W' ; kC& ■■ 3fofm Castman Clarke Htfirarp GIVEN TO BOSTON UNIVERSITY COLLEGE OF LIBERAL ARTS BY MRS. JOHN EASTMAN CLARKE MEMORY OF HER HUSBAND JULY, 19,4 Wfr* ifelS 'WflfflfEK.: - ^MW^-^^'^Mimmmm w. O ^PC"^""" if/a *£ ^. £k : m W- -*«•* : MISCELLANEOUS SCIENTIFIC PAPERS. WORKS BY PROFESSOR RANKINE. In crown 8uo, cloth. I. A MANUAL OF APPLIED MECHANICS. Ninth Edition, L2s. 6d. II. A MANUAL OF CIVIL ENGINEERING. Thirteenth Edition, 16a. III. A MANUAL OF MACHINERY AND MILLWORK. Fourth Edition, L2s. 6d. IV. A MANUAL oF THE STEAM-ENGINE AND OTHEE PRIME MOVERS. Ninth EdMm, 12s. 6d. Y. USEFUL RULES AND TABLES for Engineers, Surveyors, and Others. Fifth Edition VI. A MECHANICAL TEXT-BOOK; or, Introduction to the Study of Mechanics. By Pkop. RanKINE and E. F. Bambeb, C.E. Second Edition, 9s. Chables Griffin and Company, London. College of Liberal Arts Boston University MISCELLANEOUS SCIENTIFIC PAPERS: W. .1. MACQUORN EANKINE, (J.E.. LL.I).. F.R.S.,V LATE REGIUS PROFESSOR OF CIVIL ENGINEERING AND MECHANICS IN THE UNIVERSITY OF GLASGOW. FROM THE TRANSACTIONS AND PROCEEDINGS OP THE ROYAL AND OTHER SCIENTIFIC AND PHILOSOPHICAL SOCIETIES, AND THE SCIENTIFIC JOURNALS. WITH -A. nVCEJMIOXK, OIF 1 THE ATJTHOB By P. G. TAIT, M.A., **' PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH. EDITED BY W. J. MILLAR, C.E., SECRETARY TO THE INST. OF ENGINEERS AND SHIPBUILDERS IN SCOTLAND. titfe |0thwit, f h\Us, attbf iiapams. LONDON: CHARLES GRIFFIN AND COMPANY, stationers' hall court. MDCCCLXXXI. Sc\*f\C* ^ ^S AV \S GLASGOW PRINTED BY BELL AND BAIN, 41 MITCHELL STKEET °-r^> s c 3 C*C V\ PREFACE. The republication of a Selection of the Papers of the late Professor Macquorn Rankine was originally projected by several of his personal friends shortly after his decease, the object being to combine, in a suitable volume-form, papers which were to be found only in the Records of Scientific Societies, and in the Scientific and Engineering Journals, and thus to present to the many admirers of the talented author a memorial of his great worth and ability. \ Introductory to the Selection of Papers now published is an exhaustive Memoir by Professor Tait, who kindly consented to embody \n this form the main features of Rankine's life, together with his recollections of one with whom he had been intimately associated as a fellow- worker. There remains, therefore, for the Editor merely to point out, briefly, the principle that has guided him in making the selection from the papers placed in his hands for the purpose of republication. The object, then, kept in view has been the preservation of such papers as are most characteristic of their author in his capacity of a scientific and mathematical inquirer. Professor Rankine was not, nor did he claim to be, a popular writer; his command of mathematical expression was such, that he naturally embodied his reasoning and conclusions in symbols. All his writings, however, are marked by a power of state- ment so clear and logical, that the reader, even should he fail entirely to follow the demonstrations, cannot but be benefited in the attempt to master them. Besides the papers in the present Collection, Professor Rankine con- tributed many others to Scientific and Engineering Societies and Journals. A number of these, from their nature, (descriptions of works and of machines,) were but of passing interest. The papers now published are b BOSTON UNIVERSITY LIBRARIES vi PREFACE. of permanent value, and deal rather with general scientific principles and their applications to practice. The leading characteristics of Professor Rankine's writings are too well-known to require comment here. One special feature, however, in his method of treatment, may be pointed out — viz., the carefully arranged division of the subject into sections. Starting with a general statement of the object of the paper, he advances by degrees in the argument, giving full reference to the subordinate parts of the paper, which arc marked by numerals or letters to distinguish them, thus progressing with logical pre- cision until the conclusion is reached. Another noteworthy feature may also be referred to — viz., that of the introduction of new and suitable words or phrases, proposed by the author, to convey more dearly his meaning. Lastly, we observe throughout all Professor Rankine's writings the most minute accuracy of statement, and the mosl scrupulously honourable care to give to all fellow-workers in the same held with himself their just due, whilst pointing out what he considers to be original on his own part. The principal papers in the volume are those relating to Thermo- dynamics and to Hydrodynamics, where such subjects as the Action of ileat in the Steam-Engine, and the Forms of Waves and "Water-Lines of Ships, arc discussed at length — the scientific and mathematical investi- gation of these questions being perhaps most eminently characteristic of sor Rankine. The papers have been grouped into three Divisions, so as to bring as nearly as possible kindred subjects together; and in every case the name of the Society or Journal through which the paper was originally brought forward, together with the date of its publication, has been given. It need hardly be added, that the papers appear without change of any kind. In their present form they stand precisely as they finally left their author's pen; and no pains have been spared to ensure perfect accuracy in the reproduction. The Editor, like many others, has a grateful remembrance of Professor Rankine, having enjoyed the great privilege of being one of his students in the old College of Glasgow, and having afterwards had the advantage of his friendship in many ways. Rankine's lectures, although simpler than his text-books, were marked by the same clearness of arrangement, and were enforced by his distinct and vigorous enunciation, and admirably illustrated by carefully prepared diagrams. As chairman of the meetings of the societies of which he was president, his methodical habits and business qualifications were of marked service. These qualities were also evidenced in the drawing up of reports in committee; one of the last services which he rendered to engineering science being in connection with an experimental inquiry on safety valves on behalf of the Institu- PREFACE. Vli tion of Engineers and Shipbuilders in Scotland — his decease, unfortunately, occurring before the completion of the experiments. The Papers classed under Part I. relate more or less to Temperature and Elasticity. The First, originally published in 1849, gives an approximate equation for the elasticity of vapour in contact with its liquid, (this equation appears to have been one of the results obtained by Professor Eankine whilst investigating the molecular constitution of matter,) and concludes with the statement of a proposition borne out by experiment, which " may be safely and usefully applied in practice." In the Second Paper, published in the same year, is given a formula for calculating the expansion of liquids by heat. This formula, the author states, he had found useful whilst considering the comparative volumes of liquids at various temperatures. The Third Paper, on " The Centrifugal Theoiy of Elasticity, as applied to Gases and Vapours," published in 1851, shows how the laws of the pressure and expansion of gases may be deduced from the hypothesis of molecular vortices. The investigation was begun in 1842, but was laid aside on account of the want of experimental data ; it was again resumed after the publication of the results of M. Kegnault's experiments on gases and vapours, and laid before the Royal Society of Edinburgh in February, 1850. The hypothesis is defined to be " that which assumes that each atom of matter consists of a nucleus or central point enveloped by an clastic atmosphere, which is retained in its position by attractive forces ; and that the elasticity due to heat arises from the centrifugal force of these atmospheres revolving or oscillating about their nuclei or central points." After showing that some- what similar ideas had been entertained by philosophers at different times, there follows a supposition which, he says, " so far as I am aware, is peculiar to my own researches. It is this, that the vibration which, according to the, undulatory hypothesis, constitutes radiant light and heat, is a motion of the atomic nuclei or centres, and is propagated by means of their mutual attractions and repulsions.'" Tables are given showing the closeness of agreement between the formula? made use of, and the experimental results. In the Fourth Paper we have an extension of the preceding one ; by means of a fresh investigation the complete applicability of the hypothesis of molecular vortices to all substances in all conditions is shown, and this demonstration is followed by the deduction from that hypothesis of the law of the equivalence of heat and power. The Fifth Paper deals with the Laws of Elasticity in reference to the strength of structures — the relations between pressures and strains ; it shows how the Laws of Elasticity are simplified by adopting the supposition of atomic centres of force, but also points out that this nil PREFACE. supposition requires modification, and by means of the hypothesis of molecular \ orl ici Bimplifie I he i ion of the In the Sixth Paper the distinction between Strain and Stress is made clear, and a nomenclature adopted (from Greek equivalents) di icriptive of their relations. Cr talline fo] a idered, to pith their ad ion on li; The Seventh Paper treats of "The Vibrations of Plane-Polari ed Light." The principles laid down in the paper are shown to be incompatible with the "idea of a luminiferous ether enveloping ponderable particles;" much as thai "the Luminiferous medium is a system of atomic nuclei or centres of force, whose • give form to matter; while the atmo n by which they are surrounded give of themselves merelj extension." In the Eighth Paper an atl mpl is made to diminish the difficul attending the undulatory theory of light, hy proposing a theory of oscillations round axes, in tead of the theory of vibratii In ill" Ninth Paper, we have :i mathematical investigation into the relations existing between the velocity- of sound in tances and ; he elastic] of the m Pari II. relates principally to Energy and the Mechanical Action of The first Paper of this division | entitli d " The Etecon- ition of the Mechanical : Universe" a remarkable speculation, laid before the Briti iation al Belfast, and published in 1852, alluded to by Professor Jevons hi bis Principles of Science, and acknowledi I by Sir Wm. Thomson f ;. characteristic of Rankine. This paper, after referring to all experimental evidenci being in favour of the doctrine of the mutual convertibility of the different kind i of I he pi of the universe, ami of the tendency irds a uniform temj of matte] a to point out how it is eivable that ultimately the diffused energy may he gathered into foci, and renewed chemical power produced from the now inert compounds when passing through Hie intense heat of these foci. Tie' next Paper deals with the Law of the Transformation of Enei and the ratio <>f Work done {,, Energy expended in various forms of engines; and is followed by a comprehensive paper entitled " Outlines of Science of Energetics," treating of the laws of physical phenomena — the Sri, 'lice of Mechanics (" the Onlj example yet existing of a complete physical theory") — the use of hypotheses— definitions of Energy, Work,&c, and Efficiency of Engines. The Thirteenth Paper is mainly descriptive of the use of the term Energy, and gives Professor Rankine's reason for his introduction of the phra c " I'ui, ntial Energy." PREFACE. ix Xo. XIV., with its supplements XV.-XIX., on "The Mechanical A.ction of Heat," is an important contribution of considerable length to the Royal Society of Edinburgh, extending over the years 1850-1853. It is based upon the hypothesis of molecular vortices, and relates chiefly to the " mutual conversion of heat and mechanical power, by means of the expansion and contraction of gases and vapours." In the Twentieth Paper the various conditions existing in a heated substance are shown geometrically by curves, a method adopted first, it is stated, by James "Watt in his Steam-engine Indicator. The efficiency of thermodynamic engines is considered, and Stirling and Ericsson's hot- air engines described. The next Paper is on "Formulae for the Maximum Pressure and Latent I [eat of Vapours," followed by one on " The Density of Steam," in which the general equation of thermodynamics, stated in paper Xo. XIV., is again given, to show the connection existing between it and the law of the density of steam. In Paper Xo. XXIII. the Two Laws of Thermodynamics are stated, and it is shown that the derivation of the Second Law from steady molecular motion (e.g., in circular streams or in circulating streams of any figure), as given in previous papers, may lie more simply effected than by the methods adopted in these papers. The following Paper, published in The Engineer in 1867, refers to the want of popular illustrations of The Second Law of Thermodynamics, and explains the nature of the two laws, with the particular questions to which they are respectively applicable. "The First Law informs us that when mechanical work is done by means of heat, a quantity of heat disappears. ... To calculate this disappearance of heat, the work done must be sensibly external, and subject to direct measurement." " The Second Law informs us how to deduce the whole amount of work done, internal and external, from the knowledge which we have of the external work." An illustration is given by the expansion of a perfect or sensibly perfect gas ; but it is pointed out that it is different when we have to deal with fluids in the act of evaporating, instanced by a case showing where the second law is applicable. Paper Xo. XXVI. is on "The Working of Steam in Compound Engines," defines such engines, and states their advantages, with rules for the construction of indicator-diagrams. This is followed by Papers on "The Theory of Explosive Gas-Engines," and on " The Explosive Energy of Heated Liquids." In the first of these it is shown that, in calculations respecting the practical use of heat- engines, it is convenient to use pressures and volumes rather than X PREFACE. temperatures. The mixtures of gas and air most suitable for gas-engines are also given. The Papers in Part III. relate to Wave-Forms, Propulsion of Vessels, Stability of Structures, &c. The first of the series, No. XXIX., which is entitled " On the Exact Form of Waves near the Surface of Deep Water," shows that the form of such waves is trochoidal, and states that this form was first pointed out by Mr. Scott Eussell. The next Paper, "On Plane Water-Lines," investigates the curves suitable for the water-lines of a ship. Water-line curves are designated Neoids, Cyclogenous Neoids (or water-line curves generated by circles), Oogenous Neoids (or those generated by oval bodies), and Lissoneoids, or waterdines of smoothest gliding. It is noticed that although, from lengthened practice in the art of shipbuilding, the forms of water-lines have attained a high degree of excellence, jet that this is due rather to empirical means, than to a knowledge of general principles. The system of Chapman is shown to be wholly empirical, consisting of parabolic forms ; and Mr. Scott Russell's is instanced as the first useful theory of ships' water-lines, being based on wave figures. The various forms of water-lines are then considered in reference to their fitness for different classes of vessels. Other two Papers follow, the first of which is intended to assist those who are not familiar with the higher mathematics in understanding the subject of Stream-Lines. A stream-line is defined as the line traced by a particle in a current of fluid, and an elementary method is given for determining circular stream-lines, a subject mathematically investigated in the preceding paper. The other Paper is an investigation " to determine the relations which must exist between the laws of the elasticity of any substance, whether gaseous, liquid, or solid, and those of the wave-like propagation of a finite longitudinal disturbance in that substance." A Paper on " The Theoretical Limit of the Efficiency of Pro} tellers " follows, showing the theoretical limit of efficiency which improvements in pro- pellers may attain ; states the formulae for reaction and effective power, shows at what relative velocities the propeller is most effective, and compares the advantages of various forms of propeller, with numerical examples. Paper No. XXXIV., " On the Design and Construction of Masonry Dams," originally consisted of a report to the municipality of Bombay, made in 1870, in reference to proposed extensions of the Water- Works there, and was afterwards published in The Engineer. This Paper enters into the question of stability of structures, showing the best and most economical form which a high masonry reservoir wall PREFACE. Xi should have, and is of value to the civil engineer when proposing to adopt a masonry or concrete wall, instead of an ordinary embankment, for reservoir purposes. Papers Xos. XXXV. and XXXVI. are extensions of methods adopted by Professor Eankine in his Manual of Applied Mechanics, in connection with the stability of structures of various figures, such as Frames and Arches ; and the last paper of the series, Xo. XXXVII., is a mathematical demonstration of a property of certain curves, bearing on the forms of the slopes of wave-crests. In conclusion, the Editor desires to acknowledge the courtesy shown by the executive officers of the various Societies and Journals to which the papers, thus brought together, were originally contributed: and to express his thanks — not only for the permission, readily accorded, to republish — but also for the kindness which supplied, in many instances, copies of the Papers selected. W. J. MILLAP. Glasgow, October, 18S0. CONTENTS. PART I. PAPERS RELATING TO TEMPERATURE, ELASTICITY, AND EXPANSION OF VAPOURS, LIQUIDS, AND SOLIDS. I'AGE I. ON AN EQUATION BETWEEN THE TEMPERATURE AND THE MAXIMUM ELASTICITY OF STEAM AND OTHEE VAPOURS. Table of Vapour of Water— Formulae for calculating Temperature and Pressure — Values of Constants— Tables of Vapour of Alcohol, Ether, Turpentine, Petroleum, and Mercury, 1 — I'- ll. ON A FORMULA FOR CALCULATING THE EXPANSION OF LIQUIDS BY HEAT. Table of Constants — Tables of Expansion of Water, Mercury, Alcohol, and Sulphuret of Carbon, ........... 13 — 15 III. ON THE CENTRIFUGAL THEORY OF ELASTICITY, AS APPLIED TO GASES AND VAPOURS. Hypothesis of Molecular Vortices— Relations between the Heat and Elasticity of a Gaseous Body— Temperature— Specific Heat— Coefficients of Elasticity and Dilatation of Gases — Table of Coefficients of Dilatation- Elasticity of Vapour, and Tables of Constants for the same— Mixtures of Gases and Vapours, 16— 4S IV. ON THE CENTRIFUGAL THEORY OF ELASTICITY AND ITS CONNECTION WITH THE THEORY OF HEAT. Relations between Heat and Expansive Pressure— Hypothesis of Molecular Vortices— Elastic Pressures— Temperature and Specific Heat— Heat and Expansive Power— Latent Heat— Joule's Law— Carnot's Law, . . 49—66 V. LAWS OF THE ELASTICITY OF SOLID BODIES. Science of Elasticity as applied to Strength of Structures— Strains and Molecular Pressures— Homogeneous Bodies— Eigidity— Hypothesis of Atomic Centres— Hypothesis of Molecular Vortices— Coefficients of Pli- ability, Extensibility, and Compressibility— Modulus of Elasticity- Tables of Coefficients of Elasticity— Rigidity, Extensibility, and Com- pressibility—Application of Method of Virtual Velocities to the Theory of Elasticity— Proof of the Laws of Elasticity by the Method of Virtual Velocities, 67-118 XIV CONTENTS. VI. ON AXES OF ELASTICITY AND CRYSTALLINE FORMS. PAGE Definition of Axes of Elasticity — Strain — Stress -Potential Energy — Coefficients of Elasticity— Symmetry— Polarised J ,ight, .... 110 149 VII. ON THE VIBRATIONS OE PLANE-POLARISED LIGHT. Theories — Hypothesis of Molecular Vortices Experiments on Light, . 150—155 VIII. GENERAL VIEW OE AN OSCILLATORY THEORY OE LIGHT. Various Hypotheses — Polarised Light— Hypothesis of Oscillations — Diffrac- tion — Wave-Surface in Crystalline Bodies Reflexion — Refraction — Dispersion, 150 — 107 IX. ON THE VELO( [TY OF SOUND IX LIQUID AXI) SOLID BODIES OF LIMITED DIMENSIONS, ESPECIALLY ALONG PRISMATIC MASSES OF LIQUID. Application to Elasticity of Materials— Vibratory Movement in Homo- ueous Bodies Transmission of a Definite Musical Tone— Propagation of Sound, 168—199 PART II. PAPERS RELATING TO ENERGY AND ITS TRANSFORMATIONS, THERMODYNAMICS, MECHANICAL ACTION OF HEAT TN THE STEAM-ENGINE, .\r. X. ON THE RECONOENTRATION OF THE MECHANICAL ENERGY OF THE UNIVERSE. Convertibility of the different kinds of Physical Energy — Interstellar Medium— Diffusion, an 1 probable Reconcentration of Radiant Heat, . 200—202 XL ON THE GENERAL LAW OF THE TRANSFORMATION OF ENERGY. Actual or Sensible Energy— Potential or Latent Energy— Conservation of Energy— Transformation of Energy, 203—208 XII. OUTLINES OF THE SCIENCE OF ENERGETICS. Physical Theory— Science of Mechanics— Mechanical Hypotheses— Science of Energetics— Substance— Property— Mass— Accident— Effort— Work- Equivalence of Energy and Work— Equality of Action and Reaction- Potential Equilibrium— Transformation of Energy — Rate of Transforma- tion — Metabatic Function — Carnot's Function— Efficiency of Engines- Diffusion of Actual Energy— "Frictional Phenomena "—Measurement of Time, 209—228 CONTENTS. XV XIII. ON THE PHRASE " POTENTIAL ENERGY," AND ON THE DEFINITIONS OF PHYSICAL QUANTITIES. PAGE Potential Energy— Sir John HerschePs Views— Newton's Method— Force — The word "Energy" substituted by Dr. Thomas Young for Vis-viva— Extension of Application by Sir William Thomson— Qualification of the Noun "Energy" by the Adjectives "Actual" and "Potential," pro- posed by Eankine, and adopted by others— Carnot's "Force Vivi Virtudlc "—Definitions of the Measurement of Time, Force, and Mass, . 229—233 XIV. ON THE MECHANICAL ACTION OF HEAT, ESPECIALLY IN GASES AND VAPOURS. Hypothesis of Molecular Vortices— Mutual Conversion of Heat and Expan- sive Power— Joule's Experiments— Latent Heat— Conservation of Vis- viva— Carnot's Theory of Heat— Specific Heat— Thermal Unit— Total Heat of Evaporation— Power of the Steam-Engine— Pambour's Principle — Numerical Examples from Cornish Engines — Efficiency of Engines- Tables of Pressure and Volume of Steam, 234-284 XV. NOTE AS TO THE DYNAMICAL EQUIVALENT OF TEM- PERATURE IN LIQUID WATER, AND THE SPECIFIC HEAT OF ATMOSPHERIC AIR AND STEAM : Being a Supple- ment to a Paper On the Mechanical Action of Heat. Joule's Experiments— Specific Heat of Air and Steam, .... 285—287 XVI. ON THE POWER AND ECONOMY OF SINGLE-ACTING EXPANSIVE STEAM-ENGINES: Being a Supplement to the Fourth Section of a Paper On the Mechanical Action of Heat. Cornish Engines — Wicksteed's Experiments, 288— 299 XVII. ON THE ECONOMY OF HEAT IN EXPANSIVE MACHINES. Carnot's Law— Clausius and Thomson's Investigations— Action of Steam in Engine, 300—306 XVIII. ON THE ABSOLUTE ZERO OF THE PERFECT GAS THERMOMETER. Measurement of Temperature— Absolute Zero of Temperature, . . . 307-309 XIX. ON THE MECHANICAL ACTION OF HEAT. Properties of Expansive Heat— Heat Potential— Properties of Temperature —Hypothesis of Molecular Vortices— Sir Humphry Davy's Researches- Thermic Phenomena of Currents of Elastic Fluids, 310—338 XX. ON THE GEOMETRICAL REPRESENTATION OF THE EXPANSIVE ACTION OF HEAT, AND THE THEORY OF THERMODYNAMIC ENGINES. Introduction of Indicator-Diagram and Diagrams of Energy— Isothermal Curves— Curves of No Transmission of Heat— Total Actual Heat— Latent Heat of Expansion— Law of the Transformation of Energy— Thermo- dynamic Functions— Curves of Free Expansion— Thomson and Joule's Experiments— Velocity of Sound— Efficiency of Thermodynamic Engines XVI CONTENTS. PAGB — Use of Economiser or Regenerator— Stirling and Ericsson's Air-Engines Hypothesis of Molecular "Vortices— Relation between Temperature and Actual Heat — Efficiency of Air-Engines- Liquefaction of Vapour by Ex- pansion under Pressure — Efficiency of a Vapour-Engine — Heat in a I 'ound of Coal — Wire-drawn Steam — Composite Vapour-Engines- < iurves of Eree Expansion for Nascent Vapour- -Total Heat of Evaporation — Approximate Law for a Vapour which is a Perfect Gas, .... 339—409 XXI. ON FORMULAE FOB THE MAXIMUM PRESSURE AND LATENT HEAT OF VAPOURS. Elasticity of Vapours— Regnault's Experiments— Comparison of Formulae with Experiments, 410—416 XXII. ON THE DENSITY OF STEAM. General Equation of Thermodynamics, and its Application to the Latent Heat and Density of Steam Fairbairn and Tate's Experiments, . . 417—426 XXIII. ON THE SECOND LAW OF THERMODYNAMICS. First Law Defined Second Law Defined Absolute Temperature — Steady Motion — Metamorphic Function, 427 — 431 XXIV. ON THE WANT OF POPULAB [LLUSTRATIONS OF THE SECOND LAW OF THERMODYNAMICS. First and Second Laws— Exl rnaland Internal Wori r ofaPer- fect Heat Engine— Hypothe i of Molecular Vortices, .... 432—438 XXV. EXAMPLES OF THE APPLICATION OF THE SECOND LAW OF THERMODYNAMICS TO A PERFECT STEAM-ENGINE AND A PERFECT AIR-ENGINE. Definition of a "Perfect Engin B aerator— Examples from Steam- Engine and Air-Engine Economical Action of Regenerator in anAir- — ( iomparison of Economy of Steam and Air-En in< I [eat due to Combustion of a Pound of Coal, 139 — 153 XXVI. OX THE WORKING OF STEAM IN COMPOUND ENGINES. Principal Kinds of Compound Engines, and their Advantages— Number of Cylinders used— Diagrams of the High and Low Pressure Cylinders- Rates of Expansion— Theoretical Expansion Diagram— The Common Hyperbola a good Approximation to the true Expansion Curve — Calcula- tion of Mean Absolute Pressure and of Indicated Work — Theoretical Diagrams of Compound Engines with and without Reservoirs, . . ;."J4 — 4G3 XXVII. OX THE THEORY OE EXPLOSIVE GAS-ENGINES. Thermodynamical Propositions — Rules as to Heat and Expansion— Indicator Diagram of an Explosive Engine — Total and Available Heat of Explosion — Mixture of Air and ( J-as— Adiabatic Curves— Indicated Work— Efficiency, 4G4— 470 XXVIII. OX THE EXPLOSIVE EXERGY OF HEATED LIQUIDS. General Formulas for all Fluids — Efficiency of Projection— Full-pressure Steam Gun— Expenditure of Water and Heat-Efficiency— Full- pressure Dry Steam Gun, • 471-480 CONTENTS. PART III. PAPERS RELATING TO WAVE FOP MS, PROPULSION OF VESSELS, STABILITY OF STRUCTURES, ; \r. XXIX. OX THE EXACT FORM OF WAVES NEAR THE SURFACE OF DEEP WATER. PAGE Trochoidal Form of Waves first stated by Mr. Scott Russell— Demonstra- tion of Trochoidal Form— Height of Crests— Cycloidal Waves— Friction between a Wave and a Wave-shaped Solid— Mr. Stokes' Investigations, . 481—494 XXX. ON PLANE WATER-LINES IN TWO DIMENSIONS. Plane Water-Lines Defined— Flow of Liquid past a Solid— ^Water-Line Functions— Water- Line Curves generated by Circles and Ovals— Velocities of Gliding— Lines of Smoothest Gliding— Orbits of the Particles of Water —Trajectory of Transverse Displacement— General Problem of the Water- Line of Least Friction— Previous Systems of Water-Lines— Chapman's System— Scott Russell's System— Preferable Figures of Water-Lines— Lissoneo'ids compared with Trochoids— Combination of Bow and Stern- Provisional Formula for Resistance • • 495— 5_1 XXXI. ELEMENTARY DEMONSTRATIONS OF PRINCIPLES RELATING TO STREAM-LINES. Stream-Lines Explained— Condition of Perfect Fluidity— Circular Stream- Lines— Straight Stream-Lines— Compound Stream-Lines, . . . 522—529 XXXII. ON THE THERMODYNAMIC THEORY OF WAVES OF FINITE LONGITUDINAL DISTURBANCE. Mass- Velocity— Waves of Sudden Disturbance— Thermodynamic Conditii his — Adiabatic State— Absolute Temperature— Previous Investigations, . 530 - 543 XXXIII. ON THE THEORETICAL LIMIT OF THE EFFICIENCY OF PROPELLERS. Slip, Reaction, and Efficiency of a Propeller— Energy Expended— Useful and Lost Work— Forms of Screw Propellers— Examples from Practice, . 544-54'.) XXXIV. REPORT ON THE DESIGN" AND CONSTRUCTION OF MASONRY DAMS. Material and Mode of Building— Form and Limits of Pressure— Mathemati- cal Principles of the Profile Curves, 550 501 XXXV. ON THE APPLICATION OF BARYCEXTRIC PERSPEC- TIVE TO THE TRANSFORMATION OF STRUCTURES. Parallel Projection — Application to Skew Arches, &c, .... 562—563 XXXVI. PRINCIPLE OF THE EQUILIBRIUM OF POLYHEDRAL FRAMES, ~ lM XXXVII. ON A PROPERTY OF CURVES FULFILLING THE CONDITION^ + 5^ = 0, 5G5 - n6T MEMOIE. The life of a genuine scientific man is, from the common point of view, almost always uneventful. Engrossed with the paramount claims of inquiries raised high above the domain of mere human passions, he is with difficulty tempted to come forward in political discussions, even when they are of national importance ; and he regards with surprise, if not with contempt, the petty municipal squabbles in which local notoriety is so eagerly sought. To him the discovery of a new law of nature, or even of a new experimental fact, or the invention of a novel mathematical method, no matter who has been the first to reach it, is an event of an order altogether different from, and higher than, those which are so profusely chronicled in the newspapers. It is something true and good for ever, not a mere temporary outcome of craft or expediency. "With few exceptions, such men pass through life unnoticed by, almost unknown to, the mass of even their educated countrymen. Yet it is they who, far more than any autocrats or statesmen, are really moulding the history of the times to come. Man has been left entirely to himself in the struggle for creature comforts, as well as for the higher appliances which advance civilisation ; and it is to science, and not to so-called statecraft, that he must look for such things. Science can and does provide the means; statecraft can but more or less judiciously promote, regulate, or forbid their use or abuse. One is the lavish and utterly unselfish furnisher of material good, the other the too often churlish and ignorant dispenser of it. In the moral world their analogues are charity and the relieving officer! So much it is necessary to say for the sake of the general reader; to the world of science no apology need be made. In it Rankine's was and is a well-known name. It is high eulogy, but strictly correct, to say that Eankine holds a prominent place among the chief scientific men of the last half century. He was one of the little group of thinkers to whom, after the wondrous Sadi Carnot, the world is indebted for the pure science of modern thernio- xx MEMOIR. dynamics. "Were this all, it would be undoubtedly much. But his services to applied science -were relatively even greater. By his admirable teaching, his excellent text books, and his original memoirs, he has done more than any other man of recent times for the advancement of British Scientific Engineering. He did not, indeed, himself design or construct gigantic structures, but he taught, or was the means of teaching, that invaluable class of men to whom the projectors of such works entrust the calculations on which their safety as well as their efficiency mainly depend. For behind the great architect or engineer, and concealed by his portentous form, there is the real worker, without whom failure would be certain. The public knows but little of such men. Not every von Moltke has his services publicly acknowledged and rewarded by his imperial employer! But he who makes possible the existence of such men confers lasting benefit on his country. And it is quite certain that Rankine achieved the task. William John Macquorn Rankine was born in Edinburgh on the nth July, 1820. lie was a Scot of Scots. His father was descended from the Rankines of Carrick and the Cochranes of Dundonald. His maternal grandfather was Grahame of Drumquhassle, a descendant of tin; Grahams of Dougalston. In Rankine's .MSS. there is to be found a tracing of the various steps of his pedigree from Robert the Bruce. His father, David Rankine, was in youth Lieutenant in the 21st Regimenl (RiHc Brigade) ; but, as will be seen by what follows, was also a man of great general information, especially in practical matters. He was employed in later life in constructing railways, ami afterwards became Secretary to the Caledonian Railway Company. Rankine repeatedly notes in his journal the hints and instruction he had received from his father. He was profoundly attached to his parents; and one of the most touching notes in his journal is the brief record of his lasting obligations to them for early instruction in the fundamental principles of the Christian religion and the character of its Founder. It will be convenient to give, in the first place, a brief sketch of Rankine's career, and to reserve for a time such comments upon his more important investigations and treatises as would materially interfere with the continuity of the sketch. From Rankine's private journal it appears that his first introduction to arithmetic and elementary mathematics, mechanics, and physics was obtained mainly from his father. He attended Ayr Academy in 1828-9, and Glasgow High School in 1830. After this he seems to have been for some years privately instructed in Edinburgh, his state of health preventing his being sent to a public school. In December, 1831:, his uncle, Archibald : MEMOIR. XX i Grahame, presented him with a copy of Newton's Frincijiia, which he read carefully. He remarks — " This was the foundation of my knowledge of the higher mathematics, dynamics, and physics. My knowledge of the higher mathematics was obtained chiefly by private study." About this period he paid much attention to the theory of music. In 183G he studied Practical and Theoretical Chemistry under David Boswell Eeid ; and in November of the same year entered Edinburgh University. He there attended the Natural Philosophy course under Professor Forbes, and gained (before completing his seventeenth year) the Gold Medal for an essay on the "Undulatory Theory of Light." In the summer of 1837, he studied Natural History under Professor Jameson, and Botany under Professor Graham. He attended the Natural Philosophy Class a second time in 1S37-8, and obtained an extra prize for an essay on " Methods in Physical Investigation." He records in his journal, that in 1836-8, during leisure, he read much metaphysics, chiefly Aristotle, Locke, Hume, Stewart, and Degerando. I have learned from himself that about this period he " wasted " a great deal of time in the fascinating but too often delusive pursuit of " Theory of Numbers." In 1837-8 he made his first accjuaintance with the practice of engineer- ing, having assisted his father in superintending the works of the Leith branch of the Edinburgh and Dalkeith Eailway. In the latter year he became a pupil of the late Sir John Macneill, C.E., having among his fellow-pupils many who have since risen to eminence. His journal records the names of G. W. Hemans, J. W. Bazalgette, W. E. Le Fanu, Matthew Blackiston, John Moffat, and Jonas S. Stawell. During the succeeding four years he was employed by Macneill on various surveys, and schemes for river improvements, water-works, and harbours for Ireland. Also, for some time, on the Dublin and Drogheda Railway. "While engaged on this railway in 1841, he contrived the method of setting out curves " by chaining and angles at circumference combined," which has since been known as " Eankine's method." In 1842 appeared his first published work, a pamphlet entitled An Experimental Inquiry into the advantage of Cylindrical Wheels on Railways. This was based upon experiments suggested to him by his father, and carried out by them together. In the same year Queen Victoria visited Edinburgh for the first time, and Eankine was charged with the superintendence of the erection of the huge bonfire which blazed on the top of Arthur's Seat. He constructed it with radiating air passages under the fuel, and succeeded, as he com- placently records, in partially vitrifying the rock ! During this and the succeeding year he sent several papers to the Institute of Civil Engineers, for some of which prizes were awarded to him. He records that most of them were based on suggestions by his XX11 MEMOIR. tather, especially that on the " Fracture of Axles." He showed that such /'iclures arose through gradual deterioration or fatigue, involving the gradual extension inwards of a crack originating at a square-cut shoulder. In this paper the importance of continuity of form and fibre was first shown, and the hypothesis of spontaneous crystallisation Avas disproved. In 1844-5 he was employed under Locke and Errington on the Clydes- dale Junction Railway project; and afterwards, till 1S48, on various schemes promoted by the Caledonian Railway Company. In 1845-6 he engineered a project for Edinburgh and Leith Water-works, which was defeated by the opposition of the Edinburgh AVater Company. About 1848 he seems to have commenced that extensive series of researches on molecular physics which occupied him at intervals during the rest of his life, and which constitutes bis chief claim to distinction in the domain of pure science. The first paper he published on the subject, with the title "Elasticity of Steam," appeared in the Edinburgh New Philosophical Journal in July, 1849; and at the end of that year he sent to the Royal Society of Edinburgh his great paper on the "Mechanical Action of Heat," It was not, however, read to the Society till February, 1850. On the contents of this, and his subsequent papers dealing with similar subjects, some remarks will lie made below. In duly, 1850, he read to the British Association at Edinburgh another paper on a closely connected subject, "Elastic Solids." In 1852 the Loch Katrine Water-works scheme for the supply of Glasgow was revived by Rankine and John Thomson. This scheme, now successfully carried out, was first proposed by Lewis Cordon and Laurence Hill, Junior. In 1853, one of Rankine's most characteristic papers in pure science, •• On the General Law of Transformation of Energy," was read by him to the Glasgow Philosophical Society. In the same year, along with the late J. R. Xapier, he projected and patented a new form of air-engine. The patent was afterwards abandoned. He was now elected to the Fellowship of the Royal Society, and sent to that body his next great paper on Thermodynamics — viz., " On the Geometrical Representation of the Expansive Action of Heat," which was printed in the Philosophical Transactions. From January to April, 1855, he acted in Glasgow University as substi- tute for Professor Lewis Gordon, on whose resignation he was appointed to the Chair of Engineering, which he held till his death. His inaugural discourse, delivered on Dec. 10 of the last-mentioned year, bore the title " De concordld inter Scientiarum Machinalium Contem/plationem et Usum." In this year he wrote, among several contributions to Kichol's Cyclopcedia, an article on " Heat, Mechanical Action of," the earliest formal treatise on Thermodynamics in the English language. In 1856, the preparation of memoir. xxiii his course of lectures led him to the invention of some remarkable methods connected with Transformation of Structures. These are based on the discovery of " reciprocal diagrams " of frames and forces, since greatly extended and simplified by Clerk-Maxwell. The remarkable storm which occurred on February 7 of this year, directed his inquiries to the " Stability of Chimneys," on which he has published a valuable article. In 1857, he resigned the associateship of the Institute of Civil Engineers ; and shortly afterwards, on the establishment of the Institute of Engineers in Scotland, he delivered the opening address as first President. At this time he was busily engaged on a Treatise on Ship- building, his Manual of Applied Mechanics, an article on the same subject for the Encyclopedia Britannica, and an investigation (based on J. R. Napier's experiments) of the theory of skin-resistance of ships. He also sent to the French Academy of Sciences a memoir, — " Be TE<£uililre intdrieure d'un corps solide, elastigue, ct homogene." In July, 1859, an offer of service was sent to the Lieutenancy by the " Glasgow University Rifle Volunteers." It was accepted in October, and Rankine received his commission as Captain. He spent the greater part of November at the Hythe School of Musketry, and, on his return, instructed the officers and sergeants of his corps. In the same year appeared his valuable Ma mud of Ihc Steam-Engine and other Prime Movers. In 1860, he was made Senior Major, and commanded the second battalion of his regiment at the memorable Volunteer Review, held by the Queen in the Queen's Park, Edinburgh. In 1861, he finished his Manual of Civil Engineering, which was pub- lished early in the following year. At the International Exhibition in 1862, at London, he was a Juror in Class VIII, " Machinery in General." In 1864, he resigned his commission in the Volunteers, "finding it impossible to attend at once to duties as field-officer and as professor, to engineering business, and to literary work, especially Treatise on Ship- building." In 1865, he was appointed Consulting Engineer to the Highland and Agricultural Society of Scotland ; and became a regular contributor to the Engineer, in which many excellent articles of his appeared. In 1866, was published his Treatise em Shipbuilding, Theoretical and Practiced. Though four names were announced on the title page as joint authors, by far the greater part of the work was written by Rankine; but the proofs were revised by all four. 1869 produced Machinery and Millwork, the fourth of Rankine's great engineering treatises. The other three had then reached their eighth, sixth, and fifth editions respectively. In his journal for this year occurs the following note : — " Sept. 16. Thomas Graham, Master of the Mint, died [son of a cousin xxiv MEMOIR. of my mother's father], I applied for vacancy to Chancellor of Exchequer (Lowe). Application well supported by friends, and civilly received ; but the office was virtually abolished, being conjoined with the Chancellorship of the Exchequer." Eankine lost his father in 1870, and his mother in the following year. Both had passed the age of seventy. The loss of his parents, to whom he was so fondly attached, seems to have accelerated the development of the illness which had for some j - ears been growing upon him. He was well enough, in 1871, to contribute most valuable matter to the proceedings of the t; Committee on Designs for Ships of War," which was appointed shortly after the loss of the " ( laptain." He investigated for that Committee the "Stability of Unmasted Ships of Low Freeboard," and the "Stability of Ships under Canvas." In February, 1872, Rankine completed his memoir of his friend, John Elder, and in July reported on the cause of the explosion of the Tradeston Flour Mill. In May the increase of the endowment of his chair, which he had in vain sought from Government, was given by Mr. Elder's widow; and the income of Rankine's post was at last made sufficient to maintain its occupant. But by this time his energy was fast failing, the simplest work fatigued him; and he died on December 24, of a general decline rather than of any special disease. He had been for some years liable to violent headaches, and towards the close of his life these affected his sight. They were probably symptoms of heart disease, which ultimately developed paralysis. The gradual decay of his physical powers is painfully evident in the last pages of his journal, where, though the substance is correct and to the point, the handwriting becomes more and more irregular at each succeeding entry. Such are the more prominent events in the life of this great and good man. Even now, after the lapse of eight years, it is difficult to realise the fact of his death. He was so many-sided, and yet so complete in himself, that the mental image of him formed by each of his friends remains almost as clear and distinct as if it had been formed but a few days ago. Of the man himself it is not easy to speak in terms which, to a stranger, would appear unexaggerated. His appearance was striking and prepos- sessing in the extreme, and his courtesy resembled almost that of a gentle- man of the old school. His musical taste had been highly cultivated, and it was always exceedingly pleasant to see him take his seat at the piano to accompany himself as he sang some humorous or grotesquely plaintive song — words and music alike being generally of his own composition. Some of the best of these songs have been collected in a small volume, Songs and Fables (Second Edition; Glasgow, Maclehose, 1874). We MEMOIR. XXV extract one which give-, in a very telling form, one point of view of a much-debated semi-scientific question : — THE THEEE-FOOT EULE. When I was hound apprentice, and learned to use my hands, Folk never talked of measures that came from foreign lands : Xow I 'm a British workman, too old to go to school ; So whether the chisel or file I hold, I '11 stick to my three-foot rule. Some talk of millimetres, and some of kilogrammes, And some of decilitres, to measure beer and drams ; But I 'm a British workman, too old to go to school ; So by pounds I '11 eat, and by quarts I '11 drink, and I '11 work by my three-foot rule. A party of astronomers went measuring of the earth, And forty million metres they took to be its girth ; Five hundred million inches, though, go through from pole to pole ; So let 's stick to inches, feet, and yards, and the good old three-foot rule. The great Egyptian pyramid 's a thousand yards about ; And when the masons finished it, they raised a joyful shout ; The chap that planned that building, I 'm bound he was no fool ; And now 'tis proved beyond a doubt, he used a three-foot rule. Here 's a health to every learned man that goes by common sense, And would not plague the workman on any vain pretence ; But as for those philanthropists, who 'd send us back to school, Oh, bless their eyes, if ever they tries to put down the three-foot rule. When the "Red Lions" met during the British Association week of 1871, in Edinburgh, Eankine was hailed with universal acclaim as the Lion-King. His versatility in that singular post was very much akin to that of Pro- fessor Edward Forbes; though their paths in science were widely different. His conversation was always interesting, and embraced with equal seeming ease all topics, however various. He had the still rarer qualification of being a good listener also. The evident interest which he took in all that was said to him had a most reassuring effect on the speaker; and he could turn without apparent mental effort from the prattle of young children to the most formidable statement of new results in mathematical or physical science. Then his note-book was at once produced, and in a few lines he xxvi _ MEMOIR. jotted down the essence of the statement, to be pondered over at leisure, provided it did not ut once appear to him how it was to be verified. The questions which he asked on such occasions were always almost startlingly to the point, and showed a rapidity of thought not often met with in minds of such calibre as his, where the mental inertia, which enables them tn overcome obstacles often prevents their being quickly set in motion. His kindness, shown in the readiness with which he undertook to read proof sheets tor a friend, or even to contribute a portion of a chapter (where the subject was one to which he had paid special attention), was. for a man so constantly at work, absolutely astonishing. The writer of this brief notice has several times availed himself of such assistance. It was given almost as soon as asked, and it was invariably of sterling value. Nothing is more precious to a writer on scientific subjects (especially when questions of priority are involved ) than the assistance of a friendly -though, if necessary, seven — critic, such as was Rankine. AVe must not refrain from pointing out, in connection with his scientific merits, how very good and how exemplary for scientific writers and investigators his character was. I Ie was ambitious ; that is obvious from the number and variety of his books and papers, and the quite unnecessary display of symbols in several of his less popular writings. But he was tie' very soul of honour in respect to giving all credit to others, ami in never attempting in anything, small or great, to go a hairbreadth beyond the line of right as to his own claims. He showed a particularly good and generous temper in cases of difference on scientific questions — a temper which proved the true metal, unalloyed by any mean quality. Rankine was, in many subject-., an almost self-taughl man, and the direction of his earlier scientific wort seems not to have been a very pro- fitable one. But, once on the right tack, his progress became very rapid. Every mathematician worthy of the name has made himself: some, as Rowan Hamilton, by attacking at an early age the grander works of Lagrange and Laplace; others by attempting original flights without the assistance of books. Rankine published only one or two papers on sub- jects of pure mathematics; ami even these, though not containing any direct allusion to physics, were connected somewhat closely with kine- matical or physical investigations, such as the deformation of an elastic solid. The number of Rankine's scientific papers seems absolutely enormous, when we consider the minute and scrupulous care with which he attended to every point of detail in the writing and printing of them. How he managed, in addition to these, to find time for the composition of his many massive (not heavy) and elaborate volumes — all marked with the most striking stamp of originality — for his memoirs, and his almost weekly communications to The Engineer and other professional papers, must memoir. xxvii always remain matter for conjecture. In the Royal Society's splendid Catalogue of Scientific Paper* we find that from 1843 to 1872 (both inclusive) he published, in recognised scientific journals alone, upwards of a hundred and fifty papers— many of these being exhaustive essays on mathematical or physical questions, and all, save one or two, contain- ing genuine contributions to the advance of science. Leaving out of account the more strictly professional of these papers, we find among the titles of the rest such heads as the following : — Molecular Vortices, Elasticity of Solids, Isorrhopic Axes, Compressibility of Water at Different Temperatures, Centrifugal Theory of Elasticity, Oscillatory Theory of Light, (!. neral Law of Transformation of Energy, Plane Water-Lines, Oogenous Neo'ids. To indicate even briefly the nature and importance of the varied contents of these papers alone, would require vastly more time and space than are at present at our disposal. The more important of them are included in the present volume; others of less importance, or of less characterised originality, may be consulted by the reader in the scientific publications where they originally appeared. Unquestionably the greatest pure scientific work of Rankine'sis contained in his numerous papers bearing on the Dynamical Theory of Heat, and on Energy generally. As Sir "William Thomson has remarked, even the mere title of his earliest paper on this subject, "Molecular Vortices," is an important contribution to physical science. The mode in which Rankine, in 1819, attacked the true theory of heat, which had just been recalled to the attention of scientific men by the admirable experiments and numerical determinations of Joule, was quite different from that adopted by any one of his concurrents; and though objections may fairly be raised to certain parts, even his first paper constituted a remarkable contribu- tion to our physical knowledge. The essential characteristic of his method is the introduction of a hypothesis as to the nature of the motions and displacements (of the ultimate parts of bodies) upon which temper- ature depends, and in which heat, whether latent or sensible, consists. He thought it necessary to defend this mode of investigation, and did so in a remarkable address to the Philosophical Society of Glasgow, from which we extract the following passages, which are valuable not alone from their intention, but also from the insight^they give us into the character of the man : — " In order to establish that degree of probability which warrants the reception of a hypothesis into science, it is not sufficient that there should be a mere loose and