of the two principles, with air as the + medium for receiving the heat at a very high temperature from the + furnace; and a second medium, alternately in the state of saturated + vapor and liquid water, to receive the heat, discharged at an + intermediate temperature from the air, and transmit it to the coldest + part of the apparatus. It is possible that a complex arrangement of + this kind might be invented which would enable us to take the heat at + a higher temperature, and discharge it at a lower temperature than + would be practicable in any simple air-engine or simple steam-engine. + If so, it would no doubt be equally possible, and perhaps more + convenient, to employ steam alone, but to use it at a very high + temperature not in contact with water in the hottest part of the + apparatus, instead of, as in the steam-engine, always in a saturated + state. + +Footnote 65: + + It is probably this invention to which Carnot alludes in the following + passage: “Il a été fait, dit-on, tout récemment en Angleterre des + essais heureux sur le développement de la puissance motrice par + l’action de la chaleur sur l’air atmosphérique. Nous ignorons + entièrement ne quoi ces essais ont consisté, si toutefois ils sont + réels.” + +Footnote 66: + + From this point of view, we see very clearly how imperfect is the + steam-engine, even after all Watt’s improvements. For to “push the + principle of expansion to the utmost,” we must allow the steam, before + leaving the cylinder, to expand until its pressure is the same as that + of the vapor in the condenser. According to “Watt’s law,” its + temperature would then be the same as (actually a little above, as + Regnault has shown) that of the condenser, and hence the steam-engine + worked in this most advantageous way has in reality the very fault + that Watt found in Newcomen’s engine. This defect is partially + remedied by Hornblower’s system of using a separate expansion + cylinder, an arrangement the advantages of which did not escape + Carnot’s notice, although they have not been recognized extensively + among practical engineers, until within the last few years. + +Footnote 67: + + I am indebted to the kindness of Professor Gordon of Glasgow for the + information regarding the various cases given in the text. + +Footnote 68: + + In different Cornish engines, the pressure in the boiler is from 2½ to + 5 atmospheres; and, therefore, as we find from Regnault’s table of the + pressure of saturated steam, the temperature of the water in the + boiler must, in all of them, lie between 128° and 152°. For the better + class of engines, the average temperature of the water in the boiler + may be estimated at 140°, the corresponding pressure of steam being 3½ + atmospheres. + +Footnote 69: + + This number agrees very closely with the number corresponding to the + fall from 100° to 0°, given in Table II. Hence, the fall from 140° to + 30° of the scale of the air-thermometer is equivalent, with reference + to motive power, to the fall from 100° to 0°. + +Footnote 70: + + It being assumed that the temperatures of the boiler and condenser are + the same as those of the Cornish engines. If, however, the pressure be + lower, two atmospheres, for instance, the numbers would stand thus: + The temperature in the boiler would be only 121. Consequently, for + each pound of steam evaporated, only 614 units of heat would be + required; and therefore the work performed for each unit of heat + transmitted would be 160.3 foot-pounds, which is _more_ than according + to the estimate in the text. On the other hand, the range of + temperatures, or the fall utilized, is only from 131 to 30, instead of + from 140 to 30°, and consequently (Table II.), the theoretical duty + for each unit of heat is only 371 foot-pounds. Hence, if the engine, + to work according to the specification, requires a pressure of only 15 + lbs. on the square inch (i.e., a total steam-pressure of two + atmospheres), its performance is (160.3)/(371) or 43.2 per cent of its + theoretical duty. + +Footnote 71: + + If, in this case again, the pressure required in the boiler to make + the engine work according to the contract were only 15 lbs. on the + square inch, we should have a different estimate of the economy, for + which see Table B, at the end of this paper. + +Footnote 72: + + These engines are provided with separate expansion cylinders, which + have been recently added to them by Mr. M‘Naught of Glasgow. + +Footnote 73: + + [Note added March 15, 1881. Total work for thermal unit, 1390 (Joule), + 377.1 corrected by the dynamical theory, March 15, 1851. + + 377.1 = .2713 × 1390, + 253 = .1820 × 1390 = (1)/(5.49) × 1390.] + +Footnote 74: + + Pressure 15 lbs. on the square inch. + +------------------------------------------------------------------------ + + + + + TRANSCRIBER’S NOTES + + + Page Changed from Changed to + + 110 no appreciable change. (See no appreciable change. (See + Annales de Ohimie et de Annales de Chimie et de + + 246 If, to abridge, we call _N_ the If, to abridge, we call _N_ the + quantity (_P_)/(726), the quantity (_P_)/(267), the + + ● Fixed typos; non-standard spelling and dialect retained. + ● Renumbered footnotes and moved them all to the end of the final + chapter. + ● Enclosed italics font in _underscores_. + ● Enclosed blackletter font in =equals=. + ● The caret (^) is used to indicate superscript, whether applied to a + single character (as in 2^d) or to an entire expression (as in + 1^{st}). + ● Subscripts are shown using an underscore (_) with curly braces { }, + as in H_{2}O. + ● Images without captions use HTML alt text. + +*** END OF THE PROJECT GUTENBERG EBOOK 78610 *** |