This is a reproduction of a library book that was digitized by Google as part of an ongoing effort to preserve the information in books and make it universally accessible. https://books.google.com THE MECHANICAL THEORY OF HEAT, WITH ITS APPLICATIONS TO THE STEAM-ENGINE AND TO THE PHYSICAL PROPERTIES OF BODIES. jXi <- t iA 0-(sl - J J - - - - - k ' - - : . - _ R^CLAUSIUS, hiofeWoI! nr mrum m -I-ihTunivhrsity or iurich. 4 , KD1TED BY T., ARCHER HIRST, F.R.S., PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON. WITH AN INTRODUCTION BY PROFESSOR TYNDALL. LONDON: JOHN VAN VOORST, 1 PATERNOSTER ROW. MDCCCLXVII. f mm , , mi HARVARD UNlVERSIiY LIBRARY PRINTED BY TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET. INTRODUCTION. Nearly seventeen years ago I translated for the Philosophical Magazine the first of this series of Memoirs, by Professor Clausius, on the Mechanical Theory of Heat. A short time afterwards the Essay of Professor Helmholtz, Ueber die Erhal- tung der Kraft, was placed in my hands : I translated it, and had it published in the continuation of 'Taylor's Scientific Memoirs.' It wa3 thus my fortune to introduce to the sci entific public of England the earliest writings of two of the most celebrated contributors to the great theory in question. For many years subsequent to the period here referred to, I was careful to translate, or to have translated, every paper published by these two writers; and the fact that the fol lowing series of these Memoirs is thought worthy of being presented in a collected form to the English public, proves that I did not overestimate their importance. I have been asked by its publisher to write a line or two of introduction to the present volume. This I could not refuse to do, though I feel how superfluous it must be ; for the name and fame of Professor Clausius stand as high in this country as in his own. My Introduction therefore shall be confined to this brief statement of my relationship to his writings. They fell into my hands at a time when I knew but little of the Mechanical Theory of Heat. In those days their author was my teacher ; and in many respects I am proud to acknowledge him as my teacher still. John Tyndall. London, May 1867. a2 AUTHOR'S PREFACE. It has been repeatedly represented to me (and from very dif ferent quarters) that the memoirs on the Mechanical Theory of Heat which, since the year 1850, 1 have published from time to time, principally in Poggendorff's Annalen, are not easily acces sible to all who wish to read them, the interest taken in the Mechanical Theory of Heat having in recent times greatly augmented in circles where physical Journals are not usually found. Accordingly I have thought it advisable to collect and republish those Memoirs. In so doing I have also sought to remedy certain defects which have hitherto diminished their utility. My memoirs " On the Mechanical Theory of Heat " are of different kinds. Some are devoted to the development of the general theory and to the application thereof to those properties of bodies which are usually treated of in the doctrine of heat. Others have reference to the application of the mechanical theory of heat to electricity. The latter contain many exposi tions peculiar to the doctrine of electricity, and they form a separate group, the study of which is not requisite for under standing the former. Other memoirs, again, have reference to the conceptions I have formed of the molecular motions which we call heat. These conceptions, however, have no necessary connexion with the general theory, the latter being based solely on certain principles which may be accepted without adopting any particular view as to the nature of molecular motions. I have therefore kept the consideration of molecular motions quite distinct from the exposition of the general theory. VI AUTHOR 8 PREFACK. The memoirs constituting these three different groups did not, however, appear exactly in their present order; partly in consequence of the direction of my own studies, and partly for other reasons, I found it desirable to pass during their publication from one group to another. Hence has arisen the disadvantage that a reader desirous of becoming acquainted only with the theory, freed as much as possible from hypotheses, cannot know in advance which memoirs are requisite, and which are unnecessary for his purpose. This disadvantage is remedied in the present reprint by simply separating the memoirs into groups, as above explained. The present collection contains the memoirs which belong to the first group ; in them the mechanical theory of heat is deve loped from certain simple axiomatic principles, and is applied to a series of phenomena depending upon heat. I have also in cluded the application of the theory to steam-engines, because this application may be conveniently associated with the ex positions occurring in these memoirs, and especially with those which have reference to vapours*. The memoirs