general- agreement between its results and those of experiment. Any ingenious and imaginative person can frame such hypotheses by the dozen. The agreement should be mathematically exact to that degree of precision which the uncertainty of experimental data renders possible, and should be tested in particular cases by numerical XXVlll MEMOIR. calculation. The highest degree of probability is attained when a hypo- thesis leads to the prediction of laws, phenomena, and numerical results which are afterwards verified by experiment" as when the wave-theory of light led to the prediction of the true velocity of light in refracting media, of the circular polarisation of light by reflexion, and of the previously unknown phenomena of conical and cylindrical refraction; and as when the hypothesis of atoms in chemistry led to the prediction of the exact proportions of the constituents of innumerable compounds. I think I am justified in claiming for the hypothesis of mole- cular vortices, as a means of advancing the theory of the mechanical action of heat, the merit of having fulfilled the proper purposes of a Mechanical hypothesis in physical science, which are to connect the laws of molecular phenomena by analogy with the laws of motion, and to suggest principles such as the second law of thermodynamics and the laws of the elasticity of imperfect gases, whoso conformity to fact may after- wards be tested by direct experiment. And I make that claim the more confidently, that 1 conceive the hypothesis in question to be in a great measure the development, and the reduction to a precise form, of ideas concerning the molecular condition which constitutes heat, that have been entertained from a remote period by the leading minds in physical science . . . I wish it, however, to be clearly understood, that although I attach great value and importance to sound mechanical hypotheses as means of advancing physical science, 1 firmly hold that they can never attain the certainty of observed facts; and accordingly, I have laboured assiduously to show that the two laws of thermodynamics are demon- strable as facts, independently of any hypothesis; and in treating of the [aactical application of those laws, I have avoided all reference to hypothesis whatsoever." The application of the doctrine, that heat "ml work an ble, to the discovery of new relations among the properties of bodies, was made about the same time by three scientific men — W. Thomson, Rankine, and Clausius. Of these, Thomson cleared the way for the new theory by his account of the almost forgotten work of Carnot on the " Motive Power of Heat." This excessively important investigation was published in 182-f, when the world of science was not prepared for its reception, and had been allowed to drop out of notice. Thomson gave a very full abstract of its contents in the Transacts ras of the Royal Society of Edinburgh, 1849, and pointed out that they would require modification if the new theory were adopted, as Carnot had throughout assumed that heat is a substance, and therefore indestructible. He showed, besides, that Carnot's method was capable of giving an absolute definition of temperature ; independent, that is, of the properties of any particular substance. He also experimentally MEMOIR. xxix verified a deduction made by his brother, James Thomson (from Carnot's theory), as to the alteration of the freezing point of water by pressure. Eankine (late in 1849) and Clausius (early in 1850) took the first step towards the formation of a true theory of the action of heat on bodies, by showing (by perfectly different modes of attacking the question) the nature of the modifications which Carnot's theory required. The recent publica- tion of Carnot's MSS. proves that that verj r remarkable man had himself recognised the necessity for such modifications (and had all but succeeded in making them) before his premature death. Thomson, in 1851, put the foundations of the theory in the form they have since retained. In RanMne's paper of 1849, he applied the theory to the determination of the relation between the latent heat of steam and its density, and made a very noteworthy prediction of the true value of the specific heat of air, at a time when the experimental results which were considered the best were far from the truth. [Rankine's results were soon after verified by the experimental researches of Joule and Eegnault.] He also showed that saturated steam, pressing out a piston in a vessel impervious to heat, must cool so as to keep constantly at the temperature of satura- tion ; and that, besides, a portion of it licmefies. A very excellent statement of the claims of Eankine in thermodynamics is given in the following cmotation from an article by Clerk-Maxwell {Nature, 1878, Vol. XVIL, p. 257) :— " Of the three founders of theoretical thermodynamics, Eankine availed himself to the greatest extent of the scientific use of the imagination. His imagination, however, though amply luxuriant, was strictly scientific. Whatever he imagined about molecular vortices, with their nuclei and atmospheres, was so clearly imaged in his mind's eye, that he, as a practical engineer, could see how it would work. "However intricate, therefore, the machinery might be which he imagined to exist in the minute parts of bodies, there was no danger of his going on to explain natural phenomena by any mode of action of this machinery which was not consistent with the general laws of mechanism. Hence, though the construction and distribution of his vortices may seem to us as complicated and arbitrary as the Cartesian system, his final deductions are simple, necessary, and consistent with facts. " Certain phenomena were to be explained. Eankine set himself to imagine the mechanism by which they might be produced. Being an accomplished engineer, he succeeded in specifying a particular arrangement of mechanism competent to do the work, and also in predicting other properties of the mechanism which were afterwards found to be consistent with observed facts. "As long as the training of the naturalist enables him to trace the action only of particular material systems, without giving him the power of XXX MEMOIR. dealing with the general properties of all such systems, he must proceed by the method so often described in histories of science — he must imagine model after model of hypothetical apparatus, till he finds one which will do the required work. If this apparatus should afterwards be found capable of accounting for many of the known phenomena, and not demon- strably inconsistent with any of them, he is strongly tempted to conclude that his hypothesis is a fact, at least until an equally good rival hypothesis has been invented. Thus Elankine,* long after an explanation of the properties of gases had been founded on the theory of the collisions of molecules, published what he supposed to be a proof that the phenomena of heat were invariably due to steady closed streams of continuous fluid matter. "The scientific career of Rankine was marked by the gradual develop- ment of a singular power of bringing the most difficult investigations within the range of elementary methods. In his earlier papers, indeed, 1m- appears as if battling with chaos, as he swims, or sinks, or wades, or creeps, or flies, ' Ami through the palpable obscure finds out I Ik uncouth way •.' but he soon begins to pave a broad and beaten way over the dark abyss; and his Latest writings -how such a power of bridging over the diiliculties oi science, that his premature death must have been almost as great a loss to the diflusion of Bcience as it was to its advancement. "The chapter on thermodynamics in his book on the steam-engine was the first published treatise on the subject, and is the only expression of his views addressed directly to students. " In (hi- boot he has disencumbered himself to a great extent of the hypothesis of molecular vortices, and builds principally on observed facts, though he, in common with Clausius, makes several assumptions, some expressed as axioms, others implied in definitions, which seem to us anything but self-evident. A- an example of Kankine's best style Ave may .take the following definition : — "'A PERFECT Gas is a substance in such a condition that the total pressure exerted by any number oi' portions of it, at a given temperature, against the sides of a vessel in which they are enclosed, is the sum of the pressures which each portion would exert if enclosed in the vessel separately at the same temperature.' " Here Ave can form a distinct conception of every clause of the definition; but when we come to Eankine's Second Law of Thermodynamics we find * "On the Second Law of Thermodynamics," Phil. Mo;/., Oct., 1865, § 12, p. 244 ; but in his paper "On the Thermal Energy of Molecular Vortices," Trans. U.S. Edin., XX \"., p. 557 (1S(50), he admits that the explanation of gaseous pressure by the impacts of molecules has beeu proved to be possible. MEMOIE. XXXI that though, as to literary form, it seems cast in the same mould, its actual meaning is inscrutable. " 'The Second Law of Thermodynamics. — If the total actual heat of a homogeneous and uniformly hot substance be conceived to be divided into any number of equal parts, the effects of those parts in causing work to be performed are equal.' " We find it difficult enough, even in 1878, to attach any distinct meaning to the total actual heat of a body, and still more to conceive this heat divided into equal parts, and to study the action of each of these parts; but as if our powers of deglutition were not yet sufficiently strained, Eankine follows this up with another statement of the same law, in which we have to assert our intuitive belief that " 'If the absolute temperature of any uniform^ hot substance be divided into any number of equal parts, the effects of those parts in causing work to be performed are equal.' " The student who thinks that he can form any idea of the meaning of this sentence is quite capable of explaining, on thermodynamical principles, what Mr. Tennyson says of the great Duke — ' Whose eighty wiuters freeze with one rebuke All great self-seekers trampling on the right.' " Ivankinc's researches on heat were for the most part connected, as we have already said, with a theory of the constitution of bodies, and a specula- tion as to the physical nature of a hot body, to which he gave the name of Theory of Molecular Vortices. In this theory, the invisibly small parts of bodies apparently at rest are supposed to be in a state of motion, the rapidity of which may be compared with that of a cannon ball. It was distinguished from other theories, which attribute motion to bodies apparently at rest, by the further assumption that this motion is like that of very small vortices, each whirling about its own axis, and that the centri- fugal force of this motion contributes to the elasticity of the body. A theory of a similar kind has since been applied by Clerk-Maxwell to the explanation of magnetic phenomena ; and Sir W. Thomson has made the rigorous investigation of vortices possible by his paper on " Vortex Motion," and has also contributed to the philosophy of speculation by his theory of " Vortex Atoms." Eankine's researches on the general theory of elastic bodies are charac- terised by the fact that while, in laying the foundation of the theory, he confines himself to the use of rigorous methods, and does not shrink from any mechanical difficulty in their application, he always prepares the way for the application of the results to practice, by making the definitions so clear, the methods so simple, the results so definite, that they can be XXXU MEMOIR. mastered by the exercise of a little thought, without special mathematical training. This quality is prominent also in his researches on fluid motion, three of which are of special importance. I. The theory of the propagation of waves, such as those of sound in elastic media, is generally supposed to belong to the most abstruse depart- ments of mathematical science. Even Newton made some oversights in liis investigation, and it required more than a century of hard mathematical development before the theory reached its present state — which is still very imperfect. Rankine, by the introduction of a few new conceptions in the elementary part of the investigations, has rendered it possible for any one acquainted with elementary dynamics to follow the theory up to the point at which it was left by Laplace ami almost as much further as it has yet been carried II. The tlnory of waves on the surface of water, when their height is not regarded as infinitely small, is still more difficult than that of sound waves. Stokes has, indeed, in a masterly series of investigations, arrived at a second, and for some purposes a third, approximation. An solution, however, but of a particular ease only, was arrived at by Rankine. He was not aware that it had been given by Gerstner in 1802, having been deduced from an assumption, generally erroneous, but true for this special case. Unfortunately, as this theory essentially involves rota- tion of fluid elements, it is not a solution of the usual problem of waves at the surface of a perfect liquid, for it implies a kind of motion which could not be produced in such a substance if originally in a state of rest. III. Rankine's third investigation is that of lines of motion of water flowing past a ship. lie begins with the mathematical theory of such lines, but soon applies his results to the determination of good forms of the "line-; of a ship, and the investigation of the principal causes of the resistance to the motion of the ship, and the means of diminishing that nice. Xo other person has done so much as either Rankine or William Froude to promote naval dynamics, and the application of science to the shaping of ships, and to the estimation of their performances. To Rankine, tin' Scientific Sub-Committee of the late Admiralty Com- mittee on Designs owed most of its reports, and a very large proportion of their effectiveness. Even those most disposed to disparage that Committee and its work, have made exception as regards the Reports of tlie Scientific Sub-Committee, and, in particular, Eankine's contributions. (It seems to us that only ignorance or unfavourable bias could attempt to disparage the committee at all ; for it undoubtedly did. though in an unostentatious manner, very good sendee indeed.) Eankine's works on Applied Mechanics, on the Steam-Engine, and on Engineering, contain many valuable and original methods ; and while the MEMOIK. xxxiii publication of any one of them would have established the fame of one of our average scientific men, that on the steam-engine could not have been produced by any but an original discoverer of a high order. Some of the investigations contained in this series of volumes are as remarkable for the material aid they afford to the man of practice as for the light they throw upon his work. The following gives a striking instance of Eankine's tact under a novel and somewhat puzzling combination of circumstances. In August, 1858, he wrote to the Philosophical Magazine the annexed short letter, which was printed in the September number of that journal : — " In the course of last year there were communicated to me, in con- fidence, the results of a great body of experiments on the engine power required to propel steamships of various sizes and figures, at various speeds. From those results I deduced a general formula for the resistance of ships, having such figures as usually occur in steamers, which, on the 23rd of December, 1857, I communicated to the owner of the experimental data, and he has since applied it to practice with complete success. " As the experimental data were given to me in confidence, I am for the present bound in honour not to disclose the formula which I deduced from them ; but as I am desirous not to delay longer the placing it upon record, I have recourse to the old fashion of sending it to you in the form of an anagram, in which the letters that occur in its verbal statement are arranged in alphabetical order, and the number of times that each letter occurs is expressed by figures:— 20 A. 4 B. 6 C. 9 D. 33 E. 8 F. 4 G. 1G H. 10 1. 5 L. 3 M. 15 N. 14 0. 4 P. 3 Q. 14 R. 13 S. 25 T. 4 U. 2 V. 2 W. 1 X. 4 Y. (219 letters in all). I hope I may soon be released from my present obligation to secrecy." There could be no doubt that this refers to a remarkable investigation which Eankine carried out for his friend James E. Xapier, who had asked him to estimate the horse-power necessary to propel at a given rate a vessel which he was about to construct. Guided by this consideration, I found, in 1872, the following sentence of Eankine's in The Civil Engineer and Architects Journal (October 1, 18G1), but without any reference what- ever to the anagram or to the Plulosopldcal Magazine : — " The resistance of a sharp-ended ship exceeds the resistance of a current of water of the same velocity in a channel of the same length and mean girth, by a quantity proportional to the square of the greatest breadth, divided by the square of the length of the bow and stern." Curiously enough, Eankine seems to have made an arithmetical mistake, or a mis-spelling, because this sentence exactly fits all of the above numerical data, with the exception that it contains just one E too much, and has, therefore, 220 letters in all. Eankine's private MSS., to which I XXXIV MEMOIR. have recently had access, show that my guess was correct, but Jo not enable me to find how the numerical error just noticed arose. Mr. Napier informed me that in all his business relations with Rankine, nothing had so much impressed him as the rapid and keen insight with which he seemed at once to seize upon the most essential points in the solution of a practical question, though stated to him for the first time; how he first shook himself free from the petty complications, and gave almost immediately an approximate estimate embracing all the larger bearings of the question; and then, much more formally and deliberately, and with the minute accuracy and system of a man of business, proceeded to work the question with the desired exactness. Mr. Napier said that on the occasion of his first consultation with Rankine on the matter referred to in the anagram above, Rankin e's very first words pointed out to him what a large proportion of the resistance to a vessel's motion is due to friction, and how ill-considered was the then growing demand for long and narrow ships. Kankine's text-books on engineering subjects are in many respects the most satisfactory that have been published in any country. At the time of their publication they have always been in advance of the professional knowledge of the day, but they possess much greater merits than that of mere novelty. Rankine was peculiarly happy in discriminating between those; branches of engineering knowledge which grow from daily experi- ence, and those which depend on unchangeable scientific principles. In his books he dealt almost exclusively with the latter, which may, and certainly will, be greatly extended, but so far as they have been established can never change. Hence his books are a mine which smaller men may work for many years, rendering his knowledge more generally available by giving it a popular setting of their own. By the bulk of the engineer- ing profession the books are considered hard reading, but as engineering education improves they will more and more be recognised as both wonderfully complete and essentially simple. Rankine, by his education as a practical engineer, was eminently qualified to recognise the problems of which the solution is required in practice; but the large scope of his mind would not allow him to be content with giving merely the solution of those particular cases which most frequently occur in engineering as we now know it. His method invariably is to state the problem in a very general form, find the solution, and then apply this solution to special cases. This method does not make the books easy reading for students, nor does it give the most convenient work of reference for the practical man ; but it has produced writings the value of which is permanent, instead of being ephemeral. In his Applied Mechanics we have the best existing work on the applica- tion of the doctrines of pure mechanics to general engineering problems. MEMOIR. XXXV No specious reasoning has been detected in this great work, a fact which should for ever dispel the old and false antithesis between theory and practice — a contrast drawn by practical men who never understood fully any theory, and assented to by scientific men who were not candid enough to point out where their theories were incomplete. In the A\'ork named Civil Engineering, Rankine applied the general doctrines of applied mechanics to the special problems which the civil engineer of to-day meets with in his practice, and his volume contains much valuable statistical information. In his work on Prime Movers we have a most thoroughly original statement of the thermodynamic theory, so far as it bears on the design and use of the steam-engine. This work especially shows Rankine's clear discrimination of what is permanent and can be taught, from that which must vary from day to day, and can only be acquired by personal experience; the distinction between the science and the art of the engineer. His treatise on Machinery and Millvorl gives the mechanical engineer instruction of a kind analogous to that which the civil engineer may derive from the book called Civil Engineering. The problems stated generally in the Applied Mechanics are in it applied to the special cases which arise in the design of machinery. Several of these .Manuals have been recently translated ; Prime Movers into French, Civil Engineering into German, &c. ; and Machinery and Millwork will soon appear in Italian. Most of the common treatises on engineering subjects are mere rechauffes or compilations ; and no library becomes sooner worthless than that of the engineer, the practice of this year being wholly different from that of five years since. Really original papers and monographs rapidly lose their interest and importance, except as historical landmarks, but Eankine's works will retain their value after this generation has passed away. In concluding the scientific part of this brief notice of a true man, we need scarcely point out to the reader how much of Rankine's usefulness was due to steady and honest work. The unscientific are prone to imagine that talent (especially when, as in Rankine's case, it rises to the level of genius) is necessarily rapid and off-hand in producing its fruits. No greater mistake could be made. The most powerful intellects work slowly and patiently at a new subject. Such was the case with Newton, and so it is still. Rapid they may be, and in general are, in new applications of processes long since mastered ; but it is only your pseudo-scientific man who forms his opinion at once on a new subject. This truth was pro- minently realised in Rankine, who was prompt in reply when his know- ledge was sufficient, but patient and reticent when he felt that more knowledge was necessary. With him thought was never divorced from work — both were good of their kind — the thought profound and thorough, XXXVI MEMOIR. the work a workman-like expression of the thought. Few. if any, practical engineers have contributed so much to abstract science, and in no case has scientific study been applied with more effect to practical engineering. RanMne's name will ever hold a high place in the history of science, and will worthily be associated with those of the great men we have recently lost. And, when we think who these were, how strangely does such a list — including the names of Babbage, Boole, Brewster, Leslie Ellis, Faraday, Forbes, Herschel, Rowan Hamilton, Clerk-Maxwell, Rankine, and others, though confined to physical or mathematical science alone— contrast with tfa inning utterance of the Prime Minister of Cieat Britain and Ireland, to the effect that the present is by no means an age abounding in minds of the first order ! Tin such men lost by this little country within the last dozen years or so — any one of whom would have made himself an enduring name had he lived in any pre- ceding it that of Hooke and Newton, or that of Cavendish and Watt: Nay more, even such losses as these have not extinguished the hopes of science amongst i . E ry one of these great men has, by some mysterious influence of hi . kindled the sacred thirst for new knowledge in yonngei but kindred spirits, many of whom will certainly rival, some even may excel, their teachers ! For the dates and statements of fact in this Memoir, I am indebted mainly to RanMne's private MSS., access to which has been given me by his relative-. Some special details I have had from his own lips. I have also to acknowledge my obligations to Sir William Thomson, to Professor Jcnkin, and specially to the late Mr. J. R. Napier, who was one of Kankine's most enthusiastic admirers, lie furnished me with much of the more technical part of the materials for a notice of Kankine's scientific work, which I wrote immediately after his death for the Glasgow U'.niJJ. December 28, 1872, and of which I have made considerable use in what precedes. Of Kankine's purely scientific work I have spoken from actual acquaint- ance with his writings ; but I have found it necessary to apply for assistance while attempting to discuss the merits of his more practical investigations. P. G. TAIT. College, Edinbetegh, October, 1880. PART I. PAPERS RELATING TO TEMPERATURE, ELASTICITY, AND EXPANSION OF VAPOURS, LIQUIDS, AND SOLIDS. PAET I. PAPERS RELATING TO TEMPERATURE, ELASTICITY, AND EXPANSION OF VAPOURS, LIQUIDS, AND SOLIDS. I.— OX AN EQUATION BETWEEN THE TEMPERATURE AND THE MAXIMUM ELASTICITY OF STEAM AND OTHER VAPOURS. (See Plate I.)* In the course of a series of investigations founded on a peculiar hypothesis respecting the molecular constitution of matter, I have obtained, among other results, an equation giving a very close approximation to the maxi- mum elasticity of vapour in contact with its liquid at all temperatures that usually occur. As this equation is easy and expeditious in calculation, gives accurate numerical results, and is likely to be practically useful, I proceed at once to make it known, without waiting until I have reduced the theoretical researches, of which it is a consequence, to a form fit for publication. The equation is as follows : — Log.P = a-£-J 2 , . . . (1.) where P represents the maximum pressure of a vapour in contact with its liquid ; /, the temperature, measured on the air-thermometer, from a point which may be called the absolute zero, and which is — 274 0- 6 of the Centigrade scale below the freezing point of water ; 462° - 28 of Fahrenheit's scale below the ordinary zero of that scale, * Originally published in the Edinburgh New Phibsophical Journal for July, 1849. A 2 ON THE ELASTICITY OF VAPOURS. supposing the boiling point to have been adjusted under a pressure of 29 - 922 inches of mercury, so that 180° of Fahrenheit may be exactly equal to 100 Centigrade degrees j 461° - 93 below the ordinary zero of Fahrenheit's scale, when the boiling point has been adjusted under a pressure of 30 inches of mercury, 180° of Fahrenheit being then equal to 100 o- 0735 of the Centigrade scale. The form of the equation has been given by theory; but three constants, represented by a, [3, and y, have to be determined for each fluid by experiment. The inverse formula, for finding the temperature from the pressure, is of course M -l0£ P + 0* 4 7 - /3 -V (2.) It is obvious that for the determination of the three constants, it is sufficient to know accurately the pressures corresponding to three tem- peratures ; and that the calculation will be facilitated if the reciprocals of those temperatures, as measured from the absolute zero, are in arithmetical progression. In order to calculate the values of the three constants for the vapour of water, the following data have been taken from 3L Regnault's experi- ments : — Temperatures in Cen- tigrade Degrees. Common Logarithms of the Pressure in Millimetres of Mercury. Remarks. Above the Freezing Point. Above the Absolute Zero. o 220- 100- 26-86 o 494-6 374-6 301-46 o 4-2403 2-S80S136 1-419S ( Measured by M. Eegnault on his curve, < representing the mean results of his ex- ( periments. Logarithm of 760 millimetres. 1 Calculated by interpolation from M. Eeg- { nault's general table. These data give the following results for the vapour of water, the pressures being expressed in millimetres of mercury, and the temperatures in Centierade decrees of the air-thermometer : — Log. 7 = 5-0827176 Log. (5- a=7'831247 3-1851091 ON THE ELASTICITY OF VAPOURS. 111 <d > a PiO'B a-s o § d P O OO O O O OOOO OOOO _J Ma- «; BOP oortTfcowaTfiMH-iwoo 00090090090000 —. 6 000 0000 6 6 00 I ++ -l + + + + I + + 1 C- Differences be- tween Calcula- tion and Experi- ment iu Loga- rithms. ONflLmN-jdMHHWOil 00000000000000 00000000000000 -~-~ oooopopoopoooo ^ b bb b b b bo b b b b bb — ' + l l + 1 1 l 1 + l l + rithms of the Millimetres j ng to M. Roguault's Experiments. OOMNM«N1QN-*NOOO ■— » QOMNOMOtSNl^NOO-ttl «! KOh«^i"CN00OOh«M -ii cicococococococococo-*'*"*-* Common Loga Pressures in accord the Formula. CCN^HO-ttOMOWtCOOM ^h r- CO IO O O ~TI co CO CO 01 O O CO CO O -t* 01 1>- co O ■ — - OHWt^nMOTtn^^NOOH -• OCWNOWnONt^NOlO-fCl i_- 00 O -h co -* O O r- co O O r-i Ol co dcoeocococococococo-^-^-'*-* &0a- ■ a o g IP 8 a •- © © £* Hta a S- o D = H 01 10 O CO CO — 1 10 101 •■+ CO >-< O CO 01 01 01 -t< CO O t~- O O CI CO ■* 01 pcp-Hpoppppppppppppppppppppp ,_, 00666006606000066660666060600 CO I+ + + + + lllllll 1 1 + I 1 + + + 1 1 1 1 "-' Difference be- tween Calcu- lation and Ex- periment in Millimetres. H«-J«^e0M^»00C150IM OOO'-irtOOOHeiNK^C'lOrtHOeOia^ODMiS _ ooooooooooooooo^^-hocococo— < >n r~» tei + 111 11 + + + + + + -+- ++•++' <7+++ + "" [illimetres of cording to M. Regnault's Experiments. «*-hcOOOOOO-hCOOOtHiOOO noooHMti 10 00 r- © •**< © t— 00000000 OOtl^eiN'JSHiiHOJMTllOOMOOKflNOnWOMOlO -7 HNMOO'l'nLOtlONiK'NHN-t'O-r'M'OOOH _u h ci n lo t> ■# n « 3 c. >o -;< co co c^ 1-1 1— 1 01 01 CO -* 10 t- O — 1 -P t^ HHr-lCl Pressures in 1 Mercury, ac the Formula. «oi-.nw«oi> 10 onanooci cocpp^pcpq<lippo^7H^pi^pcp-7Hpcp>pp01t>>0 6 6 Ol ■* b l-- b ^h b 01 b co 10 b 6 <* 6 co co co ^-1 eo 00 »o 10 • Hcmoo"f mioci'onoci- u>-#w«on — -p jo rH Ol CO O C- O Tfl t- l-O Ci l.O -* CO co ^ 1-1 1— lOlOlCO^Ot-O^-^r-O i— 1 r— 1 1— 1 01 es in Centi- reos of the motor from the Abso- lute Zero. O OOOOOO-^OOOOOOOOOOOOOOOOOOOO© ^_, ^OOM»C!OOHiMW#OOM»OOHCin-)<lOOI->tOOO ^~- oioioio^o-ic^coeocococococococococoT]HTj<-*TjH-^iTi(T|H-^-*Triin Temperatur grade Deg Air-Thormo the Freez- ing Point. CO 0000000 b 000000000000000000000 ^ CO 01 --( HCKIM^OOt-OOOO-i^M^OOIXBOO-'JlM • 4 ON THE ELASTICITY OF VAPOURS. Table I. exhibits a comparison between the results of the formula and those of M. Eegnault's experiments, for every tenth degree of the Centi- grade air-thermometer, from 30° below the freezing point to 230° above it, being within one or two degrees of the whole range of the experi- ments. M. Eegnault's values are given, as measured by himself, on the curves representing the mean results of his experiments, with the exception of the pressures at 26°*86, one of the data already mentioned, and that at - 30°, which I have calculated by interpolation from his table, series h. Each of the three data used in determining the constants is marked with an asterisk *. In the columns of differences between the results of the formula and those of experiment, the sign + indicates that the former exceed the latter, and the sign — the reverse. Beside each such column of differences is placed a column of the cor- responding differences of temperature, which would result in calculating the temperature from the pressure by the inverse formula. These are found by multiplying each number in the preceding columns by dt . -dt — z= or by -7i = as the case may require. dP J d log. P J n In comparing the results of the formula with those of experiment, as exhibited in Table I., the following circumstances are to be taken into consideration : — First, That the uncertainty of barometric observations amounts in general to at least one-tenth of a millimetre. Secondly, That the uncertainty of thermometric observations is from one-twentieth to one-tenth of a degree under ordinary circumstances, and at high temperatures amounts to more. Thirdly, That, in experiments of the kind referred to in the table, those two sorts of uncertainty are combined. The fifth column of the table shows that, from 30° below the freezing point to 20° above it, where the minuteness of the pressures makes the barometric errors of most importance, the greatest difference between experiment and calculation is y 1 ^ of a millimetre, or giro °f an mcn °^ mercury, a very small quantity in itself, although, from the slowness with which the pressure varies at low temperatures, the corresponding difference of temperature amounts to -££$ of a degree. The sixth and tenth columns show that, from 20° to 230° above the freezing point, the greatest of the discrepancies between experiment and observation corresponds to a difference of temperature of only jfa of a degree, and that very few of those discrepancies exceed the amount corresponding to -^ of a degree. ON THE ELASTICITY OF VAPOURS. 5 A comparison between the sixth and tenth columns shows that, for four of the temperatures given— viz., 120°, 150°, 200°, and 210°— the pressures deduced from M. Eegnault's curve of actual elasticities, and from his logarithmic curve respectively, differ from the pressures given by the formula in opposite directions. If the curves represented by the formula were laid down on M. Regnault's diagram, they would be almost undistinguishable from those which he has himself drawn, except near the freezing point, where the scale of pressures is very large, the heights of the mercurial column being magnified eight-fold on the plate. In the case of the curves of logarithms of pressures above one atmosphere, the coincidence would be almost perfect. The formula may, therefore, be considered as accurately representing the results of all M. Regnault's experiments throughout a range of temperatures from 30° of the Centigrade scale below the freezing point to 230° above it, and of pressures from o^ott °f an atmosphere up to 28 atmospheres. It will be observed that equation (1.) bears some resemblance to the formula proposed by Professor Roche in 1828, viz. : — ■p W.P=A T + C where T represents the temperature measured from the ordinary zero point, and A, B, and C, constants, which have to be determined from three experimental data. It has been shown, however, by M. Regnault, as well as by others, that though this formula agrees very nearly with observation throughout a limited range of temperature, it errs widely when the range is extensive. I have been unable to find Professor Roche's memoir, and I do not know the reasoning from which he has deduced his formula. The use in computation of the equations I have given, whether to calculate the pressure from the temperature, or the temperature from the pressure, is rapid and easy. In Table II. they are recapitulated, and the values of the constants for different measures of pressure and temperature are stated. In calculating the values of a, the specific gravity of mercury has been taken as 13'596. Temperatures measured by mercurial thermometers are in all cases to be reduced to the corresponding temperatures on the air-thermometer, which may be done by means of the table given by M. Regnault in his memoir on that subject. ON THE ELASTICITY OF VAPOURS. Table II. — Vapour of Water. Formula for calculating the Maximum Elasticity of Steam (P), from the Temperature on the Air-Thermometer, measured from the Absolute Zero (0 :— Inverse Formula for calculating the Temperature from the Maximum Elasticity of Steam : — 1 /q-log.P ffl" _ g_ t > 7 4 7 2 2y Values of the Constants depending on the Thermometric Scale : For the Centigrade scale : — Absolute zero 274°'6 below the freezing point of water. Log. /3 = 3-1851091 Log. 7 = 5-0827176 £- = 0-00G3L".) i -ft- = 0-00004006 For Fahrenheit's scale; boiling point adjusted at 29 - 922 inches: — Absolute zero 462°*28 below ordinary zero. Log. |3 - 3-4403816 Log. y = 5-5932626 -£=0-0035163 -ft= 0-000012364 - y 4y- For Fahrenheit's scale ; boiling point adjusted at 30 inches : — Absolute zero 461°*93 below ordinary zero. Log. /3 = 3-4400625 Log. y = 5-5926244 ft= 0-0035189 -ft = 0-000012383 2-y 47^ Values of the Constant a depending on the Measure of Elasticity : For millimetres of mercury, .... a =7-831247 English inches of mercury, .... 6-426421 ON THE ELASTICITY OF VAPOURS. Atmospheres of 760 mil. = 29*922 inches^ = 14-7 lbs. on the square inch=l-0333 kil. v . on the centimetre 2 , . . . . . ) Atmospheres of 30 inches = 761 mil. "99 -\ = 14'74 lbs. on the square inch=r T036 kil. V on the centimetre 2 , . . . . .J Kilogrammes on the square centimetre, Pounds avoirdupois on the square inch, N.B. — All the Constants are for common logarithms. 4-950433 4-949300 4-964658 6-117817 I have applied similar formula? to the vapours of alcohol and ether, making use of the experiments of Dr. Ure. In order to calculate the constants, the following experimental data have been taken, assuming that, on Dr. Ure's thermometers, 180° were equal to 100 Centigrade degrees. For Alcohol of the specific gravity O'Slo, For Ether, boiling at ) 105° F., under 30 [ inches of pressure, ) For Ether, boiling at ) 104° F., under 30 [ inches of pressure, ) Temperatures on Fahrenheit's Scale from the Ordi- the Abso- nary Zero, iuto Zero. 250 173 111-02 200 14S-8 105 104 G6-7 34 7123 635-3 573 32 662-3 611-1 507-3 566-3 529-0 496-3 Pressures in Inches of Mer- cury. 132-30 30 00 6-30 142-8 66-24 30-00 30-00 1376 6 20 From Dr. Ure's Table. Do. Interpolated in the same Table. From Dr. Ure's Table. Interpolated. From the Table. From Dr. Ure's Table. Interpolated. From the Table. The values of the constants in equation (1.), calculated from these data, are as folloAvs, for inches of mercury and Fahrenheit's scale : — Alcohol, specific gravity 0-813, Ether, boiling point 105°F., Ether, boiling point 104°F., 6-16620 5-33590 5-44580 Log. £. 3-3165220 3-20S4573 3-2571312 Absolute zero 462° '3 below ordinary zero. Log. y. 5-7602709 5-5119893 5-3962460 8 ON THE ELASTICITY OF VAPOURS. The curves represented by the formulae for those three fluids are laid down on the diagram which accompanies this memoir (Plate I.), and which has been reduced to one-fourth of the original scale. The horizontal divisions represent the scale of Fahrenheit's thermometer, numbered from the ordinary zero ; the vertical divisions, pressures of vapour, according to the scales specified on the respective curves. The points corresponding to the experimental data are surrounded by small circles. The curve for alcohol extends from 32° to 264° of Fahrenheit. It is divided into two portions, having different vertical scales, suitable to high and low pressures respectively. The curve for the less volatile ether extends from 105° to 210°; that for the more volatile ether, from 34° to 104°. The results of Dr. Ure's experiments are marked by small crosses. The irregular and sinuous manner in which those crosses are dis- tributed, indicates that the errors of observation, especially at high temperatures, must have been considerable. This does not appear surprising, when we recollect how many causes of uncertainty affect all the measurements required in such experiments, especially the ther- mometric observations, and how little those causes have been understood until very recently. The data from which the constants have been calculated arc, of course, affected by the general uncertainty of the experiments. When those circumstances are taken into account, it is obvious, from inspection of the diagram, that the curves representing the formulae agree with the points representing the experiments as nearly as the irregularity of the latter and the uncertainty of the data permit ; and that there is good reason to anticipate that, when experiments shall have been made on the vapours of alcohol and ether with a degree of precision equal to that attained by M. Kegnault in the case of the vapour of water, the equation will be found to give the elasticities of those two vapours as accurately as it does that of steam. Although the diagram affords the best means of judging of the agreement between calculation and experiment, three tables (III., IV., and V.) are annexed, in order to show the numerical amount of the discrepancies at certain temperatures. The data, as before, are marked with asterisks. It is worthy of remark, in the case of alcohol, that although the lowest of the experimental data is at the temperature of lll o- 02, the formula agrees extremely well with the experiments throughout the entire range of 79 degrees Mow that point. y ./ / / / 1. <■ >M P.-\ UJ O.V ( ? I OH Mt'l.A]' / 1 fit hp - ; l LA 5T err * FT P E VAPOBS AL Oh OL, El HE R. / 7 1 TU RP E >> Tl NE N P t TR 01 E M " J i «Jt L - ' i E \l EI ill II VI S fr •/ ' /' fc / ' -,' / •? _i 4 V ^ / •" V ' j y ^ ■ ;V •■- V rf I ( */ 1,^ y ,/^ i- > <V .' / t ,^y^y ■? V ' IS? ;1 " * -"''/i.* / uS 1*" .' n < ."'■ /•:, In; A\ *v ..U l'< \W X «*"> H (*!!£ 5*2 tit ' r • ^ K-" . >• * "' > S> * e* ^ X l- 1 - t To *fci / Sea felt 1. //., <n (P, rpZ< .«> ,. ,„• Degrees ofJFafavnheits TIhti neiev.^Scaie. Ihr Alcohol ,uui Etlwt- BOSTON U _E.GE OF I LIBR I ON THE ELASTICITY OF VAPOURS. Table III. — Vapour of Alcohol, of the Specific Gravity 0-813. Temperatures in Pressures in Inches of Mercury Differences be- Degrees of Fahrenheit from the Ordinary Zero. accord the Formula. ing to Dr. Ure's Experiments. tween Calculation and Experiment in Inches. Corresponding Differences of Temperature. 32 41 40 + 01 -0-5 40 57 56 + 001 -0-4 50 0-84 0S6 -0-02 + 0-7 60 1-22 1-23 -001 + 02 70 1-74 1-76 -0 02 + 0-3 80 2 43 2-45 -0 02 + 0-2 90 3 36 3 40 -004 + 04 100 456 4 50 + 006 -0-5 110 6-12 6 00 + 0-12 -0-7 *lll-02 6 30 6 30 00 120 8-10 8-10 000 00 130 1061 10 60 + 01 -00 140 1373 13 90 -0-17 + 05 150 17-60 18-00 -0-40 + 0-9 160 22-32 22-60 -0-28 + 0-5 170 2S-06 28-30 -0 24 + 04 *173 30-00 30-00 00 180 34 96 34 73 +0-23 -03 190 43-21 43-20 + 01 -0 200 52-96 5300 -0 04 +o-o 210 64 47 65 00 -0 53 + 0-5 220 77 92 78-50 -0-58 + 0-4 230 93-54 9410 -0-56 + 04 240 111-58 111-24 + 034 -0-2 *250 132 30 132-30 o-oo 260 155-98 155-20 + 0-78 -0-3 264 165 -5S 166-10 -0 52 +02 (1.) (2.) (3.) (4.) (5.) Table IV. — Vapour of Ether; Boiling Point, 105°F. Temperatures in Pressures in Inches of Mercury Differences be- Degrees of according to tween Calculation Corresponding Fahrenheit above and Experiment Differences of the Ordinary Zero. the Formula. Dr. Ure's Experiments. in Inches of Mercury. Temperature. *105 30O0 30 00 o-oo 00 no 33 -OS 32 54 + 54 -09 120 39-98 39 47 + 0-51 -0-7 125 43-83 43-24 + 0-59 -OS 130 47-95 47-14 + 0-S1 -10 140 57-10 56 90 + 0-20 -02 ♦148-8 66 24 66-24 00 150 67 53 67 60 -0 07 + 0-1 160 79 35 SO -30 -0 95 + 09 170 92-68 92 80 -012 + 0-1 175 99-94 99-10 + 0-84 -0-6 180 107-62 108-30 -0 68 + 04 190 124-29 124 80 -051 + 03 *200 142-80 142-80 00 o-o 205 152 -78 151-30 + 1-48 -0-7 210 163-27 166 00 l 1 -2-73 + 11 10 ON THE ELASTICITY OF VAPOURS. Table V. — Vapour of Ether; Boiling Point, 104°F. Temperatures in Pressures in Inches of Mercury Differences be- Degrees of according to tween Calculation Corresponding Fahrenheit above and Experiment Differences of the Ordinary Zero. the Formula. Dr. Dre's Experiments. in Inches of Mercury. Temperature. *34 6-20 6 20 o-oo 44 S-02 s-io -0-08 + 0-4 54 10-24 10 30 -0 06 + 0-2 64 12-94 13 00 -0-06 +0-2 *66-7 13 76 13-76 00 74 16-19 16-10- + 09 -02 84 20 06 20 00 + 0-06 -01 94 24-64 24-70 -0 06 + 01 *104 30-00 30-00 00 00 (1.). (2.) (3.) (4.) (5.) The results of Dr. Ure's experiments on the vapours of turpentine and petroleum are so irregular (as the diagram shows), and the range of temperatures and pressures through which they extend so limited, that the value of the constant y cannot be determined from them with precision. I have, therefore, endeavoured to represent the elasticities of those two vapours approximately by the first two terms of the formula only, calculating the constants from two experimental data for each fluid. The equation thus obtained Los. P a — is similar in form to that of Professor Roche. The data and the values of the constants are as follows : — Temperatures ( Scalo the Ordinary Zero. m Fahrenheit's from the Absolute Zero. Pressures in Inches of Mercury. Values of the Constants for Fahrenheit's Scale and Inches of Mercury. 360 304 370 316 S22-3 766-3 S32-3 778-3 Turpentine. 60 -SO 30-00 Petroleum. 60-70 30-00 * = 5-981S7 Log. |3 = 3 -5380701 *=6-19451 Log. |3 = 3 -5648490 Although the temperatures are much higher than the boiling point of water, I have not endeavoured to reduce them to the scale of the air- thermometer, as it is impossible to do so correctly without knowing the nature of the glass of which the mercurial thermometer was made. ON THE ELASTICITY OF VAPOURS. 11 The diagram shows that the formula agrees with the experiments as well as their irregularity entitles us to expect. The followinc; tables crive some of the numerical results : — Table VI. — Vapour of Turpentine. Temperatures in Degrees of Fahrenheit from the Ordinary Zero. Pressures in In accord the Formula (of two terms). ehes of Mercury ing to Dr. lire's Experiments. Differences be- tween Calculation and Experiment in Inches of Mercury. Corresponding Differences of Temperature. *304 30-00 30-00 o-oo o 310 32 52 33-50 -0-9S + 2-3 320 37 09 37-06 +0 03 -o-o 330 42-16 42-10 + 0-06 -o-i 340 47-78 47-30 + 0-48 -0-9 350 53 98 53-80 + 0-18 -0-3 *360 60-80 60 -SO o-oo o-o 362 62-24 62-40 -0-16 +o-o Table VII. — Vapour of Petroleum. Temperatures in Pressures in Inches of Mercury Differences be- Degrees of according to tween Calculation Corresponding Fahrenheit, from and Experiment Differences of the Ordinary the Formula Dr. Ure's in Inches of Temperature. Zero. (of two terms). Experiments. Mercury. *316 30 00 30-00 o-oo o o-o 320 31*71 31-70 +001 -00 330 36-35 36-40 -0-05 +0T 340 41-52 41-60 -0-08 + 0-2 350 47-27 46-86 + 0-41 -0-7 360 53-65 53-30 + 0-35 -0-5 *370 60-70 60-70 o-oo 375 64-50 64-00 + 0-50 -0-7 (1.) (2.) (3.) (±.) (5.) I have also endeavoured, by means of the first two terms of the formula, to approximate to the elasticity of the vapour of mercury, as given by the experiments of M. Kegnault. The data and the constants are as follows : — Temperatures Degree the Freezing Point. in Centigrade s from the Absolute Zero. Pressures in Millimetres of Mercury. Values of the Constants in the Formula Log. P = i-f 358 177-9 632-6 452-5 760-00 10-72 a. for millimetres = 7 "5305 ,, for English inches 6T259 Log. /3 Centigrade scale 3 -4685511 ,, Fahrenheit's scale, \ *£SF S*ii(»w" inches, . . . . ) 12 ON TOE ELASTICITY OF VArOURS. The following table exhibits the comparative results of observation and experiment : — Tu'.i.r. VIII. — Vapour of Mercury. TatnjiiTiiturrs in Centigrade Degreei i i area In Millimetres or Mercury ding to DIfferenoea iiotwoou Oaloulatlon and from Hi" l''i I Point tin> Formula (.if two tenna), M. ltognaull's Experiments. i 1 raerlment in Millimetres. 7274 0-115 0-183 -0 068 10011 0-480 ((•107 +0-073 100 6 0-49 one, -0-07 146-3 3-49 3-46 + 03 •1779 10*72 10-72 o-oo 200-5 2 1 -85 22-01 -016 »358 760 00 760 00 00 The discrepancies are obviously of the order of errors of observation, and the formula may be considered correct for all temperatures below 200°C, and for a short range above that point From its wanting the third term, however, it will probably be found to deviate slightly from the truth between 200° and 358°; while above the latter point it must not be relied oil. I have not carried the comparison below 72°, because in that part of the scale the whole pressure! becomes of the order of errors of observation. In conclusion, it appears to me that the following proposition, to which 1 have been led by the theoretical researches referred to at the commence- ment of this paper, is borne out by all the experiments I have quoted, especially by those of greatest accuracy, and may be safely and usefully applied to practice. If the maximum elasticity of any vapour in contact with its liquid be tained for three points on the scale of the air-thermometer, then the constants of an equation of the form may be determined, which equation will give, for that vapour, with an accuracy limited only by the errors of observation, the relation between the temperature (t), measured from the absolute zero (27 4'6 Centigrade degrees beloiu the freezing point of ivatcr), and the maximum elasticity (P), at all temperatures between those three points, and for a considerable range beyond them. THE EXP„ ? LIQUIDS EY HI 13 EL— OS A FORMULA FOB CALCULATING THE EXPAJ : LIQUIDS BY HEAT Hating been lately much engaged in researches involving the comparative volumes of liquids at various temperatures, I have found the following formula very useful : Log. Y represents the common logarithm of the volar. .--.ass of liquid, as compared with its volume at a certain standard temper; which, for wa: temperature of its maximum density, or 4**1 i ie, and for other liqui 1 rigrade. ? is the temperature measured from the absolute zero mentioned in paper on the of Vapours, in * oi for July. 184 as preat&mg Vapct . ari u found by a/1 - .• " : B. and C, are three constants, depending on the natur : liquid, whose values for the Centigrade scale, corresponding to water, Watg*, a:-..-/--.\ . Log.B_ : 1 ■ I ■ :7"4 ' - (HB29130 ' .- ' . 15033 ' - 1-289X86 -■ ' - 1-2192054 HaHstrom : for mercury, from those of Begnault : and for alcohol and n.~:.-:~; .: :i: .-. :r;n :._:.- .: .--.;-- :'.-..' As t:.^ :.~; :.v-.\ : : 1L Gay-Lossae give only the apparen t expansion of the liquids in glass, I have assumed, in order to calrnlate the true expansion, that the dilatatio n " '. :.. .l: . -I - -- EL. %-.:—.': ':■ . • - :' -".:..-. ':-■' 14 THE EXPANSION OF LIQUIDS BY HEAT. of the glass used by him was -0000258 of its volume for each Centigrade degree. This is very nearly the mean dilatation of the different kinds of glass. M. Kegnault has shown that, according to the composition and treatment of glass, the coefficient varies between the limits "000022 and •000028. Annexed are given tables of comparison between the results of the formula and those of experiment. The data from which the constants were calculated are marked with asterisks. The table for water shows, that between 0° and 30° Centigrade, the formula agrees closely with the experiments of Hallstrom, and that from 30° to 100° its results lie betAveen those of the experiments of Gay-Lussac and Deluc. The experiments of Gay-Lussac originally gave the apparent volume of water in glass as compared with that at 100°. They have been reduced to the unit of minimum volume by means of Hallstrijm's value of the expansion between 4 0, 1 and 30°, and the coefficient of expansion of glass already mentioned. In the fifth column of the table of comparison for mercury, it is stated which of the experimental results were taken from M. Eegnault's own measurements on the curve, representing the mean results of his experi- ments, and which from his tables of actual experiments, distinguishing the series. In the experimental results for alcohol and sulphuret of carbon, the respective units of volume are the volumes of those liquids at their boiling points, and the volumes given by the formula have been reduced to the same units. Expansion of Water. Temperature on Volume as compared with that at Difference between Authorities for the Centigrade 4"1 C. ac( oraing to Calculation and the Scale. the Formula. the Experiments. Experiment. Experiments. o 1-0001120 1 -00010S2 + -000003S HallstrUm. *4'1 1-0000000 1-0000000 0000000 Do. 10 1 -0002234 1-0002200 + 0000034 Do. 20 1 0015668 1-0015490 + 00001 7S Do. *30 1 0040245 1-0040245 0000000 Do. 1 -0041489 - 0001244 Deluc. 40 100750 1-00748 + 00002 Gay-Lussac. 1-00774 _ 00024 Deluc. 60 1 -01718 1 -01670 + 00048 Gay-Lussac. ... 1-01773 _ 00055 Deluc. 80 1-03007 1-02865 + 00142 Gay-Lussac. 1-03092 _ 00085 Deluc. 100 1-04579 1-04290 + 00289 Gay-Lussac. ... 104664 - -00085 Deluc. THE EXPANSION OF LIQUIDS BY HEAT. 15 Expansion of Mercury. Volume as compared with that at Temperature on the Centigrade 0° C. ace ording to M. Regnault's Experiments. Difference between Calculation and Remarks. Scale. the Formula. Experiment. o *0 1-000000 1-000000 ■oooooo Curve. 90-22 1-016333 1-016361 - -000028 Series I. 100 00 1-018134 1-018153 - -000019 • Curve. 100-52 1-018230 1-018267 - -000037 Series I. *150-00 1-027419 1027419 •oooooo Curve. 198-79 1-036597 1-036468 + -000129 Series 11. 205 07 1-037786 1-037805 - -000019 Series IV. 205-57 1 -037905 1-037910 - -000005 Series III. *300-00 1-055973 1 055973 •oooooo Curve. Expansion of Alcohol. Temperature on the Volume as compared with that at 78° - 41 C. according to Difference between Calculation and Centigrade Scale. the Formula. m. way-l-iussac s Experiments. Experiment. o 3 41 •91795 •91796 - -ooooi *1S-41 •93269 •93269 •ooooo 33-41 •94803 •94799 + -00004 *48-41 •96449 •96449 •ooooo 63-41 •9S183 •98210 - -00027 *7S-41 1-00000 1-00000 •ooooo Expansion of Sulphuret of Carbon. Temperature on the Centigrade Scale. Volume as compared accorc the Formula. with that at 4G o -60C. ing to M. Gay-Lussac's Experiments. Difference between Calculation and Experiment. * - 13-40 + 1-60 *16-60 31-60 *46-60 •93224 •9476S •96417 •98163 1-00000 •93224 •94776 •96417 •98163 1-00000 •ooooo - -00008 •ooooo •ooooo •ooooo 1G ELASTICITY OF GASES AND VAPOURS. HI.— ON THE CENTRIFUGAL THEORY OF ELASTICITY, AS APPLIED TO GASES AND VAPOURS.* 1. The following paper is an attempt to show how the laws of the pressure and expansion of gaseous substances may be deduced from that which may be called the hypothesis of molecular vortices, being a peculiar mode of conceiving that theory which ascribes the elasticity connected with heat to the centrifugal force of small revolutions of the particles of bodies. The fundamental equations of this theory were obtained in the year 1842. After having been laid aside for nearly seven years, from the want of experimental data, its investigation was resumed in consequence of the publication of the experiments of M. Regnault on gases and vapours. Its results having been explained to the Royal Society of Edinburgh in February, 1850, a summary of them was printed as an introduction to a paper on the Mechanical Action of Heat in the twentieth volume of the Transactions of that body. I now publish the investigation in detail in its original form, with the exception of some intermediate steps of the analysis in the second and third sections, which have been modified in order to meet the objections of Professor William Thomson, of Glasgow, to whom the paper was submitted after it had been read, and to whom I feel much indebted for his friendly criticism. This paper treats exclusively of the relations between the density, heat, temperature, and pressure of gaseous bodies in a statical condition, or when those quantities are constant. The laws of their variation belong to the theory of the mechanical action of heat, and are investigated in the other paper already referred to. The present paper consists of six sections. The first section explains the hypothesis. The second contains the algebraical investigation of the statical relations "between the heat and the elasticity of a gas. The third relates to temperature and real specific heat. The fourth treats of the coefficients of elasticity and dilatation of gases, and compares the results of the theory with those of M. Regnault's experiments. * Eead before the Royal Society of Edinburgh, February 4, 1850, and published in the Philosophical Magazine for December, 1851. ELASTICITY OF GASES AND VAPOURS. 17 The fifth treats of the laws of the pressure of vapours at saturation. The sixth relates to the properties of mixtures of gases of different kinds. I have endeavoured throughout this paper to proceed as directly as possible to results capable of being compared with experiment, and to carry theoretical researches no further than is necessary in order to obtain such results with a degree of approximation sufficient for the purpose of that comparison. Section I. — On the Hypothesis of Molecular Vortices. 2. The hypothesis of molecular vortices may be defined to be that which assumes — that each atom of matter consists of a nucleus or central point enveloped by an elastic atmosphere, which is retained in its position by attractive forces, and that the elasticity due to heat arises from the centrifugal force of those atmospheres, revolving or oscillating about tlmr nuclei or central points. According to this hypothesis, quantity of heat is the vis viva of the molecular revolutions or oscillations. Ideas resembling this have been entertained by many natural philoso- phers from a very remote period; but, so far as I know, Sir Humphry Davy was the first to state the hypothesis I have described in an intelli- gible form. It appears since then to have attracted little attention, until Mr. Joule, in one of his valuable papers on the Production of Heat by Friction, published in the London and Edinburgh Philosophical Magazine for May, 1845, stated it in more distinct terms than Sir Humphry Davy had done. I am not aware, however, that any one has hitherto applied mathematical analysis to its development. 3. In the present stage of my researches, there are certain questions connected with the hypothesis as to which I have not found it necessary to make any definite supposition, and which I have therefore left indeter- minate. Those questions are the following : — First, Whether the elastic molecular atmospheres are continuous, or consist of discrete particles. This may be considered as including the question, whether elasticity is to a certain extent a primary quality of matter, or is wholly the result of the repulsions of discrete particles. Secondly, Whether at the centre of each molecule there is a real nucleus having a nature distinct from that of the atmosphere, or a portion of the atmosphere in a highly condensed state, or merely a centre of condensation of the atmosphere, and of resultant attractive and repulsive forces. There- fore, although the word nucleus properly signifies a small central body, I shall use it in this paper, for want of a better term, to signify an atomic centre, whether a real nucleus or a centre of condensation and force. I B 18 ELASTICITY OF GASES AND VAPOURS. assume, however, that the volume of the nucleus, if any. is inappree: . small as compared with that of the atmosphere. ■4. I have now : . supposition, which, so far as I am aware, is peculiar to my own researches. It is this : — that the vibration ichich, according to the undulatory hypothesis, t radiant light and heat, is a motion of the atomic nuclei or centres, and by means of then mutual attractions and repulsions. It will be perceived at once, that from the combination of this sup- position with the hypothesis of molecular vortices, it follows that the absorption of light and of radiant heat consists in the transference of motion from the nuclei to their atmospheres; and conversely, that the emission of light and of radiant heat is the b ace of motion from the atmospheres to the nuclei It appears to me that the supposition I have stated advantages over the ordinary hypothesis of a luminiferous ether pervad- ing the spaces between ponderable particles, especially in the following respects : — . ue propagation of transverse vibrations requires the operation of forces, which, if not altogether attractive, are of a very different nature from those capable of producing gaseous and which it is difficult to ascribe to such a substance as the ether is suppo; whii between the atomic centres are perfectly with their "being kept asunder by the ela~" Leir atm nd. The immense velocity of light and radiant heat is a natural consequence of th: ition, according to which the vibrating m be extremely small as compared with the forces exerted by them. rding to the most probable view of the theory of dispersion, the unequal refrangibility of undulations of different lengths is a c quence of the distances between the pa: the vibrating medium having an appreciable magnitude as compared with tl _ v.s of the undulations. This is scarcely conceivable of the ether, but easily con- ceivable of the atomic n The manner in which the propagation of light and of radiant heat is affected by the molecular arrangement of crystalline bodi-- I much more intelligible if the vibrations are supposed to be t of the atomic nuclei, on whose mutual forces and positions the form of .llisation must depend. — The consequences of this supposition, in the theory of double :tion and polarisation, are pointed out and shown to be corroborated by Pi . 5l kest experiments on diffraction, in a paper read to the of Edinburgh on the 2nd of December, 1850, and publ. in the 1 fkkai Magazine for June, 1851.] ELASTICITY OF GASES AND VAPOURS. 19 Section II— Investigation of the General Equations between the Heat and the Elasticity of a Gas. 5. I now proceed to investigate the statical relations between the heat and the elasticity of a gaseous body ; that is to say, their relations when both are invariable. The dynamical relations between those phenomena which involve the principles of the mutual conversion of heat and mechanical power by means of elastic fluids, and of the latent heat of expansion and evaporation, form the subject of another paper. 6. It is obvious that, in the condition of perfect fluidity, the forces resulting from attractions and repulsions of the atomic centres or nuclei upon their atmospheres and upon each other, must be considered as being sensibly functions merely of the general density of the body, and as being either wholly independent of the relative positions of the particles, or equal for so many different positions as to be sensibly independent of them ; for otherwise a certain degree of viscosity would arise, and con- stitute an approach to the solid state. For the same reason, in the state of perfect fluidity each atomic atmosphere must be considered as being sensibly of uniform density in each spherical layer described round the nucleus with a given radius, and the total attractive or repulsive force on each indefinitely small portion of an atmosphere must be considered as acting in a line passing through its nucleus ; that force, as well as the density, being either independent of the direction of that line, or equal for so many different and symmetrical directions as to be sensibly independent of the direction. 7. An indefinite number of equal and similar atoms, under such con- ditions, will arrange themselves so that the form of their bounding surfaces will be the rhombic dodecahedron, that being the nearest to a sphere of all figures which can be built together in indefinite numbers. 8. I may here explain that by the term bounding surfaces of the atoms, I understand a series of imaginary surfaces lying between and enveloping the atomic centres, and so placed that at every point in these surfaces the resultant of the joint actions of all the atomic centres is null. To secure the permanent existence of each atom, it must be supposed that the force acting on each particle of atomic atmosphere is centripetal towards the nearest nucleus or centre. The variation of that force in the state of perfect fluidity must be so extremely small in the neighbourhood of those surfaces, that no appreci- able error can arise, if, for the purpose of facilitating the calculation of the elasticity of the atmosphere of an atom at its bounding surface, the form of that surface is treated as if it were a sphere, of a capacity equal to that of the rhombic dodecahedron. 9. If the several atoms exercised no mutual attractions nor repulsions, 20 ELASTICITY OF GASES AND VAPOURS. the total elasticity of a body would be equal to the elasticity of the atomic atmospheres at their bounding surfaces. Supposing such attractions and repulsions to exist, they will produce an effect, which, in the state of perfect fluidity, will be a function of the mean density of the body ; and which, for the gaseous state, will be very small as compared with the total elasticity. Therefore, if p be taken to represent the superficial elasticity of the atomic atmospheres, P the actual or total elasticity of the fluid, and D its general density, P-P+/(D), .... (1.) where /(D) is a function of the density, which may be positive or negative according to the nature of the forces operating between distinct atoms. 10. The following relations must subsist between the masses of the atmosphere and nucleus, and the density and volume of each atom : Let R represent the radius of the sphere already mentioned, whose capacity is equal to the volume of an atom, that volume being equal to t es - Let /i denote the mass of the atmosphere of an atom, m that of the nucleus, and M = fi + m the whole mass of the atom (so that if there is no real nucleus, but merely a centre of condensation, m = 0, and M = /u). Then D being the general density of the body, ^-D is the mean density of the atomic atmosphere, and M=— E 3 D. o If ull be taken to denote the distance of any spherical layer of the atmosphere from the nucleus, the density of the layer may be repre- sented by and the function ifm will be subject to this equation of condition, ix=\ du\ J u-0 V M which is equivalent to iTrW^Du^u), f du(u*rpu) (2.) 11. So far as our experimental knowledge goes, the more substances are rarefied — that is to say, the more the forces which interfere with the operation of the elasticity of the atomic atmospheres are weakened — the more nearly do they approach to a condition called that of perfect gas, in ELASTICITY OF GASES AND VAPOURS. 21 which the elasticity is simply proportional to the density. I therefore assume the elasticity of the atomic atmosphere at any given point to be represented by multiplying its density at that point by a constant coefficient b, which may vary for different substances, but, as I have already stated, without deciding whether that elasticity is a primary quality or the result of the repulsion of particles. Consequently, the superficial atomic elasticity P ^bp)m, .... (3.) ^(1) being the value of \pu, which corresponds to the bounding surface of the atom, where u = 1. 12. Let an oscillatory movement have been propagated from the nuclei to every part of their atmospheres, the size of the orbits of oscillation being everywhere very small as compared with the radii of the atoms, and let this movement have attained a permanent state, which will be the case when every part of each atmosphere, as well as each nucleus, moves with the same mean velocity, v — mean velocity signifying that part of the velocity which is independent of periodic changes. It is necessary to suppose that the propagation of this movement to all parts of a molecular atmosphere is so rapid as to be practically instantaneous. We shall conceive all the masses and densities referred to, to be measured by weight. Then taking g to represent the velocity generated by the force of gravity at the earth's surface in unit of time, the whole mechanical power to which the oscillatory movement in question is equivalent in one atom will be represented in terms of gravity by V= 2; {L) that is to say, the weight of the atom, M, falling through the height — due to the velocity v ; and this is the mechanical measure of the quantity of heat in one atom in terms of gravity. 13. Any such motion of the particles of a portion of matter confined in a limited space, will in general give rise to a centrifugal tendency with respect to that space. In order to obtain definite results with respect to that centrifugal tendency in the case now under consideration, it is necessary to define, to a certain extent, the general character of the supposed movement. In the first place, it is periodical ; secondly, it is similar with respect to so large a number of radii drawn in symmetrical directions from the atomic centre, as to be sensibly similar in its effects with respect to all directions round that centre. Thi3 symmetry exists in the densities of the different particles of the atomic atmosphere in a gas, and in the 22 ELASTICITY OF GASES AND VAPOURS. forces which act upon them ; and we are therefore justified in assuming it to exist in their motions. Two kinds of motion possess these characteristics : First, Radial oscillation, by which a portion of a spherical stratum of atmosphere surrounding an atomic centre, being in equilibrio at a certain distance from that centre, oscillates periodically to a greater and a less distance. This forms part of the vis viva of the molecular movements ; but it can only affect the superficial atomic elasticity by periodic small variations, having no perceptible effect on the external elasticity. Second, Small rotations and revolutions of particles of the atomic atmosphere round axes in the direction of radii from the atomic centre, by which each spherical layer is made to contain a great number of equal and similar vortices, or equal and similar groups of vortices having their axes at right angles to the layer, and similarly situated with respect to a great many symmetrical directions round the atomic centre. Let us now consider the condition, as to elasticity, of a small vortex of an atmosphere whose elasticity is proportional to its density, inclosed within a cylindrical space of finite length, and not affected by any force at right angles to the axis except its own elasticity. Let Z denote the external radius of the cylinder, p x its external density, p its mean density, p the density at any distance z from the axis (all the densities being measured by weight), w the uniform velocity of motion of its parts. The condition of equilibrium of any cylindrical layer is, that the difference of the pressures on its two sides shall balance the centrifugal force ; con- sequently (b being the coefficient of elasticity) (jz dz ' The integral of this equation is ,>■'■ p' = az b 'S The coefficient a is determined by the following relation, analogous to that of equation (2), between the densities : Z 2 p -=J dz(pz)= ^—Z» whence V+ 2 ■ti +I And the general value of the density is ,,2 £+i)®* ' (5 ° ELASTICITY OF GASES AND VAPOURS. 23 Making z = Z, and multiplying by the coefficient of elasticity b, we obtain for the elasticity of the atmosphere, at the cylindrical surface of the vortex, b Pi = b P + -l£-> • • • W which exceeds the mean elasticity bp by a quantity equivalent to the weight of a column of the mean density p, and of the height due to the velocity u; and independent of the radius of the vortex. Supposing a spherical layer, therefore, to contain any number of vortices of any diameter, in which the mean density is equal, it is . necessary to a permanent condition of that layer that the velocities in all these vortices should be equal, in order that their lateral elasticities may be equal. Although the mean elasticity at the plane end, or any plane section at right angles to the axis of a vortex, is simply = bp, being the same as if there were no motion, yet the elasticity is variable from point to point, and the law of variation depends on the velocity. Therefore, if two vortices are placed end to end, it is necessary to a stable condition of the fluid, not only that their terminal planes should coincide, and that their mean elasticities should be in cquilibrio, but also that their velocities should be equal, or subject only to periodical deviations from a state of equality. Therefore, the mean velocity of vortical motion, independent of small periodic variations, is the same throughout the whole atomic atmosphere; and the mean total velocity, independent of small periodic variations, being uniformly distributed also, the vis viva of the former may be expressed as a constant fraction of that of the latter, so that v? = ^, . . . . (5b.) - being the mean value of a coefficient which is subject to small periodical k variations only * * As it has been represented to me that I have, without stating sufficient grounds, assumed the velocity of revolution w to be constant throughout each individual vortex, I add this note to assign reasons for that supposition. First, Unless w, the velocity of revolution of a particle, is independent of z, its radius vector, the atomic atmosphere cannot be in a permanent condition. For if w is a function of z, the external elasticity of a vortex will be a function of its diameter. If the whole atmosphere is in motion, vortices of different diameters must exist in the same spherical layer ; and if their external elasticities are different, their condition cannot be permanent. ,Second, Whatsoever may be the nature of the forces by which velocity is communi- cated throughout the atmosphere, the tendency of those forces must be to equalise that velocity, and thus to bring about a permanent condition. 24 ELASTICITY OF GASES AND VAPOURS. This coefficient, being the ratio of the vis viva of motion of a peculiar kind to the whole vis viva impressed on the atomic atmospheres by the action of their nuclei, may be conjectured to have a specific value for each substance, depending, in a manner "as yet unknown, on some circumstance in the constitution of its atoms. It will afterwards be seen that this circumstance is the chemical constitution. Let the entire atmosphere of an atom be conceived to be divided into a great number of very acute pyramids meeting at the centre, and having even numbers of faces, equal and opposite in pairs ; and let one of these pyramids, intersecting a spherical layer whose distance from the nucleus is R?i and thickness Tulu, cut out a frustum, containing and surrounded by vortices. Consider one pair of the faces of that frustum ; their length being TMu, let their breadth be h, and their distance asunder/. Then they make with each other the angle at the apex of the pyramid o s ; n -i ; J . their common area is KRdu ; and the sum of the volumes of the two triangular frusta of the spherical layer, included by diagonal planes drawn between their radial edges, is fKRflll 2 ' the sum of all such triangular frusta being the whole volume of the spherical layer. The additional pressure due to the centrifugal force of vortices, viz. — 2gV acts on the two lateral faces, its total amount for each being 2gk The transverse components of this pair of forces balance each other. Their radial components, amounting to / v 2 p 1 - rt7 iPpfhdu Hu 2gk 2gku constitute a centrifugal force relatively to the atomic centre, acting on the pah* of triangular frusta whose mass is pfJiRdu 2 ' The condition of permanent, or periodical, equilibrium of this pair of frusta, requires that this centrifugal force shall be balanced by the varia- tion of the mean elasticity of the atmosphere at the two surfaces of the ELASTICITY OF GASES AND VAPOURS. 25 spherical layer, combined with the attraction of the nucleus. The action of the former of these forces is represented by 7 dp y ill — b^au X — . cm 2 Let the accelerating force of attraction towards the nucleus be repre- sented by <f>(Rn) E, ' ^ being a function, which, by the definition of an atomic bounding surface in article 8, is null at that surface, or when u = 1. Then the attraction on the pair of frusta is pfhdu(p(Ru) 2 " Add these three forces together; let the sum be divided by ■^pfhdw, and let the density p be denoted, as in article 10, by then the folloAving differential equation is obtained as the condition of a permanent state of the atomic atmosphere : -T-77T --r^-0 = O. . . . (5c.) gku \p(u) du r This equation will be realised for each layer at its mean position, on each side of which its radial oscillations are performed. The variation of this expression being of opposite sign to the variation of , shows that any small disturbance of the density produces a force tending to restore that distribution to the state corresponding to the position of equilibrium of the layers, and therefore that the state indicated by equation (5 c) is stable. 1 4. The integral of equation (5c) is xPu=u^e a ~"f U 1 du -<P. . . . (6.) The arbitrary constant a is determined from the equation of condition (2) in the following manner : — 2G ELASTICITY OF GASES AND VAPOURS. Substituting for $11 in equation (2) its value as given above, we obtain or r- r w -•=s/*(« 3B+ V*/ 1 * , *) J . . . (7.) which integration having been effected, we shall obtain for the value of the superficial elasticity of the atomic atmospheres, *=OWl) = 6£D* • (8.) M rv J M To obtain an infinite series for approximating to the value of the integral in equation (7), let the following substitutions be made : — Lo" u=X 1 gkb + 3 = 30 J l — (O, (9.) and let the values of the successive differential co- efficients of 10 with respect to A, when A = 0, $ = 0, and w=r 1, be denoted by (a/), (to"), (w'"), &c. Then e- a =3J d\.e sex u>. - -so The value of which (when the function (p is such as to admit of its having a finite value) is 6 -0V 36 + 90- 270 s + &C '/' whence i 1 + W + ^90^ + 27 3 + &c.) Now, because (a/) = — t0 ( m = d = 0, ^(l) = e« = e(l-^]+&c.) j ' which may be represented by Gi +l )( l - r < D ^)' ' (10.) (10a.) ELASTICITY OF GASES AND VAPOURS. 27 F(D,0) being a quantity which becomes continually less as the density becomes less and the heat greater. The complete expression for the elasticity of a gas is therefore, according to equations (1), (8), and (10a), P^+/(D)-^D(^ + j)(l-F(D,e))+/(D);. (11.) when each atom contains a quantity of heat measured by the mechanical power corresponding to the velocity v in the weight M, or Mr 2 according to equation (4). Section III. — Of Temperature, and of Real Specific Heat. 15. The definition of temperature consists of two parts: — First, the definition of that condition of two portions of matter when they are said to be at the same temperature ; and" second, the definition of the measure of differences of temperature. Two bodies are said to be at the same temperature when there is no tendency for one to become hotter by abstracting heat from the other ; that is to say (calling the two bodies A and B), when there is either no tendency to transmission of heat between them, or when A transmits as much heat to B as B does to A. Now it is known by experiment, that any surface or other thing which affects the transmission of heat being placed between B and A, has exactly the same influence upon the same quantity of heat passing in either direction; therefore, to produce equilibrium of temperature between A and B, the powers of their atoms to communicate heat must be equal. 15a. If we apply to vortices at the surface of contact of the atmo- spheres of two atoms of the same or different kinds, the conditions of permanency laid down in article 1 3 for vortices in the same atmosphere, these conditions take the folloAving form : — First, The superficial atomic mean elasticities must be the same ; in other words, the superficial atomic mean densities must be inversely as the coefficients of elasticity of the atmospheres. This is the condition of equilibrium of pressure. Second, The law of variation of the elasticity from the centre to the circumference of a vortex, as expressed in equation (5), must be the same .2 2 for both atoms; and this law depends on the quantity ~ =y h ; therefore the condition of equilibrium of heat is, that the square of the velocity of vortical motion, divided by the coefficient of atmospheric elasticity, shall 28 ELASTICITY OF GASES AND VAPOURS. 1)0 the same for each atom. Of this quantity, therefore, and of constants common to all substances, temperature must be a function. Taking the characteristics (A) and (B) to distinguish the quantities proper to the two atoms, we have the following equation : — jgD*(l)(A>=8|D*(l)(B)' >=>),[ • • (12.) temperature = f — , universal constants ) . 16. In sx perfect gas, equation (11) is reduced to the pressure being simply proportional to the mean elasticity of the atmo- spheric part of the gas multiplied by a function of the heat, which, as equation (12) shows, is a function of the temperature, from its involving only — and universal constants. kb Therefore, in two perfect gases at the same pressure and temperature, the mean elasticities of the atmospheric parts arc the same, and consequently — The mean specific gravities (if the atmospheric parts of all perfect gases are inversely proportional to the coefficients of atmospheric elasticity. Let n therefore represent the number of atoms of a perfect gas which fill unity of volume under unity of pressure at the temperature of melting ice, so that nM is the total specific gravity of the gas, and njx that of its atmospheric part ; then bn/n = constant for all gases, . . (12b.) and consequently kb k { ' } Therefore, Temtperatwre is a function of universal constants, and of the vortical vis viva of the atomic atmospheres of so much of the substance as would, in the condition of perfect gas, fill unity of volume under unity of pressure at some standard temperature. The equation (12a) further shows, that in any two perfect gases the respective values of the quotient of the pressure by the density corresponding to the same temperature, bear to each other a constant ratio for all temperatures, being that of the values of the coefficient b ~. ° M ELASTICITY OF GASES AND VAPOURS. 29 Therefore the pressure of a perfect gas at a given density, or its volume under a given pressure, is the most convenient measure of temperature. Let P represent the elasticity of a perfect gas of the density D at the temperature of melting ice, P that of the same gas at the same density, at a temperature distant T degrees of the thermometric scale from that of melting ice, and C a constant coefficient depending on the scale employed; then the value of T is given by the equation P-Pol T=C Po (13.) or r • T-fC=C~j The value of the constant C is found experimentally as follows: — Let Pj represent the elasticity of the gas at the temperature of water boiling under the mean atmospheric pressure, T 1 the number of degrees, on the scale adopted, between the freezing and boiling points of water; then P — P 1 rp p r l X 1 1 5 and 1- (14.) c = t 'pFpJ C is in fact the reciprocal of the coefficient of increase of elasticity with temperature, or the reciprocal of the coefficient of dilatation, of a perfect gas at the temperature of melting ice. 17. As it is impossible in practice to obtain gases in the theoretical condition referred to, the value of C can only be obtained by approxima- tion. From a comparison of all M. Eegnault's best experiments, I have arrived at the following values, which apply to all gaseous bodies. For the Centigrade scale, C = 274 0, 6, being the reciprocal of 0-00364166. For Fahrenheit's scale, if adjusted so that 180° are equal to 100° Centigrade, — C for temperatures measured from the freezing-point of water = 494°-28. C for temperatures measured from the ordinary zero = 494°-28-32° = 462°-28. The point C degrees below the ordinary zero of thermometric scales may be called the absolute zero of temperature ; for temperatures measured from that point are proportional to the elasticities of a theoretically perfect gas of constant density. 30 ELASTICITY OF GASES AND VAPOURS. Temperatures so measured may be called absolute temperatures. Through- out this paper I shall represent them by the Greek letter t, so that T + C. (15.) It is to be observed, that the absolute zero of temperature is not the absolute zero of heat. 18. If we now substitute for P in equation (13) its value according to equation (12a), we obtain the following result: — '= T + C = c £i(£*+ J ) o V: V/''- Let n represent, as before, the theoretical number of atoms in unity of volume under unity of pressure, at the temperature of melting ice, of the gas in question, supposing the disturbing forces, represented by - F(D,0) and /(D), to be inappreciable; then nM is the weight of unity of volume under those circumstances, and it is evident that Consequently D AT 1 *{jf k + h (16.) being the complete expression for that function of heat called temperature. It follows that the function 0, which enters into the expressions for the elasticity of gases, is given in terms of temperature by the equation oglcb Cnjub' (16a.) If, according to the expression 4, for the quantity of heat in one atom we substitute -^- for v 2 in equation (16), we obtain the following equa- tions : Mv 2 3£M/ r A (17.) and if Q represent the quantity of heat in unity of weight, 19. The real specific heat of a given substance is found by taking the differential coefficient of the quantity of heat with respect to the tempera- ELASTICITY OF GASES AND VAPOURS. 31 ture. Hence it is expressed in various forms by the following equations, in which the coefficient j is supposed not to vary sensibly with the tem- perature. Eeal specific heat of one atom, dq _ 3&M dr ~2CV real specific heat of unity of weight, dQ__U_ dT~2CV < 18 -> real specific heat of so much of a perfect gas as occupies unity of volume under unity of pressure at the temperature of melting ice, dq _ 3&M n dr~~2Cjx- The coefficient - — , representing the ratio of the total vis viva of the motions of the molecular atmospheres to the portion of vis viva which pro- duces elasticity, multiplied by the ratio of the total mass of the atom to that of its atmospheric part, is the specific factor in the capacity of an atom for heat. The view which I have stated as probable in article 1 3 — that the first factor of this coefficient is, like the second, a function of some permanent peculiarity in the nature of the atom — is confirmed by the laws discovered by Dulong : that the specific heats of all simple atoms bear to each other very simple ratios, and generally that of equality ; that the same property is possessed by the specific heats of certain groups of similarly constituted compound atoms ; and that the specific heats of equal volumes of all simple gases, at the same temperature and pressure, are equal. The coefficient — varies in many instances to a great extent for the A* same substance in the solid, liquid, and gaseous states. So far as experi- ment has as yet shown, it appears not to vary, or not sensibly to vary, with the temperature j and this I consider probable a priori, except at or near the points of fusion of solid substances. Apparent specific heat differs from real in consequence of the con- sumption and production of certain quantities of heat by change of volume and of molecular arrangement, which accompany changes of temperature. This subject belongs to the theory of the mechanical action of heat. 32 ELASTICITY OF GASES AND VAPOURS. Section IV. — Of the Coefficients of Elasticity and Dilatation of Gases. 1) T 20. If in equation (11) we substitute f or — -? + b its value ~— we obtain the following value for the elasticity of a gas, D ^1_F(D,0))+/(D); . . (19.) P ~~ nM C in which - denotes the ratio of the actual weight of unity of volume to wM the weight of unity of volume under unity of pressure, at the absolute temperature C, in the theoretical state of perfect gas ; t is the absolute temperature ; - F(D, 6) is a function of the temperature and density, representing the effect of the attraction of the atomic nucleus or centre in diminishing the superficial elasticity of its atmosphere ; And /(D) is a function of the density only, representing the effect of the mutual attractions and repulsions of the atoms upon the whole elasticity of the body. From this equation are now to be determined, so far as the experiments of M. Eegnault furnish the requisite data, the laws of the deviation of gases from that theoretical state in which the elasticity is proportional to the density multiplied by the absolute temperature. 21. The value of -F(D, 0) is given by the infinite series of equations T (10), (10fl), substituting in which for its value » — = we obtain the following result : — A., A 2 , A 3 , &c, being a series of functions, the value of which is given by the following equation : — ~K= 31^ ®l + mi ' ■ ( 20 -) 35 1+7n being the coefficient of z 1+m in the development of the reciprocal of the series 1 - (w')x + (a/> 2 - (a/ ")a? + &c, when (d)') &c. have the values given in equation (9). Equation (19) is thus transformed into p =a(e-v-?- &c -W< D >-- • ^ The series in terms of the negative powers of the absolute temperature ELASTICITY OF GASES AND VAPOURS. S3 converges so rapidly, that I have found it sufficient, in all the calculations I have hitherto made respecting the elasticity of gaseous bodies, to use the first term only, . T 22. Instead of making any assumption respecting the laws of the attractions and repulsions which determine the functions A and /(D), I have endeavoured to represent those functions by empirical formulae, deduced respectively from the experiments of M. Kegnault on what he terms the coefficient of dilatation of gases at constant volume, which ought rather to be called the coefficient of increase of elasticity with temperature, and from his experiments on the compressibility of elastic fluids at constant temperature. From the data thus obtained I have calculated, by means of the theory, the coefficients of dilatation of gases under constant pressure, which, as a test of the accuracy of the theory, I have compared with those deduced by M. Kegnault from experiment. 23. The mean coefficient of increase of elasticity with temperature at constant volume between 0° and 100° of the Centigrade thermometer is found by dividing the difference of the elasticities at those two tempera- tures by the elasticity at 0°, and by 100°, the difference of temperature. It is therefore represented by P — P E — -^ — °- (22/) ioo°p ' • K } where E represents the coefficient in question, and P and P x the elasti- cities at 0° and 100° Centigrade respectively. Now by equation (21), neglecting powers of - higher than the first, we have ^=30 -£)+'<« D_ /100° + C _ A \ 1_ »M\ C 100° + C/ +M h whence D /l A \ , . _ 7iMP VC + C(C + 100 o )/• ' ' ' t ■' Supposing the value of ... ■ to be known, this equation affords the 11 ° wMP ' * means of calculating the values of the function A corresponding to various densities, from those of the coefficient E as given by experiment. C 34, ELASTICITY OF GASES AND VAPOURS. As a gas is rarefied _ approximates to unity, A diminishes without limit, and the value of E consequently approximates to ~, the reciprocal of the absolute temperature at 0° Centigrade. This conclusion is verified by experiment ; and by means of it I have determined the values already given— viz., C = 274°6 Centigrade, and ^ = -003641 6G for the Centigrade scale. C 24. In order to calculate the values of D //-Ml',, 1 have made use of empirical formulae, deduced from those given by M. Regnault in his memoir on the Compressibility of Elastic Fluids. In M. Regnault's formulae, the unit of pressure is one metre of mercury, and the unit of density the actual density corresponding to that pressure. In the formula? which I am about to state, the unit of pressure is an atmosphere of 760 millimetres of mercury, or 29 - 922 inches; and the unity of density, the theoretical density in the perfectly gaseous state at 0° Centigrade, under a pressure of one atmosphere, which has been found from M. Regnault's M?iP formula' by making the pressure = in the value of . — °. M. Regnault's experiments were made at temperatures slightly above the freezing point, but not sufficiently so to render the formula; inaccurate for the purpose of , , . , . . . D calculating the ratio in question, ^ p . The formula; are as follows : — Supposing — given, (Vdi* rD L> a D\ 2 M/ : which, when T is small, or - nearly = C, gives an approximate value of - . Supposing P given, -I) CwMP = l+ 7 P -f £ P 2 ; which, when T is small, gives an approximate value of wMP ' , -4. The values of the constants a, /3, 7, e, and of their logarithms, are giverj, together with the mean temperatures above the freezing point at which ELASTICITY OF GASES AND VAPOUES. 35 M. Eegnault's experiments were made, for atmospheric air, carbonic acid gas, and hydrogen. ft Atmospheric Air. Constants. - 7 = - -000860978 + -000011182 - -000009700 4°-75. Logarithms. 4-9349920 5-0485140 6-9867717 Carbonic Acid Gas. T = 3°-27. a = - y = - -00641836 3-8074242 /3 = _ -0000041727 6-6204126 £ = + -0000865535 5-9372846 Hydrogen. T = 4°-75. = + -000403324 + -0000048634 _ -0000044981 4-6056546 6-6869401 6-6530291 The three substances above-mentioned are the only gases on which experiments have yet been made, under circumstances sufficiently varied to enable me to put the theory to the test I have described in article 22. 25. M. Regnault has determined the values of the coefficient of elasticity E for carbonic acid at four different densities, and for atmospheric air at ten. By applying equations (23) and (24) to those data, I have ascertained that the function A for these two gases may be represented empirically, for densities not exceeding that corresponding to five atmospheres, by the formula? given below, which lead to formulae for the coefficient E. For Carbonic Acid, A = a D where log. a = 0-3344538, and consequently «MP CV " r C+100° mM a lo C + 100 1 = 3-7608860. (25.) 3G ELASTICITY OF GASES AND VAPOURS. For Atmospheric Air, A = a[ —T7 r, 10 . where log. «=0-3176168, and consequently E = »MP 'C S0+ l0£ "C + 100° 1 c + ioo° 3-7440490 *) (26.) The value of log. ~ is 3-5012995. The following table shows that those empirical formula; accurately represent the experiments, the greatest differences being less than one- half of -00001 3 C, which M. Kegnault, in the seventy-first page of his memoir, assigns as the limit of the errors of observation due to barometric measurements alone. As the coefficient E for hydrogen has been determined for one density only, it is impossible to obtain an empirical formula for that gas. The single ascertained value of E is nevertheless inserted in the table. Table of Coefficients of Increase of Elasticity with Temperature at Constant Volu.m i Pressure at Density Coefficient E Coefficient E Difference be- Atmospheres = £„. D nM" according to the Formula. according to Experiment. Formula; and Experiment. CARBONIC ACID. I. 9980 1-00448 •003G865 •0036856 + -0000009 II. 1 -1857 1-19487 •0036951 •0036943 +•0000008 III. 2 2931 2-327S8 •0037465 •0037523 - -0000058 IV. 4-7225 4-87475 •003S647 •0038598 + -0000049 ATMOSPHERIC AIR. I. 01444 0-1444 •0036484 •0036482 + '0000002 II. 0-22'J4 0-2294 •0036507 •0036513 - -0000006 III. 0-3501 0-3502 •0036535 •0036542 - -0000007 IV. 0*4930 0-4932 •0036564 •0036587 - -0000023 V. 0-4937 0-4939 •0036564 •0036572 - -oooooos VI. 1-0000 1-000S5 •0036652 •0036650 + -0000002 VII. 2-2084 2-2125 •0036810 •0036760 + -0000050 VIII. 2-2270 2-2312 •0036812 •0036800 + -0000012 IX. 2-8213 2S279 •0036880 •0036894 - -0000014 X. 4-8100 4-8289 •0037081 •0037091 - -oooooio ] iYDROGEN. 1-0000 0-9996 No formula. •0036678 1 ELASTICITY OF GASES AND VAPOURS. 37 26. The empirical formula? (24), representing the experiments of M. Eegnault on the compressibility of carbonic acid gas, atmospheric air, and hydrogen at certain temperatures, give for these temperatures the values of a function which is theoretically expressed by CraMP CA CwM/(D) rD T f tD * (27.) -r ■ 1 • C»MP p . ■ . , ., It is evident, that supposing the value of — =— tor any given density TXJ to be known by experiment, and that of A to be calculated from the value of the coefficient E, or from the empirical formulae (25) and (26), nMf(D) the corresponding value of the function 1) may be determined by means of equation (27). By this method I have obtained the following empirical formulae for calculating the values of that function : — For Carbonic Acid, «M/(D) _ D D mM' where log. A =3*1083932. Far Atmospheric Air, "M/(D) _ / D D where log. h= 3-8181545. nMf(D) * »il (28.) As only one value of I) for hydrogen can at present be ascer- tained, it is impossible to determine a formula for that gas. The single value in question is — For P = 1 atmosphere, wM ^ D ) = -01059. . (29.) 27. I now proceed to determine theoretically, from the data which have already been obtained, the mean coefficients of dilatation at constant pressure, between 0° and 100° of the Centigrade scale, for the three gases under consideration, at various pressures. Let E' represent the coefficient required: S and S x the respective %M values of -=r for 0° and 100° under the pressure P, that is to say, the volumes occupied by the weight nM at those temperatures ; A and A v 38 ELASTICITY OF GASES AND VAPOURS. / and f v the corresponding values of A and /(D). Then from equation (21) we deduce the following results: — So = f(l-^ Q +S /o) s -V14- 1000 - A < +S f V • ' ~i'V r C C + lOO ' ' lJl )' and consequently E' = s i _ ,s ,> 100 s 1 /l + S P\C ' 100C 100(C+100) 100 )• • (30.) In applying the empirical formuhe (25), (26), and (28), to determine the values of Aj and S, /, in the above equation, it will produce no appreciable error to use TTTfJ^o as an approximate value of D x for that purpose only. By making the necessary substitutions, the following formula? arc obtained : — For Carbonic Acid, wMPVC" 1 " »M/ J where log. 0= 5*5189349. For Atmospheric Air, L ",»Ml' Vc +a V»M/ ~P'\nW ) where log. a =5*4717265 log. /3 = 6*9759738 J (31.) 28. The following table exhibits a comparison between the results of the formulae and those of M. Eegnault's experiments. It is not, like the preceding table (article 25), the verification of empirical formulae, but is a test of the soundness of the theoretical reasoning from which equations (30) and (31) have been deduced. It is impossible, from the want of a sufficient number of experimental data, to give a formula similar to (31) for hydrogen. I have calculated, however, the value of the coefficient E' for that gas, corresponding to the pressure of one atmosphere, on the assumption that at that pressure a formula similar to that for carbonic acid gas is applicable without sensible error. ELASTICITY OF GASES AND VAPOURS. 39 The table shows only one instance in which the difference between the result of the theory and that of experiment exceeds '0000136; the limit, according to M. Eegnault, of the errors of observation capable of arising from one cause alone, — the uncertainty of barometric measurements. That discrepancy takes place in one of the determinations of the coefficient E' for carbonic acid gas under the pressure of one atmosphere. In the other determination the discrepancy is less than the limit. The agreement between theory and experiment is most close for the highest pressures; and M. Eegnault has shown that the higher the pressure the less is the effect of a given error of observation in producing an error in the value of the coefficient. The theory is therefore successful in calculating the coefficients of dilatation of gases, so far as the means at present exist of putting it to the test. Table of Coefficients of Dilatation under Constant Pressure, showing a Comparison between Theory and Experiment. Pressure in Atmospheres. Coefficient E' according to the Theory. Coefficient E' according to M. Kegnault's Experiments. Difference between Theory and Experiment. CARBONIC ACID GAS. 1-000 3-316 •00369S8 ■0038430 First Memoir. •0037099 Second Memoir. •003719 First Memoir. •0038450 - 0000111 - -0000202 - -0000020 ATMOSPHERIC AIR. 1-0000 3-3224 3-4474 •0036650 ■0036955 •0036969 First Memoir. •0036706 Second Memoir. 003663 •003667 First Memoir. •0036944 •0036965 - -0000056 + -0000020 - -0000020 + •0000011 + -0000004 HYDROGEN. 1-0000 •0036598 •0036613 - -0000015 40 ELASTICITY OF GASES AND VAPOURS. Section V. — Of the Elasticity of Vapour in Contact with the same Substance in the Liquid or Solid State. 29. As the most important phenomena of evaporation take place from the liquid state, I shall generally use the word liquid alone through- out this section in speaking of the condition opposed to the gaseous state ; but all the reasonings are equally applicable to those cases in which a substance evaporates from the solid state. 30. In considering the state of a limited space entirely occupied by a portion of a substance in the liquid form, and by another portion of the same substance in the form of vapour, both being at rest, the most obvious condition of equilibrium is, that the total elasticity of the substance in each of the two states must be the same; that is to say, P=^4V(D )=i J i+/(D 1 ), • • • (32.) where p represents the superficial atomic elasticity in the liquid state, p x that in the gaseous state, and /(D ), /(DJ, the corresponding values of the pressures, positive or negative, due to mutual actions of distinct atoms. 31. A second condition of equilibrium is, that the superficial elastici- ties of two contiguous atoms must be equal at their surface of contact. Hence, although there may be an abrupt change of density at the bounding surface between the liquid and the vapour, there must be no change of superficial atomic elasticity except by inappreciable degrees; and at that bounding surface, if there is an abrupt change of density (as the reflexion of light renders probable), there must be two densities corresponding to the same superficial atomic elasticity. 32. A third condition of equilibrium is to be deduced from the mutual attractions and repulsions of the atoms of liquid and of vapour. In a gas of uniform density, those forces, acting on each individual particle at an appreciable distance from the bounding surface, balance each other, and have accordingly been treated as merely affecting the total elasticity of the body by an amount denoted by /(D) ; but near the bounding surface of a liquid and its vapour, it is obvious that the action of the liquid upon any atom must be greater than that of the vapour. A force is thus produced which acts on each particle in a line perpendicular to that bounding surface, and which is probably attractive towards the liquid, very intense close to the bounding surface, but inappreciable at all distances from it perceptible to our senses. Such a force can be balanced only by a gradual increase of superficial atomic elasticity in a direction towards the liquid. Hence, although at perceptible distances from the surface of the liquid the density of vapour is sensibly uniform, the layers close to that surface are probably in a state of condensation by attraction, analogous to that of the earth's atmosphere under the influence of gravity. ELASTICITY OF GASES AND VAPOURS. 41 Professor Faraday has expressed an opinion, founded on his own experi- ments and those of MM. Dulong and Thenard, that a state of condensation exactly resembling that which I have described is produced in gases by the superficial attraction of various substances, especially platinum, and gives rise to chemical actions which have been called catalytic. To express this third condition algebraically, let the boundary between the liquid and the vapour be conceived to be a plane of indefinite extent, perpendicular to the axis of x ; and let positive distances be measured in a direction from the liquid towards the vapour. Let x, x + dx represent the positions of two planes, perpendicular to the axis of x, bounding a layer whose thickness dx is very great as compared with the distance between two atomic centres, but very small as compared with any perceptible distance, and let a portion of the layer be considered whose transverse area is unity. Let p represent the mean density of the layer. Then it is acted upon by a force -pXrfx, the resultant of the actions of all the neighbouring atoms, which has the negative sign, because it is attractive towards the liquid, X being a func- tion of the position of the layer in question, and of the densities and positions of all the neighbouring layers. The superficial atomic elasticity behind the layer being p, and in front of it p + -j- dx, it is also acted on by the force -to 6 *'* hence its condition of equilibrium is !+» x =° ( 33 -> In order to integrate this equation, so as to give a relation applicable at perceptible distances from the surface, let x , x x represent the positions of two planes perpendicular to the axis of x, the former situated in the liquid, the latter in the vapour, and so far asunder that the densities beyond them are sensibly uniform, and equal respectively to D for the liquid and D 1 for the vapour, the corresponding superficial atomic elastici- ties being p and p v Then dividing equation (33) by p, and integrating between the limits x Q and x v the result obtained is o l J o Had we a complete knowledge of the laws of molecular forces in the solid, liquid, and gaseous states, this equation, taken in conjunction with the 42 ELASTICITY OF GASES AND VAPOURS. two conditions previously stated, would be sufficient to determine formulae for calculating the total elasticity and the respective densities of a liquid and its vapour when in contact in a limited space, at all temperatures. 