which treat of the application to electricity, and those which relate to my conceptions of molecular motions, I intend subsequently to collect in like manner. The memoirs contained in this collection, however, are quite independent of the others, and form in themselves a complete and con nected whole. Another disadvantage which, as I frequently found, dimi nished the usefulness of my memoirs, arose from the fact that many passages therein were with difficulty understood. The mechanical theory of heat has introduced new ideas into science, differing from the earlier accepted views, and accordingly re quiring special mathematical treatment. An instance of this, especially worthy of mention, is a certain kind of differential equations which I have used in my researches, and which differ from the ordinary ones in one essential point : misconception * [The ninth memoir of the present edition having been published in Ger many subsequent to the appearance there of the First Part of the Collected Memoirs, was not included therein. It is now published for the first time in English ; and, at the Author's suggestion, its appropriate place in the entire series of Memoirs is here assigned to it. — T. A. H.] author's preface. vn might easily arise if this difference were not sufficiently observed. The signification of, and the mode of treating these differential equations have, indeed, long been known to mathematicians through the researches of Monge ; but, from the fact that an energetic attack on my theory originated in a misconception of the true nature of these equations, it would appear that they have not been sufficiently well studied. In order to avoid similar misunderstandings in future, I gave at the time a more detailed explanation of the subject ; as this, however, was not published in PoggendorfPs Annalen, in which my other memoirs appeared, but in Dingler's Polytechnic Journal (which contained that attack), it may possibly have been seen by few of my readers. In order, once for all, to remove any difficulty of this nature, the present collection is preceded by a mathematical intro duction, in which the treatment of the differential equations in question is discussed in a manner similar to that adopted in Dingler's Journal. I have also in many places added notes and appendices, in order to elucidate passages in the text. The memoirs are reprinted verbatim in their original form. The mechanical theory of heat, to the establishment and develop ment of which these memoirs have, as I believe, essentially con tributed, is of so great importance that it has already frequently given rise to discussions on priority. Under these circum stances it appeared to me advisable to allow myself no altera tions ; for even unimportant ones, having reference solely to modes of expression, might possibly give rise to the thought that I intended thereby either to take credit, ultimately, for some thing which did not appear in the original memoirs, or to sup press something which was there inserted*. The notes and appendices now given for the first time are plainly recognizable as such. In order to distinguish these notes from those which were previously published, the former * [It is scarcely necessary to state that in the present English edition this rule has not been adhered to. The translations of the original memoirs, which are here reprinted from the Philosophical Magazine, were made by different persons ; and in order to secure the necessary uniformity in termi nology, verbal alterations were frequently requisite. All such alterations, however, have been made with Prof. Clausius's sanction, to whom the proofs have all been submitted for revision. — T. A. IL] viii author's preface. are enclosed in square brackets ; and to every note containing more than a mere reference is added the date. To the appen dices also dates have been affixed. Should apparently superfluous repetitions be here and there detected, it must be remembered that the memoirs were published at different times during the course of fourteen years, and that often, between two memoirs which directly follow each other in this edition, I had published several others bearing upon different subjects. It was necessary in such cases to recapitulate such portions of the antecedent memoirs as were deemed essential to the comprehension of the new one, or requisite for bringing the reader into the proper train of thought. R, Clausius. Zurich, August 1864. CONTENTS. MATHEMATICAL INTRODUCTION*. On the treatment of differential equations WHICH ABE NOT DIRECTLY INTEGRARLE, pp. 1-13. Page Notation 1 Condition of immediate integrability, and treatment of a differential equation when this condition is not fulfilled 4 Example from analytical mechanics 7 Difference between the results obtained in the two cases 9 Generalizations relative to the form of the equation and to the mode of treating it 10 Differential equations involving more than three variables 11 FIRST MEMOIR.