33. In the absence of that knowledge, I have used equation (34), so as to indicate the form of an approximate equation suitable Re- calculating the elasticity of vapour in contact with its liquid at all ordinary temperatures, the coefficients of which I have determined empirically, — for water and mercury, from the experiments of M. licgnault, and for alcohol, ether, turpentine, and petroleum, from those of Dr. Ure. It has been shown (equation 19) that the superficial atomic elasticity is expressible approximately in terms of the density and temperature for gases by f , / r \\ where the function F is a very rapidly converging series, in terms of the negative powers of the absolute temperature, the coefficients being functions of the density. It is probable that a similar formula is applicable to liquids, the series being less convergent. It follows that the density is expressible approximately in terms of the superficial atomic elasticity by P=i ,^(i+*( P ,^)), the function 4> being also a converging scries in terms of the negative powers of the absolute temperature, and the coefficients being functions of p. Making this substitution in the first side of equation (34), and abbreviating <p (j), T J into <£, we obtain the following result : — ?1 j 1 _r_ |», _J_ ]p o 1K p CnMJp Q ^'j»(l+$) = cii( l0 - * " lo s-^o -// *£(ifl)) = -(Vx; (35.) o from which, making log- tPo+l 1 d P ■ n\*\ = *> and CnM I * ch ■ X = Q ' 8c/0 Jj; x p(i+*) )\ the following value results for the hyperbolic logarithm of the super- ficial atomic elasticity of the vapour at sensible distances from the surface of the liquid : — ~ log., ^ = ¥ - -. . • • (36.) ELASTICITY OF GASES AND VAPOURS. 43 In the cases which occur in practice, the density of the vapour is very small as compared with that of the liquid. Hence it follows that in such cases the value of ^F depends chiefly on the superficial atomic elasticity of the liquid, and that of Q on its density. The density is known to diminish with the temperature, but slowly. The superficial atomic elasticity, according to equation (32), is expressed by Po = Pi +f(Pi) -/(*U where p x and f(D x ) are obviously small as compared with /(D ), a function of the density of the liquid, so that the variations of p and of M* with the temperature are comparatively slow also. Therefore, when the density of the vapour is small as compared with that of the liquid, the principal variable part of the logarithm of its superficial atomic elasticity, and consequently of its whole pressure, is negative, and inversely proportional to the absolute temperature; and ft a (a and /3 being constants) may be regarded as the first two terms of an approximate formula for the logarithm of the pressure. A formula of two terms, similar to this, was proposed about 1828 by Professor Eoche. I have not been able to find his memoir, and do not know the nature of the reasoning from which he deduced his formula. It has since been shown, by M. Eegnault and others, to be accurate for a limited range of temperature only. The quantity corre- sponding in it to t is reckoned from a point determined empirically, and very different from the absolute zero. Thus far the investigation has been theoretical. The next step is to determine empirically what other terms are requisite in order to approxi- mate to the effect of the function /(D), and of the variation of the functions ^P and Q. The analogy of the formula for the dilatation of gases, the obvious convenience in calculation, and the fact that the deviations of the results of the first two terms from those of experiment are greatest at low temperatures, naturally induced me to try, in the first place, the effect of a third term inversely proportional to the square of the absolute temperature, making the entire formula for the logarithm of the pressure of vapour in contact with its liquid a — r 7" and the inverse formula, for calculating the abso- lute temperature from the pressure, 1 _ ja - log. P g 2 _ (3_ y 4y 2 2y (37.) 44 ELASTICITY OF GASES AND VAPOURS. the values of the constants a, /3, 7, being determined by the ordinary methods from three experimental data for each substance. 34. The agreement of those formulae with the results of experiment proved so remarkable, that, as they are calculated to be practically useful, I thought it my duty not to delay their publication until I should have an opportunity of submitting my theoretical researches to the Royal Society of Edinburgh. I therefore communicated the formulas to the Edinburgh New Philosophical Journal for July, 1849, together with the full details of their comparison, graphic and tabular, with the experiments of M. Regnault upon water and mercury, and with those of Dr. Ure upon alcohol, ether, turpentine, and petroleum, but without giving any account of the reasoning by which 1 had been led to them. Without repeating those details here, I may state that the agreement between the results of the formula' and those of observation is in every case as close as the precision of the experiments renders possible. This is remarkable, especially with respect to the experiments of M. Regnault on the elasticity of steam, which extend throughout a range of tempera- tures from 30° below zero of the Centigrade scale to 230° above it, and of pressures from ytgv °f an atmosphere to 28 atmospheres, and which, from the methods of observation adopted, especially those of measuring temperature, necessarily surpass by far in precision all other experiments • if the same kind. From 20° to 230° Cent, the greatest discrepancy between calculation and experiment corresponds to a difference of jfa of a Centigrade degree, and very few of the other differences amount to so much as -^ of a degree. Below 20°, where the pressure varies so slowly with the temperature that its actual value is the proper test of the formula, the greatest discrepancy is -j 3 ^ of a millimetre of mercury, or 2-jjxy of an inch. If the curves representing the formula? were laid down on M. Regnault's diagram, they would be scarcely distinguishable from those which he has himself drawn to exhibit the mean results of his experiments. Annexed is a table of the values of the constants a, log. /3, log. y, ly Ay- existing experiments on mercury, turpentine, and petroleum, are not sufficiently extensive to indicate any precise value for the coefficient y (which requires a great range of temperatures to evince its effect), I have used for these fluids, as an approximation, the first two terms of the formula only, a — — . T For different measures of pressure, the contact a evidently varies equally with the complement of the logarithm of the unit of pressure. For different thermometric scales /3 varies inversely as the length of a „ for the fluids for which they have been calculated. 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For all the fluids except water, it will probably he found necessary to correct more or less the values of the constants, when more precise and extensive experiments have been made, especially those for the more volatile ether, and for turpentine, petroleum, and mercury, which have all been determined from data embracing but a small range of pressures. In reducing the constants for the Centigrade scale to those for Fahren- heit's scale, 180° of the latter have been assumed to be equal to 100° of the former. In order that this may be the case, the boiling point of Fahrenheit's scale must be adjusted under a barometric pressure of 760 millimetres, or 29"922 inches, of mercury, whose temperature is 0° Centigrade. In the ninth and tenth columns of tin- table are given the limits on the scales of temperature and pressure between which the formulae have been compared with experiment. It is almost certain that the formula for the pressure of steam may be employed without material error for a consider- able range beyond, and probably also that for the pressure of vapour of alcohol; but none of the formula' are to be regarded as more than approximations to the exact physical law of the elasticity of vapours, for the determination of which many data are still wanting, that can only be supplied by extensive series of experiments. The following are some additional values of the constant a for steam, corresponding to various units of pressure used in practice : — I'nits of Pressure. Values of a. Atmospheres of 760 millimetres of mercury — = 29-922 inches of mercury = 14*7 lbs. on the square inch = 1-0333 kilog. on the square centim., 4-950433 Atmospheres of 30 inches of mercury — = 761-99 millimetres = 14-74 lbs. on the square inch = 1-036 kilog. on the square centim., 4-949300 Kilogrammes on the square centimetre, . . 4-964658 Kilogrammes on the circular centimetre, . 4 "859 748 Pounds avoirdupois on the square inch, . . 6-117662 Pounds avoirdupois on the circular inch, . 6-012752 Pounds avoirdupois on the square foot, . . 8-276025 All the numerical values of the constants are for common logarithms. ELASTICITY OF GASES AND VAPOURS. 47 35. According to the principles which form the basis of calculation in this section, every substance, in the solid or liquid state, is surrounded by an atmosphere of vapour, adhering to its surface by molecular attrac- tion ; and even when the presence of vapour is imperceptible at all visible distances from the body's surface, the elasticity of the strata close to that surface may be considerable, and sufficient to oppose that resistance to being brought into absolute contact, which is well known to be very great in solid bodies, and perceptible even in drops of liquid. It is possible that this may be the only cause which prevents all solid bodies from cohering when brought together. The action of an atmosphere of vapour, so highly dense and elastic as to operate at visible distances, may assist in producing the spheroidal state of liquids. If the particles of clouds are small vesicles or bubbles (which is doubt- ful), the vapour within them may, according to these principles, be considerably more dense than that which pervades the external air, and may thus enable them to preserve their shape. Section VX — Of Mixtures of Gases and Vapours of Different Kinds. 36. The principle stated in Section II. article 11, that the elasticity of the atomic atmosphere is proportional to its density, might be otherwise expressed by saying, that the elasticity of any number of portions of atomic atmosphere, compressed into a given space, is equal to the sum of the elasticities which such portions would respectively have, if they occupied the same space separately. If the same principle here laid down for portions of atomic atmosphere of any one kind of substance, be considered as true also of portions of atomic atmosphere of substances of different kinds mixed, and if it be supposed that when two or more gases are mixed there is no mutual force exerted between atoms of different kinds, except the elastic pressure of the atomic atmospheres, it will then evidently follow. — First, that the mixed gases will only be in equilibria when the particles of each of them are diffused throughout the whole space which contains them. Secondly, that the particles of each gas taken separately will be in the same condition as to density, elasticity, arrangement, and mutual action, and also as to gravitation, or any other action of an external body, as if that gas occupied the space alone. Thirdly, that the joint elasticity of the mixed atomic atmospheres at any given point, will be the sum of the elasticities which they would respec- tively have had at that point if each gas had occupied the space alone. 48 ELASTICITY OF GASES AND VAPOURS. Fourthly, that the value of the elasticity, positive or negative, resulting from the attractions and repulsions of separate atoms, will be the sum of the values it would have had if each gas had occupied the space alone ; and, Fifthly, that the total elasticity of the mixed gases will be the sum of the elasticities which each would have had separately in the same space. If there are any mutual actions between the particles of different gases except the elasticity of the molecular atmospheres, these conclusions will no longer be rigidly true • but they will still be approximately true if the forces so operating are very small. This is probably the actual condition of mixed gases. 37. On applying the same principle to the case of a gas mixed with a vapour in contact with its liquid, it is obvious that if the attractions and repulsions of the particles of the gas upon those of the vapour are null or inappreciable, the direct effect of the presence of the gas upon the elasticity assumed by the vapour at a given temperature will also be null or inappreciable. The gas, however, may have a slight indirect influence, by compressing the liquid, and consequently increasing its superficial atomic elasticity and its 4 attractive power, on which the functions ¥ and £2 in equation (3G) depend. The probable effect of this will be to make the elasticity of the vapour somewhat less than if no gas were present. There appear to be some indications of such an effect ; but they are not sufficient to form a basis for calculation. Supposing the gas, on the contrary, to exercise an appreciable attraction on the particles of vapour, the elasticity of the latter will be increased. Traces of an effect of this kind are perceptible in M. Regnault's experi- ments on the vapour of mercury in which air was present. 38. I have already referred to the property ascribed by Professor Faraday to various substances, of attracting, and retaining at their surfaces, layers of gas and vapour in a high state of condensation. Sup- posing a solid body to acquire, in this manner, a mixed atmosphere, consisting partly of its own vapour and partly of foreign substances, the total elasticity of that atmosphere at any point will be equal, or nearly equal, to the sum of the elasticities which each ingredient would have had separately; and thus solid metals, glass, charcoal, earthy matters, and other substances, may acquire a great power of resisting cohesion, although producing no perceptible vapours of their own at ordinary temperatures. THE CENTRIFUGAL THEORY OF ELASTICITY. 49 IV.— ON THE CENTRIFUGAL THEORY OF ELASTICITY AND ITS CONNECTION WITH THE THEORY OF HEAT.* Section I. — Relations between Heat and Expansive Pressure. 1. In February, 1850, I laid before the Royal Society of Edinburgh a paper in which the laws of the pressure and expansion of gases and vapours were deduced from the supposition that that part of the elasticity of bodies which depends upon heat arises from the centrifugal force of the revolutions of the particles of elastic atmospheres surrounding nuclei or atomic centres. A summary of the results of this supposition, which I called the Hypothesis of Molecular Vortices, was printed in the Transactions of this Society, Vol. XX., as an introduction to a series of papers on the Mechanical Action of Heat; and the original paper has since appeared in detail in the Philosophical Magazine. In that paper the bounding surfaces of atoms were defined to be imaginary surfaces, situated between and enveloping the atomic nuclei, and sym- metrically placed with respect to them, and having this property — that at these surfaces the attractive and repulsive actions of the atomic nuclei and atmospheres upon each particle of atomic atmosphere balance each other. The pressure of the atomic atmospheres at those imaginary boundaries is the part of the total expansive pressure of the body which varies with heat, the effect of the centrifugal force of molecular vortices being to increase it. In the subsequent investigation it was assumed that, owing to the sym- metrical action of the particles of gases in all directions, and the small amount of those attractive and repulsive forces which interfere with the elasticity of their atmospheres, no appreciable error would arise from treating the boundary of the atmosphere of a single atom, in calculation, as if it were spherical, an assumption which very much simplified the analysis. An effect, however, of this assumption was to make it doubtful whether the conclusions deduced from the hypothesis were applicable to any sub- stances except those nearly in the state of perfect gas. I have, therefore, * Read before the Royal Society of Edinburgh on December 15, 1S51, and published in Vol. XX. , Part iii. , of the Transactions of that Society. D 50 THE CENTRIFUGAL THEORY OF ELASTICITY. in the present paper, investigated the subject anew, without making any assumption as to the arrangement of the atomic centres, or the form of the boundaries of their atmospheres. The equations deduced from the hypothesis between expansive pressure and heat, are therefore applicable to all substances in all conditions; and it will be seen that they are identical with those in the original paper, showing that the assumption that the atomic atmospheres might be treated, in calculation, as if spherical, did not give rise to any error. By the aid of certain transformations in those equations I have been enabled, in investigating the principles of the mutual transformation of heat and expansive power, to deduce Joules law of the equivalence of heat and mechanical power directly from them, instead of taking it (as I did in my previous papers) as a consequence of the principle of vis viva. Carnofs law is also deduced directly from the hypothesis, as in one of the previous papers. 2. Classification of Elastic Pressures. — The pressures considered in the present paper are those only which depend on the volume occupied by a given weight of the substance, not those which resist change of figure ill solids and viscous liquids. Certain mathematical relations exist between those two classes of pressures ; but they do not affect the present investigation. To illustrate this symbolically, let V represent the volume occupied by unity of weight of the substance, so that is the mean density; Q, the V quantity of heat in unity of weight, that is to say, the vis viva of the mole- cular revolutions, which, according to the hypothesis, give rise to the expansive pressure depending on heat ; and let P denote the total expan- sive pressure. Then, P = F(V,Q)+/(V),. . . . (1.) In this equation, F(V, Q) is the pressure of the atomic atmospheres at the surfaces called their boundaries, which varies with the centrifugal force of the molecular vortices as well as with the mean density ; and /( V) is a portion of pressure due to the mutual attractions and repulsions of distinct atoms, and varying with the number of atoms in a given volume only. If the above equation be differentiated with respect to the hyperbolic logarithm of the density, we obtain the coefficient of elasticity of volume V V V where ft denotes the cubic compressibility. THE CENTRIFUGAL THEORY OF ELASTICITY. 51 The latter portion of this coefficient, — -==f(V), consists of two parts, T one of which is capable of being resolved into forces, acting along the lines joining the atomic centres, and gives rise to rigidity or elasticity of figure as well as to elasticity of volume, while the other, which is not capable of being so resolved, gives rise to elasticity of volume only. The ratio of each of those parts to their sum must be a function of the heat, the former part being greater and the latter less, as the atomic atmosphere is more concentrated round the nucleus ; that is to say, as the heat is less ; but their sum, so far as elasticity of volume is concerned, is a function of the density only. That is to say, as in equation (12) of my paper on the Laws of the Elasticity of Solids {Cambridge and Dublin Mathematical Journal, February, 1851), let the total coefficient of elasticity of volume be denoted thus — ~ = J + cp(C v C 2 ,C s ), . • (IB.) C 1? C 2 , C 3 , being the coefficients of rigidity round the three axes of elas- ticity, and J a coefficient of fluid elasticity ; then J = -^F(V,Q)-^(V,Q,)-^/(V) T V d y • (ic.) ^(C 1; C 2 ,C 3 ) = -(l-^V,Q))-^/(V) V j For the present, we have to take into consideration that portion only of the expansive pressure which depends on density and heat jointly, and is the means of mutually converting heat and expansive power ; that is to say, the pressure at the boundaries of the atomic atmospheres, which I shall denote by Pressures throughout this paper are supposed to be measured by units of weight upon unity of area ; densities by the weight of unity of volume. 3. Determination of the External Pressure of an Atomic Atmosphere. — Let a body be composed of equal and similar atomic nuclei, arranged in any symmetrical manner, and enveloped by an atmosphere, the parts of which are subject to attractive and repulsive forces exercised by each other and by the nuclei. Let it further be supposed that this atmosphere, at each point, has an elastic pressure proportional to the density at that point, multiplied by a specific coefficient depending on the nature of the BOSTON UNIVERSITY C.ni i FGF or LIBERAL ARTS 52 THE CENTRIFUGAL THEORY OF ELASTICITY. substance, which I shall denote by h. (This coefficient was denoted by b in previous papers.) Let p and p denote the density and pressure of the atomic atmosphere at any point ; then / = h P be the accelerative forces operating on a particle of atomic atmosphere in virtue of the molecular attractions and repulsions which I have made explicitly negative, attractions being supposed to predominate. The property of the surfaces called the boundaries of the atoms is this (^=0,(1)^0,(^=0, the suffix , being used to distinguish the value of quantities at those surfaces. Hence $ x is a maximum or minimum. Those surfaces are symmetrical in form round each nucleus, and equidistant between pairs of adjacent nuclei. Their equation is $ - $! = 0. Let M denote the total weight of an atom ; /x that of its atmospheric part, and M — fi that of its nucleus; tl en MV is the volume of the atom, — the mean density of the atmospheric part, measured by weight, the nucleus being supposed to be of insensible magnitude; — and we have the following equations MV = //./( 1 )' / ''' / M = m v j 1 i Q d x '' y '' "■ = 1 1 1 d) p dxd y (l -■ (2.) J The suffix (j) denoting that the integration is to be extended to all points within the surface ($-$ 1 = 0). According to the hypothesis now under consideration, Meat consists in a revolving motion of the particles of the atomic atmosphere, com- municated to them by the nuclei. Let v be the common mean velocity possessed by the nucleus of an atom and the atmospheric particles, when the distribution of this motion has been equalised. I use the term mean velocity to denote that the velocity of each particle may undergo small periodic changes, which it is unnecessary to consider in this investigation. THE CENTRIFUGAL THEORY OF ELASTICITY. 53 Then the quantity of heat in unity of weight is 2g being equal to the mechanical power of unity of weight falling through the height — . The quantity of heat in one atom is of course MQ, and in the atmospheric part of an atom, pQ. I shall leave the form of the paths described by the atmospheric par- ticles indeterminate, except that they must be closed curves of permanent figure, and included within the surface (<!> -c&^O). Let the nucleus be taken as the origin of co-ordinates, and let a, )3, y, be the direction- cosines of the motion of the particles at any point (x, y, z). Then the equations of a permanent condition of motion at that point, are P dx ax \ dx ' ay dz/ 1 dp P 1 dp' d$ 2 p' dy dy 1 dp' d& p' dz dz 1 di ■f.+pi+^ ■ (3.) rt / d ~. d d\ „ Let r be the radius of curvature of the path of the particles through (x, y, z), and o' /3' 7', its direction-cosines; then the above equations obviously become 1 dp_ p dx 1 dp' p dy 1 rf_p' p dz d$ dx d$> dy d$ dz a r - 2Q _ 2 q£-=C 2Q^ r (3A.) If these equations are integrable, — dx + $- dy +^-dz r r r must be an exact differential. Let — <p be its primitive function ; the negative sign being used, because a, /3', y' must be generally negative. Then the integral of the equations (3) is lo SvP = dp' 1 , - £ - = -=-(2 Q0 — <P) + constant: h 54 THE CENTRIFUGAL THEORY OF ELASTICITY. or taking f> y to denote the pressure at the bounding surface of the atom : — P = Pi e ■ (4.) Our present object is to determine the superficial-atomic density, p v and thence the pressure 2 )= hpv ^ u terms °f tne mean density and heat Q. For this purpose we must introduce the above value of p into equation (2), giving ^t^p^ J J f/>fi dxdydz whence \ ( f h dxdydz. (5.) Let the volume of the atom be conceived to be divided into layers, in each of which <p has a constant value. Then we may make the following transformations. p = h Pl =hfi+J j | ( f f Id x d y d z=kMvfe ~ d<J> — 1 (*_* 1 ) h(<p-<Pi) > (C) dxdydz=kMYje ^d<p k being a specific constant, and xp and w functions of cf>, and of the nature and density of the substance. The lower limit of integration of $ must be made — cc , that it may include orbits of indefinitely small magnitude described round the atomic centre. The nature of the function \p is limited by the following condition, l=k\ Vl e -T-d<h J — GO Wt (7.) 20 Let^+1 = hie Then these transformations give the following result for the pressure at the bounding surface of an atom : — p=h Pl Jl/UL MV + £ h<p x fl*(0-0i) kdcp hfx MV Wl w \ i w l w l , p ► . (8.) e ■ e 2 THE CENTRIFUGAL THEORY OF ELASTICITY. 55 to' v &c, being the successive differential coefficients of w with respect to k([>, when <p = (p v 4. The following transformation will be found useful in the sequel. Let X be the indefinite value of log. e V, and \ its actual value in the case under consideration. Let G be the same function of A which w is of k<p, and let G', G", &c, be its successive differential coefficients with respect to A. Let „ f\ *(*-*!)„ ,. g x g; , g;' , Then p-uv^ {J -> The function H has the following properties, which will be afterwards referred to : — < ^ + 6li 1 -G 1 =0 \ mix=- d ^ — oo a v (10.) 5. Case of a Perfect Gas. — As a substance is rarefied, it gradually approaches a condition in which the pressure, under like circumstances as to heat, varies proportionally to the density. This is because the effect of the molecular attractions and repulsions on the pressure diminishes with the density, so that <P, w, and G approximate to constant quantities. In the limiting or perfectly gaseous condition, therefore, G, H i=7T and hnQ hfi /2Q \ P =MY=MvKhk +1 )' * • (1L 6. Equilibrium of Heat : Nature of Temperature and Real Specific Heat. — When the atmospheres of atoms of two different substances are in contact at their common bounding surface, it is necessary to a per- manent condition that the pressure in passing that surface should vary continuously. Let (a) and (b) be taken as characteristics, to distinguish the specific quantities peculiar to the two media respectively. Let dm denote the volume of an indefinitely thin layer, close to the bounding surface. Then the following equations must be fulfilled to ensure a permanent condition : — i;(a) ^ ;(6); ^ (a)= ^ (J)when/ ^ * * (12 ° 56 THE CENTRIFUGAL THEORY OF ELASTICITY. By making the proper substitutions in equation (4), it appears that j/=pe 7- Hence *•? £p=^=^-dM + dm dm Now, j) is the same for both media : & • _ --*<•-•>> / = c is either a maximum or a minimum, so that its differential is null ; and dm is a continuous function of lc<p, so that dm v dm v There remains only the function of heat : 20 Therefore the condition of a permanent state of molecular motion, that is to say, the condition of equttibrivm of heat, is that this function shall be the same for the two substances ; or that ?%=|a as.) Hence, temperature depends on the above function only ; for the definition of temperature is, that bodies at the same temperature are in a permanent condition as to heat, so far as their mutual action is concerned. The ratio of the real specific heat of (a) to that of (b) is obviously KK ■■ *A- • • • ( u -> 7. Measure of Temperature ami Specific Heat. — The function 6 is proportional to the pressure of a perfect gas at a constant density. That pressure, therefore, is the most convenient measure of temperature. Let t denote absolute temperature, as measured by the pressure of a perfect gas at constant density, and reckoned from a certain absolute zero, 274 0- G Centigrade, or 494° - 28 Fahrenheit below the temperature of melting ice. Let k be a constant which depends on the length of a degree on the thermometric scale, and is the same for all substances in nature. Then 2^, 1 M hi + K Q = (r-«) C I '2« J THE CENTRIFUGAL THEORY OF ELASTICITY. 57 and the real specific heat of the substance, that is to say, the depth of fall, under the influence of gravity, which is equivalent to a rise of one degree of temperature in the body, is represented by fe=P (1G.) The pressure of a perfect gas is represented in terms of temperature by h U. T ,-. „ \ It may also be expressed thus : let r denote the absolute temperature of meltiDg ice in degrees of the scale employed, and V the volume of unity of weight of the substance in the theoretical state of perfect gas, at the temperature of melting ice and pressure unity : — then *-%P ■ ■ < 18 -> v T o On comparing this with equation (17) we see that kfx _ V _ 1 h flT h JUL K (19.) MV 'MV r J Xow h is the specific elasticity of the atomic atmosphere of the substance ; =-£=- is the mean specific gravity of that atmosphere, when the body is in M V the theoretical state of perfect gas; and k and r are the same for all substances in nature. Therefore, for every substance in nature, the mean specific gravity of the atomic atmosphere in the theoretical state of perfect gas is inversely proportional to the specific elasticity of that atmosphere. Real specific heat may also be thus expressed : — k== y Bi *M (20) V 1 -, ^M, 3£M in which -°- corresponds to t^-tv m my former papers, and — — to -r— - 1 or =r. N The latter factor appears to depend on the chemical constitution of the substance, being the same for all simple gases. 8. Total Pressure of Substances in general, expressed in terms of Temperature. In equation (9) let - be put for 9 : then 58 THE CENTRIFUGAL THEORY OF ELASTICITY. v= P +/( v) =/(V) + % £»,+ {«.- K ^+^' : - M where A x = — — ; A 2 = — ^ 2 (G/ 2 — G/') a 3 = - j£ 3 (<V 3 - 20/6/'+ G/"); &c. This formula is identical with that which I employed in my former paper to represent the pressure of an imperfect gas, and which I found to agree with M. Regnault's experiments, when the coefficients A and the function /(V) had been calculated empirically. Section II. — Eelations between Heat and Expansive Power. 9. Variations of Sensible and Latent Heat : Fundamental Equation of the Theory. — If the forms, positions, and magnitudes of the paths described by the revolving particles of the atomic atmospheres be changed, whether by a variation of mean density, or by a variation of temperature, an increase or diminution of the vis lira of their motion, that is to say, of the heat of the body, will take place in virtue of that change of the paths of motion — an increase when they are contracted, and a diminution when they are dilated. Let S.Q represent, when positive, the indefinitely small quantity of heat which must be communicated to unity of weight of a substance, and when negative, that which must be abstracted from it, in order to produce the indefinitely small variation of temperature S t simultaneously with the indefinitely small variation of volume 2V. Let S.Q be divided into two parts SQ + SQ'=S.Q, of which SQ, being directly employed in varying the velocity of the particles, is the variation of the actual or sensible heat possessed by the body ; while SQ', being employed in varying their orbits, represents the amount of the mutual transformation of heat with expansive power and molecular action, or the variation of what is called the latent heat; that is to say, of a molecular condition constituting a source of power, out of which heat may be developed. (§Q' in this paper corresponds to — 2Q' in my former papers.) THE CENTRIFUGAL THEORY OF ELASTICITY. 59 The variation of sensible heat has evidently this value, ZQ = \\St (22.) Let dx, By, $z, be the displacements of the orbit of the particles of atomic atmosphere at the point (x, y, z) . A molecule p dxclydz is acted upon by the accelerative forces (see equation 3A). parallel to the three axes respectively. The sum of the actions of those forces on the molecule pdxdydz during the change of temperature and volume, is <>+^+^-V-^--< dy = — 2Q<$<ppdxd yd :. The sum of such actions upon all the particles in unity of weight is equal in amount and opposite in sign to the variation of latent heat ; that is to say, W = ^fff {i) p$<pdxdydz. . . (23.) To determine the value of the variation Bcp, let it be divided into two parts, thus : — t(p — ccp l + oAcp, where A<j> = <p — (p r First, With respect to $<f> v it is obvious that because, according to equations (6, 7), • ,-0i ks<p j. MY = IMY e -f-dd>, J -co \p x r we must have BV=:kYB(p 1 and $<Pi = t^- Hence the first part of the integral (23) is =&-4- • • • < 23A -> To determine the second part of the integral we have the condition that the quantity of atomic atmosphere inclosed within each surface at which A(f> has some given value is invariable; that is to say, GO THE CENTRIFUGAL THEORY OF ELASTICITY. Hence ftp. M Ve — , The value of the second part of the integral (23) is now found to be: — l- e±$ M [ j f f QP SA<pdxdydz= ^ k Pl M v/* x . ' ^ SA0^ In the double integral, let X = log.,, V be put for fc0, G for w, and H for the single integral, as in equation (9). Then the double integral becomes G, dr' Also,because Pl MV = M ' : by Eq. (9), and ^ = - (t- k), the second part of the integral (23) is found to be hfx M ('-o(»'c +,v ^)©- • • (33R) Hence, adding together (2 3 A) and (23B) we find for the total variation of latent heat so'- hfL ( r *\ i £r ^ lo S> H i ■ SV ■ (1 4-- 10 -' H A 1 (24 ) 8Q-^(r- K )|Sr.— ^- +6V ^ y + dTd y)$-W To express this in terms of quantities which may be known directly by experiment, we have, by equations (10) and (9) — ^ + 0-^ = 0, that is to say, d\og., TL 1 _ Gj _ r M _ t_ dV ~H X V K\~hfx P kY' and, therefore, TYT f — log. e H x = — h?fZ V- -log v V+/(r) + constant. Il [A J K /(r) is easily found to be = — log. e r for a perfect gas, and, being indepen- dent of the density, is the same for all substances in all conditions ; hence we find (the integrals being so taken that for a perfect gas they shall = 0) THE CENTRIFUGAL THEOEY OF ELASTICITY. 61 dr J Vi/mclr kVJ t < PlogH 1= Mf^ | 1 dr 2 A^c.' c?r 2 t 2 ' d?\og. e n x ■_ M dp _ ± drdV h/u. dr kV and, therefore, W-(r-.){*.(& + /glv) + «7.4} . (3 ,) is the variation of latent heat, expressed in terms of the pressure, volume, and temperature ; to which, if the variation of sensible heat, 3Q = feSr, be added, the complete variation of heat, SQ + SQ' = S.Q, in unity of weight of the substance, corresponding to the variations SV and Sr of volume and temperature, will be ascertained. It is obvious that equation (25), with its consequences, is applicable to any mixture of atoms of different substances in equilibria of pressure and 7 72 temperature; for in that case r, J-, and — are the same for each substance. We have only to substitute for ~ the following expression : — X M X 2 M 2 where n v «.„ &c, are the proportions of the different ingredients in unity of weight of the mixture, so that n^ + n 2 + &c. = 1. Equation (25) agrees exactly with equation (6) in the first section of my original paper on the Theory of the Mechanical Action of Heat. It is the fundamental equation of that theory ; and I shall now proceed to deduce the more important consequences from it. 10. Equivalence of Heat and Expansive Power: Joule's Law. — From the variation of the heat communicated to the body, let us subtract the variation of the expansive power given out by it, or PSV={j>+/(V)}SV. The result is the variation of the total power exercised upon or com- municated to unity of weight of the substance, supposing that there is no chemical, electrical, magnetic, or other action except heat and pressure ; and its value is — * This coefficient corresponds to - E in the notation of my previous paper on the Mechanical Action of Heat. * G2 THE CENTRIFUGAL THEORY OF ELASTICITY. ^SQ + SQ'-P^Sr.{k+^-^)+(r-K)/0^v} + SV.{(t-k)-^-P-/(V)}. ■ . . (2G.) This expression is obviously an exact differential, and its integral is the following function of the volume and temperature : — ¥=k(r-K)+^(log,T+^) +j{(r- K )^ T - P }dY-ff(Y)dV. (27.) Accordingly, the total amount of power which must be exercised upon unity of weight of a substance, to make it pass from the absolute temperature r and volume V to the absolute temperature r x and volume Y,, is <P(V 1 ,r 1 )-¥(V fll r ). This quantity consists partly of expansive or compressive power, and partly of heat, in proportions depending on the mode in which the intermediate changes of temperature and volume take place; but the total amount is independent of these changes. Hence, if a body be made to pass through a variety of changes of temperature and volume, and at length be brought back to its primitive volume and tempera- tun', the algebraical sum of the portions of power applied to and evolved from the body, whether in the form of expansion and compression, or in that of heat, is equal to zero. This is one form of the law, proved experimentally by Mr. Joule, of the equivalence of heat and mechanical power. In my original paper on the Mechanical Action of Heat, I used this law as an axiom, to assist in the investigation of the equation of latent heat. I have now deduced it from the hypothesis on which my researches are based — not in order to prove the law, but to verify the correctness of the mode of investigation which I have followed. Equations (26) and (27), like equation (23), are made applicable to unity of weight of a mixture, by putting 2«ft for ft, and 2ra -^ for =£. The train of reasoning in this article is the converse of that followed by Professor William Thomson of Glasgow, in article 20 of his paper on the Dynamical Theory of Heat, where he proves from Joule's law that the quantity corresponding to S^ is an exact differential. 11. Mutual Conversion of Heat and Expansive Poiver : Carnot's Law of the Action of Expansive Machines. — If a body be made to pass from the volume V and absolute temperature r to the volume V x and absolute temperature t v and be then brought back to the original volume and THE CENTRIFUGAL THEORY OF ELASTICITY. Go temperature, the total power exerted (¥) will have, in those two operations, equal arithmetical values, of opposite signs. Each of the quantities ¥ consists partly of heat and partly of expansive power, the proportion depending on the mode of intermediate variation of the volume and temperature, which is arbitrary. If the mode of variation be different in the two operations, the effect of the double operation will be to transform a portion of heat into expansive power, or vice versa, Let (a) denote the first operation, (b) the reverse of the second. Then The terms of ¥ which involve functions of r only, or of V only, are not affected by the mode of intermediate variation of those quantities. The term on which the mutual conversion of heat and expansive power depends, is therefore f{(r-K)f T -p}dY(b) = f{(T-K)^-p}dY(a), /(^)"©=/(^-')"» f^dY(a)-f^dY(b) =jpdY(a)-fpd Y (b), or, Hence, which last quantity is the amount of the heat transformed into expansive power, or the total latent heat of expansion in the double operation. Let J dr J t — k a \ dY=( T - K )dF, Then because dV we have "V V -F F J jnlY(a)-f 1 pdY(b) = j \t-k) d F («)- f '(i— *)tfF(&) = f Fl (r a -r b )dF= f "LlZH ^ d Y. . (28.) •'F Jv r a -K d\ In which r a and r^ are the pair of absolute temperatures, in the two operations respectively, corresponding to equal values of F. This equation gives a relation between the heat transformed into expansive power by a given pair of operations on a body, the latent heat of expansion in the first operation, and the mode of variation of tempera- ture in the two operations. It shows that the proportion of the original latent heat of expansion finally transformed into expansive power, is a 04 THE CENTRIFUGAL THEORY OF ELASTICITY. function of the temperatures alone, and is therefore independent of the nature of the body employed. Equation (28) includes Carnot's law as a particular case. Let the limits of variation of temperature and volume be made indefinitely small. Then , 7A . dr dq 7 ,. and dividing by drdY dp = _J_ d<$ dr~ t— k' dV This differential equation is also an immediate consc luence of equation (25). If - be put for , and JM for ( V? T , it becomes identical with the J i — k d V equation by which Professor William Thomson expresses Carnot's law, as deduced by him and by Mr. Clausius from the principle, that it is impossible to transfer heat from a colder to a hotter body without expenditure of mechanical pov: if. The investigation which I have now given is identical in principle with that in the fifth section of my paper on the Mechanical Action of Heat ; but the result is expressed in a more comprehensive form. Equation (28), like (25), (26), and (27), is applicable to a mixture, composed of any number of different substances, in any proportions, dp, '/-/'. provided the temperature, the pressure, and the coefficients j- -5 ._, are the same throughout the mass. 12. Apparent Specific Heat. — The general value of apparent specific heat of unity of weight is „ dQ dq dq dY • ( hfi [cPp -] y (29.) ,dY dp I "*" dr'dr\ J agreeing with equation (13) of my previous paper. The value in each particular case depends on the mode of variation of volume with temperature. Specific heat at constant volume is *,-*+('-«>(& + /£«).• • w When the pressure is constant, we must have dV .dp w clY+-dr = 0, THE CENTRIFUGAL THEORY OF ELASTICITY. and, consequently, dp dV_ _th L dr~ dF' dV therefore specific heat at constant pressure is 65 K p = K V + (t — k) (ML _dY dY (31.) J. This agrees with equation (10) of Professor Thomson's paper, if — in his notation — r — k. If the body be a perfect gas, then r \2 / x" r r K. = ^ + 5 ►r t—k d Y r 2 + V ' d *> T Q \Zfl y (32.) K p = K v + _ *) = lg* + 1 _ /' The fact that the specific heats of all simple gases for unity of weight are inversely proportional to their specific gravities, shows that is the same for them all. -> 13. Velocity of Sound in Fluids. — Let a denote the velocity of sound in a fluid, and d . P the total differential of the pressure. Then « = V(,-^)=V{,V<- dj dY dV_ d dr' dY ;)}■ (33.) If it were possible to maintain the temperature of each particle of the fluid invariable during the passage of sound, this velocity would be simply v(,-5> d. y But Ave have reason to believe that there is not time, during the passage of sound, for an appreciable transfer of heat from atom to atom, so that for each particle E G6 THE CENTRIFUGAL THEORY OF ELASTICITY. dQ + dQ' = 0; or, K = in equation (29). To fulfil this condition, we must have dr i — k dp dV = ~ K v ' It' Consequently, or, by equation (31), ,/ d? K P \ , nt . That is to say, the action of heat increases the velocity of sound in a fluid beyond what it ivould be, if heat did not act, in the ratio of the square root of the specific heat at constant pressure, to the square root of the specific heat at constant volume. This is Laplace's law of the propagation of sound, which is here shown to be applicable, not only to perfect gases, but to all fluids whatsoever. LAWS OF THE ELASTICITY OF SOLID BODIES. 67 Y.— LAWS OF THE ELASTICITY OF SOLID BODIES.* Introduction. 1. The science of the elasticity of solid bodies, considered with reference to its most important application, the determination of the strength of structures, consists of three parts : First, The investigation of what may be specially termed the Laws of Elasticity ; that is to say, the mutual relations which must exist between the elasticities of different kinds possessed by a given solid, and between the different values of those elasticities in different directions. Secondly. The integration of the equations of equilibrium and motion of the particles of an elastic solid. The results of this process enable us to determine the relative displacements of the particles from their natural positions in a solid body of a given material and figure, subjected to a given combination of forces. Thirdly. The application of the results derived from the first two branches of the theory to our experimental knowledge of the pressures and relative displacements to which the particles of known materials may safely be subjected in practice. This enables us to compute the strength of actual structures. 2. Notwithstanding the great amount of attention which has been paid to the strength of materials, and the numerous and elaborate experiments which have been made respecting it, few examples exist of the sound application of physical and mathematical principles to practice in con- nection with this subject. This has arisen chiefly from the fact, that the first and second branches of the inquiry have to a great extent been carried on without reference to their application to the third, and the third conducted without regard to the principles of the first and second. The results of investigation, on correct principles, into the theory of elasticity have been limited in their applications, with a few exceptions, to the laws of the propagation of vibratory movements ; and those few exceptions relate almost exclusively to bodies of equal elasticity in all directions — a class which excludes many of the most useful materials of * Eead before the British Association at Edinburgh, on August 1, 1S50, and published in the Cambridge and Dublin Mathematical Journal, May, 1851. G8 LAWS OF THE ELASTICITY OF SOLID BODIES. construction. On the other hand, when it has been found necessary to adopt theoretical principles, for the purpose of reducing the results of experiments on the strength and elasticity of materials to a system, assumptions have often been made, with a view chiefly to simplicity in calculation, of a kind inconsistent with the real nature of elastic bodies. 3. The present inquiry relates to the first part of the theory of elasticity — viz., the laws of the relations which must exist between the elasticities of different kinds possessed by a given substance, and between their various values in different directions. Section I. — Composition and Resolution of Strains and Molecular Pressures. 4. At the outset of the inquiry two preliminary problems present them- selves : the composition and resolution of relative molecular displacements, and the composition and resolution of pressures such as the parts of elastic bodies exert upon each other. The former is a question of pure geometry ; the latter, of pure statics. They are usually considered simultaneously, on account of the analogy which exists between their solutions. This is not the result of the physical connection between the two classes of phenomena, and it would still exist although there were no such physical connection ; it is merely a consequence of the analogy between forces in statics and straight lines in geometry. Those two problems have been so fully investigated by MM. Cauchy, Lam£, and Clapeyron, as to leave nothing further to be done. The theorems and formulae which they have obtained are many and important. In the present paper I shall state those principles and results only to which there will be occasion to refer in the sequel. 5. It is desirable that some single word should be assigned to denote the state of the particles of a body when displaced from their natural relative positions. Although the word strain is used in ordinary language indiscriminately to denote relative molecular displacement, and the force by which it is produced, yet it appears to me that it is well calculated to supply this want. I shall therefore use it, throughout this paper, in the restricted sense of relative displacement of particles, whether consisting in dilatation, condensation, or distortion ; while under the term pressure I shall include every kind of force which acts between elastic bodies, or the parts of an elastic body, as the cause or effect of a state of strain, whether that force is tensile, compressive, or distorting. The nature and magnitude of a simple and uniform strain are defined by three things : LAWS OF THE ELASTICITY OF SOLID BODIES. 69 First. The direction of the lines along which the particles of the body are displaced from their natural position. Secondly. The direction along which the rate of variation of the displace- ment from point to point is a maximum. This direction is normal to a series of planes of equal displacement, and may be called the strain-normal. Thirdly. The amount of that rate of variation, being the differential coefficient of the displacement with respect to distance along the strain- normal. C. A strain may be resolved into three components, in which the directions of displacement shall be respectively parallel to three rectangular axes, while the strain-normal remains unchanged, by multiplying its amount by the direction-cosines of the total displacement. Each of these three components may itself be resolved into three com- ponents, in which, the direction of displacement remaining unchanged, the strain-normals are respectively parallel to the three axes, by multiplying its amount by the direction-cosines of the original strain-normal. Thus every strain is reducible to nine components. These nine components, however, are equivalent to but six distinct strains. If we consider the strains as thus reduced to three rectangular axes, we shall find that they are of two kinds : longitudinal, that is to say, strains of linear extension or condensation, where the displacements are parallel to the strain-normals ; and transverse, or strains of distortion, when these directions are at right angles. Thus, if x, y, z, denote the three rectangular axes, and £, jj, £, small molecular displacements respectively parallel to them, then dE, dr\ dZ, </ .!•' dif d z' are longitudinal strains, which are dilatations when positive, and condensa- tions when negative. I shall denote them respectively by N^ N 2 , N 3 ; their sum, when positive, is the cubic dilatation of the particles, and when negative, the cubic condensation. Transverse strains, or distortions, are represented by the six differential coefficients of the displacements with respect to axes at right angles to them — viz., dr\ dZ, , dZ, f?s. d% dr) dz' dy' dx' dz' dy' dx' Let the axis of x be perpendicular to the plane of the paper. Let .ABCD be the section, by the plane yz, of a prism which in its natural state is square, and has its faces normal to the axes of y and z. A distor- tion in the plane yz, relatively to these axes, is measured by the deviation from rectangularity of this originally square section, that deviation being 70 HWS OF THE ELASTICITY OF SOLID BODIES. considered positive which makes the angles B and D acute. Now, so far as the positions of the particles in this prism relatively to each other are concerned, it is immaterial whether that devia- tion from rectangularity is produced by keep- ing the sides AD and BC parallel to their original positions, and giving angular motion to AB and DC — a change represented by + y A n D y e ch) dz ■ or by keeping AB and DC parallel to their original positions, and giving angular motion to AD and BC — a change repre- 7 V sented by — - ; or by combining those two oper- dy ations: so that the total transverse strain in the plane yz is represented by the sum of these two coefficients, dr\ dZ, _ T Tz + dy~ JiLv Similar reasoning gives, for the total distortion in the plane zx, and in the plane x y, dz dz dj, dn _ T dy + dx " a ' The factor 2 is used in these expressions for the sake of convenience in the employment of certain formula?, to be afterwards quoted. The halved-differences of the pairs of differential coefficients, 2\dz dyP 2 \dx dz)'2\dy dxF represent rotations of the prism as a whole round the axes of x, y, z, respec- tively, which have no connection with the positions of its particles relatively to each other. The component strains into which all others can be resolved with respect to a given set of axes are thus reduced to six, three longitudinal and three transverse. 7. A pressure, like a strain, is defined by three things : 1st. The direction of the pressure. 2nd. The position of the surface at which the pressure is exerted. 3rd. The amount of the pressure as expressed in units of force pes unit of area of the surface of action. A pressure on a plane, in whatsoever direction it may act, may be resolved into three rectangular components, one normal to the plane LAWS OF THE ELASTICITY OF SOLID BODIES. 71 and two tangential. The normal pressure may be compressive or tensile: when compressive, it is considered as positive; when tensile, negative. In an elastic solid which is in equilibrio, let a cube be conceived to exist with its faces normal to the axes of co-ordinates, and let the pressures throughout its extent be uniform. This cube exerts on the matter round it, and is reacted on by three pairs of normal pressures, at the faces respectively normal to the axis of x, y, z, which may be denoted by Pj, P q , P 3 , the pressures at opposite faces being equal. Let A B C D represent the section of this cube by the plane yz. On the faces A B and C D, parallel to x z, let a pair of tangential forces act in the directions denoted by the order of the letters, tending to produce dis- tortion by making the angles B and D acute and A and C obtuse. Let a pair of forces of similar tendency act on the faces C B and A D, parallel to zy. These two pairs of forces are equal and opposite to those which the cube, in consequence of the transverse displacements of its particles, exerts on the surrounding portion of the solid. No displacement of the relative situations of a system of particles can give the system a tendency to revolve as a whole round an axis. Such a tendency must exist in the cube unless the tangential forces on the faces A B, CD are equal to those on the faces C B, A D. Therefore, the tangential pressure parallel to z, on a plane normal to y, is equal to the tangential pressure parallel to y, on a plane normal to z: a theorem first proved by Cauchy. The common value of those forces may be denoted by Q 1 , as they are both perpendicular to x. Similar reasoning shows that the two pairs of tangential forces perpen- dicular to y have one common value, Q 2 . In like manner, those perpendicular to z may be denoted by Q 3 . Thus the pressures exerted by and on the cube are reduced to six, three normal and three tangential. 8. The composition of pressures applied to different planes, and their reduction to new axes, depends on the following principle : — Conceive a small triangular pyramid, with its apex at the origin of rectangular co-ordinates, its sides being formed by the three co-ordinate planes, and its base by a plane in any given direction intersecting them. Let pressures in one given direction act on the three sides, and be balanced by a pressure in the same direction on the base. Each of the three sides is equal to the base multiplied by the cosine of the angle between the normal to the base and the normal to the side in question. Therefore, the total pressure on the base is equal to the sum of the pressures on the sides, each multiplied by the cosine of the angle between the normal to the side in question and the normal to the base. If the normal to this base is one of three new axes of rectangular co-ordinates, the 72 LAWS OF THE ELASTICITY OF SOLID BODIES. total pressure thus found may be reduced to normal and tangential pressures, by multiplying it by its direction-cosines with respect to the new axes. 9. I annex, for convenience of reference, the general formulae which have been deduced from this principle. Let x, y, g, be rectangular axes of co-ordinates, and P^ P 2 , P 3 , Q x , Q , < v > . normal and tangential pressures which act as shown in the following table : Pressuros parallel to Normals. Planes. X jf Z x yz P x Q 8 Q 2 y z* Q 3 P- Qi z xy Q 2 Qj P s Let Ej, R 2 , R 3 , be the rectangular components of the total pressure at a plane, the direction-cosines of whose normal are a v a 2 , a s . Then R x = a y Pj + a 2 Q s + « a Q 2 ,->> R 2 = a 1 Q 3 + « 2 P 2 + « 3 Q 1 , I (i.) E 3 = a x Q 2 + a.Xi i + « 3 P 3 , J Let this normal be taken as the axis of x in a new set of rectangular axes x, y', z, which make with the original axes the angles whose cosines are given in the following table : — Original Axes. New Axes. x' y z' h c 2 VDirection-cosines. c,J y 3 L 3- Let P/, P 2 ', P 3 ', Q/, Q 2 ', Q 3 ', be the normal and tangential pressures, un- reduced to the new axes : then P/^P^ + P^ + p^ + 2Q 1 a 2 a 3 + 2Q 2 a 3 aj, + 2Q^a 1 a 2 , + 2Q 1 b 2 b 3 + 2Q. 2 b s b 1 + 2Q.J) 1 b 2 , P3'=P lCl 2 + P 2 ^ + P 3 c 3 2 + 2Q lC2 c 3 + 2Q 2 c 3 c 1 + 2Q 3 c 1 r 2 , Q/ = P 1 5 1 c 1 + P 2 b 2 c 2 + P 3 6 3 c 3 + Qi(^ 3 + b 3 c 2 ) + Q 2 (6g Cl + b^) 4- Q 3 (V 2 + 6 2 cj). Qo — Px^i^x + P 2 C 2 a 2 + P 3 C 3 a 3 + Qj(c 2 a 3 + c 3 a 2 ) + Qo(c 3 a 1 + c x « 3 ) + Q 3 (c 1 « 2 + c./^), Q 3 ' = P 1 a 1 6 1 + P 2 «/; 2 + P 3 « 3 6 3 + Q^ao&a + a z b 2 ) + Q 2 (a 3 6 x + a^ 3 ) + Q^a-ft., + a.p x ), y (2.) LAWS OF THE ELASTICITY OF SOLID BODIES. 73 By the substitution of N for P and T for Q, the formulae given above are made applicable to the reduction of strains to new axes of co-ordinates. I shall not here recapitulate the many elegant and important theorems which MM. Cauchy and Lame and Clapeyron have deduced from those equations, as they do not relate to the branch of the theory of elasticity of which this paper treats. I may mention that in their memoir in the seventh volume of Crelle's Journal, MM. Lame and Clapeyron have used X and T to denote pressures, d ? and have expressed strains simply by the differential coefficients — , &c. Section II.— Physical Relations between Pressures and Strains, SO FAR AS THEY ARE INDEPENDENT OF HYPOTHESES RESPECTING THE Molecular Constitution of Matter. 10. In almost all investigations which have hitherto been made respecting the elasticity of bodies which have different degrees of elasticity in different directions, it has been the practice to take some hypothesis as to the molecular constitution of solid bodies as the basis of calculation from the outset of the inquiry. It appears to me, however, that the more philosophical course is, to ascertain, in the first place, what conclusions can be attained as to the laws of elasticity without the aid of any such hypothesis, and afterwards to inquire how far the theory can be simplified, and what additional results can be gained by introducing suppositions respecting the ultimate constitution of matter. For the present, therefore, I shall make no assumption as to the questions whether bodies are systems of physical points, or of atoms of definite bulk and figure, or are continuous, or have a constitution intermediate between those three; and I shall use the word particle in its literal sense of a small part 11. I shall restrict the present inquiry to homogeneous bodies possessing a certain degree of symmetry in their molecular actions, which consists in this : that the actions upon any given particle of the body of any two equal particles situated at equal distances from it within the sphere of molecular action, in opposite directions, shall be equal and opposite. Substances may possess higher degrees of molecular symmetry, but this is the lowest. The statement that a body is homogeneous means, when applied to- molecular action, that the mutual actiou of a pair of particles, situated at a given distance from each other in a given direction, shall be equal to that of any other pair of particles equal to the first, situated at an equal distance from each other in a parallel direction. 12. It is known by observation, that strains and pressures are physically 74< LAWS OF THE ELASTICITY OF SOLID BODIES. connected. It is also known by observation, that the pressure with which a strain is connected consists in a tendency of the body to recover its natural state, and is opposite or nearly opposite in direction to the strain ; thus longitudinal condensation is accompanied with positive normal, or nearly-normal pressure ; longitudinal dilatation, with negative normal, or nearly-normal pressure ; and distortion in a given plane, with tangential pressure in the same plane, of opposite sign. It is known by experiment, that when a pressure and the strain with which it is connected are given in direction, and when the strain does not exceed a certain limit, being in most cases the utmost limit to which a structure can be strained without danger to its permanency, the pressure and the strain are sensibly proportional to each other. The quantity by which a strain is to be multiplied to give the corresponding pressure is a coefficient of elasticity, and is expressed, like a pressure, by a certain number of units of force per unit of surface. I have said that a strain and the corresponding pressure referred to the same plane are opposite or nearly opposite in direction ; for they are not of necessity exactly opposite for all directions of strain, except in sub- stances which are possessed of the highest degree of molecular symmetry ; that is to say, which are equally elastic in all directions. For those having lower degrees of symmetry, the following proposition is true : — Theorem I. In an elastic substance which is homogeneous and symmetrical with respect to molecular action, there are three directions at right angles to each other in which a longitudinal strain produces an exactly normal pressure on a plane at right angles to the direction of the drum. Those three directions are called Axes of Elasticity. The proposition is equivalent to an assertion, that the lowed degree of symmetry of molecular action necessarily involves symmetry with respect to three rectangular co-ordinate planes. This theorem has been often demonstrated for systems of atoms. But it is easily seen that the truth of these demonstrations de- pends, not on the special hypotheses which they involve, but on the fundamental con- dition of symmetry. The following demonstration involves no hypothesis. Let a point in the interior of a body be assumed as the origin of rectangular co-ordinates, the axes being considered as fixed, and the body as movable angularly in all directions about the origin. Space round is divided by the co-ordinate planes into eight similar indefinitely-extended LAWS OF THE ELASTICITY OF SOLID BODIES. 75 rectangular three-sided pyramids. Let those pyramids be designated as follows, according to the signs of the co-ordinates comprised in them. Signs of Designation of x 'i I z Pyramid. . A . B . C . D . E . F . G . H To express the relative situations of these pyramids, as taken in pairs, let the following terms be used : Diametrically opposite — when the pyramids touch at the apex only; comprising the following pairs, A, G ; B, H ; C, E ; D, F. Diagonally opposite — when they touch at an edge ; comprising the pairs A, H ; D, E ; B, G ; C, F ; A, F ; B, E ; D, G ; C, H ; A, C ; B, D ; E, G ; F, H. Contiguous — when they touch in a face ; comprising the pairs A, B ; D, C ; H, G ; E, F ; A, D ; B, C ; F, G ; E, H ; A, E ; B, F ; C, G ; D, H. Each pair of contiguous pyramids forms a rectangular wedge, which has an opposite wedge touching it along the edge, and a contiguous wedge touching it at each of its two faces. The pairs of opposite wedges are A B, G H ; C D, E F ; AD,FG; BC,EH; A E, C G ; B F, D H. The pairs of contiguous wedges are AB, CD; CD, GH; GH, EF; E F, AB; AD, BC; BC, FG; FG, EH; EH, AD; AE, BF; BF, CG; CG, DH; DH, AE. 76 LAWS OF THE ELASTICITY OF SOLID BODIES. According to the condition of symmetry already stated, the portions of matter comprised in any pair of diametrically opposite pyramids must be symmetrical in their actions on a particle placed at 0, or on any pair of equal particles symmetrically placed with respect to 0, whatsoever may be the angular position of the body with respect to the axes. Suppose the body to receive a longitudinal strain in the direction of the axis of z. Let a small circular area w be conceived to exist in the plane of xy, with its centre at 0; and let this area be the base of a cylinder extending indefinitely in a negative direction along the axis of z, and denoted by to z. The pressure on the plane x y is proportional and parallel to the resultant of the actions of the four pyramids A, B, C, D, on the cylinder u> ~, divided by the area w. The action of each of those pyramids consists of a normal component parallel to z, and a tangential component parallel to the plane xy. In order that the total pressure may be normal, those tangential actions must balance each other, which can only be the case when the tangential action of the wedge AB, parallel to the axis of y, is equal and opposite to that of the contiguous wedge CD, and the tangential action of the wedge B C, parallel to the axis of x, is equal and opposite to that of the contiguous wedge A D. The pair of contiguous wedges A B, C D, touch in the plane of xz, having the axis of x for their common edge. If the actions of this pair of wedges on w~, when longitudinally strained along .:, are unsymmetrical, this cannot arise from the form, position, or strain of these wedges, which are exactly symmetrical with respect to each layer of particles in w:, but from the nature of the particles occupying tin- wi dges. Now by rotating the body through a right angle about the axis of x, we can bring the particles which formerly occupied C D into A B, and the particles which formerly occupied G H (which consists of two pyramids diametrically opposite, and therefore molecularly symmetrical to A and B) into C D. In this new situation of the body with respect to the axes of co-ordinates, the resultant of the tangential actions, parallel to y, of the wedges A B and C D, on wz, though not necessarily eqml, will be opposite in direction to the original resultant; and this change will have been produced, not abruptly, but continuously, so that the value of the resultant must have passed through zero. Therefore, whatsoever may be the situation of the axis of x amongst the particles of the body, it is possible, by rotating the body about that axis, to find a position in which the tangential actions of the wedges AB and CD, parallel to y, on the cylinder wz, shall balance each other. And by similar reasoning it may be proved, that whatsoever may be the situation of the axis of y amongst the particles of the body, it is possible, by rotating the body about that axis, to find a position in which the tangential actions of the Avedges B C and A D, parallel to x, on the cylinder w z, shall balance each other. LAWS OF THE ELASTICITY OF SOLID BODIES. 77 Therefore, by combining rotations about the axes of x and y, it is possible to find a position of the solid with respect to the axes of co-ordinates such that the tangential actions of the four pyramids A, B, C, D, on the cylinder wz, arising from a longitudinal strain along ,~ shall be in equilibrio, and that the total pressure on x y shall be normal. The direction, with respect to the solid, which fulfils this condition, is called an axis of elasticity. Let — z O + z, being now an axis of elasticity, be considered as fixed in the solid. From the manner in which the two pairs of wedges AB, CD and BC, AD, are composed of the four pyramids A, B, C, D, it is clear that the actions of the pair of diagonally opposite pyramids A, C, are symmetrical, and also those of the diagonally opposite pyramids B, D. From this and the symmetry of the actions of diametrically opposite pyramids it follows, that the actions of the four pairs of contiguous pyramids, A, E ; D, H ; B, F; C, G, are symmetrical, and also those of the two pairs of diagonally opposite pyramids, E, G ; F, H. This symmetry of action (subject to the condition of symmetry of strain) is not disturbed by rotation about the axis of z. Let the small circular area to be now conceived to exist in the plane y~, and let the cylinder of which it is the base extend in a negative direction along the axis of x, and be called the cylinder wx. Let the solid receive a longitudinal strain along the axis of x. The action of A on iox is sym- ' metrical to that of E, and the action of D to that of II ; therefore, the tangential actions of the wedges A D, E H, parallel to z, balance each other. It remains only to make the tangential actions of the wedges A E, D H, parallel to y, on wx, balance each other, which is to be done by rotation about the axis of :. The solid is now in such a position that a-, as well as z, is an axis of elasticity. The pairs of contiguous pyramids are now all molecularly symmetrical about their common faces. Therefore the pairs of contiguous wedges AB, E F ; A E, B F, are symmetrical in their actions on a cylinder to y, when longitudinally strained along y. Therefore y also is an axis of elasticity, and the theorem is proved. It is not necessary to the existence of rectangular axes of elasticity that the body should be homogeneous (in the sense in which I have used the word) throughout its whole extent, but only round each point throughout a space which is large as compared with the sphere of appreciable mole- cular action of each particle. Hence the rectangular axes of elasticity may vaiy in direction at different points of the same body ; and some, or all of them, may follow the course of a system of curves, as they do in a rope, a piece of bent timber, or a curved bar of fibrous metal. LXIS. Plane. X yz y zx z xy 78 LAWS OF THE ELASTICITY OF SOLID BODIES. 13. The axes of elasticity are evidently those which ought to be selected as axes of co-ordinates, for the resolution of all pressures and strains, in researches on the laws of elasticity. The strains and pressures being so resolved, we shall have the expression - A x N x for part of the normal pressure on the plane yz : A x being the coefficient of longitudinal elasticity for the axis of x. But this is not the whole of that pressure ; for it is known by observation, that the normal pressure on a given plane is augmented by condensation, and diminished by dilatation of the particles, in a direction parallel to the given plane as well as normal to it. The normal pressure P x on yz, therefore, depends not only on the longitudinal strain N x along x, but also on the longitudinal strains N 2 and N alon"- y and z. Applying similar reasoning to the other normal pressures, they are found to be represented as follows : P x = - A X N X - B 3 N - Bo'Ng "] p^-b^-aX-bX r • ( 3 -) P 3 =-B 2 N 1 -B 1 'N 2 -A 8 N S J The tangential pressures are represented, in terms of the distortions, in the following manner : Tlane. yz . . Q 1= - 20^ zx . . Q„= -2C.X y . . . (4.) xy . . Q 3 =-2C 3 T 3 J These six equations are merely the representation of observed facts, framed with regard to the principle of axes of elasticity. They contain twelve coefficients of elasticity, which may be thus classified : A x , A 2 , A 3 , are the coefficients of longitudinal elasticity for the axes of x, y, z, respectively ; B 1? B x ', are the coefficients of lateral elasticity in the plane of yz : the former expressing the effect of a strain along z in producing normal pres- sure parallel to y : the latter, the effect of a strain along y in producing normal pressure parallel to z. B 2 , B 2 ', are the coefficients of lateral elasticity in the plane of zx; and B 3 , B 3 ', in the plane of x y. C 1; C 2 , C 3 , are the coefficients of transverse or tangential elasticity, or of rigidity, in the planes of yz, zx, and xy, respectively. The possession of this species of elasticity is the property which distinguishes solids from fluids, and is that upon which the strength and stability of solid structures entirely depend. When a beam, or any other portion of a solid structure, takes a set, as it is called (or undergoes permanent alteration of figure), it LAWS OF THE ELASTICITY OF SOLID BODIES. 79 is the rigidity which has been overstrained and has given way. So far as I am aware, however, it has not hitherto been directly referred to in researches on the strength of materials, except in those relative to torsion. The principal object of the present inquiry is to determine what mutual relations must necessarily exist amongst those twelve coefficients of elasticity in each substance. 14. The three coefficients of rigidity, so far as we have as yet seen, represent the elasticity called into play by three kinds of distortion, measured respectively by the alteration of the angles of the three rectan- gular sections of a cube whose faces are normal to the three axes of elasticity. I shall now, however, prove that the tangential pressures pro- duced by equal distortions are equal, so long as the plane in which the distortion takes place is unchanged, and are not altered by any change of the direction, in that plane, of the sides of the figure on which the distor- tion is measured ; that is to say — Theorem II. The coefficient of rigidity is the same for all directions of dis- tortion in a given plane. Let ABCD be the section at right angles to the edges of a rhombic prism having any angles ; and G E and F H two lines normal respectively to the faces of the prism. Let this prism undergo a small alteration in the ^ angles of its section ABCD. Whether we estimate the distortion so produced as a transverse displacement of the particles in lines parallel to AB, and varying along the strain- normal G E, or as a transverse displacement of the particles in lines parallel to A D, and vary- ing along the strain-normal F H, the result, so far as the relative transverse displacements of the par- G tides are concerned, will be the same. Also, the tangential pressures are the same at the pair of faces A B and C D, and at the pair of faces B C and A D ; for otherwise a relative dis- placement among the particles of a body would produce a force tending to make it revolve as a whole round an axis, which is impossible. Therefore, the tangential forces produced by equal transverse displace- ments relatively to two strain-normals which make any angle with each other are equal, provided the displacements are in the same plane with the normals ; therefore the coefficient of rigidity is the same for all directions in a given plane. 15. This theorem leads to another, which expresses the relations between the twelve coefficients of elasticity, as far as it is possible to <S0 LAWS OF THE ELASTICITY OF SOLID BODIES. determine them independently of hypotheses respecting the constitution of matter. Theorem III. In each of the co-ordinate planes of elasticity the coefficient of rigidity is equal to one-fourth part of the sum of the two coefficients of longitudinal elasticity for the axes which lie in that plane, diminished by one-fourth part of the. sum of the two coefficients of lateral elasticity in tin' same plane. For example, let the plane be that of yz, in which the coefficient of rigidity is C v those of longitudinal elasticity A 2 and A 3 , and those of lateral elasticity B x and B/. Let 2 ly represent a distortion in the plane yz, relative to two new A .axes y, ,:', in the same plane, and let the angle yy' = 9. Let this distortion be resolved, with respect to the original axes, according to equation (2). Then N x = 0; N 2 = - 2 TV °o s sin 6>; X., = 2 T/ cos 6 sin 0; 2 T, = 2 T/ (cos- e - sin' 2 0); T 2 = 0; T 3 = 0. The corresponding pressure.-- referred to the original axes arc T 1 = - 2 T/ (II. - IV) cos 6 sin 0, P 2 = - 2T X ' (- A.+ 15,) cos 6 sin 0. p 3 = - 2 t; (- ]y + A3) cos e sin e, Q 2 = - 2 C^T/ (cos 2 B - sin 2 0), Q, = 0, Q, = 0. Let us now determine, according to equation (2), from the above pressures, the tangential pressure Q/ as referred to the new axes. Then Q/ = - 2 T/ {C x + cos 2 e sin 2 6 (X, + A 3 - B, - B,' - 4 CJ}. But by the preceding theorem we have also 0'= -2TT for all values of 0, which cannot be true unless the coefficient of cos 2 sin 2 in the first value of Q/ is = 0. Consequently C i = \ (A 2 + A 3 - Bj - B/). % applying similar reasoning to the planes of sx and xy it is also proved that 1 C 2 = I (A 3 +A 1 -B 2 -B 2 '), C 3 = |(A 1 + A 2 -B 3 -B 3 , being the algebraical statement of the theorem enunciated. (50 LAWS OF THE ELASTICITY OF SOLID BODIES. 81 Thus the number of independent coefficients of elasticity is reduced to nine, of which the other three are functions ; and this is the utmost reduc- tion of their number which can be made without the aid of suppositions as to the constitution of matter. The determination by experiment of nine constants for each substance is an undertaking almost hopeless ; it is therefore desirable to ascertain whether, by the introduction of some probable hypothesis, their number can be further reduced. Section III. — Eesults of the Hypothesis of Atomic Centres. 16. Almost all the investigations of the laws of elasticity which have hitherto appeared, are founded on the hypothesis of Boscovich : that matter consists of physical points or centres of force, or of atoms acting as if their masses were concentrated at their centres; which physical points or atoms occupy space, and produce the phenomena of elasticity, because the forces which act between them, and which depend on their relative distances and positions, tend to make them remain in certain relative positions, and at certain distances apart. Although the results of this supposition are not verified by all solid substances, still it seems probable that its errors are to be corrected, not by rejecting it, but by combining it with another, to which I shall afterwards refer. I shall now, therefore, show to what extent the laws of elasticity are simplified by adopting Boscovich's supposition of atomic centres of force, acting on each other by attractive and repulsive forces along the lines joining them. It will be seen, that in consequence of the course adopted, of determining, in the first place, the necessary relations between the coefficients of elasticity which must exist independently of all special hypotheses, this investigation is almost entirely freed from the algebraical intricacy in which it would otherwise be involved. 17. All the consequences peculiar to this hypothesis flow from the following single theorem, in which the term 'perfect solid is used to denote a body whose elasticity is due entirely to the mutual attractions and repulsions of atomic centres of force. Theorem IV. In each of the co-ordinate planes of elasticity of a perfect solid, the tivo coefficients of lateral elasticity, and the coefficient of rigidity, are all equal to each other. Take, for example, the plane of y z. The proposition enunciated is equivalent to the assertion, that the tangential pressure parallel to y at F S2 LAWS OF THE ELASTICITY OF SOLID BODIES. the plane of xy, produced by a given transverse strain 2T X = -5-, which consists in a displacement of the atomic centres parallel to y and varying with z, is equal to the normal pressure parallel to : at the same plane xy, d produced by a longitudinal strain N 8 = , . which consists in condensing or dilating the atomic centres in a direction parallel to y. provided that longitudinal strain is equal in amount to the transverse strain. The pressure on a given area of the plane xy, is the effect of the joint actions of the atomic centres on the negative side of that plane upon the atomic centres on the positive side. In the natural or unstrained condition of the body, this pressure is null, showing that those forces neutralise each other. "When the body is strained, therefore, the pressure is the resultant of the variations of all those forces, arising from the displacements of the atomic centres from their natural relative positions.* Let m and //. denote a pair of atomic centres, m being situated on the positive side of the plane xy, and ju on the negative side. The force acting between m and /j. is supposed to act along the line joining them, and to be a function of its length. When the relative displacement of the atoms is very small as compared with their distance, the variation of this force will be sensibly proportional to the variation of distance, multiplied by some function of the distance. It may therefore be denoted by <pr.dr, where /• denotes the distance (jim). Let this line make with the axes the angles a. /3. y. Let the strain to be considered, in the first place, be transverse, the displacements being parallel to y and varying with .:, the rate of variation being dz ~ * L » and the force to be estimated being in the direction y. Then the dis- placement of m relatively to /x is A ?; = 2 T, r cos y. The variation of their distance apart is £ r = cos j3 A ij = 2 T 1 r cos /3 cosjy. The variation of the force acting between them is (pr .^r = 2 T 1 r (pr . cos ft cos y. * Small quantities of the second order relatively to the strains Tj, kc, are here nedeeted. LAWS OF THE ELASTICITY OF SOLID BODIES. 83 And the component of that variation parallel to y, which forms the part of the tangential pressure due to the action of li on m, is cos /3 <pr . S r = 2 T x r <p r . cos 2 /3 cos y. . . (a.) Next, let the strain be longitudinal, parallel to y, and denoted by N =^ dy Then the displacement of m relatively to fx is A tj = N 2 r cos |3. The variation of their distance apart is 8r = cosj3ATj = N 2 rcos*j3. The variation of the force acting between them is (f>r .Sr = ~N 2 r <pr cos 2 /3. And the component of that variation parallel to z, which forms the part of the normal pressure on the plane xy due to the action of fx on m, is cos y (j) r . S r = N 2 r (j> r cos 2 /3 cos y. . . (b.) On comparing the expressions (a) and (b) it will be seen that the quantities by which 2l\ and N 2 are multiplied are identical. Therefore, the tangential force in the direction y on the plane xy produced by a distortion in the plane yz, and the normal force in the direction z produced by a longitudinal strain along y, are equal when the strains are equal, for each pair of atomic centres. They are therefore equal for a perfect solid, because its elasticity is wholly due to the mutual actions of atomic centres; and the theorem is proved for the plane yz, and may in the same manner be proved for the other co-ordinate planes of elasticity. It is expressed algebraically as follows : Plane. yz . . . B^B^G^ z x . . . B 2 = B 2 ' = C 2 xy . . . B 3 = B 3 ' = C 3 (6.) 18. The combination of these equations with the equations (5) of Theorem III. leads immediately to the following results : 84 LAWS OF THE ELASTICITY OF SOLID BODIES. p A 2 + A 3 "I L i- 6 p A 3 + A l p A l + A 2 ^ — 6~ j A 2 = 3 (C 2 + C s - < 3^ A 2 = 3 (C 8 + C x - DJ A 3 = 3 (Cj + c 2 - D 8 )^ • (7.) • (8.) that is to say, Theorem V. In each of the three co-ordinate planes of elasticity of a perfect solid, the coefficient of rigidity is equal to one-sixth part of the sum of the two coefficients of longitudinal elasticity ; and consequently, For each axis of elasticity of a perfect solid, the coefficient of longitudinal elasticity is equal to three times the sum of the two coefficients of rigidity for the co-ordinate planes which pass through that axis, diminished by three times the coefficient of rigidity for the plane normal to that axis. We have now arrived at the conclusion, that in a body whose elasticity arises wholly from the mutual actions of atomic centres, all the coefficients of elasticity are functions of the three coefficients of rigidity. Rigidity being the distinctive property of solids, a body so constituted is properly termed a perfect solid. When the three coefficients of rigidity are equal, the body is a perfect solid, equally elastic in all directions. The equations 6 and 8 become A = 3 C ; B = C, agreeing with the results deduced by various mathe- maticians from the hypothesis of Boscovich. Section IV. — Results of the Hypothesis of Molecular Vortices. 19. The great and obvious deviations from the laws of elasticity, as deduced from the hypothesis of atomic centres, which many substances present, render some modification of it essential. Supposing a body to consist of a continuous fluid, diffused through space with perfect uniformity as to density and all other properties, such a body must be totally destitute of rigidity or elasticity of figure, its parts having no tendency to assume one position as to direction rather than another. It may, indeed, possess elasticity of volume to any extent, and LAWS OF THE ELASTICITY OF SOLID BODIES. 85 display the phenomena of cohesion at its surface and between its parts. Its longitudinal and lateral elasticities will be equal in every direction ; and they must be equal to each other by equation (5), which becomes = A - B; C = 0. If we now suppose this fluid to be partially condensed round a system of centres, there will be forces acting between those centres greater than those between other points of the body. The body will now possess a certain amount of rigidity; but less, in proportion to its longitudinal and lateral elasticities, than the amount proper to the condition of perfect solidity. Its elasticity will, in fact, consist of two parts, one of which, arising from the mutual actions of the centres of condensation, will follow the laws of perfect solidity ; while the other will be a mere elasticity of volume, resisting change of bulk equally in all directions. In a paper on the Mechanical Action of Heat in connection with the Elasticity of Gases and Vapours (Trans, of the Royal Society of Edinburgh, Vol. XX., Part I.), I have attempted to develop some of the consequences of a supposition of this kind, called the hypothesis of molecular vortices."" It assumes that each atom of matter consists of a nucleus or central physical point, enveloped by an elastic atmosphere, which is retained in its position by forces attractive towards the atomic centre, and which, in the absence of heat, would be so much condensed round that centre as to produce the condition of perfect solidity in all substances; that the changes of condition and elasticity due to heat arise from the centrifugal force of revolutions among the particles of the atmospheres, diffusing them to a greater distance from their centres, and thus increasing the elasticity which resists change of volume alone, at the expense of that which resists change of figure also; and that the medium which transmits light and radiant heat consists of the nuclei of the atoms, of small mass, but exerting intense forces, vibrating independently, or almost independently of their atmospheres; absorption being the communication of that motion to the atmosphere, so that it is lost by the nuclei. 20. A body so constituted, in which the rigidity is considerable, may be called, in general, an imperfect solid ; and it is obvious that its various elasticities may be represented in the following manner : Theorem VI. In an imperfect solid, according to the hypothesis of molecular vortices, each of the coefficients of longitudinal and lateral elasticity is equal to the same function of the coefficients of rigidity which would be its value in a perfect solid, added to a coefficient of fluid elasticity which is the same in all directions. * An abstract of that paper is published in PoggendorfF's Annalen for 1850, No IX. 86 LAWS OF THE ELASTICITY OF SOLID BODIES. Denoting this fourth coefficient by J, we have the following equations, giving the values of the coefficients of longitudinal and lateral elasticity in terms of the coefficient of fluid elasticity, and of the three coefficients of rigidity. A x = 3 (C 2 + C, - C x ) + J A, = 3 (C 3 + Oj - C 2 ) + J A 3 = 3 (C x + 2 - C 3 ) + J B x = C x + J B 2 = C, + J B 3 = C 3 + J (0.) The utmost number of independent coefficients is thus increased to four. If the coefficients of rigidity be progressively diminished without limit, as compared with the coefficient of fluid elasticity, the body will pass through every stage of the gelatinous state; and when the coefficients of rigidity vanish, its condition will be that of a perfect fluid, in which the longitudinal and lateral elasticities are all equal, and represented by the single coefficient J. It is to be observed, that in this condition the independent actions of the nuclei or physical points at the atomic centres upon each other, which are the means of radiation, may be very great; their sensible effect on the elasticity of the body being neutralised by other forces, exerted by the parts of the atmospheres. If two of the coefficients of rigidity are equal (as C 9 = C 3 ), the body is equally elastic in all directions round an axis, which in this case is that of x ; and equations (9) become A x = 6 C, 2 - 3 C x + J A 2 = A 3 = 3 C x + J B 2 = B 3 = C 2 + J j (9A.) When the three coefficients of rigidity are all equal, the body is an imperfect solid equally elastic in all directions. The results of this condition have been investigated by Professor Stokes, M. Wertheim, and Mr. Clerk Maxwell. Equations (9) in this case become C + J; B = C + J. (9B.) LAWS OF THE ELASTICITY OF SOLID BODIES. 87 Note respecting Previous Investigations. (18 a.) The investigations of Poisson (Mem. de VAcad. des Sciences, XVIIL), of M. Cauchy (Exercices des MatMmatiques, passim), and of Mr. Haughton (Trans. Boy. Irish Acad., XXL), respecting the elasticity of substances unequally elastic in different directions, are all founded on the hypothesis of atomic centres. So far as they relate to substances possessed of rectangular axes of elasticity, they agree in expressing the elasticity of such bodies by means of six coefficients, corresponding respectively to those which I have denoted by A v A 2 , A 3 , C 15 C 2 , C 3 . None of those investigations indicate any relations amongst these six coefficients. The researches of Mr. Green on the propagation of vibratory movement (Camb. Trans., VII.) differ materially from those which preceded them, inasmuch as they are applicable, not merely to systems of atomic centres or physical points, but to solid substances constituted in any manner whatsoever.* So far as they are applicable to bodies possessed of axes of elasticity, they involve nine coefficients: three of longitudinal, three of lateral, and three of transverse elasticity. The following table exhibits a comparison between Mr. Green's notation and that of this paper: Coefficients of Elasticity. In the Notation of Longitudinal. Lateral. Transverse. Mr. Green, . This Paper, G H I K A 2 A 3 P Q & B x B 2 B 3 L M N C\ C 2 C 3 There is nothing, however, in the researches of Mr. Green to indicate any mutual relations amongst those nine coefficients; and to establish such relations, indeed, it appears to me that the subject must be in- vestigated, not dynamically, but statically. It may here be observed, that Mr. Green's equations contain three additional coefficients, to represent the effect of a strained condition of the medium on the propagation of vibratory movement; but those three * A second paper by Mr. Haughton (Trans. Roy. Irish Acad., XXII.) is equally comprehensive. 88 LAWS OF THE ELASTICITY OF SOLID BODIES. quantities, being foreign to the subject of this paper, have no expressions corresponding to them in its notation. In the equations of the propagation of light, Mr. Green effects an apparent reduction in the number of coefficients by introducing the supposition that the vibrations are of necessity wholly tangential to each wave-front. But this supposition is quite at variance with the nature of elastic solids, and is obviously intended by the author as merely an assumption for the purpose of facilitating calculation, and obtaining approximately true results, in the case of luminiferous undulations. Mr. M'Cullagh's researches on the propagation of light (Trans. Roy. Irish Acad., XXI.) involve a similar assumption. The result peculiar to the investigations contained in the present paper is the establishment of certain mutual relations amongst the different elasticities of a given substance, whereby the six coefficients of Poisson and Cauchy are reduced to functions of three, and the nine coefficients of Mr. Green to functions of four ; the former representing the condition of a medium whose elasticity is wholly due to the mutual actions of atomic centres, the latter that of a su bstance whose condition is intermediate between those of a system of centres of force, and of a continuous and uniformly diffused fluid. General equations of vibratory movement, in the particular case of uncrystallised media, agreeing with those of Mr. Green, are given by Professor Stokes in his memoir on Diffraction (Camb. Trans. IX.) His two coefficients of elasticity have the following values in the notation of this paper : — Prof. Stokes. This Paper. 6 2 D" g denotes the accelerating force of gravity; and D, the weight of unity of volume of the vibrating medium. a and b, in Professor Stokes's paper, are the velocities of propagation of normal and tangential vibrations respectively. In the researches of Poisson, Navier, Cauchy, Lam6, and others, on the elasticity of bodies equally elastic in all directions, the coefficients are often expressed in terms of two quantities, denoted by k and K, in the following expression for a normal pressure on the plane yz : P 1 = -^N 1 -K(N 1 + N 2 + N 3 ): k represents a species of longitudinal elasticity, under the condition that LAWS OF THE ELASTICITY OF SOLID BODIES. 89 the volume remains unchanged; and K, an elasticity resisting change of volume. Their values in the notation of this paper are as follows : — & = A-B = 2Cj K = B = C + J. It is evidently impracticable to apply an analogous notation to bodies unequally elastic in different directions. M. Wertheim has recently made a most elaborate and valuable series of experiments on the elasticity of brass, glass, and caoutchouc, according to a method suggested by M. Eegnault, for the purpose of determining the laws of elasticity of uncrystallised substances (Ami. de Chun, ct de Phys., Ser. III., torn. XXIII.) He concludes that for brass and glass, and for caoutchouc moderately strained, the following equation is nearly if not exactly true, in the notation to which I have just referred, h = K, which, in the notation of this paper, is equivalent to the following : J = C; B = 2C; A = 4 C. M. "Wertheim has investigated the consequences which must follow in the solution of several problems connected with elasticity, if this law be universally true for solid bodies. This supposition must be regarded as doubtful ; and it is not, indeed, advanced by M. Wertheim as more than a conjecture. So far as our present knowledge goes, it seems more probable that the relations between C and J may be infinitely varied. If the effect of heat is to diminish C and increase J, there maybe some temperature for each substance at which M. Wertheim's equation is verified. In the sequel I shall consider more fully the consequences to be deduced from M. Wertheim's experiments on this subject. Section V. — Coefficients of Pliability, and of Extensibility and Compressibility, Longitudinal, Lateral, and Cubic. Examples of their Experimental Determination. 21. Coefficients of elasticity serve to determine pressures from the corresponding strains. We have now to consider the determination of strains from pressures. To determine a distortion from the corresponding tangential pressure, it is sufficient to multiply, using the negative sign, by the reciprocal of the proper coefficient of rigidity. This reciprocal may be called a coefficient of pliability. A similar process, however, cannot be applied to the calculation of longitudinal strains from normal pressures; because, as each normal 90 LAWS OF THE ELASTICITY OF SOLID BODIES. pressure is a function of all the three longitudinal strains, so each longi- tudinal strain is a function of all the three normal pressures. Let the longitudinal strains be represented, in terms of the normal pressures, by the following equations : N 1 = -;r 1 P 1 + b ;2 P, + ^P 3 ^ N 2 = h,^- BgPss + JbrjPa L . . (10.) N 3 = ^Pi + ^Ps-KgPj Then the coefficients in these equations are found, by a process of elimination, to have the following values in terms of the coefficients of elasticity. Let K - 24(C 2 2 C 3 + C 2 C 3 2 + C 3 2 C x + C 8 (V + C^C, + (W -C,» - C 8 « - C 3 3 ) - 52C 1 C 2 C 3 + J{8(C 2 C 3 + 0,0x4- 0,0.,) - 4(C X 2 + C 2 2 + C 3 2 )}. Then 1 {80^-9(0, -03)2 + 40^} %=^{8C 2 2-9(C 3 -C 1 ) 2 + 4C 2 Jj a 3 =g{8C 3 2 -9(C 1 -C 2 ) 2 + 4C,.l; 1 fr 3 =^{3(c x +c 2 --c,)C8-c 1 c 2 +2(c 1 +c 2 -c,)j} (ii.) The above coefficients may be thus classified : tt v cl 2 , n 3 , are the coefficients of longitudinal extensibility and compressibility parallel respectively to the three axes of elasticity. b v l) 2 , I) 3 , are the coefficients of lateral extensibility and compressibility for the three co-ordinate planes of elasticity, serving to determine the effect of a normal pressure on those dimensions of a body which lie at right angles to its direction. From the manner in which the coefficient J enters into the common denominator K, it is obvious that when the coefficients of rigidity diminish without limit as compared with that of fluid elasticity, the six coefficients of linear extensibility and compressibility increase ad infinitum. In a body whose three coefficients of rigidity are different, the coefficient LAWS OF THE ELASTICITY OF SOLID BODIES. 91 of cubic compressibility, that is to say, the quotient of the sum of the three longitudinal strains by the mean of the three normal pressures, with the sign changed, has no fixed value unless some arbitrary relation be fixed between those pressures. Let them be supposed, then, to be all equal ; let their common value be P, and let the coefficient of cubic com- pressibility in this case be denoted by & : then h = _ *Ti+% + 3gT. = Ri + , + , 3 _ 2(Irx + ,b 2 + *,) = 4" { 8 (° 2 C 3 + C a C i + CxC 2 ) - 4(<V + C 2 s + C 3 2)} y a 2 -) Hence i = J + 6(C 1 +C, + C 3 ) lDGC^CgC. ~ 8 (C 2 3 + O^ + C X C^ - 4(0/ + C 2 2 + C 3 *) So that this coefficient is the sum of the three longitudinal coefficients of compressibility, diminished by twice the sum of the three lateral coefficients. It does not, like them, increase ad infinitum when the rigidity vanishes ; its ultimate value in that case being 1 J' the reciprocal of the coefficient of fluid elasticity, as might have been expected. If C 2 = C 8 , so that the body is equally elastic in all directions round the axis of x, equations (11) and (12) take the following forms : K = 4C 1 {12C 1 C 3 -6C 1 2- a 2 + J(4C 2 -C 1 )} »i =g ( 8C i 2 + 4ClJ) = UQ X G, - bCV - C„ 2 + J(4C 2 - i\) R 2 = »,«^{8C 1 «-9(0 1 -C i y + 40 a J} b ± = ~ {6CVC, -30^-a 2 + J(4C 2 - 2CJ} 1 ( ( Co + J & 2 = I>3 = g(2C 1 C 2 + 2C X J) = j MCi0g-1 2C 1 *-30 8 * + J(SC 2 - 2C X ) ir = s (16C 1 C 2 -4C 1 2 ) = ^C^o - 6C X 2 - C 2 2 + J(4C 2 - C~) 1 490 2 £ = J + 12CO + 6C!- jp ' M12A) 4Co-C x 92 LAWS OF THE ELASTICITY OF SOLID BODIES. For bodies equally elastic in all directions the coefficients of com- pressibility and extensibility take the following values : 2C + J . C + J ^-^=50^ ••■r J+|a In substances of this kind the coefficient of cubic compressibility is the same, whether the three normal pressures arc equal or unequal, being equal to the sum of the three longitudinal strains divided by the mean of the three normal pressures with the sign changed : that is to say, *~ " " i\ + p 2 + iv One of the most frequent errors in investigations respecting the elasti- city and strength of materials, and the propagation of sound, has been to confound the coefficients of longitudinal elasticity with the reciprocals of the coefficients of longitudinal compressibility. The equations of this section show clearly how widely these two classes of quantities may differ. The reciprocal of the longitudinal extensibility, , is what is commonly termed the Weight of the modulus of elasticity. 22. The following formula may be found useful in the determination of the coefficient J of fluid elasticity from experimental data. Let us suppose that the three coefficients of rigidity of a substance, C v C 2 , C 3 , have been determined by experiments on torsion, and that some one of the coefficients of compressibility and extensibility in equa- tion (11), or those derived from it, has also been determined by experi- ment. Let the actual value of this coefficient be called f, and the value which it would have had, had J been = 0, f . Also let K denote the value which the denominator K would have had, had J been = 0, and let n be the factor by which J is multiplied in the numerator of f, and in, in the denominator. Then J = K .-^L C (13.) u mi — n When applied to coefficients of longitudinal extensibility, this formula labours under the disadvantage that a comparatively slight error in the experimental data may cause a serious error in the determination of J. Let us take, for example, an uncrystallised substance, and make succes- sively the two following suppositions, J = 0, J = C: LAWS OF THE ELASTICITY OF SOLID BODIES. 93 it will be found that the results are respectively, a = - x 0-4, n = r - X 0375, being in the ratio of 1 6 : 1 5 ; so that any uncertainty in the experiments is in this case increased fifteenfold in computing the value of — . Hence it appears, that without very great precision in the experiments, the coefficient of fluid elasticity cannot be satisfactorily determined by a comparison of the effects of longitudinal tension with those of torsion. It is especially desirable that the two sets of experiments should be made on the same piece of the material. The best data for calculations of this kind would be experiments on cubic compressibility, in conjunction with experiments on torsion ; for, as equations (12), (12 A), and (12B) show, in order to determine J we have simply to subtract a certain symmetrical function of the rigidities from the reciprocal of the cubic compressibility. In the process of calculation, the errors in the experiments on rigidity are multiplied, on an average, by ~ only, while those of the experiments on compressibility sustain no augmentation whatsoever. Next after data of this kind may be ranked experiments on longitudinal extensibility, as compared with the cubic extensibility or compressibility of the same piece of material. Of this method, suggested by M. Eegnault, and carried into effect by M. Wertheim, I shall presently speak more fully. Were it possible to ascertain the velocity of sound in an unlimited mass of an elastic material along each of the axes of elasticity, the coefficients of longitudinal elasticity could be determined with great precision by the formula where v is the velocity of sound, D the weight of unity of volume of the substance, and g the accelerating force of gravity. But it is only practi- cable to determine the velocity of sound along prismatic or cylindrical rods; and, as I shall show in a subsequent paper, it is impossible, in the present state of our knowledge of the molecular condition of the superficial particles of solid bodies, to assign theoretically the ratio in which the velocity of sound along a rod is less than its velocity in an indefinitely extended mass. That ratio is only known empirically in a few cases, having various values lying between 1 and ^/f . 23. The experiments of M. Wertheim, on longitudinal and cubic extensibility {Ann. de Chim. et de Phys., Ser. III., Tom. XXIII.) were made 94 LAWS OF THE ELASTICITY OF SOLID BODIES. upon brass and crystal, the results being calculated on the supposition that those substances are homogeneous and equally elastic in all directions. There can be no doubt of the correctness of this supposition with respect to well annealed crystal ; and with respect to brass, it is probably very near the truth. In those experiments, a cylindrical tube of the substance to be examined was strained by longitudinal tension. The increase of length was observed directly. The increase of bulk was found by observing the depression in a capillary tube connected with the summit of the strained tube, of a column of liquid with which they were filled. Let R denote the tensile force reduced to unity of surface ; let L be the original length of a given portion of the tube, U its original volume, A L and A U the increase of those quantities by the tension R ; and let the axis of x be that of the tube. Then we have N = AL - X +N +N =^ J - P l = -R; P 2 = 0; P 3 = 0: and consequently, for uncrystallised substances, 1 AL . 3 AU a =B,'^r> * = r- it- • • < 14 -> To determine the coefficients of rigidity and fluid elasticity from these data, we have the following formulae : C = 3 a - £ & „AL AU 6 T~ IT Vn / VL AU / (14A.) J The experiments of M. Wertheim were made on three tubes of brass and five of crystal. In the following table those tubes are designated as M. Wertheim has numbered them. The coefficients a and Jr, transcribed or calculated from his statement of the mean results of numerous experiments on each tube, express the fraction by which the material is elongated, or increased in bulk, by tension at the rate of one kilogramme on the square millimetre; that is to say, 1422-34 pounds avoirdupois on the square inch. (The common logarithm of this number is 3-153004.) The reciprocals of those coefficients, and the coefficients of elasticity, as calculated by equation (14A), are given in kilogrammes on the square millimetre. LAWS OF THE ELASTICITY OF SOLID BODIES. 95 H S3 55 H X! H K O H a »— t ■ m m X 1-1 5,«j o Hi H O P5 ■ I q pq 05 a H < t3 O iJ o H 55 SI E=4 o Eh O H a£ <!- — 3 lap O 01 g •S = W «|J a a 23 S-l 3 "-» a g" -|o ^H O ^H •-5 a a © o cS os m >o o Tfl CO CO T* CO CO C5 CM t— lO CO Ci o o in CO CO CM CM o ~H CM <M CM <M CM o o CO O o o o O o o o o o o Q o a o o o o o o O o h1 o o o o o © o o H 96 LAWS OF THE ELASTICITY OF SOLID BODIES. The various degrees of elasticity of the brass tubes are ascribed by M. Wertheim to the relative frequency with which they were subjected to wire-drawing, to reduce the thickness of metal. It may be observed, that this operation seems to increase the rigidity more than the fluid elasticity, a fact which might naturally have been expected. The means of the three sets of results for brass are given in the following table : — Coefficients of Rigidity, Fluid Elasticity, . Longitudinal Elasticity, Lateral Elasticity, Reciprocals of Extensibilities. Longitudinal (or Weight of the ) 1 Modulus of Elasticity), . ) a 1 Cubic, ..... t Kilogrammes on the Sq. Millim. 3745-3 4389-0 15G25-0 8134-3 10D.V1--1 10G31-0 Lbs. Avoird. on the Square Inch. 5327100 6242700 22224000 11579000 14301000 15121000 Coefficients of Extensibility Per Kilog. on the Per lb. on the Square and Compressibility. Square Millim. Inch. Longitudinal, ;t 0-00009946 0-0000000699 Cubic, b 0-00009406 0-0000000661 Latei'al, b 0-00003405 0-0000000239 The following result is calculated from the experiments of M. Savart on the torsion of brass wire (Ann. dc Chim. et de Phys., August. 1829):— Coefficient of Rigidity, The difference beins Kilog. on the Square Millim. C 3682 63-3 Lbs. on the Square Inch. 5237100 900U0 Hence we see that the rigidity of wire-drawn brass, as determined directly by torsion, differs from that calculated from the longitudinal and cubic extensibilities by only one-sixtieth part, being a very small discrepancy in experiments of this kind. The following table gives the means of the four sets of results, I., III., IV., V., for crystal : — LAWS OF THE ELASTICITY OF SOLID BODIES. 07 Kilog. on the Square Millim. Lbs. on the Square Inch c . 1518 2159100 J . 1438 2045300 A . 5992 8522600 B . 2956 4204400 1 4039 5746100 l 3968 5643800 Per Kilog. on the Square Millim. 0-0002476 0-0002520 0-0000818 Per lb. on the Square Inch. 0-0000001740 0-0000001772 0-0000000575 It is obvious that the above mean values for crystal are not to be relied upon as equally accurate with those for brass ; for the wide dis- crepancies between the results of the experiments on the five crystal tubes show that this substance, like every kind of glass, is subject to great variations in the physical properties of different specimens. 24. So far as I am aware, there is no substance whose elasticity varies in different directions, for which experimental data as yet exist, adequate to determine the three coefficients of rigidity, and the coefficient of fluid elasticity. Supposing the three coefficients of rigidity of a substance of this kind to be known by experiments on torsion, the process of MM. Regnault and AVertheim would readily furnish data for calculating the fluid elasticity. For example : let a tension E per unit of area be applied to the ends of a tube whose axis is one of the axes of elasticity, say that of x. Let —==- De the fraction by which its volume is increased, as before. Then AU_ N 1 + N 2 + N 3 _ "\ = 1{8C;--2C 1 (C 2 + C 3 )-6(C 2 - - C V 21 r v 3/ / 1 y (is.) Let the above equation be abbreviated into A U _ 0(C) R U ~ K + m J ' G 98 LAWS OF THE ELASTICITY OF SOLID BODIES, where K = K + mJ, as in equation (13). Then The formula? corresponding to equation (15) for tubes whose axes are parallel to y and z, are easily found by permutations of the indices 1, 2, 3. The sum of the three values of ~ TT thus obtained is obviously = tf. h U [It may be remarked, with reference to Sect. 1 7 of the preceding paper, that the effect of alterations of direction in the lines joining pairs of particles is not taken into account in the investigation of the elastic forces arising from the states of strain which are there considered. It appears to me that this effect, except for particular laws of force, will be of the same order as that which depends on the alterations of the mutual distances between the particles ; and that if it be taken into account, the demonstration of Theorem IV. fails. This objection occurred to me after the whole of the paper was in type, and I immediately suggested it to the author ; but, as he was not con- vinced of the correctness of my view, he desired that the paper should be published as it stands, reserving additional explanations or modifications, if necessary, for the next number of the journal. — W. T.] Supplementary Paper to Section III., Article 17. In the portion above referred to of my paper on the Elasticity of Solids, published in the Cambridge and Dublin Mathematical Journal for February, 1851, the theorem is laid down, that in a given plane in an elastic solid consisting entirely of atoms acting on each other by attractions and repulsions between their centres, the coefficients of rigidity and of lateral elasticity are equal. The proof of this proposition depends on the principle that the elastic force in such a solid, called into play by a strain, in which the relative displace- ments of the atoms arc very small as compared with tlieir distances apart, is sensibly the resultant of the variations of force due to the variations of distance only, the variations of relative direction producing no appreciable effect. This principle being granted, it is easily shown that the portion of that resultant for each pair of atoms is the same for a given amount of strain in a given plane, whether lateral or transverse with respect to the plane on which elastic pressure is estimated. LAWS OF THE ELASTICITY OF SOLID BODIES. 99 In the paper referred to, I assumed this principle without demonstration. The editor of this journal, however, has since shown me, that my having done so may be considered as causing a defect in the chain of reasoning. I shall now, therefore, proceed to prove it. Let it be possible for a solid to exist in an unstrained condition, consisting entirely of atomic centres of force acting on each other along the lines joining them, with forces which are functions of the lengths of these lines. Then must the pressure, estimated in any direction, on any portion of any plane in that solid be null. That pressure is the resultant, in the direction assumed, of the mutual actions of all the atoms whose lines of junction pass through the given portion of the given plane. Let the given portion be indefinitely small, and let it be called w, being situated in the arbitrarily-assumed plane P Wj p, which divides the solid into two por- +x tions, A and B. Let — X o> + X be an arbi- trary axis along which pressure is to be esti- mated. The pressure exerted by the portion A upon the infinitesimal area 10 of the portion B, is the resultant, reduced to the direction — X to + X, of all the forces exerted by the atoms in A on the atoms in B, in lines passing through (o; and the body being unstrained, this resultant must be null. Assume a new position, Vuyp', for the plane of separation, making an equal angle P'w + X=+XwPon the opposite side of the axis to the original position. The same letters applying to the two portions of the solid, the pressure of A on the area 10 of B along -XwX must still be null. The two planes divide the solid into two pairs of opposite wedges. The action of A on B along X through w in the original position of the plane, may be divided into two parts, viz. — The resultant of the actions of the atoms in the wedge P top' on those in the opposite wedge Ywp ; The resultant of the actions of the atoms in the wedge p u>p on those in the wedge P w P'. In the new position of the plane, the pressure on u> is made up as follows : The resultant of the actions of the atoms in the wedge Pw/on those in the wedge P'wp, which is the same as hi the original position of the plane; The resultant of the actions of the atoms in the wedge PwP' on those in the wedge p'wjp, being identical in amount but opposite in direction to 100 LAWS OF THE ELASTICITY OF SOLID BODIES. that of the atoms in p' up on those in P to P', which formed part of the pressure in the original position of the plane. Now the pressures in the two positions of the plane of separation cannot both be null, unless the resultant of the mutual actions of the atoms in each pair of opposite wedges is separately null; for we see that the action of a pair of wedges can be reversed in direction without affecting the nullity of the total resultant. The position of the pair of opposite wedges is arbitrary; so also is their angular magnitude, which may be indefinitely small. Therefore, no mere change of angular position of a pair of opposite elementary wedges can produce a pressure. Every strain in which the relative displacements of the particles are small as compared with their relative distances, may be reduced to angular displacements of pairs of opposite elementary wedges, and varia- tions of the mutual distances of the particles contained in them. The angular displacements can produce no pressure of themselves; the variations of distance are therefore the sole cause of that portion of the pressure which is of the same order of small quantities with the strain: being the principle to be proved. The combination of the angular displacements with the variations of distance will give rise to pressures of the second and higher orders of small quantities as compared with the strain; but for the small strains to which the present inquiry is limited, those are inappreciable, and may be nesdected. Note respecting Mr. Clerk Maxwell's Paper <; On the Equilibrium of Elastic Solids." (Trans. Hoy. Soc. Edin., Vol. XX., Part I.) I HAVE already referred to the researches of Mr. Clerk Maxwell, of the general nature of which only I was aware at the time of the publication of my paper on this subject in the Cambridge and Dublin Mathematical Journal for February, 1851. Since then I have had an opportunity of reading Mr. Maxwell's paper, so as to compare his notation with my own. Mr. Maxwell's investigations relate to such solids only as are equally elastic in all directions. He expresses their elasticity by means of two coefficients, fx and m, having the following properties: LAWS OF ELASTICITY. 101 M = 1 P x + P 9 + P, 3 d£ dv dV dx dy d z m = P — P P - d £ dr\ dr\ d x dy dy - P., d z dZ, dz~ -Pi _dj dx From which it is clear, that those coefficients have the following values in the notation of the paper which I have published. ju = ^. = -JC + J = reciprocal of the cubic compressibility, m = 2 C = twice the rigidity ; consequently C = ^ m, J = fi — 4 m. The particular problems solved by Mr. Clerk Maxwell are of a very interesting character, especially those relative to the optical changes produced in transparent bodies by straining them. ON THE LAWS OF ELASTICITY.* Section VI.— Ox the Application of the Method of Virtual Velocities to the Theory of Elasticity. 25. Lagrange's method of virtual velocities having been applied to the problems of the equilibrium and motion of elastic media by Mr. Green (Camb. Trans., VII.) and by Mr. Haughton (Trans. Royal Irish Acad., XXL, XXIL), it is my purpose in this and the following section to point out the mutual correspondence between the coefficients in the formulas arrived at by these gentlemen, and the coefficients of elasticity which form the sub- ject of the previous portion of this paper, and also to show how far the laws of relation between the nine coefficients of elasticity of a homogene- ous body, which I originally proved by a method chiefly geometrical, are capable of being deduced symbolically from equations found according to Lagrange's method. 26. The principle of virtual velocities, as applied to molecular action, it * Originally published in the Cambridge and Dublin Mathematical Journal, Nov., 1S52. 102 LAWS OF ELASTICITY. as follows • Let X, Y, Z, denote the total accelerativc forces applied to any particle, whose mass is m, of an elastic medium, through agencies dis- tinct from molecular action (such as the attraction of gravitation); let u v to be the components of the velocity of m; let dx, By, Bz, denote indefinitely small virtual variations of x, y, z; let S be the total accelera- tive molecular force applied to m, Bs an indefinitely small virtual variation of the line along which it acts j then the following equation »[-{(x-^.. + (y-50«» + (--^.}]\ w + S(mS8s) = J (the summation 2 being extended to all the particles of the medium), expresses at once all the conditions of equilibrium and motion of every particle of the medium. 27. In applying this principle to the theory of elastic media, both Mr. Green and Mr. Haughton assume the following postulates : First. That in calculation Ave may treat each particle m as if it were a small rectangular space, dx dy dz, filled with matter of a certain density p : so that for the symbol 2 m we may substitute that of a triple integration. fffpdxdydz. Secondly. That the virtual moment m S B s of the total molecular force acting on any particle m, is capable of being expressed by the product of the small rectangular space dx dy dz into the variation, B V, of a certain function V of the relative position of m and the other particles of the body* Equation (16) is thus transformed into the following: — + (z - ~^Bz\dxdydz +ff/BYdxdydz = (17.) J 28. The term elasticity properly comprehends those molecular forces only whose variations are produced by, and tend to produce, variations in the volume and figure of bodies. There are, therefore, conceivable kinds of molecular force, which are not included in the term elasticity. For example, let us take the forces which Mr. MacCullagh ascribed to the particles of the medium which transmits light. * This amounts, in fact, to the assumption that no part of the power developed by a variation of the relative positions of the particles is permanently converted into heat, or any other agency: in other words, that the body is perfectly elastic. LAWS OF ELASTICITY. 103 Let £, 17, £, denote displacements of a point in the medium parallel respectively to x, y, z. Then Mr. MacCullagh supposes the molecular forces to be functions of dr\ dZ, dZ, d£ d£ dr\ dz dy } d x dz ' dy dx' which are proportional to the rotations of an element dxdydz from its position of equilibrium about the three axes respectively. This amounts to ascribing to the particles of the medium a species of polarity, tending to place three orthogonal axes in each particle parallel respectively to the three corresponding axes in each of the other particles : the rotative force acting between the corresponding axes in each pair of particles being a function of the projection of the relative angular displacement of the axes on the plane passing through them, of the position of that plane, and of the distance between the particles. A portion of a medium endowed with such molecular forces only would transmit oscillations; but it would not tend to preserve any definite bulk or figure, nor would it resist any change of bulk or figure. It would be a medium or system, but not a body. Molecular forces of this kind, therefore, are not comprehended under the term elasticity ; and the limits of the present investigation exclude those forms of the function V which represent the laws of their action. 29. The inquiry being thus restricted to molecular forces dependent on the variations of the bulk and figure of bodies, there is to be intro- duced a Third Poshdate: That supposing the body to be divided mentally into small parts, which, in the undisturbed state of the body, are rectangular and of equal size, those parts, in the disturbed state, continue to be sensibly of equal bulk and similar figure, throughout a distance round each point at least equal to the greatest extent of appreciable molecular action. This assumption has been made in all previous investigations, except those respecting the dispersion of light; and it seems, indeed, to be perfectly consistent with the real state of tangible bodies. Its advantage in calculation is, that it enables us to treat the variations of the molecular forces acting on a given particle, as functions simply of the variations of bulk and figure of an originally rectangular element situated at that particle: seeing that the adjoining elements throughout the extent of appreciable molecular action continue always to undergo sensibly the same variations of bulk and figure as the element under consideration. Let x , y , z be the co-ordinates of any physical point in a homogeneous 104 LAWS OF ELASTICITY. body in equilibrio, and whose particles are not operated upon by any extraneous forces, X, Y, Z. In this condition it is evident that SV = at every point, and that we may also make V = 0. In the disturbed condition, let £, ?j, Z,, be the displacements of the point whose undisturbed position is (x , y , z ), so that x = o: + £, &c. Then all the variations of bulk and figure which can be undergone by an originally rectangular element, consistently with the third postulate, may be expressed by means of the following six quantities, which I have elsewhere called strains : d% _ dt) _ r. ell, _ Tz~ a > Ty- pm ' Tz~ 7 ' dj ,dX_ d% &%, _ cZj; djj _ ,. dz + dy~ ' dx + dz~ fM} dy + dx~ V ' of which a, ft, y, are longitudinal extensions if positive, compressions if negative, and X, ju, v, are distortions in the planes perpendicular to x, y, z, respectively. Hence it appears that Y = <f>(a,ft, 7> \,fx,v). . . . (18.) 30. The first assumption, that we may treat the body in calculation as composed of rectangular elements p d x d y d z, involves the consequence that we may express all the molecular forces which act on each such element by means of pressures, normal and tangential, exerted on its six faces. Taking yz, zx, xy, to denote the position of the faces of such an element, P to denote generally a normal pressure expressed in units of force per unit of area, and Q a tangential pressure similarly expressed, let the nine component pressures on unity of area of those faces be thus denoted : * This notation is substituted for Ni, N 2 , N 3 , 2Tn, 2T a 2T 3 , as being more convenient. LAWS OF ELASTICITY. 105 Position of Face. Direction of Pressure. X y Z y~ Pi % <¥ nx q; p. Qi xy Q 2 Qi' p 3 By the definition of elasticity all pressures are excluded, except those whose variations produce and are produced by variations of volume and figure of the parts of the body. Hence the pressures Qi-Q/; Q«-Qii Q3-Q/; whose tendency is to make the element dxclydz rotate about its three axes respectively, without change of form, must be null; and therefore Qi = Qi'; Q 2 = Q 2 '; Q 3 = Q 3 '- • ■ (19-) Mr. Haughton correctly remarks that this often quoted theorem of Cauchy is not true for all conceivable media. It is not true, for instance, for a medium such as that which Mr. MacCullagh assumed to be the means of transmitting light. It is true, nevertheless, for all molecular pressures which ' properly fall under the definition of elasticity, if that term be confined to the forces which preserve the figure and volume of bodies. Let us now express the sum of the virtual moments of the molecular forces acting on the element d x d y d ~, in terms of the pressures P 1? &c. ; to do which, we must multiply each pressure by the virtual variation of the effect which it tends to produce in its own direction. Thus we obtain the following result : — SV = P 1 Sa + PoS/3 + P 3 £7 + Q 1 gA + Q 2 S y u + Q 3 Si/. . (20.) Hence the function V bears the following relations to the normal and tangential pressures at the faces of a rectangular element : 1_ da' 2 ~ d(}' 3_ dy ^~d\> ^-df,' ^~Jv) (21.) 31. A Fourth Postulate, generally assumed in investigations of this kind, is that the pressures are sensibly proportional simply to the strains with which they are connected. This assumption must be approximately true of any law of molecular action, when the pressures and strains are suffi- ciently small. It is known to be sensibly true for almost all bodies, so 106 LAWS OF ELASTICITY. long as the pressures and strains are not so great as to impair their power of recovering their original volume and figure. According to this postulate, the pressures P 1? &c., are algebraic functions of the first order of the strains a, &c. ; and consequently V is an algebraic function of the second order of those strains. The constant part of V, as we have already seen (Art. 29), is null. Following the notation adopted by Mr. Haughton, let (a) denote the 2 coefficient of a in V, (a 2 ) that of — — , (/3y) that of — (5y, &c. Then V = (a)a + (/3)/3 + (7)7 + W ^ + (/*)/* + W» « 2 P2 .,2 \2 ..2 ,,2 - (« 2 ) 2 - (0 2 ) f- ~ M I - (X 2 ) y " M J - M> •> — ($y) fiy — (yet) ya — (a/3) a/3 — (fiv) fxv — (vX) v\ — (\/ul) \/jl — (aX) a\ — (a/u) a/j. — (av) av -(P\)P\--(M(5f*-((3v)pv -(y\)y\ — (yfi)yfA—(yv)yv. . . (22.) The six coefficients of the terms of the first order. in this equation obviously represent the pressures, uniform throughout the whole extent of the body, to which it is subjected when its particles arc in those positions from which the displacements are reckoned: that is to say, when I = 0; r, = Oj Z = 0. Let P 1>0 &c, denote those pressures. Then («) = i\,o; 0)-p m j (7) = p,,o| W = Qi,oJ (m)=Q 2 , ; W^Q^J The twenty-one coefficients of the terms of the second order are the coefficients of elasticity of the body, as referred to the three axes selected. The negative sign is prefixed to each, because it is essential to the stability of a body that molecular pressures should be opposite in direction to the strains producing them. The transformation of the quantities in equation (22) for any set of rectangular axes, is effected by means of equation (2) of Sect. I, Art. 9, by making the following substitutions : — for PPP20'>O'>O x 1' -"" 2' 3' A H'ij * k'01 - V3J substitute a, (3, y, X, p, v, and make similar substitutions for the accented symbols. By multiplying the six equations referred to together by pairs, twenty-one equations are LAWS OF ELASTICITY. 107 obtained, serving to transform the squares and products of a, (5, 7, X, jjl, v. Formulae similar to those which transform the strains a, &c., and their half-squares and products, serve also to transform the respective coefficients of those quantities in equation (22).* It is shown by Mr. Haughton, that by properly selecting the axes of co-ordinates, the number of independent coefficients of elasticity may always be reduced to three less than when the axes are indefinite ; and by Mr. Haughton and Mr. Green, that when the body has orthogonal axes of elasticity at each point, then, if those axes be taken as the axes of co-ordinates, the coefficients of elasticity are reduced to the first nine. The latter proposition is obvious, because if molecular action be sym- metrical about three orthogonal planes, and those be taken for co-ordinate planes, then the value of that part of the function V which is of the second order cannot be altered by a change in the sign of either of the dis- tortions A, ju, v; so that the coefficients of the last twelve terms of equation (22) must each be null. The nine coefficients of elasticity of a body in those circumstances have the following values, in terms of the notation of the previous sections : Coefficients of Longitudinal Elasticity. (a 2 ) = A i; m=A 2 ; ( 7 2 )=A 3 (24.) Coefficients of Lateral Elasticity. (/3 7 ) = B i; ( 7 a) = B 2 ; (a/3)=B 3 . Coefficients of Eigidity. 32. Def. Let the term Perfect Fluid be used to denote the state of a body, which under a given uniform normal pressure, and at a given temperature, tends to preserve, and if disturbed to recover, a certain bulk; but offers no resistance to change of figure. In such a body, if the element whose original bulk was dxdydz, becomes of the bulk (1 + 0-) dxdydz (o- being a small fraction), we shall have a = a + /3 + J, and the function V must be of the form Y = Y „-(o»-) a ~ . . . (25.) where P is the uniform normal pressure when the particles are not dis- placed, and (o- 2 ) a coefficient of elasticity, whose value, in the notation of the previous sections, is : * See the Note at the end of this paper. 108 LAWS 0F ELASTICITY. Coefficient of Fluid Elasticity. («*) = (a 2 ) = (/3 2 ) = ( 7 2 ) = (/3y) = (y«) = («0) = J- (26.) The normal pressure in the disturbed state, which is the same in all directions, is obviously The tangential pressures are each = 0. Section VII.— On the Proof of the Laws of Elasticity by the Method of Virtual Velocities. 33. Having thus followed very nearly the steps of the researches of Mr. Haughton and Mr. Green, so as to compare their coefficients with those used in the previous part of this paper, I shall now investigate how far the method ■ of Lagrange can be used to establish those relations between the coefficients of elasticity of different kinds in homogeneous solid bodies, which I have elsewhere deduced from geometrical and physical considerations. The fluid elasticity considered in the last article cannot arise from the mutual actions of centres of force; for such actions would necessarily tend to preserve a certain arrangement amongst those centres, and would therefore resist change of figure. Fluid elasticity must arise either from the mutual actions of the parts of continuous matter, or from the centri- fugal force of molecular motions, or from both those causes combined. On the other hand, it is only by the mutual action of centres of force that resistance to change of figure and molecular arrangement can be explained, that property being inconceivable of a continuous body. The elasticity peculiar to solid bodies is, therefore, due to the mutual action on centres of force. Solid bodies may nevertheless possess, in addition, a portion of that species of elasticity which belongs to fluids. The investigation is simplified by considering, in the first place, the elasticity of a solid body as arising from the mutual action of centres of force only, and afterwards adding the proper portion of fluid elasticity. It is known that solid bodies are capable of preserving bulk and figure, although their surfaces are acted upon by no sensible pressure, normal or tangential. We may take the positions of the particles in this condition as points from which to measure their displacements. Thus we cause the coefficients of all the terms of the first order in equation (22) to vanish. To investigate the properties of the coefficients of elasticity, the function SV is to be expressed in a new form, — viz., as the sum of the virtual moments of the actions exerted upon each of the centres of force in the LAWS OF ELASTICITY. 10D particle under consideration, by the centres of force in all the other particles. Mr. Haughton, in his first memoir, having performed this process, shows by means of its results, that in a body composed entirely of centres of force acting along the lines joining them, the number of independent coefficients of elasticity for any system of orthogonal axes is reduced to fifteen, which, by properly selecting those axes, may be reduced, for bodies in general, to twelve, and for those having axes of elasticity, to six. I shall now endeavour to prove by the method of virtual velocities, what I have in the third section proved by other modes of reasoning, that in a homogeneous body constituted of centres of force only, the independent coefficients of elasticity are reducible to three, of which, and of the position of the axes, the twenty-one in equation (22) are functions. A fourth independent coefficient is to be added in solids possessing a portion of fluid elasticity; that is to say, in all known solids. 34. It is known that a homogeneous solid can exist, with its particles in an unstrained condition, bounded by plane surfaces in any direction. In this condition, therefore, the total molecular action upon a particle situated at any bounding plane must be null. Conceive the bounding plane still to pass through the same particle, but to have its position shifted through any angle. The molecular action on the particle will still be null. Now the effect of the shifting of the bounding plane is to take away a wedge of matter from one side of the particle, and to substitute an equal and similarly constituted wedge, lying in a diametrically opposite direction. Hence, in the unstrained condition of a solid body, the action exercised upon any particle, by a wedge of matter bounded by any two planes passing through the particle, is null. This shows that the action of a wedge of solid matter on a particle situated at its edge, is not altered by varying the angular position of the wedge; and consequently, that the molecular actions which produce elasticity are not directly functions of the relative angular positions of the centres of force which act on each other, but merely of their distances apart, so that if the actions of the several equal wedges into which a body may be conceived to be divided, round a given particle, are different, this does not arise directly from the angular positions of the wedges, but from the different distribution of their centres of force as to distance from those of the particle operated upon. (I have proved this, in a manner slightly different in form, in a supplementary paper to Sect. III., Art. 17.) See p. 98. This further shows, that the mutual action of two centres of force in a solid must be directed along the line joining them; for otherwise it would tend to bring that line into some definite angular position, and would be a function of the direction of the line. HO LAWS OF ELASTICITY. It finally results, from what has been stated, that the action of an indefinitely slender pyramid of a solid body upon a particle at its apex must be a direct attraction or repulsion along the axis of the pyramid, which is a function of the several distances of the centres of force in the pyramid from those in the particle at the apex, and which, in the unstrained condition of the body, must be null. The principles stated above have to a greater or less extent been taken for granted in previous investigations, but have not hitherto been demon- strated. They may all be regarded as the necessary consequences of the following : — Def. Let the term Elastic Solid be used to denote the condition of a body, which, when acted upon by any given system of pressures, or by none, and at a given temperature, tends to preserve, and if disturbed, to recover, a definite bulk and figure; and such that, if lohile in an unstrained condition it be cut into parts of any figure, those parts, when separate, will tend to preserve the same bulk and figure as they did when they formed one body. Experience informs us that bodies sensibly agreeing with this definition exist; its consequences are, therefore, applicable to them in practice. 35. Let r denote the distance apart of two centres of force in an unstrained solid, and let <pr be proportional to their mutual action. Then dxdydzd 2 a . 2i<pr — may be taken to represent the total action of an indefinitely slender pyramid which subtends the element of angular space d 2 <o upon a particle at its apex d x d y d z. In consequence of a strain, let each of the distances r become (1 +*)r, £ being a very small fraction. Then the total action of the pyramid becomes d x d y d z d 2 o> . 2 (<p r + tr<p'r) = dx dy dzd 2 w . e 2 r <p' r; for by the third postulate, e is uniform throughout the extent of appreci- able molecular action. The quantity which the force acting between two centres of force tends to vary, is their relative displacement along the line joining them, or e r. Hence the sum of the virtual moments of the actions of all the slender pyramids into which the solid is conceived to be divided, that is to say, the total virtual moment of its molecular action upon the particle dx dy dz, is SV dxdy dz = dx dy dzffeSt.S (r 2 <f,' r)~.d 2 u, LAWS OF ELASTICITY. Ill the double integration extending to all angular space. Consequently, we obtain as a new value of the function V, V = ff\ £ 2 2 (r 2 <f>' r) .cPw. . . (28.) Let a, b, c be the direction-cosines of the axis of a given slender pyramid. Then it is easily seen that the strain e along that axis has the following value, in terms of the six strains as referred to the axes of co-ordinates : e = a a 2 + /3 b 2 + 7 c 2 + X b c + fx c a + v a b, and, consequently, that 9 9 /")f> £ o 4. , P Tj , C 4 + y% 2 C 2 + £*rf + .£rfP + /3 7 6 2 c 2 + 7 a c 2 ft 2 + a j3 ft 2 & 2 + /tv(i 2 J(;+ v\«i 2 c-|- \ jul a be 2 + a \ or be + a fj. a s c + av a 3 b + fi\b 3 c + /3m«^ 2 c+ /3va& 3 + 7 X k 3 + 7 /a a c 3 + 7 v a 5 c 2 . If this value of 1 1 2 be substituted in equation (28), and the result compared with equation (22), it is at once obvious that the twenty-one co- efficients of elasticity have the following values (putting S (r 2 <p' r) = — R, which is negative that equilibrium may be stable) : (a 2 )=//VRd 2 W (/3 2 )=//S*E^< u (7 2 )=yyVRrZ 2 a, (\ 2 )^(/3 7 ) =//-5 2 c 2 Rrf 2 (u (/M 2 ) = (7 a) =ffc 2 a 2 TLd 2 (o („*)= ( a j3) = //a 2 b 2 Ud 2 (o (fjiv) = (oX) =ffa 2 bc~Rd 2 u> (vA) = (pV) =//aS 2 C R f Z 2 w (X/*)= (tv) = ffabc i B,d 2 <a (0A) =ffV i cnd*to (yX) =//ic 3 EfZ 2 fc > iir).= //***&» (a fx) = f/c a 3 ~Rd 2 to (av) =ffa s b-Rd 2 to C/3v) = //*a& 3 Rtf 2 w 112 LAWS OF ELASTICITY. In the above equations, which agree with those given by Mr. Haughton, the number of independent coefficients is fifteen. 36. Their reduction to a smaller number arises from the nature of the function E = - 2 (r 2 r/>' r). This quantity is a function of the distances of the centres of force in a given indefinitely slender pyramid from those in a particle at its apex, and can vary with the direction-cosines a, b, c of the axis of the pyramid, solely because those distances vary with them. Now in a homogeneous solid, that is, one composed of a succession of similar and regularly placed groups of centres of force, those distances depend upon a quantity which may be called the mean interval between the centres of force jn a given direction : a quantity of such a nature that the product of its three values for any three orthogonal directions is a constant quantity; being the space occupied by a centre of force, or by a definite group of such centres. To have this property, the mean interval must be a quantity of this form: • __ ,f+ga" + hh' 2 + Jcc- + lbc+mca + nab. (30) that is to say, its logarithm must be proportional to the reciprocal of the square of the radius of an ellipsoid, whose axes are those of molecular arrangement, and therefore of molecular action, and of elasticity. Let the axes of this ellipsoid be taken as axes of co-ordinates. Then I = 0, m = 0, n = 0; and the above equation is reduced to i = cf+gat + hv + i-c^ m t m (3QA.) and because the quantity R = F (0 =^(f+rj a? + h & + k c 2 ), . (31.) does not change its sign or value by any change of the signs of the cosines a, I), c, it follows that all the coefficients in (29) containing odd powers of those cosines, that is to say, all except the first six, disappear when the axes of molecular arrangement are taken for axes of co-ordinates. These six, for all known homogeneous substances, are reducible to three, by the following reasoning : Let us assume as a Fifth Postulate, what experience shows to be sensibly true of all known homogeneous substances — viz., that their elasticity varies very little in different directions. Those substances, such as timber, whose elasticity in different directions varies much, are not homogeneous, but composed of fibres, layers, and tubes of different substances. If this be assumed, it follows that, in the expression (31), for the quantity E, the variable terms g a 2 + h 6 2 + k r, LAWS OF ELASTICITY. 113 are very small compared with the constant term /, and that E may be developed in the form R = f (/)+ f. (/). (ga* + h& + Ice") + &c. If this value of R be introduced into equation (29), and if small quantities of the second order be neglected, it is easily seen, on performing the inte- grations, that the following relations exist amongst the six coefficients already specified: (/3 2 ) + ( 7 2 ) = 6(\2) = 6(/3 7 ) ( 7 2) + («2) = 6 ( M 2) = 8 ( r «) (a 2 )+(/3 2 )=6(v 2 ) = 6(«/3) or, by transformation, S~ . (32.) (a 2 ) = 3 {(& + (v 2 ) - (X 2 )} (/3 2 ) = 3 {(v 2 ) + (X 2 ) - 0. 2 )} (r) = 3 {(x^) + (tf) - (v 2 )} These equations reduce the number of independent coefficients of elas- ticity, arising from the actions of centres of force, to three. They are identical with the equations (7) and (8), embodied in the fifth theorem in Sect. III., although arrived at by a different process. 37. Let us suppose the solid under consideration to possess a portion of fluid elasticity, represented by the coefficient J. Then the coefficients of elasticity have evidently the following relations : (a 2 ) = 3 (Ox 2 ) + („ 2 ) - (X 2 )} + J 1 (/3 2 )=3{(v 2 )+(X 2 )-( M 2 )} + J (7 2 ) = 3{(X 2 )+a t 2 )-(v 2 )} + J (/3y) = (A 2 )+J ( 7 «)=(,x 2 ) + J (a/3) = (v 2 ) + J which are identical with the six equations (9) comprehended under the sixth theorem, in Sect. IV. 38. The laws of elasticity stated in this paper are the necessary con- sequences of the definitions of elasticity and of fluid and solid bodies, given in Arts. 28, 32, and 34, respectively, when taken in conjunction with five postulates or assumptions, which, however, may be summed up in two — viz. H y (33.) 114 LAWS OF ELASTICITY. First, That the variations of molecular force concerned in producing elasticity are sufficiently small to be represented by functions of the first order of the quantities on which they depend ; and, Secondly, That the integral calculus and the calculus of variations are applicable to the theory of molecular action. It is thus apparent that the science of elasticity is, to a great extent, one of deduction a priori. The functions of perceptive experience in connection with it are two- fold : first, by observation, to inform us of the existence of substances, agreeing to a greater or less degree of approximation with the definitions and postulates; and, secondly, by experiment, to ascertain the numerical value of the coefficients of elasticity of each substance. Note to Sections VI. and VII. of Preceding Paper. On the Transformation of Coefficients of Elasticity, hy the aid of a Surface of the Fourth Order. (The following note contains no original principle, and is designed merely to put on record, for the sake of convenient reference, a series of equations which will be found useful in future investigations.) It has been pointed out by Mr. Haughton, in his first paper, that if we take into consideration that part only of the elasticity of a solid which arises from the mutual actions of centres of force, so that the function V shall contain at most but fifteen unequal coefficients (viz., those whose values are given in equation 29), and if, with those fifteen coefficients, we construct a surface of the fourth order, whose equation is the following, U= (« 2 K + (/3 2 )^+( 7 2 )^ + 6 (X 2 ) f$ + 6 Ou 2 ) z^ + G (v 2 ) x}if + 12 («\) x^yz + 12 (j3 M ) xfz + 12 (yv) xyz* + 4 (a i<) :c" y + 4 (a n) x z z + i(fi\)fz+ 4:(i3v)y*z + 4:(yn)£x+4:(y\)z*y = l ■ • • •■ (A.) then will U be the same function of the six quantities a?, y 2 , z\ 2yz, 2zx, 2xy, LAWS OF ELASTICITY. 115 that — 2 V is of the six strains «> /3, 7, A, ix, v, which are known to be transformed by the same equations "with the above functions of the second order of x, y, g; and consequently, the same equations which serve to transform the coefficients of the surface U = 1, into those suitable for a new set of rectangular co-ordinates, will also serve to transform the coefficients of elasticity in the function V. Now, it is obvious that if the equation ^ (», y, z) = xp (x, y, z) be true for two sets of rectangular co-ordinates having the same origin, then must the equation <1> f d d d \ \dx' dv' Tz) i ? ) (B.) d d d dx' dy' dz J ~~ r \ d aj" d y" d be true also. It follows that the fifteen coefficients of elasticity (a 2 ), &c, which are proportional to the differential coefficients of U of the fourth order with respect to x, y, z, are transformable hj means of the same equations which serve to transform the fifteen algebraical functions of the fourth order of x, y, z, by which they are respectively multiplied in the value of U. The following is the investigation of those fifteen equations of the fourth order, as well as of the six equations of the second order, from which they are formed by multiplication. Let the relative direction-cosines of the two sets of rectangular axes be expressed as follows: 1 Original Axes. N ew Axes. V z X a i h c l V tt.y h c 2 z a z h c '3 \- cosines. J Let the following notation be used for functions of those cosines. (It is the same which is employed by Mr. Haughton.) q 1 = c l a 1 ; p 2 — b 2 c 2 ; q 2 = c 2 a 2 ; r„ — a„ k 2 ; % — H a z> r* = a, \\ 116 LAWS OF ELASTICITY. l 1 = b 2 c 3 + b 3 c 2 ; l 2 = b 3 c x + \ c 3 ; l s — \ c 2 + \ c x ; m 1 — c 2 a 3 + c z a 2 ; rn 2 = c z a x + c x a z ; m 3 = c x a 2 + c 2 a x ; n x = a 2 b 3 + a 3 \; n 2 = a 3 b x + a x b 3 ; n z = a x b 2 + a 2 b x ; then the following are the six equations of transformation of the second order for the surface U = 1, x 2 = x 2 a 2 + y 2 a 2 2 + z 2 a z 2 + 2y z a 2 a z + 2 z x a z a x + 2 x y a x a 2 , (for y' 2 , z' 2 , similar equations in b, c, respectively). y'z' = x 2 p x + fv 2 + z 2 p 3 + yzl x + zxl 2 + xyl 3 , (for z x, a similar equation in q and m), (for x y, a similar equation in r and n), . . . (C.) Those equations are made applicable to the transformation of strains by the following substitutions : for x 2 , y 2 , z 2 , 2yz, 2zx, 2xy, substitute a, (5, j, A, p, v; and to that of pressures, by the following : for o: 2 , f, z 2 , yz, zx, xy, substitute P 1? P 2 , P 3 , Q v Q 2 , Q 3 , and similar substitutions for the accented symbols. The following are the fifteen equations of transformation of the fourth order: a/ 4 = re* a x * + y* a 2 * + z * a 3 * + 0>y 2 z 2 « 2 2 a 3 2 + Qz 2 x 2 a 2 a 2 + 6 x 2 y 2 a x 2 a* + 12 x 2 y z a x 2 a 2 a 3 + 12 x y 2 z a x a 2 2 a 3 + 12 xy z 2 a x a 2 a z 2 + 4 x 3 y a x 3 a 2 + 4 x 3 z a x 3 a 3 + 4 y 3 z a 2 3 a z + Ay 3 x a 2 3 a x + 4 c 3 x a z 3 a l + 4 s 3 y a z 3 a 2 ; (for y ri , z ri , similar equations in b, c, respectively). y' 2 z' 2 = a*p* + y'P-f + s> 3 2 + y 2 # (h 2 + 2p 2 p z ) + z 2 x 2 {l 2 + 2p sPx ) + * 2 y 2 (h 2 + 2 PiPi) + 2«?yz{lhh + hh) + %xy 2 z{2hk + h l d + 2 xyz 2 (p z l 3 + l x l 2 ) + 2x 3 yp x l z + 2 x 3 zp x L + 2y 3 zp 2 l x + 2y 3 xp 2 l 3 + 2z 3 xp z l 2 + 2z 3 yp z ( x ; (for z' 2 x' 2 , a similar equation in q and m) ; (for x' 2 y' 2 , a similar equation in r and n) ; LAWS OF ELASTICITY. 117 x 1 y z' = x* a 2 p x + y* a 2 p 2 + s 4 a 3 2 p 3 + z 2 x 2 {a 3 2 Pi + a \Vz + 2 a 3 a x ? 2 ) + x 2 y 2 {a 2 p 2 + a 2 p x + 2a t a 2 l 3 ) + x 2 y z (a x 2 l x + 2 a 2 a 3 p t + 2 a s a x l 3 + 2 a x a. 2 I.) + xy 2 z (a.? 1. 2 + 2a 3 a x p 2 + 2 a x a. 2 l L + 2 a 2 a 3 l 3 ) + xy z 2 (a 3 2 l 3 + 2 a x a. 2 p 3 + 2 a 2 a 3 l 2 + 2 a 3 a x l x ) + x 3 y(a l 2 l 3 4- 2a 1 a 2 p l ) + x 3 z(a 1 2 l 2 + 2a 1 a 3 p 1 ) 4- 2/ 3 z (a. 2 2 l x + 2 a 2 a 3 p 2 ) + y 3 x (a. 2 2 l 3 + 2 a 2 a x a. 2 ) + z 3 x (a 2 l 2 + 2a 3 a x p 3 ) + & y (a. 2 l x + 2 a 3 a. 2 p s ) ; (for x y' 2 z', a similar equation in b, q, m); (for x y' z' 2 , a similar equation in c, r, n) ; Ay' - x^a^r x + y i a 2 2 r 2 + z i a 3 2 r 3 + 3 y 2 z 2 a 2 a 3 ?i x + 3 z 2 x 2 a 3 a x n + 3 x 2 y 2 a x a 2 n 3 + 3 x 2 y z (a x 2 n x + 2 a. 2 a 3 r x ) + 3xy 2 z (a 2 n 2 + 2 a 3 a x r 2 ) + 3 x y z 2 (a 3 2 n 3 + 2 a t a 2 r 3 ) + x 3 y (a x 2 n 3 + 2 a x a. 2 r x ) + x 3 z (a-, 2 n 2 + 2 a 3 a x r y ) + y 3 z (a 2 2 n x + 2a 2 a 3 r 2 ) + y 3 x (cc 2 2 n 3 + 2a x a 2 r 2 ) + z 3 x (a 3 2 n 2 + 2 a 3 a x r 3 ) + z 3 y (« 3 2 n x + 2a 2 a 3 r 3 ) ; (for x' 3 z', a similar equation in a, q, and m) ; (for y' 3 z', a similar equation in b, p, and I) ; (for y' 3 x', a similar equation in b, r, and n); (for z' 3 x', a similar equation in c, q, and m) ; (for z' 3 y', a similar equation in c, p, and I); . (D.) The above equations are made applicable to the transformation of the ■coefficients of elasticity arising from the mutual actions of centres of force only, by the following substitutions: for % A , f, z 4 , y 2 z 2 , z 2 x\ x 2 y 2 , substitute (a 2 ), (/3 2 ), ( 7 2 ), (j3 7 ) = (X 2 ), (y a) = (ft), (a/3) = (v 2 ); for x 2 yz, xy 2 z, xyz 2 , substitute (a X) = (// v), (/3 jui.) = (v X), (7 v) = (\fi); 118 LAWS OF ELASTICITY. for a 3 y, x 3 z, y 3 z, f x, z 3 x, z 3 y, substitute (a v), (aim), (/3 A), (/3 v), (y ft), (7 A); and similar substitutions for the accented symbols. Should the substance under consideration be endowed with a portion of fluid elasticity in addition to that which arises from the mutual action of centres of force, the coefficient of that fluid elasticity J must be sub- tracted from the coefficients into which it enters, viz. — (a-), (/3 2 ), (r), (/3 7) = (X 2 ) + J > (7 «) = 0" e ) + J > (°/ 3 ) = O 2 ) + J > before effecting the transformation. The results of the transformation for those six coefficients, being in- creased by the same, quantity J which was previously subtracted, will give their entire values for the new axes. If the original axes of co-ordinates are those of elasticity, each of the fifteen equations of transformation is reduced to its first six terms, in which the following substitutions are to be made for the unaccented symbols : for x*, y 4 , z 4 , fz\ z 2 x 2 , x 2 f y substitute Aj- J, A 2 -J, A 3 -J, Bj-J^C^ B 2 -J=C 2 , B 3 -J = C 3 . AXES OF ELASTICITY AND CRYSTALLINE FORMS. 119 VI.—ON AXES OF ELASTICITY AND CRYSTALLINE FORMS.* Section 1. — General Definition of Axes of Elasticity. As originally understood, the term " axes of elasticity " was applied to the intersections of three orthogonal planes at a given point of an elastic medium, with respect to each of which planes the molecular actions causing elasticity were conceived to be symmetrical. If the elasticity of solids arose either wholly from the mutual attractions and repulsions of centres of force, such attractions and repulsions being functions of the mutual distances of those centres, or partly from such mutual actions and partly from an elasticity like that of a fluid, resisting change of volume only, it is easy to prove that there would be three such orthogonal planes of symmetry of molecular action in every homogeneous solid. But there is now no doubt that the elastic forces in solid bodies are not such as can be analysed into fluid elasticity and mutual attractions between centres simply; and though there are, as will presently be shown, orthogonal planes of symmetry for certain kinds of elastic forces, those planes are not necessarily the same for all kinds of elastic forces in a given solid. The term " axes of elasticity," therefore, may now be taken in a more extended sense, to signify all directions with respect to which certain hinds of clastic forces are symmetrical ; or speaking algebraically, directions for which certain functions of the coefficients of elasticity are null or infinite. The theory of axes and coefficients of elasticity is specially connected with that branch of the calculus of forms which relates to linear trans- formations, and which has recently been so greatly advanced by the researches of Mr. Sylvester, Mr. Cayley, and Mr. Boole. In such applications of that calculus as occur in this paper, the nomenclature of Mr. Sylvester is followed ; f and by the adoption of the " Umbral * Read before the Royal Society of London, on June 21, 1855. t See Cambridge and Dublin Mathematical Journal, Vol. VII. ; and Philosophical Transactions, 1853. 120 AXES OF ELASTICITY AND CRYSTALLINE FORMS. Notation" of that author immense advantages are gained in conciseness and simplicity. * Section 2. — Strains, Stresses, Potential Energy, and Coefficients of Elasticity. In this paper, the word " Strain " will be used to denote the change of volume and figure constituting the deviation of a molecule of a solid from that condition which it preserves when free from the action of external forces ; and the word " Stress " will be used to denote the force, or combination of forces, which such a molecule exerts in tending to recover its free condition, and which, for a state of equilibrium, is equal and opposite to the combination of external forces applied to it. In framing a nomenclature for quantities connected with the theory of elasticity, OXtyt-Q i s adopted to denote strain, and raaig to denote stress. It is well known that the condition of strain at any given point in the interior of a molecule may be completely expressed by means of the following six elementary strains, in which £, )j, Z, are the components of the molecular displacement parallel to three rectangular axes, x, y, :. PI ,. dl£ dr\ r. dZ, Elongations, . — = a; -j-= ft; -^ = y; dj = dx a; dr\ dy~ ft; dZ> dz dy^ dz = A; dl dz + T dx Distortions, . ~ + j^ = A; "p - ^ = ft; -p + — = v . dy dz dz dx dx ay It is also well known that the condition of stress at a given point may be completely expressed, relatively to the three rectangular co-ordinate planes, by means of six elementary stresses, viz. — Normal Pressures, . . P 1? P 2 , P 3 , Tangential Pressures, . . Q lt Q 2 , Q 3 ; these quantities being estimated in units of force per unit of surface. Let each elementary stress be integrated with respect to the elementary strain which it tends directly to diminish, from the actual amount of that strain, to the condition of freedom; the sum of the integrals is the potential energy of elasticity of the molecule dxdydz, expressed in units of work per unit of volume ; viz. — + j > Q 1 d\ + jq. 2 d^+ j°Q 3 dv. . . (1.) * See the Note at the end of the paper. AXES OF ELASTICITY AND CRYSTALLINE FORMS. 121 The condition that the function U shall have the same value, in what order soever the variations of the different elementary strains take place, amounts to supposing that no transformation of energy of the kind well distinguished by Professor Thomson as frictlonal or irreversible, takes place during such variations ; in other words, that the substance is perfectly elastic. Each of the elementary stresses being sensibly a linear function of the six elementary strains, the potential energy of elasticity is, as Mr. Green first showed, a function of those strains of the second degree, having twenty-one constant coefficients, which are the coefficients of elasticity of the body, and will in this paper be called the Tasinomic Coefficients; that is to say, adopting Mr. Green's notation for such coefficients, — ' U = („ 2 )| 2 + (/3 2 )f + ( 7 2 )| 2 + (X 2 )^ + (m 2 )| 3 + if)t + (/37)/3y+( 7 «)7« + («£W3 + (fX v) /H V + (v X) V X + (XjUi) XyU + (a \)a X + (/3/x) /3/x + (7 v) y v + (/3X) /3X + ( yf x) yn + (a v) a v + (y\)y\+(a[i)an+(Pv)Pv . . (2.) From a theorem of Mr. Sylvester it follows, that every such function as U is reducible by linear transformations to the sum of six positive squares, each multiplied by a coefficient. The nature and meaning of this reduction have been discussed by Professor William Thomson. The following classification of the tasinomic coefficients will be used in the sequel : — Designation of Coefficients. Elasticities. Symbols. C Euthytatic, Direct or Longitudinal, (a 2 ) (8 2 ) (7 2 ) Orthotatic -l Platytatic, Lateral, . . (/3 7) (7 a) (a /3) I Goniotatic, Rigidities, . . . (X 2 ) (;u 2 ) (v 2 ) Plagiotatic, . . . Unsymmetrical, . . (jut v), &c, &c. The twenty-one equations of transformation by which the values of these coefficients, being known for any one set of orthogonal axes, are found for any other, are founded on the following principles. It is well known, that for rectangular transformations, the operations d d d dz' dy' dz 122 AXES OF ELASTICITY AND CRYSTALLINE FORMS, are respectively covariant with •'•, y, ■"-> from which it is easily deduced, that because the displacements are respectively covariant with &, y, z, therefore the elementary strains, a, /3, y, A, fx, v the operations, d d d d d d^ da' dp' d^' IX' ~d~ii' dv and the strains P P P ^ O ° 2 ■*• 1' 2' 3' " 'I' ~ v *-2> »-o' must be respectively covariant with the squares and products, /-, f, #, 2yz, 2zx, 2xy. Section 3. — Thlipsimetric and Tasimetric Surfaces and Invariants. Isotropic functions of the elementary strains and stresses, which may be called respectively Thlipsimetric and Tasimetric Invariants, are easily deduced from the principle, that the strains may be represented by the coefficients of the following Thlipsimetric Surface, as 5 + fitf + 7^ + \yz + ftzx + vxy=l, . (3.) and the stresses by the coefficients of the Tasimetric Surface, T> l3 ? + J> 2 y* + Y 3 z* + 2Cl 1 yz+2Q 2 zx+2%xy=l. (4.) These surfaces, and others deduced from them, have been fully discussed by M. Cauchy and M. Lame. The invariants in question may all be deduced from the following pair of contrasredient matrices : — AXES OF ELASTICITY AND CRYSTALLINE FORMS. For Stresses. Pi Q 3 Q-2 123 For Strains. V JUL (5.) <! ; fi . | yU. X 9, 9 *y Q 3 P 2 Qj ^ (5 a.) Q* Qi p 3 J The following are the primitive thlipsimetric invariants, from which an indefinite number of others may be deduced by involution, multiplication, addition, and subtraction : — a + /3 + 7 = 9 1 (the cubic dilatation) ; /3 7 + 7 « + a/3-i(A 2 + /x 2 + i/ 2 ) = 2 ; (6.) The potential energy U is what Mr. Sylvester calls a " Universal Mixed Concomitant," its value being U = - | (P l0 + P 2 /3 + P 37 + Q X A + Q,fx + Q 3 v). (7.) Section 4. — Tasinomic Functions, Surfaces, and Umbrae. If, in any isotropic function of the co-ordinates and the elementary strains, there be substituted for each square or product of elementary strains that tasinomic coefficient which is covariant with it, the result will be an isotropic function of the co-ordinates and tasinomic coefficients, called a Tasinomic Function. The following Table of Covariants is readily deduced from the prin- ciples stated at the end of Sect. 2 : — r Squares of ) „ !o- J Strains, } a ' P~> ?' A ' Co variant 1 Tasinomic "i Coefficients J '-(« 2 )> m, ir), 4 (A 2 ), 4( M 2 ), 4(v 2 ); Co- variant Products ") J Tasinomic ") y£y, 7 a, a/3, M v, vA, A//, (/3 y), ( 7 a), (a /3), 4 Gu v), 4 („ A), 4 (A M ), MS-) . Coefficients j a A, a /a, a v, /3 A, /3 /*, /3 v, 7 A, 7 /z, y v, 2(aA),2(a / x),2(av),2(/3A),2(/3 A ; 2(/3v),2( 7 A),2( 7it< ),2( 7 r).J 124 AXES OF ELASTICITY AND CRYSTALLINE FORMS. Each tasinomic function being equated to a constant, forms the equation of a Tasinomic Surface; and on the geometrical properties of such surfaces depend many of the laws of coefficients and axes of elasticity. A convenient and expeditious mode of forming tasinomic functions is obtained by the aid of an Umbral Notation analogous to that intro- duced by Mr. Sylvester in the calculus of forms. Let each tasinomic coefficient be regarded as compounded of two Tasinomic Umbrae, those umbra? being expressed by the following notation : (a), ((3), (y), (X), W, (v); then the following equation, deduced from that of the thlipsimetric surface (3), by substituting uinbraj for elementary strains according to the following Table of Covariance, Strains, ... a, /3, y, X, f.t, v, Umbrae, . . . (a), (/?), ( 7 ), 2 (X;, 2 fa), 2 („), is the equation of the Tasinomic Umbral Ellipsoid, from which, by elimi- nation, multiplication, involution, addition, subtraction, and differentia- tion, various tasinomic functions may be deduced, (a) a 2 + (/3) f + (y) ,: 2 + 2 (X) yz + 2 fa) zx + 2 ( v ) xy = (0) = 1. (8a.) Section 5. — Tasinomic Invariants and Spheres. Tasinomic invariants are constant isotropic functions of the tasinomic coefficients, which are deduced, either by substitution from thlipsimetric invariants, or directly from the Umbral Matrix, («) W fa) \ (v) 08) (X) I . (9.) G») (X) (y) ) The following invariant is umbral of the first order : — (&- + i? + £) • W = W + W> + M = W> (9«0 Invariants of the second order in umbra; are real quantities of the first order, viz. — (a 2 ) + (/3 2 ) + (y 2 ) + 2 (/3 y) + 2 (y a) + 2 (a /3) = (6J 2 (the Ctt&ic <««%) (/3y)+(ya) + (a/3)- (X 2 ) - ( M 2 ) - (* 2 ) =(0 2 ) (a 2 ) + (/3 2 ) + (y 2 ) +2(X 2 ) + 2 fa 2 ) +2(v 2 ) =(0 1 ) 2 ~2(0 2 ). . (10). AXES OF ELASTICITY AND CRYSTALLINE FORMS. 125 The equation of a Tasinomic Sphere is formed by multiplying a tasinomic invariant by a? + f + *\ or any power of that quantity, and equating the result to a constant. Section 6. — Of Two Tasinomic Ellipsoids, and their Axes, Orthotatic and Heterotatic. The equations of two ellipsoids with tasinomic coefficients are derived from that of the umbral ellipsoid (8a.), in one case by multiplying each term by the umbral invariant (0), and in the other by substituting for each umbra in the function (fa), the contravariant component of the Inverse to the umbral matrix (9). The results are as follows : — Orthotatic Ellipsoid. (e i )x(^ = {(a 2 ) + (a^) + (y«)}.r 2 +{(^) + (^) + (y«)}!/ 2 + {(y 2 )+(y«)+(£y)K + 2{(aX) + (/5X) + (yX)}^~+2{(a M ) + (/3 / >c) + (y / ,)}.r C + 2{(«v) + (/3v) + (yv)}^=l. . . (11). Heterotatic Ellipsoid. {(^)-(A 2 )!r+{(ya)-W}!/ 2 +[M)-(v 2 )}^ + 2{^v)-(a\y j yz+2{(v\)-^^}zx+2{(\^-(yv)}xy=l. (12.) The three Orthotatic Axes are three rectangular directions for which the following sums of plagiotatic coefficients are null : — (aA) + (/3X) + (yX) = 0; (apt) + (/3 *) + (y /») = 0; (a v) + (/3 v) + (y v) = 0. } (13.) It was proved by Mr. Haughton, in a paper published in the Trans- actions of the Royal Irish Academy, Vol. III., Part 2, that there are three rectangular directions having this property in a solid whose elasticity arises solely from the mutual actions of physical points, and which has but fifteen independent coefficients of elasticity. The present investi- gation shows that there are three such axes at each point of every solid, 126 AXES OF ELASTICITY AXD CRYSTALLINE FORMS. independently of call hypothesis. The physical meaning of this result is expressed by the following Theorem as to Orthotatic Axes. At each point of an clastic solid there is one position in which a cubical molecule may be cut out, such that a uniform dilatation or condensation of that molecule by equal elongations or equal compressions of its three dimensions, shall produce no tangential stress on the faces of the molecule. The properties of the Hetcrotatic Axes are expressed by the following equations : — (/*v)-(aX) = 0; (i/X)-(/3ju) = 0; (X^)-(yv) = 0; (14.) or by the following Theorem as to Heterotatic Axes. At each point of an elastic solid there is one fiosition in which a cubical molecule may be cut out, such that if there he a distortion of that molecule round x (x being any one of its three axes), and an equal distoiiion round y (y being either of its other two axes), the normal stress on the faces normal to x arising from the distortion round x shall be equal to the tangential stress round z arising from the distortion round y. The six coefficients of the heterotatic ellipsoid may be called the Heterotatic Differences. For a solid whose elasticity is wholly due to the mutual attractions and repulsions of physical points, each of those differences is necessarily null ; therefore, they represent a part of the elasticity which is necessarily irreducible to such attractions and repul- sions. There is reason to believe that part at least of the elasticity of every substance is of this kind. If this part of the elasticity of a solid be, as suggested in a series of papers in the Cambridge and Dublin Mathematical Journal, for 1851-52, a species of fluid elasticity, resisting change of volume only, the solid may be said to be heterotatically isotropic. The equations (14) will be fulfilled for all directions of axes, and also the following equations : — (/3y)-(X 2 ) = (ya)-( M 2 ) = (a / 8)-(v 2 )j . . (15.) that is to say, the excess of the platytatic above the goniotatic coefficient will be the same in every plane. In a substance orthotatically isotropic, the equations (13) are fulfilled for all directions, and also the following : — (a 2 ) + (a/3) + (y a) = (/3 2 ) + (/3y) + (a/3) = (y 2 ) + (y a) + (/3y), (1 G.) AXES OF ELASTICITY AND CRYSTALLINE FORMS. 127 that is to say, a uniform compression in all directions produces a uniform normal stress in all directions, and no tangential stress. The equations (16) may be reduced to the following form : — (a 2 ) - (/3 y) = (/3 2 ) - (y «) = (y 2 ) - (a /3). . (17.) In a substance which is at once orthotatkaUy and hetcrotatically isotropic, there may still be eleven independent quantities amongst the tasinomic coefficients, viz. — (18.) Three Euthytatic Coefficients, (a 2 ), (/3 2 ), (y 2 ), The Isotropic excess, . . (a 2 ) — (/3 y), The Isotropic excess, . . (/? y) — (X 2 ), Six Plagiotatic Coefficients, (/3X), (yX), (y/x), (a/ui), (av), (/3v), j Such a substance may, therefore, be far from being completely isotropic with respect to elasticity. Section 7. — Biquadratic Tasinomic Surface; Homotatic Coefficients; Euthytatic Axes Defined. If the equation (8a.) of the umbral ellipsoid be squared, there is obtained the following equation of a Biquadratic Tasinomic Surface. (#=(« 2 K + (/3 2 )^ + (y 2 ).: ! + 2{(/3y) + 2(X 2 )}^ 2 + 2{(ya) + 2(^ 2 )}, 2 ^ + 2{(«/3) + 2(v 2 )}^ 2 + 4{2( / xv) + {a\))xhjz + 4{2(„A) + (fti)}atf» + 4{2(X M ) + (yv)}x^ + l(PX)fz + 4( y X)//: 3 + 4( 7M )r,,: + 4( a/ /)^ 3 + i(av):>hj + 4((3v)xf =1 (19) The fifteen coefficients of this surface (which will be called the Homo- tatic Coefficients) are covariant respectively with the fifteen biquadratic powers and products of the co-ordinates, with proper numerical factors. It is obvious, that when the fifteen homotatic coefficients, and the six heterotatic differences, are known for any set of orthogonal axes, the twenty-one tasinomic coefficients are completely determined. Mr. Haughton, in the paper previously referred to, discovered the biquadratic surface for a solid constituted of centres of force. It is here shown to exist for all solids, independently of hypotheses. Those diameters of the biquadratic surface which are normal to that surface, are axes of maximum and minimum direct elasticity, and have also this property, that a direct elongation along one of them produces, on a plane perpendicular to it, a normal stress, and no tangential stress; so that they may be called Euthytatic Axes. Though such axes sometimes 128 AXES OF ELASTICITY AND CRYSTALLINE FORMS. form orthogonal systems, their complete investigation requires the use of oblique co-ordinates, and is therefore deferred till after the eighteenth section of this paper, which relates to such co-ordinates. Section 8. — Orthogonal Axes of the Biquadratic Surface. Metatatic Axes, Orthogonal and Diagonal. By rectangular linear transformations, it is always possible to make three of the terms Avith odd exponents, or three functions of such terms, vanish from the equation of the biquadratic surface. Thus are ascertained sets of orthogonal axes having special properties. To exemplify this, let the rectangular transformation be such as to make the following functions vanish : — (G8X)-(yX)}(^-tf>; {(yri-MK*-^«i {(«v)-(Pv)}(?-f)xy. A cubical molecule having its faces normal to the axes fulfilling this condition has the following property : — if there be a linear elongation along y, and an equal linear compression along z (or vice versti), no tangential stress ivill result round x on planes normal to y and z • and similarly of other pairs of axes. This set of axes may be called the Orthogonal or Principal Metatatic Axes, and their planes, Metatatic Planes. Let the suffix 1 designate co-ordinates and coefficients referred to these axes. Let Oy, 0: be any new pair of orthogonal axes in the plane y x z v Then since (|3 X) — (7 X) is co variant with (y 2 — f?)yz, it follows that (/3 X) - (y X) = {2 (0 y\ + 4 (X 2 ) x - C/3 8 ), - (y\) • ^~~ (20.) (where w = < y x y), a quantity which is = for all values of to which are multiples of 45°* There are of course similar equations for the other metatatic planes. Hence it appears that in each of the three Metatatic Planes there is a pair of Diagonal Metatatic Axes, bisecting the right angles formed by the Principal Metatatic Axes. Each pair of diagonal axes is metatatic for that plane only in which it is situated. Thus there are in all nine metatatic axes, three orthogonal axes, and three pairs of diagonal axes. The diagonal axes are normal to the faces of a regular rhombic dodecahedron. Let Oy, Oz be a pair of rectangular axes in any plane ivhatsoever ; Oy', Oz' any other pair of rectangular axes in the same plane ; and let <yOy = u/ ; AXES OF ELASTICITY AND CRYSTALLINE FORMS. 129 then (/3 X)' - (y\y = [2 O y) + 4 (X 2 ) - m - (y 2 )} ^^ + {(/3A)-(y\)}cos4 w ', . . (21.) a quantity which is null for eight values of &/, differing from each other by multiples of 45°. Hence, in each plane in an elastic solid, there is a system of two pairs of axes meiatatic for that plane, and forming with each other eight equal angles of 45°. In equation (21), make to' = — O) (/3\)'-(yX)' = (/3X) 1 -(yX) 1 = 0; then from equations (20) and (21), it is easily seen that 2 (By) + 4(X 2 )-OT-(y 2 ) = { 2 (£ 7)i + ^ (A 2 ) x - (/3 2 ) x - (y 2 X} • cos 4 «. . (22.) The trigonometrical factor cos 4 w is + 1 for all values of to which are even multiples of 45°, — 1 for all odd multiples of 45°, and = for all odd multiples of 22i°. Hence, in every plane in an elastic solid, the quantity (22), which may be called the Metastatic Difference, is a maximum for one of the two pairs of metatatic axes, a minimum of equal amount and negative sign for the other, and null for the eight intermediate directions. Section 9.— Of Metatatic Isotropy. A solid is Metatatically Isotropic when, if a cubical molecule, cut out in any position whatsoever, undergoes simultaneously an elongation along one axis, and an equal and opposite linear compression along another axis, no tangential stress will result on the faces of that molecule. For such a substance, the metatatic differences must be null for all sets of axes, viz. : — 2(/3y) + 4(X 2 )-(/3 2 )-(r) = C>n 2(ya) + 40u 2 )-(y 2 )-(a 2 ) = O; L . (23.) 2(a / 3) + 4(„ 2 )-(«s)-(/3 2 ) := 0. J In a paper in the Cambridge and Dublin Mathematical Journal, Vol. VI., this theorem was alleged of all homogeneous solids, it having been, in fact, tacitly taken for granted, that homogeneity involves metatatic isotropy, as above denned. I 130 AXES OF ELASTICITY AND CRYSTALLINE FORMS. Section 10. — Of Orthotatic Symmetry. If it be taken for granted that symmetrical action with respect to a certain set of axes, between the parts of a body under one kind of strain, involves symmetrical action with respect to the same axes under all kinds of strains, then one and the same set of orthogonal axes will be at once orthotatic, heterotatic, metatatic, and euthytatic, and for them the whole twelve plagiotatic coefficients will vanish at once, and the independent tasinomic coefficients be reduced to the nine orthotatic coefficients enumerated in Sect. 2. As long as the rigidity of solid bodies was ascribed wholly to mutual attractions and repulsions between centres of force, it is difficult to sec how, with respect to homogeneous substances, the above assumption could be avoided. It is probable that there exist substances for which it is true. Such substances may be said to be Orthotatically Symmetrical. Orthotatic symmetry requires that the equation (1 9) of the biquadratic surface should be reducible by rectangular transformations to its first six terms, and that the axes so found should also be those of the heterotatic ellipsoid. The conditions which must be fulfilled in order that a biquadratic function of three variables may be reducible by rectangular transformations to its first six terms, have been investigated by Mr. Boole.*' Section 11. — Of Cybotatic Symmetry. Let a substance be conceived which is not only orthotatically symme- trical, but for which the three kinds of orthotatic coefficients are equal for the three orthotatic axes, viz. — (a 2 ) = (/3 2 ) = (y 2 ); (/3y) = (y«) = (aft; (X 2 ) = ( M 2 ) = (v 2 ). (24.) Then, for such a substance the metatatic difference may be expressed by 2(/3y) + 4(\ 2 )-2(a 2 ); . - • (25.) and if the body be not metatatically isotropic, this difference will have equal maxima or minima for the three orthogonal axes, normal to the faces of a cube, and, conversely, equal minima or maxima for the six diagonal axes, normal to the faces of a regular rhombic dodecahedron. Symmetry of this kind may be called Cybotatic, from its analogy to that of crystals of the tessular system. * Cambridge and Dublin Mathematical Journal, Vol. VI. AXES OF ELASTICITY AND CRYSTALLINE FORMS. 131 Section 12. — Of Pantatic Isotropy. When a body fulfils the conditions of cybotatic symmetry, and at the same time those of metatatic isotropy, it is completely isotropic with respect to elasticity, or pantatically isotropic. It has but three tasinomic coefficients — viz., the euthytatic, platytatic, and goniotatic coefficients, which are equal for all sets of axes, and are connected by the following equation, expressing the condition of metatatic isotropy : (a 2 ) = (/3y) + 2(A 2 ). . . . (26.) The properties of such bodies have been fully investigated by various authors. Section 13. — Of Thlipsinomic Coefficients. If the six elementary strains, a, &c, at a given point in an elastic solid, be expressed as linear functions of the six elementary stresses, P p &c, these expressions will contain twenty-one coefficients of compressibility, extensibility, and pliability, which are the second differential coefficients of the potential energy of elasticity with respect to the six elementary stresses ; that energy being represented as follows : — p2 p2 p2 O 2 O 2 o 2 U = (« 2 )f +(5 2 )^ + (c 2 )-f + (Z 2 )-f + K)f+(^)f + (5 C )P 2 P 3 + (ca)P 3 P 1 +(a&)P 1 P 2 +( W w)Q 2 Q3+(wOQ3Qi + (MQiQ 2 -\-{(al) P 1+ (5/)P 2 +(d) P 3 }Q 1 + {(am)P 1 + (6OT)P 2 +(cm)P 3 }Q 2 + (HP 1 + (k)P 2 +(«)P 3 }Q 3 . . . . (27.) The twenty-one coefficients in the above equation may be comprehended under the general term Thlijisinomic, and classified as follows : — Designations of Coefficients. Properties expressed by them. Symbols. rEuthythliptic, Longitudinal Extensibilities, (a 2 ), (JP), (c 2 ) Orthothliptic<j Platythliptic, Lateral Extensibilities, . . (be), (ca), (ab), [Goniothliptic, Pliabilities, (Z 2 ), (m 2 ), (re 2 ), Plagiothliptic, .... Unsymmetrical Pliabilities, (mn), &c, &c. 132 AXES OF ELASTICITY AND CRYSTALLINE FORMS. Section 14. — Of Thlipsinomic Transformations, Umbrae, Surfaces, and Invariants. » The equations of transformation of the thlipsinomic coefficients are easily deduced from the principle, that the operations d d d d d d dP^ d¥ 2 ' dP 3 ' dCfc JQ 2 ' dQ 3 are respectively covariant with V *2> , 3> * ^-1' " ^-2> ~" ^v3» and these with A f, r, 2yz, 2zz, 2xy. We may regard the thlipsinomic coefficients, like the tasinomic coeffi- cients, as binary compounds of the following six Umbrcr, (a), (b), (c), (I), (m), (n), which being respectively substituted for "l> * 2» 3' * ^1' * ^-2> " tw3» in the equation of the tasimetric surface (4), produce the following- equation of the Umbral Thlipsinomic Ellipsoid, (a) :>? + (b) f + (c) r + (I) yz + (m) zx + {n)xy = 1, (28.) from which, by involution, multiplication, and other operations exactly analogous to those performed on the umbral tasinomic ellipsoid, there may be deduced the equations of Thlipsinomic Surfaces exactly correspond- ing to the tasinomic surfaces already described ; while, from the umbral matrix, (a) }(») J(m)>| too (») *© y • • • (29.) may be formed Thlipsinomic Invariants corresponding to the tasinomic invariants. Hence it appears, that every function of the tasinomic coefficients is converted into a function of the thlipsinomic coefficients with analogous AXES OF ELASTICITY AND CRYSTALLINE FORMS. 133 properties, by the substitution of thlipsinomic for tasinomic umbrae according to the following table : — Tasinomic Umbra?, Thlipsinomic Umbra?, . («), 08). (y), (X), W, (v), («), (»). (0» K0ki(»)»iW- Amongst the thlipsinomic invariants may be distinguished the Cm6/c Compressibility, which is formed by squaring the umbral invariant (a) + (^) + (c), and has the following value : O 2 ) + (J 2 ) + (r) + 2 (6 e) + 2 (c a) + 2 (a J). Section 15. — Thlipsinomic and Tasinomic Contragredient Systems. Let the following square matrices be formed with the tasinomic and thlipsinomic coefficients respectively : — (a 2 ) (a/5) (ya) (a X) (ap) (av) (a/3) OT (/3y) G8X) (/5m) (/5.) (ya) (/3y) (y 2 ) (yX) (y/t) (yv) (aX) (/3X) (yX) (X 2 ) (\ M ) (vX) (a//) (/3/x) (y/*) (X/x) (m 2 ) 0*v) (av) Q3v) (yv) (v\) (fiv) (v 2 ) , > (a 2 ) (aft) (ca) (a2) (ab) (b 2 ) (be) (bl) (ca) (be) (c 2 ) (cl) (a I) (bl) (cl) (I 2 ) (a m) (a n) (b m) (b n) (c m) (c n) (Im) (nl) (30.) (31.) (a m) (b m) (c m) (I m) (m 2 ) (m n) (an) (bn) (en) (nl) (mn) (n 2 ) Then will these matrices be mutually inverse, the two systems of coefficients arrayed in them, with their respective systems of functions mutually contragredient, and each coefficient or function belonging to one system contravariant to the corresponding coefficient or function belonging to the other system. The values of the coefficients in either of those matrices are expressed 134 AXES OF ELASTICITY AND CRYSTALLINE FORMS. in terms of those in the other matrix, in Mr. Sylvester's umbral notation, by twenty-one equations, of which the following are examples : — (« 2 ) = (ab) = 03), (y), (X), (a*), (v) 0), (y), (a), 0*), to (/3), (y), (X), (/*), W (a), (y), (X), GO, to («), (/3), (y), (X), W to («), O), (y)> ^)' 0*). to («), (/3), (y), (A), W to («). (/3), (y), (X), Gk), to V. (32.) Section 1G. — Of Thlipsinomic Axes. If, under given conditions, any symmetrical system or function of the constituents of one of the above matrices be null, then under the same conditions will the contravariant system or function of the constituents of the inverse matrix be null or infinite. Therefore, Systems of Thlipsinomic Axes coincide with the corresponding systems of Tasinomic Axes. Section 17. — Platythliptic Coefficients are Negative. It may be observed, as a matter of fact, that in consequence of the largeness of the euthytatic coefficients (a 2 ), (/3 2 ), (y 2 ), as compared with the other tasinomic coefficients, the platythliptic coefficients (be), (ca), (a b), are generally, if not always, negative. To illustrate this, the case of pantatic isotropy may be taken, for which the two matrices have the following forms : — (a 2 ) (/3y) (/3y) 0y) (« 2 ) (Pv) o o o 03 y) (/3y) (a 2 ) (A 2 ) (A 2 ) (A 2 ) (a 2 ) (be) {be) 0' (be) (a 2 ) (6c) (be) (be) (a 2 ) (P) (I 2 ) (Z 2 ) J y (33.) from which it is easily seen that the sole platythliptic coefficient has the following value : -(fly) (be) (33A) (a 2 ) 2 + (a 2 )(/3y)-2(/3y) 2, " The denominator of this fraction is always positive so long as (a 2 ) AXES OF ELASTICITY AND CRYSTALLINE FORMS. 135 exceeds (/3y); a condition invariably fulfilled by solid bodies, and, in fact, necessary to their existence. Section 18. — Of Oblique Co-ordinates and Contra-ordinates. As there are, in the relations between two systems of oblique co-ordinates, or between a system of oblique co-ordinates and a system of rectangular co-ordinates, six independent constants of transformation, it is possible, by referring the equation of the biquadratic surface (19) to oblique co-ordinates, to make the six terms vanish which contain the cubes of the co-ordinates. The conception of the physical meaning of such a transformation is much facilitated by the employment of a system of three auxiliary variables, which will be designated as Contraordinates. The relations between co-ordinates and contraordinates are as follows : — Through an origin let any three axes pass, right or oblique. Let E be any point, and let 0~R = r. Through It draw three planes, parallel respectively to the three co-ordinate planes, and intersecting the axes respectively in the points X, Y, Z. Also, on OE, as a diameter, describe a sphere, intersecting the axes respectively in U, V, W. Then will OX = <c, OY = y, OZ = z, be the co-ordinates of E, as usual, and OU = w, OV = v, OW = «e, its contra-ordinates, being, in fact, the projections of E on the three axes. For rectangular axes, co-ordinates and contra-ordinates are identical. Co-ordinates and contra-ordinates are connected by the following equation : — r 2 = u x + v y + we. . . . (34.) In the language of Mr. Sylvester, a system of co-ordinates, and the concomitant system of contra-ordinates, are mutually Contragredient ; and the square of the radius vector is their universal mixed concomitant. Let the cosines of the angles made by the axes with each other be denoted as follows: cos2/Os = q; coszOx = c i ; cos xOy = c 3 ; 136 AXES OF ELASTICITY AND CRYSTALLINE FORMS. then the contra-ordinates of a given point are the following functions of the co-ordinates: Also let u = x + c s y + c 2 z } v = c z x + y -f c 1 z \ I to = c 2 x + c x y + z J = l-cf-4-ci+2c 1 c 2 c 8 = C; (35.) then the co-ordinates are the following linear functions of the contra- ordinates : — Also, x = h x u — k 3 v — k 2 w ; y = — k, A u + h 2 v — k x w ; z = — h 2 u — l\ v + h z w ; J r = x' -f y 2 + z- + 2 c x y z + 2c 2 zx + 2c % xy (3C) (37.) = h x u 2 + /; 2 v 2 + h z to 2 -ll^vw - 2k 2 ivu- 2 fc s m v. (37a) Differentiations with respect to the contra-ordinates are obviously covariant with the co-ordinates, and vice versa; that is to say, ,, ,. d d d d d d the operations . — -, — , — , — , — , — - dx cly dz du dv dw are respectively ) covariant with ) y ■ (38.) *■> y, By making substitutions according to the above law of covariance in the equations (34), (37), (37a), three equivalent symbols of operation are obtained, which, being applied to isotropic functions of the second degree, produce invariants of the first degree. AXES OF ELASTICITY AND CRYSTALLINE FORMS. 137 Section 19. — Of Molecular Displacements and Strains as referred to oblique axes. If the displacement of a particle from its free position be resolved into three components g, r\, Z,, parallel respectively to three oblique axes, x, y, z, those components are evidently covariant respectively with the co-ordinates x, y, z. It is now necessary to find a method of expressing the strain at any particle in an elastic solid by a system of six elementary strains, which shall be covariant respectively with the squares and doubled-products of these oblique co-ordinates. This condition is fulfilled by considering the elementary strains as being constituted by the variations of the components of the molecular displacement with respect to the distances of the strained particle from three planes passing through the origin, and normal respectively to the three axes; that is to say, with respect to the contra-ordinates of the particle, as expressed in the following equations: — Elongations, . a = ^— ; j3 — ^ ) Y — T~ ■> ° du } ' dv 3 * dw' n -tv* i- ^ d %, dr l d^.dK. _dri d% Quasi-Distortions, A = -; — h t~ ; ix = -3 — h 3- , v — 5 — (-7- dv dw die du du dv y (39.) The six elementary strains, as above defined, are obviously covariant with the squares and doubled-products of the co-ordinates, according to the following table : a, /3, y, X, /*, v, ") r . . (40.) x 2 , y\ z 2 , 2yz, 2zx, 2xy. ) Section 20. — Of Stresses, as referred to Oblique Axes. It is next required to express the stress at any particle of an elastic solid by means of a system of six elementary stresses, which shall be contragredient to the system of six elementary strains defined in the preceding section. This is accomplished in the following manner. It is known that the total stress at any point may be resolved into three normal stresses on the three principal planes of the tasimetric surface. Let the direction and sign of any one of those three principal stresses be represented by those of a line K, and its magnitude, as reduced to unity of area of the plane normal to that direction, by the square of that line. OR 2 = r 2 . 138 AXES OF ELASTICITY AND CRYSTALLINE FORMS. Let u, v, iv be the contraordinates of E, as referred to the oblique axes X, Y, Z. Then will the stresses on unity of area of planes normal to those axes, in the direction E, be represented respectively by u r, v r, to r. Let the Elementary Stresses be defined to be, the projections on the three axes of co-ordinates of the total stresses on unity of area of the three pairs of faces of a parallelepiped, normal to the three axes respectively : then, if we take S to denote the summation of three terms arising from the three principal stresses, the elementary stresses will be expressed as follows : — Normal stresses on the faces normal to x, y, z, P 1 = S.tt 2 ; P 2 = S.* 2 ; P 3 = S.ir; Oblique stresses on the faces normal to (41.) Iu the directions "V These expressions fulfil the condition of making the elementary stresses P lf P 2 , P 3 , Q lf Q 2 , Q 3 contravariant respectively to the elementary strains a, ft, y> \ /"> v > so that for oblique axes, as for rectangular axes, the potential energy of elasticity is represented by ♦ U = - 1 (P ia + Po/3 + P 3 y + Q X X + Q,fx + Q 3 v), the universal concomitant; and may be expressed either by a homo- geneous quadratic function of the six elementary strains (as iu equation 2) with twenty-one tasinomic coefficients, or by a homogeneous quadratic function of the six elementary stresses, as in equation (27), with twenty- one thlipsinomic coefficients, forming a system contragredient to that of the tasinomic coefficients. axes of elasticity and crystalline forms. 139 Section 21*. — Of Tasinomic and Thlipsinomic Umbrae for Oblique Axes. The tasinomic coefficients for oblique axes may be regarded as com- pounded of umbra? (a), 08), (y), (A), (/*), (v), contravariant respectively to the elementary strains «, (3, 7> I A, I ft, hv, and consequently covariant with the squares and products of the contra- ordinates v?, v 2 , w 2 , 1)10, wu, uv; and the thlipsinomic coefficients for oblique axes may be regarded as compounded of umbra? (a), (b), (c), (/), (m), («), contravariant respectively to the stresses P lf P 2 , P s , 2Q X , 2Q 2 , 2Q 3 , and consequently covariant with the squares and products of the co-ordinaks ar, ij , (. , *></~, &%j>, *jxy. Section 22. — Of the Biquadratic Surface, and of Principal euthytatic axes. For oblique as well as for rectangular axes of co-ordinates, the char- acteristic function of the biquadratic tasinomic surface is represented by equation (19); and the fifteen homotatic coefficients are covariant respectively with suitable multiples of the fifteen biquadratic powers and products of the contraordinates. If by linear transformations a system of three axes, oblique or rectan- gular, be found which reduces the characteristic function of the biquad- ratic surface to the canonical form, consisting of not more than nine terms, viz. — (0) 2 = (a 2 K+(W+(rK + 2{(/3y) + 2(\ 2 )}^ 2 + 2{(ya) + 2( y a 2 )}^ 2 +2{(a/3) + 2(v 2 )}a ; Y + 4{2( M v) + (aA)}ry: + 4{2(^) + (|3,i)}^ 2 2 + 4{2(\ M ) H-( 7 v)}^ 2 = l; .... (42.) 140 AXES OF ELASTICITY AND CRYSTALLINE FORMS. then, for that system of axes, the following six plagiotatic coefficients are null, (|3X) = 0; (yX) = 0; (yfi) = 0; (afx) = 0; (av) = 0; ((Sv) = 0; (43.) and each of those axes is Euthytatic, according to the definition in Sect. 7, that is to say, is a direction of maximum or minimum direct elasticity (absolute or relative), and also a direction in which a direct elongation or compression produces a simply normal stress. There are necessarily three euthytatic axes at least in every solid — viz., the three Principal Euthytatic Axes, as above described, which are normal to the faces of a hexahedron, right or oblique, as the case may be; but in special cases of symmetry there are additional or secondary euthytatic axes, of which examples will now be given. Section 23. — Of Rhombic and Hexagonal Symmetry. When a solid has three oblique principal euthytatic axes making equal angles with each other round an axis of symmetry, and having equal systems of homotatic coefficients corresponding to them, viz. — (« 2 ) = Q3 2 ) = (y 2 ); (|3y) + 2(^)=:( 7 a) + 2(^) = (a/3) + 2(v 2 )1 r ( 43a 2( MV ) + (aA) = 2( v A) + (/3/0 = 2(X/0 + (7v) J it may be said to possess rhombic symmetry, because the three oblique axes are normal to the faces of one rhombohedron, and to the edges of another belonging to the same series, crystallographically speaking. It is evident in this case, that the axis of symmetry must be a fourth Euthytatic Axis. In the limiting case, when the three oblique axes make with each other equal angles of 120°, they lie in the same plane, normal to the axis of symmetry, and are normal to the faces of one hexagonal prism and the edges of another. Let y x denote the longitudinal axis of symmetry of the prism ; z x any one of the three transverse axes perpendicular to y v The equation of a section of the biquadratic surface by the Plane of Hexagonal Symmetry y x z v is as follows : — (i3 2 ) 1 ^ + (y 2 ) 1 ^ + 2{(/3y) 1 +2(X 2 ) 1 } 2 /i,1=.l. . (44.) The equation of the same section, referred to any other pair of orthogonal axes y, Oz, in the plane of y x z v is as follows : — (j3 2 ) . y* + (y 2 ) . **+ 2 {(/3y) + 2(\ 2 )} ! /V+ 4{(/3 A) / + (yA)*»Jy*= 1. (44a.) AXES OF ELASTICITY AND CRYSTALLINE FORMS. 141 From considerations of symmetry, it is evident that the coefficient (/3»/) must be null for every direction of the axis y, in the plane of y^z y ; consequently, every direction Oy in that plane, for which (/3\) = 0, is an euthytatic axis. To ascertain whether, and under what conditions, there are other euthytatic axes in the planes of hexagonal symmetry besides the longitudinal and transverse axes, it is to be considered, that for rectangular co-ordinates (/3\) is covariant with yh; hence, let Zy l Oy = w, then (£A) = sin_2<o m j- { , (/ , y ^ + 4 (x% _ ^^ _ (y2)i} cog 2 ^ - (P\ + (r)i] • (4r>.) The first factor of the above expression is null for the longitudinal and transverse axes only. The condition of there being additional euthytatic axes in the plane y x z x is, that the second factor shall vanish ; that is to say, that cos o„_ (ff)l - (V% (i6) and that the value of w which makes it vanish shall neither be 0° nor 90°; that is to say, that the second member of the above equation (46) shall lie between + 1 and — 1 ; in which case the equation is satisfied by equal values of to with opposite signs. Hence are deduced the following theorems, which are stated in such a form as to be applicable to planes of symmetry, whether hexagonal or otherwise : If, in any plane of tasinomic symmetry containing a pair of orthogonal euthytatic axes, the difference of the euthytatic coefficients for these axes be equal to or greater than the metatatic difference, there are no additional euthytatic axes in that plane. If, on the other hand, the difference of such euthytatic coefficients be less than the metatatic difference, there are, in such plane of symmetry, a pair of additional euthytatic axes making with each other a pair of angles bisected by the orthogonal euthytatic axes. 2 to is the angle bisected by the axis y v In the case of hexagonal symmetry, the additional axes thus found are normal to the faces of one pyramidal dodecahedron and the edges of another. 142 AXES OF ELASTICITY AND CRYSTALLINE FORMS. Section 24. — Of Orthorhombic Symmetry. Let a solid have one of the three principal euthytatic axes, z v normal to the other two, Ox v 0^; let the last two be oblique to each other, and have equal sets of homotatic coefficients, viz. — (« 2 )> = (/3 2 )i ; O y)i + 2 (A 2 )! = (y «)x + ^ (S\ ; 2frv) + (a\) = 2(v\) + (Pfi), • ( i7 -) then, that solid may be said to have Orthorhombic Symmetry, its principal euthytatic axes being normal to the faces of a right rhombic prism. The existence or non-existence, and the position, of a pair of additional euthytatic axes in the longitudinal planes of y x r. v s t o\, is to be determined as in the preceding section. "When such axes exist, they are normal to the faces of an octahedron with a rhombic base. Section 25. — Of Orthogonal Symmetry. If the three principal euthytatic axes be orthogonal, they are normal to the faces of a right rectangular or square prism, and to the edges of a rigid rhombic or square prism. The existence or non-existence, and position, of a pair of additional euthytatic axes in each of the principal planes of such a solid, are determined as in Sect. 23. If there be a pair of such additional axes in each of the three principal planes, they are normal to the faces of an irregular rhombic dodecahedron, and to the edges of a rhombic octahedron. If there be a pair of such additional axes in two of the three principal planes, those axes are normal to the faces of an octahedron with a rectangular or square base, and to the edges of an octahedron with a rhombic or square base. If there be a pair of such additional axes in one of the planes of orthotatic symmetry only, those axes are normal to the lateral faces of a right rhombic prism. Section 26. — Of Cyboid Symmetry. The case of Cyboid Symmetry is that in which the homotatic coefficients are equal for three orthogonal axes, viz. — (« 2 ) - m = (y 2 ); (73 y) + 2 (X 2 ) = (y a) + 2 Ou 2 ) = (a 0) + 2 (t, 2 ); 2 0uv)+(aX) = 2(vX) + (j3 j u) = 2(X i u) + ( 7 v) = O. (48.) AXES OF ELASTICITY AND CRYSTALLINE FORMS. 143 In this case, the principal metatatic axes coincide with the principal euthytatic axes, which are normal to the faces of a cube; the diagonal metatatic axes normal to the faces of a regular rhombic dodecahedron, are euthytatic also; and there are, besides, four additional euthytatic axes symmetrically situated between the first nine, and normal to the faces of a regular octahedron, making in all thirteen euthytatic axes. Section 27. — Of Monaxal Isotropy. Monaxal Isotropy denotes the case in which the homotatic coefficients are completely isotropic round one axis only. In this case, the principal euthytatic axes are, the axis of isotropy, and every direction perpendi- cular to it ; and when there are additional axes, determined as in the preceding sections, they are normal to the surface of a cone. Section 28. — Of Complete Isotropy. In the case of complete isotropy of the homotatic coefficients, every direction is a euthytatic axis. Section 29. — Probable Eelations between Euthytatic Axes and Crystalline Forms. In the preceding sections it has been shown what must be the nature of the relations between the fifteen homotatic coefficients, for various solids, having systems of euthytatic axes normal to the faces and edges of the several Primitive Forms known in crystallography. It is probable that the normals to Planes of Cleavage are euthytatic axes of minimum elasticity. It may also be considered probable, that in some cases, especially in the tessular system, which corresponds to cyboi'd symmetry, and in the case of the pyramidal summits of crystals of the rhombohedral system, euthytatic axes correspond to symmetrical summits of crystalline forms. In the icositetrahedral crystals of leucite and analcime, and the tetra- contaoctahedral crystals of diamond, there are twenty-six symmetrical summits, one pair corresponding to each of the thirteen axes of cybo'id symmetry. The following is a synoptical table of the various possible systems of euthytatic axes, arranged according to their degrees and kinds of symmetry, and of the crystalline forms to the faces and edges of which such systems of axes are respectively normal. 144 AXES OF ELASTICITY AND CRYSTALLINE FORMS. Systems of Euthytatic Axes. Crystalline Forms. Faces. Edges. I. Asymmetry. Tetarto-prismatic System. 1. Three unequal oblique axes, . Oblique hexahedron. II. Symmetry about One Plane. Hemiprismatic System. 2. Two unequal oblique axes, and ) ,.,,.,,. „.,. . _.. \ , . \ Ri<ditrhornboidal prism, Oblique rhombic prism, one rectangular axis, . ) a r ' * wo equa < ( Oblique rhombic prism, Right rhomboidal prism, oblique axis, . ) x * III. Rhombic and Hexagonal Rhomboiiedral System. Symmetry. 4. Three equi-oblique principal j axes round one axis of > Rhombohedron, . . Rhombohedron. symmetry, . . . . ) 5. Three equi-oblique principal j axes in one plane, normal > Hexagonal prism, . Hexagonal prism, to axis of symmetry, . . ) 6. Three pairs of secondary axes Pyramidal dodecahe- Pyramidal dodecahe- in planes of symmetry, . dron, . . . dron. IV. Orthoriiombic Symmetry. Prismatic and Pyramidal Systems. 7. Two equal oblique transverse \ axes normal to one longi- > Right rhombic prism, . Rectangular prism, tudinal axis, . . . ) 8. Two pairs of secondary axes in Octahedron with rhom- Octahedron with rectan- longitudinal planes, . . bic base, . . . gular base. V. Orthogonal Symmetry. 9. Three orthogonal axes, not all Rectangular and square Right rhombic and equal, prisms, . . . square prisms. ! Octahedron with rhom- bic base and rectan- gular prism. ,,_,.„ I Octahedron with square Octahedron with square 11. Two pairs of secondary axes, . j or rectangular base , . or rhombic base . Same with 7. One pair of second- ) . ' . . arv axes \ ra S ut rhombic prism, . Rectangular prism. VI. Cyboid Symmetry. Tessular System. 12. Three equal orthogonal axes, . Cube. 10 „. ,. , I Regular rhombic dode- Cube and regular octahe- 13. Six diagonal axes, . \ , j n ° ( cahedron, . . . dron. 14. Four symmetrical intermediate axes, Regular octahedron, . Rhombic dodecahedron. AXES OF ELASTICITY AND CRYSTALLINE FORMS. 145 VII. MONAXAL ISOTKOPY. 15. One axis of isotropy, . . Isotropic lamina?. 16. Innumerable transverse axes, . Istropic fibres. 17. Innumerable equi-oblique axes, Conical cleavage. VIII. Complete Isotropy. 18. Innumerable axes of isotropy, . Amorphism. Section 30. — Mutual Independence of the Euthytatic and Heterotatic Axes, and of the Homotatic and Heterotatic Coefficients. The fifteen homotatic coefficients of the biquadratic surface, on which the euthytatic axes depend, and the six heterotatic differences, coefficients of the heterotatic ellipsoid, constitute twenty-one independent quantities; so that the euthytatic axes may possess any kind or degree of symmetry or asymmetry, and the heterotatic axes any other kind or degree, in the same solid. Hence, if it be true that crystalline form depends on the arrangement of euthytatic axes, it follows that two substances may be exactly alike in crystalline form, and yet differ materially in the laws of their elasticity, owing to differences in then respective heterotatic coefficients. It may be observed, however, that this complete independence of those two systems of axes and coefficients is mathematical only, and that their physical dependence or independence is a question for experiment. Section 31. — On Eeal and Alleged Differences between the Laws of the Elasticity of Solids, and those of the Luminiferous Force. For every conceivable system of tasinomic coefficients in a solid, the plane of polarisation of a wave of distortion is that which includes the direction of the molecular vibration and the direction of its propagation, being, in fact, the plane of distortion. On the other hand, it appears to be impossible to avoid concluding, from the laws of the diffraction of polarised light, as discovered by Professor Stokes, and from those of the more minute phenomena of the reflexion of light, as investigated theoretically by M. Cauchy and experimentally by M. Jamin, that in plane-polarised light the plane of polarisation is perpendicular to the direction of vibration, or rather (to avoid hypothetical language) to the direction of some physical phenomenon K 146 AXES OF ELASTICITY AND CRYSTALLINE FORMS. whose laws of communication are to a certain extent analogous to those of a vibratory movement. This constitutes an essential difference between the laws of the elastic forces in a solid, and those of the luminiferous force. In order to frame, in connection with the wave theory of light, a mechanical hypothesis which should take that difference into account, it has been proposed to consider the elasticity of the luminiferous medium to be the same in all substances, and for all directions, or Pantatically Isotropic ; and to ascribe the various retardations of light to variations in the inertia of the mass moved in luminiferous waves, in different substances, and for different directions of motion.* Another essential difference between the laws of solid elasticity and those of the luminiferous force is, that under no conceivable system of tasinomic coefficients in a homogeneous solid, would the plane of distortion in a wave be rotated continuously round the direction of propagation. Much has been written, both recently and in former times, concerning an alleged difficulty in the theories of waves, both of sound and of light, arising from the physical impossibility of the actual divergence of waves from, or their convergence to, a mathematical point. This impossibility must be admitted; but the supposed difficulty to which it gives rise in the theories of waves is completely overcome in Mr. Stokes' paper " On the Dynamical Theory of Diffraction,"! in which that author proves, that waves spreading from a focal space, or origin of disturbance, of finite magnitude, and of any figure, sensibly agree in all respects with waves spreading from an imaginary focal point, so soon as they have attained a distance from the focal space which is large as compared with the dimensions of that space ; so that the equations of the propagation of waves spreading from imaginary focal points may be applied, without sensible error, to all those cases of actual waves to which it is usual to apply them. The physical impossibility of focal points applies to light independently of all hypotheses; for at such points the intensity would be infinite. It appears to be worthy of consideration, whether this impossibility may not be connected with the appearance of spurious disks of fixed stars in the foci of telescopes. Section 32. — On the Action of Crystals on Light. If we set aside those actions on light to which there is nothing analogous in the phenomena of the elasticity of homogeneous solids, the laws of * Philosophical Magazine, June, 1851, December, 1853. + Cambridge Transactions, Vol. IX., Part 1. AXES OF ELASTICITY AND CRYSTALLINE FORMS. 147 the refractive action of a crystal on light are in general of a more sym- metrical kind, or depend on fewer quantities than those of its elasticity. Thus, the elasticity of a homogeneous solid depends on twenty-one quantities ; its crystalline form, on fifteen (the homotatic coefficients), while its refractive action on homogeneous light in most cases is expres- sible by means of the magnitudes and directions of the orthogonal axes of Fresnel's wave-surface, making in all six quantities. Crystals which possess only rhombic or hexagonal symmetry in their euthytatic axes, are usually monaxally isotropic in their action on light ; while crystals which possess only cybo'id symmetry in their euthytatic axes, are completely isotropic in their action on light. From these remarks? however, there are exceptions, as in the case of the extraordinary optical properties discovered by Sir David Brewster in analcime, which, in its refraction as well as in its form, is cyboidally symmetrical without being isotropic. Note referred to at page 120, On Sylvestrian Umbrae. Without attempting to enter into the abstract theory of the umbral method, it may here be useful to explain the particular case of its appli- cation which is employed in this paper. Let U be a quantity having an absolute value, constant or variable, (such, for example, as any physical magnitude), and u, v, . . . &c. a set of quantities, m in number, such that tJ is of them a homogeneous rational function of the nth. degree. There are an indefinite number of possible sets of m quantities satisfying this condition; and the quantities of each set are related to those of each other set by m equations of the first degree, called equations of linear transformation. Let the Up v v . . . be two such sets. ^t (U . . . development of and let M 2 > ^2> denote the coefficient of u a v b . (u + v+ . . .)» U = 2{C a , 6 , • • • A l,a,6, • • • K V l ■ • ■} = 2{C a . & , . . . A., , ... u-, a v? . 2. a, b, • • • w 2 "2 • •}• The two sets of coefficients A v A. z , are connected by linear equations 148 AXES OF ELASTICITY AND CRYSTALLINE FORMS. of transformation, the investigation of which is much facilitated by the following process. Let two sets, each of m symbols, a v [3 V &c. . . . a 2 , /3 2 , . . . &c, be assumed such that «i «i + & v x + and that, consequently, = a 2 u 2 + fi 2 v 2 + («1 M 1+/Vl+ • ■ .)*~2{€U • . . <& & . . ufvf . • •} = (a 2 u 2 + /3 2 v 2 + ■ • -) n =^K, b , ■ ■ <A> & • • «/^ 2 6 • • •}• Then, if the m equations of transformation between the two sets of symbols a v & • • • and a 2 , (3 2 . . . be formed, and if from them be deduced the equations between the two sets of products a x re fi^ , and a 2 a (3 2 , &c, and if in the latter system of equations, there be substituted for each product a a /3 b . . . the corresponding coefficient A a _ &> . . . , the result will be the system of equations sought. Also, if -any function of the products a°/S & . . . be invariant (i e., a function whoso value, like that of the original function U, is not altered by the transformation), the corresponding function of the coefficients A will be invariant. The symbols a, (3, &c, with reference to their relation to the coefficients A, are called umbrce ; that is, factors of symbols, whose equations of transfor- mation are similar to those of the coefficients A. In the umbral notation, umbrae are usually distinguished from symbols denoting actual quantities by being enclosed in brackets thus : (a), 08), &c. . . . and each coefficient A is represented by enclosing in brackets that product of umbrae with which it is covariant; thus : a, I, = (a a /3 6 •)• The umbral notation is applied to abbreviate the expression of deter- minants in a manner of which the following are examples : — a, (3, y, &c. a, /3, y, &c. denotes (« 2 ) («/3) (ay) &c. («/3) (/3 2 ) (/3y) &c. (ay) (/3y) (y 2 ) &c. &c. &c. &c. &c. AXES OF ELASTICITY AND CRYSTALLINE FORMS. 149 a, y, S, &c. /5, y, S, &C. denotes («/3) (/3y) CSS) &c. («y) (r) (yS) &c. (« 8) (yS) (S 2 ) &c. &c. &c. &c. &c. 150 THE VIBRATIONS OF PLANE-POLARISED LIGHT. VII— ON THE VIBRATIONS OF PLANE-POLARISED LIGHT.* 1. The important question, whether the direction of vibration in plane- polarised light is normal or parallel to the plane of polarisation, is equivalent to this : — whether the velocity of propagation of a rectilinear transverse vibratory movement in the medium which transmits light in crystallised bodies is a function of the direction of vibration only, or of the position of the plane, which includes the direction of vibration and the direction of transmission. The former of these views was adopted by Fresnel, as being necessary to explain the phenomena of polarisation by reflexion; and in Mr. Green's investigation of the laws of these phenomena, in which the conclusions of Fresnel are shown to agree either exactly or approximately with the consequences of strictly mechanical principles, the same supposition is adopted. But if we follow the generally received theory, that the different velocities of differently polarised rays in crystalline bodies are due solely to the different degrees of elasticity possessed by the vibrating medium in different directions, Fresnel's supposition must be abandoned, and the opposite one adopted. For if there is any proposition more certain than others respecting the laws of elasticity, it is this : — that the transverse elasticity of a medium, or the elasticity which resists distortion of the particles, depends upon the position of the plane of distortion, being the same for all directions of distortion in a given plane. This law is impli- citly involved in the researches of Poisson, of M. Cauchy, of Mr. Green, and others on elasticity; and in a memoir read to the British Association at Edinburgh, in 1850, and published in the Cambridge and Dublin Mathematical Journal for February, 1851, I have shown that it is true independently of all hypotheses respecting the constitution of matter, being a necessary consequence of the conception of an elastic medium. Now a wave of plane-polarised light is a wave of distortion : the plane of distortion is the plane which includes the direction of transmission and the direction of vibration: the elasticity called into play depends on the position of this plane; therefore, if the velocity of propagation depends upon elasticity alone, the plane of distortion must be the plane of polaris- ation; and if a normal be drawn to that plane, the velocity of propagation * Read before the Royal Society of Edinburgh, on December 2, 1850, and published in the Philosophical Magazine, June, 1851. THE VIBRATIONS OF PLANE-POLARISED LIGHT. 151 will be a function of the position of that normal, and not, as supposed by Fresnel, of the direction of vibration itself. 2. Up to a very recent period, no experimental data existed adequate to determine which of these suppositions is supported by facts ; for the phenomena of double refraction are consistent with either ; and the theory of polarisation by reflexion is not regarded as sufficiently certain to afford the means of deciding this question. At length, however, the experimentum cruris has been made by Professor Stokes, and the result is conclusive in favour of the supposition of Fresnel. In his paper on Diffraction (Cambridge Transactions, Vol. IX., Part 1), Professor Stokes has shown, that on any conceivable theory of the propagation of undulations of light, vibrations normal to the plane of diffraction must be transmitted round the edge of an opaque body with less diminution of intensity than vibrations in that plane. Therefore, Avhen light, of which the vibrations are oblique to the plane of diffraction, is so transmitted, the plane of vibration will be more nearly perpendicular to the plane of diffraction in the diffracted ray than in the incident ray. He has found by experiment, that when light of which the plane of polarisa- tion is oblique to the plane of diffraction is transmitted round the edge of an opaque body, the plane of polarisation is more nearly parallel to the plane of diffraction in the diffracted than in the incident ray. The necessary conclusion is, that the direction of vibration in plane-polarised light is normal to the plane of polarisation; in other words, that the velocity of light in crystallised media depends on the direction of vibration, as con- jectured by Fresnel. This result of experiment is at variance with the necessary consequences of the supposition, that the velocity of light depends on elasticity alone; therefore, that supposition is inadequate to explain the phenomena of polarised light. 3. Having considered what modifications must be introduced into our hypothetical conceptions of the nature of the medium which transmits light, to make them adequate to explain the facts which have thus been established, I now offer the following suggestions. In a paper read to the Royal Society of Edinburgh, and published in their Transactions, Vol. XX., Part 1, I proposed, as a foundation for the theory of heat, and of the elasticity of gases and vapours, an hypothesis called that of molecular vortices ; and in a subsequent paper, already referred to, I deduced from the same hypothesis some principles relative to the elasticity of solids. I shall now show that Fresnel's conjecture as to the direction of vibration in plane-polarised light is a natural consequence of that hypothesis. The fundamental suppositions of the hypothesis of molecular vortices are the following: — 152 THE VIBRATIONS OF PLANE-POLARISED LIGHT. First. That each atom of matter consists of a nucleus or central physical point enveloped by an elastic atmosphere, which is retained round it by attraction; so that the elasticity of bodies is made up of two parts — one arising from the diffused portion of the atmospheres, and resisting change of volume only ; the other arising from the mutual actions of the nuclei, and of the portions of atmosphere condensed round them, and resisting not only change of volume, but also change of figure. Secondly. That the changes of elasticity due to heat arise from the centrifugal force of revolutions or oscillations among the particles of the atomic atmospheres, diffusing them to a greater distance from their nuclei, and thus increasing the elasticity which resists change of volume only, at the expense of that which resists change of figure also. Thirdly. That the medium which transmits light and radiant heat consists of the nuclei of the atoms vibrating independently, or almost independently, of their atmospheres; absorption being the transference of motion from the nuclei to the atmospheres, and emission its transference from the atmospheres to the nuclei. This last supposition is peculiar to my own researches, the first two having more or less resemblance to ideas previously entertained by others. If an indefinitely extended vibrating medium, equally elastic in all directions, consists of a system of atomic nuclei, tending to preserve a certain configuration in consequence of their mutual attractions and repulsions, it is well known that such a medium is capable of transmitting two sorts of vibrations only, longitudinal and transverse, the latter alone being supposed to be concerned in the phenomena of light. It is also well known, that the square of the velocity of propagation of transverse vibrations is directly proportional to a quantity called the transverse elasticity of the medium, arising from the mutual actions of the nuclei, and inversely proportional to its density — that is, to the sum of the masses of all the nuclei contained in unity of volume. To account for the immense velocity of light, the masses of the atomic nuclei must be sup- posed to be very small as compared with the mutual forces exerted by them. In stating the third supposition of the hypothesis, the nuclei are said to vibrate independently, or almost independently, of their atmospheres; for the absolute independence of their vibrations is probably an ideal case, not realised in nature, though approached very nearly in the celestial space, where the atomic atmospheres must be inconceivably rarefied. As a pendulum is known to be accompanied in its oscillations by a portion of the air in which it swings, so the nuclei probably in all cases carry along with them in their vibrations a small portion of their atmos- pheres, which acts as a load, increasing the vibrating mass without increas- ing in the same proportion the elasticity, and consequently retarding THE VIBRATIONS OF PLANE-POLARISED LIGHT. 153 the velocity of transmission. The amount of this load must depend on the density of the atomic atmosphere; and, accordingly, we find that, generally speaking, the most dense substances are those in which the velocity of light is least. Now, if we assume, what is extremely probable, that in crystallised media the atomic atmospheres are not similarly diffused in all directions round their nuclei, but are more dense in certain directions than in others, we must at once conclude that in such media the velocity of propagation of vibratory movement depends on the direction of vibration ; for upon that direction depends the load of atmosphere which each nucleus carries along with it. 4. Having thus shown that the conjecture of Fresnel, which has been confirmed by the experiments of Professor Stokes, is a natural consequence of the hypothesis of molecular vortices, I shall now prove that that hypothesis leads to those mathematical laws of the transmission of light in crystalline media which Fresnel discovered. Considering it desirable in this paper to avoid lengthened algebraical analysis, I shall with that view state, in the first place, certain known geometrical properties of the ellipsoid, to which it will be necessary for me to refer. I. If a curved surface be described about a centre, such that the sum of the reciprocals of the squares of any three orthogonal diameters is a constant quantity, that surface, if no diameter is infinite, is an ellipsoid. II. Every function of direction round a centre, whose variation from a given amount varies as the reciprocal of the square of the diameter of an ellipsoid described about that centre, is itself proportional to the reciprocal of the square of the diameter of another ellipsoid described about the same centre with the first, and having the directions of its axes the same. III. It follows from the last proposition, that if there be a function of direction round a centre which is proportional to the reciprocal of the square of the diameter of an ellipsoid of small excentricities, so that the range of variation of the function is small as compared with its amount, then any function of that function, whose range of variation is small also, may be represented approximately by the reciprocal of the square of the diameter of another ellipsoid, having its centre and the directions of its axes the same with those of the first. The square of the velocity of propagation of transverse vibrations is proportional to the transverse elasticity of the medium divided by the mean density; that is, by the sum of all the vibrating masses in unity of volume. That sum is the sum of the masses of the nuclei, added to the masses of atmosphere with which they are loaded. The atmospheric load of each nucleus depends on, or is a function of, the density of the 154- THE VIBRATIONS OF PLANE-POLARISED LIGHT. atmosphere adjoining the nucleus along the line in which the latter vibrates. The mode of distribution of the atmospheres depends on the attraction of the nuclei upon them, and therefore on the mode of arrange- ment of the nuclei. The mode of arrangement of the nuclei, when it is symmetrical and uniform, may be expressed by means of their mean intervals. The mean interval of the nuclei is a function of direction, of such a nature that its three values for any three orthogonal directions being multiplied together give a constant result — viz., the space occupied (not filled) by one nucleus, or the quotient of a given space by the number of nuclei contained in it. Hence the sum of the values of the logarithm of the mean interval for any three orthogonal directions is a constant quantity; and that logarithm, therefore, is proportional to the reciprocal of the square of the diameter of an ellipsoid, -whose three axes may be called the axes of atomic distribution. Therefore, the mean interval, the atmospheric load of the nuclei, and the square of the velocity of propa- gation, for a given direction of transverse vibration, are all functions of the reciprocal of the square of the diameter of an ellipsoid, and have maxima and minima corresponding to its three axes, which are those of atomic distribution. Now, in all known crystalline media, the range of variation of these quantities for different directions is very small compared with their amount. Therefore, each of them may be approximately represented by the reciprocal of the square of the diameter of an ellipsoid whose axes are parallel to those of atomic distribution. If, then, the directions of vibration in a given crystal which correspond to the greatest and least velocities of transmission are known, let these directions (which are at right angles to each other), and a third direction at right angles to them both, be taken for the axes of an ellipsoid, the lengths of those axes being inversely proportional to the corresponding velocities of transmission. Then will the velocity of transmission of any transverse vibratory movement be sensibly proportional to the reciprocal of a diameter of that ellipsoid, drawn parallel to the direction of vibration. And if a plane be drawn through the centre of the ellipsoid parallel to a series of plane-waves, the two axes of the elliptic section so made will represent, in magnitude, the reciprocals of the greatest and least normal velocities of transmission of waves parallel to that plane, and, in direction, the corresponding directions of vibration. This agrees exactly with the construction given by Fresnel, on which his entire theory of double refraction is founded. The degree of symmetry and uniformity of arrangement of the atoms which is necessary in order that the mean interval may have a definite value, and that three axes of distribution may exist, is the same which THE VIBRATIONS OF PLANE-POLARISED LIGHT. 155 is necessary to the existence of rectangular axes of elasticity in a solid body. It must extend round each point, throughout a space which is large as compared with the sphere of appreciable molecular action. The experiments of Sir David Brewster and of Fresnel on the action of compressed glass on polarised light, show that rays polarised in a plane normal to the direction of compression — that is to sa}^ vibrations parallel to that direction — are accelerated. This indicates that the atmospheric load on each vibrating nucleus in that direction is diminished, probably by the displacement of a portion of the atmosphere out of that line. 5. Though I have assumed, in the course of this investigation, that the luminiferous medium is equally elastic in all directions, I by no means intend to assert that it is necessarily so in all substances ; but merely that, in most known crystalline media, an atmospheric load on the vibrating nuclei is the predominant cause of variation in the velocity of transmission with the direction of transverse vibration. It is remarkable that Fresnel, in his theory of the intensity of reflected and refracted light, speaks of the particles of the luminiferous medium as being more or less loaded in substances of greater or less refractive power. He did not, however, apply this idea to double refraction, although he adopted a theory, which, as we have seen, results from it. The principles laid down in this paper are not compatible Avith the prevalent idea of a luminiferous ether enveloping ponderable particles. The fundamental idea from which they spring is the converse: that the luminiferous medium is a system of atomic nuclei or centres of force, whose office it is to give form to matter ; while the atmospheres by which they are surrounded give, of themselves, merely extension. 156 AN OSCILLATORY THEORY OF LIGHT. VIII— GENERAL VIEW OF AN OSCILLATORY THEORY OF LIGHT.* Section I. — Difficulties of the Present Hypothesis. Notwithstanding the perfection to which the geometrical part of the undulatory theory of light has been brought, it is admitted that great difficulty exists in framing, to serve as a basis for the theory, a physical hypothesis ■which shall at once be consistent with itself and with the known properties of matter. The present paper is a summary of the results of an attempt to diminish that difficulty. All the conclusions stated have been deduced by means of strict mathematical analysis ; and although it is impossible to read the investigations before the British Association in detail, their results can easily be verified by every mathematician who is familiar with the undulatory theory in its present form. It may be considered as established, that if we assume the supposition that plane-polarised light (out of the varieties of which all other light can be compounded) consists in the wave-like transmission of a state of motion, the nature and magnitude of which are functions of the direction and length of a line transverse to the direction of propagation, we can deduce from this supposition, with the aid of experimental data, and of certain auxiliary hypotheses, the laws of the phenomena of the inter- ference of light, of its propagation in crystalline and uncrystalline substances, of diffraction, of single and double refraction, of dispersion by refraction, and of partial and total reflexion. It has hitherto been always assumed, that the kind of motion which constitutes light is a vibration from side to side, transmitted from particle to particle of the luminiferous medium, by means of forces acting between the particles. In order to account for the transmission of such transverse vibrations, the luminiferous medium has been supposed to possess a kind of elasticity which resists distortion of its parts, like that of an elastic solid ; and in order to account for the non-appearance in ordinary cases of effects which can be ascribed to longitudinal vibrations, it has been found * Read before the British Association at Hull, September 10, 1^53, and published in the PhilosojjJdcal Magazine, December, 1853. AN OSCILLATORY THEORY OF LIGHT. 157 necessary to suppose further, that this medium resists compression with an elasticity immensely greater than that with which it resists distortion; the latter species of elasticity being, nevertheless, sufficiently great to transmit one of the most powerful kinds of physical energy through interstellar space, with a speed in comparison with which that of the swiftest planets of our system in their orbits is appreciable, but no "more. It seems impossible to reconcile these suppositions with the fact, that the luminiferous medium in interstellar space offers no sensible resistance to the motions of the heavenly bodies. A step towards the solution of this difficulty was made by Mr. MacCullagh. The equations which he used to express the laws of the propagation of light, when interpreted physically, denote the condition of a medium whose molecules tend to range themselves in straight lines, and when disturbed, to return to those lines with a force depending on the curvature of the lines into which they have been moved. But even this hypothesis requires the assumption that the elasticity of the luminiferous medium to resist compression is immensely greater than the elasticity which transmits transverse vibrations. The difficulty just referred to arises from a comparison of the hypo- thesis of transverse vibrations with the observed phenomena of the world. Another difficulty arises within the hypothesis itself. Fresnel originally assumed, that in crystalline media, where the velocity of light varies with the position of the plane of polarisation, the direction of vibration is perpendicular to that plane. This is equivalent to the supposition, that the velocity with which a state of rectilinear transverse vibration is transmitted through such a medium, is a function simply of the direction in which the particles vibrate. From this hypothesis he deduced the form of that wave-surface . which expresses completely the law of the propagation of plane-polarised light through crystalline media, and he obtained also a near approximation to the laws of the intensity of plane- polarised light reflected from singly refracting substances. But it was afterwards demonstrated, that the elastic forces which propagate a transverse movement in any medium must necessarily be functions, not merely of the direction of the movement, but jointly of this direction and the direction of propagation ; that is to say, of the position of the plane containing these two directions. Consequently, if the various velocities of variously polarised light in a doubly refracting medium, arise from variations of elasticity in different directions, the direction of vibra- tion is in the plane of polarisation, contrary to the hypothesis of Fresnel. Fresnel's wave-surface, and his approximate formulas for the intensity of reflected light, are deducible from this supposition as readily as from his original hypothesis; and Mr. MacCullagh obtained from it formula? for the 158 AN OSCILLATORY THEORY OF LIGHT. intensity of light reflected from doubly refracting substances, agreeing closely with the experiments of Sir David Brewster. On the other hand, the formula? of M. Cauchy, and those of Mr. Green, as modified by Mr. Haughton,* expressing the effects of reflexion on the intensity and phase of polarised light, all of which are founded on the supposition that the direction of vibration is perpendicular to the plane of polarisation, have been shown to be capable, by the introduction of proper constants, of giving results agreeing closely with those of the important experiments of M. Jamin {Annales de Chimie et de Physique, 3rd Series, Vol. XXIX., 1850); and it is difficult, if not impossible, to see how such formulae could have been deduced from the opposite supposition. But the true crucial experiment on this subject has been furnished by the researches of Professor Stokes on the Diffraction of Polarised Light (Camb. Trans., Vol. IX.). Whatsoever may be the nature of the motion that constitutes light, if it can be expressed by a function of the direction and length of a line perpendicular to the direction of propagation (which may be called a transversal), it is certain that this motion will be more abundantly communicated round the edge of an obstacle, when its trans- versal is parallel than when it is perpendicular to that edge ; so that the effect of diffraction is, to bring every oblique transversal into a position more nearly parallel to the diffracting edge. But it has been shown by the experiments of Professor Stokes, that the effect of diffraction upon every ray of light polarised in a plane oblique to the diffracting edge, is to bring the plane of polarisation into a position more nearly perpendicular to the diffracting edge. Therefore, the transversal of a ray of plane-polarised light (which, if light consists in linear vibratory movement, is the direction of vibration) is perpendicular to the plane of polarisation. Hence it follows, that, in a crystalline medium, the velocity of light depends simply on the direction of the transversal characteristic of the movement propagated, and not on the direction of propagation. This conclusion is opposed to the laws of the propagation of transverse vibrations through a crystalline elastic solid, or through any medium in which the velocity of propagation depends on elasticity varying in different directions. Therefore, the velocity of light depends on something not analogous to the variations of elasticity in such a medium. To solve this difficulty, the author of this paper some time since suggested the hypothesis, that the luminiferous medium consists of particles forming the nuclei of atmospheres of ordinary matter ; that it transmits transverse vibrations by means of an elasticity which is the same in all substances and in all directions ; and that the variations in the velocity of the transmission of vibrations arise from variations in the atmospheric load, which the luminiferous particles carry along with them in their vibrations, * Philosophical Magazine, August, 1S53. AN OSCILLATORY THEORY OF LIGHT. 159 and which is a function of the nature of the substance, and, in a crystalline body, of the direction of vibration. But although this hypothesis removes the inconsistency just pointed out as existing within the theory itself, it leaves undiminished the difficulty of conceiving a medium pervading all space, and possessed of an elasticity of figure, at once so strong as to transmit the powerful energy of light -with its enormous velocity, and so feeble as to exercise no direct appreciable effect on the motions of visible bodies. Section II — Statement of the Proposed Hypothesis of Oscillations. The hypothesis now to be proposed as a groundwork for the undulatory theory of light, consists mainly in conceiving that, the luminiferous medium is constituted of detached atoms or nuclei distributed throughout all space, and endowed with a peculiar species of polarity, in virtue of which three orthogonal axes in each atom tend to place themselves parallel respectively to the corresponding axes in every other atom ; and that plane-polarised light consists in a small oscillatory movement of each atom round an axis transverse to the direction of propagation. Such a movement would be transmitted through such a medium with a velocity proportional, directly, to the . square root of the total rotative force exercised by the luminiferous atoms in a given small space, upon those in a given adjacent small space lying in the direction of propagation, in consequence of a given amount of relative angular displacement round the axis of oscillation ; and inversely, to the square root of the sum of the moments of inertia round the axes of oscillation of the atoms contained in a given space, loaded with such portions of molecular atmospheres surrounding them as they may carry along with them in their oscilla- tions. Then, denoting by h, the velocity, in a given direction of plane-waves, of oscillation round transverse axes parallel to a given line; C, a coefficient of polarity or rotative force for the given directions of propagation and of axes ; M, a coefficient of moment of inertia for the given direction of axes; the above principle may be represented by this equation, M The coefficient of polarity in question is proper only to an axis of 100 AN OSCILLATORY THEORY OF LIGHT. oscillation transverse to the direction of propagation. To account for the stability of direction of the axes of the atoms, and also for the non-appear- ance, in ordinary cases, of phenomena capable of being ascribed to oscillations round axes parallel to the direction of propagation, it is necessary to suppose the corresponding coefficient for the latter species of oscillations to be much greater than the coefficient for transverse axes of oscillation. It is evident, that how powerful soever the polarity may be which is here ascribed to the atoms of the luminiferous medium, it is a kind of force which must be absolutely destitute of direct influence on resistance to change of volume or change of figure in the parts of that medium, or of any body of which that medium may form part ; and that, consequently, the difficulty, which in the hypothesis of vibrations arises from the necessity of ascribing to the luminiferous medium properties like those of an clastic solid, has no existence in the hypothesis of oscillations now proposed. * The luminiferous atoms may now be supposed to be diffused throughout all space, and, as molecular nuclei, throughout all bodies ; the distribution and motion of their centres being regulated by forces wholly independent of that species of polarity which is the means of transmitting a state of oscillation round those centres. Section III. — Of the Diffraction of Plane-polarised Light, and the Relation of Axes of Oscillation to Planes of Polarisation. In the diffraction of an oscillatory movement round transverse axes past the edge of an obstacle, a law holds good exactly analogous to that demonstrated by Professor Stokes for a transverse vibratory movement, substituting only the axis of oscillation for the direction of vibration — that is to say : The direction of the axes of oscillation in the diffracted icave is the projection of that of the axes of oscillation in the incident wave on a plane tangent to the front of the diffracted wave. Consequently, oscillations in the incident wave, round axes oblique to the diffracting edge, give rise to oscillations in the diffracted wave round axes more nearly parallel to the diffracting edge. But the experiments of Professor Stokes .have proved, that light polarised in a plane oblique to the diffracting edge, becomes, after diffraction, polarised in a plane more nearly perpendicular to the diffracting edge. Therefore, the axes of oscillation in plane-polarised light are perpendicular to the plane of polarisation. AN OSCILLATORY THEORY OF LIGHT. 161 Therefore, the velocity of transmission of oscillations round transverse axes, through the luminiferous medium in a crystalline body, is a function simply of the direction of the axes of oscillation. Now, if the variations of the velocity of transmission arose from variations of the coefficient of transverse polarity (denoted by C), they would depend on the direction of propagation as well as upon that of the axes of oscillation, so that the plane of polarisation would be that which contains these two directions. Since the velocity of transmission depends on the direction of the axes of oscillation only, it follows that its variations in a given crystalline medium arise wholly from variations of the moment of inertia of the luminiferous atoms, together with their loads of extraneous matter. Consequently, the coefficient of polarity, C, for transverse axes of oscilla- tion is the same for all directions in a given substance. To account for the known laws of the intensity and phase of reflected and refracted light consistently with the hypothesis of oscillations, it is necessary to suppose, also, that this coefficient is the same for all substances ; so that the variations of the velocities of light, and indices of refraction for different media, depend solely on those of the moments of inertia of the loaded luminiferous atoms. There is reason to anticipate, that, upon further investigation, it will appear that this condition is necessary to the stability of the luminiferous atoms. Section IV. — Of the Wave-surface in Crystalline Bodies. Let the axes of co-ordinates be those of molecular symmetry in a crystalline medium. Let M v M 9 , M 3 , be coefficients proportional to the moments of inertia of the luminiferous atoms with their loads of extraneous matter, round axes parallel to x, y, z, respectively. Let r be a radius vector of the diverging wave-surface in the direction («> A y)- Then the equation of that surface for polar co-ordinates is, ~ - - 2 • ~ { (M 2 + M 3 ) cos 2 a + (M 3 + M x ) cos 2 /3 + (M x + M 2 ) cos 2 y } + -^ {M 2 M 3 cos 2 a + M 3 M x cos 2 /3 + M x M 2 cos 2 y } = ; 162 AN OSCILLATORY THEORY OF LIGHT, and for rectangular co-ordinates, ^(x 2 + 2/ 2 + ^).(M 2 M 3 x 2 + M 3 M l2 / 2 + M 1 M 2 . 2 ) -i{(M 2 +M3)x 2 +(M 3 +M 1 ) 2 / 2 +(M 1 +M 2 )x 2 }=:l. The above equations are exactly those of Fresnel's wave-surface, with the following semi-axes: — Directions. Semi-axes. x ' Vm 2 ' Vm 3 ; y ' Vm 3 ' Vm^ the squares of the semi-axes of the wave-surface along each axis of co-ordinates being inversely proportional to the moments of inertia of the loaded luminiferous atoms in a given space round the other two axes of co-ordinates. The plane of polarisation at each point of the wave-surface is perpen- dicular to the direction of greatest declivity. The equation of the index-surface, whose radius in any direction is inversely proportional to the normal velocity of the wave, is formed from that of the wave-surface by substituting respectively, r J- ± ! ' M x ' M 2 ' M 3 ' for c , M 1? M 2 , M 3 , These equations are obtained on the supposition that the coefficient of polarity for axes of oscillation parallel to the direction of propagation, (which we may call A), is either very large, or very small, compared with that for transverse axes. By treating the ratio of these quantities as finite, there is obtained an equation of the sixth order, representing a wave-surface of three sheets, differing somewhat from that of the propaga- tion of vibrations in an elastic crystalline solid ; inasmuch as the former has always three circular sections, while the latter has none, unless it is symmetrical all round one axis at least. By increasing the ratio ~ AN OSCILLATOEY THEOKY OF LIGHT. 163 without limit, this equation is made to approximate indefinitely to the product of the equation of Fresnel's wave-surface by the following — which represents a very large ellipsoidal wave of oscillations round axes parallel to the direction of propagation. Section V. — Of Reflexion and Effraction. According to the proposed hypothesis of oscillations, the laws of the phase and intensity of light reflected and refracted at the bounding surface of two transparent substances, are to be determined by conditions analogous to those employed in the hypothesis of vibrations by M. Cauchy and Mr. Green. They are the consequences of the principle, that if we have two sets of formula? expressing the nature and magnitude of the oscillations in the two substances respectively, then either of those formulae, being applied to a particle at the bounding surface, ought to give the same results. According to this principle, the following six quantities for a particle at the bounding surface must be the same at every instant, when computed by either of the two sets of formulae : — The three angular displacements round the three axes of co- ordinates, The three rotative forces round the same three axes. There is, generally speaking, a change of phase when light undergoes refraction or reflexion. It is known that we may express this change of phase by subdividing each reflected or refracted disturbance into two, of suitable intensities and signs; one synchronous in phase with the corresponding incident disturbance, and the other retarded by a quarter of an undulation. There are thus twelve quantities to be found — viz., the amplitudes of the six components of the reflected disturbance, and those of the six components of the refracted disturbance. To determine these quantities there are twelve conditions — viz., the equality at every instant, according to the formulas for either medium, of the total angular displace- ments, and of the total rotative forces, round each of the three axes of co-ordinates, for the set of waves composed of the incident wave and those synchronous with it, and for the set of waves retarded by one quarter of an undulation. 164 AN OSCILLATORY THEORY OF LIGHT. The results of these conditions have been investigated in detail for singly refracting substances. The indices of refraction of such substances are proportional to the square roots of the moments of inertia of the loaded luminiferous atoms in a given space. Thus, if the coefficients M', M" are proportional to these moments in two given substances respectively, then the index of refraction of the second substance relatively to the first is M /M" In the case of light incident on a plane surface between two such media, the axes of co-ordinates may be assumed respectively perpendicular to the reflecting surface, perpendicular to the plane of reflexion, and along the intersection of those two planes ; and oscillations round axes normal and parallel to the plane of reflexion may be considered separately. When the axes of oscillation are normal to the plane of reflexion — that is to say, when the light is polarised in that plane — the formula} for the intensities of the reflected and refracted light agree exactly with those of Fresnel. When the reflexion takes place in the rarer medium, the reflected light is retarded by half an undulation; when in the denser, there is no change of phase, unless the reflexion is total, when there is a certain acceleration of i hase depending on the angle of incidence. In the last case, the disturbance in the second medium is an evanescent wave, analogous to those introduced into the vibratory theory by M. Cauchy and Mr. Green — that is to say, a wave in which the amplitude of oscilla- tion diminishes in proportion to an exponential function of the distance from the bounding surface (called by M. Cauchy the modulus), and which travels along that surface with a velocity less than the velocity of an ordinary wave ; the square of the negative exponent of the modulus being proportional to the difference of the squares of those velocities, divided by the square of the velocity of an ordinary wave. This is an evanescent wave of oscillation round transverse axes. How large soever the coefficient of polarity for oscillations round longitudinal axes may be, an evanescent wave of such oscillations may travel along the bounding surface of a medium with any velocity, however slow, provided the negative exponent of the modulus is made large enough. Consequently, in framing the formulas to represent oscillations round axes parallel to the plane of incidence, we must introduce in each medium two such evanescent waves of suitable exponents and indeterminate amplitudes ; one travelling along the surface with the incident wave, and the other a quarter of an undulation behind it. The maximum amplitudes of oscilla- tion in these evanescent waves constitute four unknown quantities ; the AN OSCILLATORY THEORY OF LIGHT. 1G5 amplitudes in the two ordinary reflected waves, and the two ordinary refracted waves, differing by one quarter of an undulation, constitute four more unknown quantities, making eight in all : four conditions having been fulfilled by the waves polarised in the plane of incidence, there remain to be fulfilled eight conditions — viz., the identity, as calculated by the formulae for the first and second substance respectively, of the following eight functions at the bounding surface ; the angular displace- ment, and the rotative forces, round each of the two axes in the plane of incidence, for the incident wave, and the set of waves synchronous with it, and for the set of waves retarded by one quarter of an undula- tion. These conditions are sufficient to determine the unknown quantities, and to complete the solution of the problem. The following is a general statement of the results of the solution when the second medium is the denser. They agree with the results of the experiments of M. Jamin, and are, in every respect, analogous to those deduced from the hypothesis of vibrations by M. Cauchy, Mr. Green, and Mr. Haughton. Light polarised in a plane perpendicular to the plane of incidence, suffers by reflexion at a perpendicular incidence no alteration of phase. At a grazing incidence (or when the angle of incidence differs insensibly from 90°), the phase, like that of light polarised in the plane of incidence, is retarded by half an undulation. The variation of phase with the angle of incidence is, in fact, con- tinuous; but it is, generally speaking, not appreciable by observation, except in the immediate neighbourhood of an angle, called by M. Jamin the principal incidence, where the retardation of phase is a quarter of an undulation. This angle differs by a very small amount, appreciable only in certain substances, from the polarising angle, at which the intensity of light polarised in a plane at right angles to the plane of incidence is a minimum. The " law of Brewster," that the tangent of the polarising angle is equal to the index of refraction, is, theoretically, only approximately true; but the error is quite inappreciable. When the second medium is the less dense, the phase of the reflected light is half an undulation in advance of its value when the second medium is the denser. In either case, light polarised in planes perpendicular to the plane of incidence is less retarded — that is to say, is accelerated in phase — as compared with light polarised in that plane, according to the following table : — 166 AN OSCILLATORY THEORY OF LIGHT. Angle of Relative Incidence. Acceleration. o Perpendicular incidence, . . \ undulation. Principal incidence, \ undulation. Grazing incidence, . . .90 In the case of total reflexion, light polarised in planes perpendicular to the plane of incidence, lias its phase more accelerated than light polarised in that plane, by an amount to which the formulae of Fresnel give a close approximation. The proposed hypothesis has not yet been applied to reflexion from doubly refracting crystals ; but there can be little doubt that it will be found to represent the phenomena correctly. Section VI. — Of Circular and Elliptic Polarisation. Light polarised in a plane oblique to the angle of incidence is, generally speaking, elliptically polarised after reflexion, the plane-polarised com- ponents of the disturbance being in different phases. According to the hypothesis of oscillations, circularly and elliptically polarised light, being compounded of oscillations in different phases round two transverse axes, consist in a sort of nutation of the longitudinal axis of each luminiferous atom. The direction of this nutation, and the form of the circle or ellipse described by the ends of the longitudinal axes, serve to define the character of the light. The ellipse of nutation has its axes in the same proportion with, but perpendicular in position to, those of the elliptic orbit supposed to be described by each atom according to the hypothesis of vibrations. The molecular mechanism by which certain media transmit * right and left-handed circularly or elliptically polarised light with different velocities, is still problematical, according to either hypothesis. The laws of the phenomena, however, may be represented by means of the assumption^ that in the substances in question the extraneous load on the luminiferous atoms is a function of the direction of nutation. Section VII. — Of Dispersion. If we assume the extent of sensible direct action of the polarity of the luminiferous atoms to be appreciable, as compared with the length of a wave, the velocity of propagation (precisely as with the vibratory AN OSCILLATORY THEORY OF LIGHT. 167 hypothesis) is found to consist of a constant quantity, diminished by the sum of a series in terms of the reciprocal of the square of the length of a wave. It may be doubted, however, whether this supposition is of itself adequate to explain the phenomena of dispersion, and whether it may not be necessary to assume, also, that the load upon the luminiferous atoms is a function of the time of oscillation, as well as of the nature of the substance and the position of the axes of oscillation. In conclusion, it may be affirmed, that, as a mathematical system, the proposed theory of oscillations round axes represents the laws of all the phenomena which have hitherto been reduced to theoretical principles, as well, at least, as the existing theory of vibrations ; while, as a physical hypothesis, it is free from the principal objections to which the hypothesis of vibrations is liable. 168 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. IX.— ON THE VELOCITY OF SOUND IN LIQUID AND SOLID BODIES OF LIMITED DIMENSIONS, ESPECIALLY ALONG PEISMATIC MASSES OF LIQUID.* Intkoductory Kemarks. 1. The velocity of sound in elastic substances of different kinds, solid, liquid, and gaseous, has been made the subject of numerous and careful experiments, most of which are well known. The object of this investi- gation is to determine to what extent our present knowledge of the condition and properties of elastic bodies, and of the laws of elasticity, enables us to use those experiments as data for calculating the elasticity of the materials ; and, also, to point out circumstances which, so far as I am aware, have been insufficiently attended to, if not altogether overlooked, in previous theoretical researches, and which must limit our power of drawing definite conclusions from those experiments, until our knowledge of molecular forces shall be in a more advanced state. 2. If it were possible for us to ascertain by experiment the velocities of transmission of vibratory movements along the axes of elasticity of an indefinitely extended mass of any substance, we could at once calculate the coefficients of elasticity of that material ; for in such a mass we can assign the direction of vibratory movement corresponding to each given direction of transmission, and consequently the nature of the molecular forces which are called into play, and whose intensity is indicated by the velocity of transmission. In an uncrystallised medium, for instance, the direction of vibration must either be exactly longitudinal or exactly transverse with respect to the direction of transmission, so that we can calculate from the velocity of transmission the longitudinal or the transverse elasticity, as the case may be. In a crystalline medium having rectangular axes of elasticity, the directions of vibration, though not always exactly longitudinal or transverse, unless the direction of transmission coincides with an axis, have still certain definite positions. 3. It is only in air and water, however, that such experiments are possible. For other substances, the best experiments which it is prac- ticable for us to make, are those upon the transmission of nearly longi- tudinal vibrations along prismatic or cylindrical bodies. Were we able * Eead before the British Association at Ipswich, July 3, 1851, and published in the Cambridge and Dublin Mathematical Journal, Nov., 1851. VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 169 to ensure that the vibrations of those prisms and cylinders should be exactly longitudinal, we might compute from their velocity of transmission, as from that of such vibrations in an unlimited mass, the true longitudinal elasticity. This we can do for gaseous substances, as M. Wertheim has proved (Ann. de Chim. et de Phys., Se>. III., torn, xxiii.), by making the organ-pipes in which they vibrate of proper construction. In liquid and solid columns, on the other hand, it is impossible to prevent a certain amount of lateral vibration of the particles, the effect of which is to diminish the velocity of transmission in a ratio depending on circumstances in the molecular condition of the superficial particles, which are yet almost entirely unknown. 4. It has, indeed, been sometimes supposed, that the coefficient of elasticity, as calculated from the vibrations of a solid rod, is that called the weight of the modulus of elasticity — that is to say, the reciprocal of the fraction by which the length of a rod is increased by a tension applied to its ends of unity of weight upon unity of area; that coefficient being- less than the true coefficient of longitudinal elasticity, because the lateral collapsing of the particles enables them to yield more in a longitudinal direction to a given force, than if their displacements were wholly longitudinal. This conjecture, however, is inconsistent with the mechanics of vibratory movement; and, accordingly, experiment has shown that the elasticity corresponding to the velocity of sound in a rod agrees neither with the modulus of elasticity, nor with the true longitudinal elasticity; although it is, in some cases, nearly equal to the former of those quantities, and in others to the latter. 5. In liquids, it has been shown by the experiments of M. "Wertheim (Ann. de Chim. el de Phys., Se>. III., torn, xxiii.), that the velocity of sound in a mass contained in a trough, and set in motion through an organ-pipe, bears to that in an unlimited mass the ratio of ^2 to y/3. This has led him to form the conjecture, that liquids possess a momentary rigidity for very small molecular displacements as great in comparison with their other elastic forces as that of solids. This conjecture, paradoxical as it may seem, would indeed be necessary to account for the facts, if the supposition I have already mentioned were true, that the velocity of sound in a rod depends upon the modulus of elasticity. I shall show, however, in the sequel, that if we suppose that at the free surface of every mass of liquid, an atmosphere of its own vapour is retained by molecular attraction under certain conditions of equilibrium, the ratio mJ2 : *J3 between the velocities of sound in a prism and an unlimited mass, is a consequence of the equations of motion in all cases in which the liquid has any rigidity whatsoever, even although so small as to be insensible by any means of observation; so that the supposition of a 170 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. rigidity for small displacements equal to that of solids becomes unneces- sary. 6. With respect to solids, all that theory is yet adequate to show us is, that the velocity of sound along a rod must be less than in an unlimited mass, a conclusion in accordance with experiment. The precise ratio depends on properties of the superficial particles yet unknown. General Equations of Vibratory Movement in Homogeneous Bodies. 7. Having now stated generally the objects of this paper, I shall proceed, in the first place, to the mathematical investigation of the integrals of the general differential equations of vibratory movement in homo- geneous bodies; because, although those equations have already been integrated by many mathematicians, it will be necessary in this paper to introduce functions into the integrals which have hitherto been almost totally neglected in such researches; having been applied only to the theory of waves rolling by the influence of gravity, to that of total reflexion, by Mr. Green (Camb. Trans., Vol. VI.), and by Professor Stokes to represent the gradual extinction of sound by its conversion into heat. 8. Let g represent the accelerating force of gravity : D the weight of unity of volume of a homogeneous substance, having orthogonal axes of elasticity whose directions are the same throughout its extent ; A p A 2 , A 3 , the coefficients of longitudinal elasticity for the axes of x, y, z, respectively : B 1? B 2 , B 3 , the coefficients of lateral elasticity ; and C v C 2 , C 3 , those of rigidity for the planes of yz, zx, xy, respectively ; £, 7), £, the displacements of a particle parallel to x, y, z, respectively. Then, it is well known that the differential equations of small vibratory movements are the following, when small quantities of the second order are neglected : . / D d? d? , n d? , n d?\ t g dt 2 1 dz 2 3 dy d' 2 n d 2 ? + (B 3 + C 3 ) -£4- + (B 2 + C 2 ) f-±- x •* 6I dz dy v l u dz dz _/ D d? « <? d* LP i 2 \ ~ Vlj'W + ^da* + 2 dy* + Ul rf7V (1.) VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 171 \ g df "dx l l dy "dz-/ of which the integrals are I = ^{L^i^e.t + ax + (3y + yz+ k)} n .= S {L 2 0(^ £ J + aa + j8y + y 2 + *)} £ = 2{L 3 0(Ve.* + a^; + /3y + ys + *)}• ^ • (I-) Y ■ (2.) The form of the function <£ being arbitrary, subject to a restriction to be afterwards referred to, and 2 extending to any number of terms, the coefficients of which fulfil the following conditions. Let w 1 = A x a 2 + C 3 /3 2 + C 2 y 2 w , = C 3 a 2 + A. 2J 8 2 + C 1 y 2 u, 3 = C 2 a 2 + C 1 /3 2 + A 3 y- Pl = ^ 1 .+ C 1 )i8 7 Pi — ( B 2 + C 2) 7 a p 3 = (B 3 + C 3 )«/3 9 (a.) E Then the following equations must be satisfied by the coefficients of each set of terms in equation (2) : = L x ( Wl - E) + L 2 p 3 + L 3 p 2 - = L lP3 + L 2 (a> 2 -E) + L 3/3l [ . . (3.) = L l Pi + L 2 P\ + L 3 ( W 3 — E )- „ By elimination we transform those equations as follows : let G = w x + w 2 + (o 3 , H = (o. 2 w 3 + (o s w 1 -\r w x Wo — p^ — p 2 — Pz> r* (b-) K =. w x Wo <*) z + 2 pj p 2 p 3 — w x p* — w 2 p 2 — h> z p 3 2 . J 172 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. Then for each set of values of a, /3, y, E has three values which are the roots of the cubic equation, = E 3 -GE 2 + HE-K; . . . (4.) so that ^e has six values, three positive and three negative, of equal arithmetical amount. The absolute values of L v L 2 , L 3 , are arbitrary, but their mutual ratios are fixed by the following equations : L 1 {(w 1 -E)^ 1 -p : ,p. i }=L 2 {(w 2 -E)p. 2 -p3p 1 } "J consequently, they have in general three sets of ratios for each set of values of a, ft, y, corresponding to the three values of E. 9. The condition that the motions of the particles of the body must be small oscillations restricts the variations of the displacements £, rj, £, within certain limits. Now, as the time t increases ad infinitum, this can be fulfilled only when each of those quantities is either a periodical circular function of t, or a function developable into a sum or definite integral of such functions. We may, therefore, make each of the functions <j> a trigonometrical function of t. This being the case, those functions must be either trigonometrical or exponential with respect to x, y, and z, or compounded of both, being trigonometrical so far as a, (3, y, are real, and exponential so far as they are imaginary. If we suppose each of these coefficients to consist of a real and an imaginary part, then each of their functions which enters into the equations of condition, will also consist of a real and an imaginary part. Each of the equations of condition thus becomes divided into two, which must be separately satisfied. Thus we arrive at the following results : For the symbol <£ { }, put e 2 W-l { }; so as to make £, &c, trigono- metrical with respect to t. Let X be a line of such a length that and let a? + l 2 + c 2 = 1, a = z-(+a— a J — 1), = £(T&-&V-1). y = <( + c~ cV~ !) VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 173 also, let Li = I + l's/ — 1> L 2 — m + m ' s/ — li L 3 = w + n'*/ — 1 ; so that the displacements become -{a'x + b'y + c'zdz V-l(V e - i-ax-Jy-cs)} | rj = S {terms in m, m], £ = 2 {terms in w, w'} > a) J Let the quantities in the equations of condition be thus represented «i -i\ ± iAV- !> &c - ; pi = & ± 2W- *> &c - ; g = g ± aV- i ; H = & ± &V- 1 ; K = k ± kV- 1- The equations of notation now become Pl = A x (a 2 - a' 2 ) + C 3 (6 2 - 6' 2 ) + C 2 (c 2 - c' 2 ) ^ 2 = C 3 (a 2 - a' 2 ) + A 2 (6 2 - b' 2 ) + C x (c 2 - c' 2 ) j? 3 = C 2 (a 2 - a' 2 ) + C^fi 2 - 6' 2 ) + A 3 (c 2 - c' 2 ) < ?1 = (B 1 + C 1 )(&c-6V) #2 = (^2 + ^2) ( ca ~ c ' a ') q 3 = (B 3 + C 3 )(ab-a'b') Pi = 2 (A x a a' + C 3 6 6' + Co c c') p 2 = 2 (C 3 a a' + A 2 6 6' + Cj c c') j9 3 ' = 2 (C 2 a a' + C x b b' + A 3 c c') ?1 ' = (B 1 + C 1 )(6c' + 6 / c) ? 2 ' = (B 2 + C 2 )(ca' + c'a) ? 3 '=(B 3 + C 3 )(a&' + a'£) — = Las before, or c = -^r 9 D q =Pl+P 2 + p 3 , q' =pj + 2h + #$' & = P 2 #5 +^1 + ^l2>2 ~ ^l 2 ~ ?2 2 - ?3 2 -P2P&-P3P1-P1P2 + 9i 2 + <li l + 9z 2 # =P 2 P3 +P2P3 +PzP\ + #$>i + Pi Pz+PlPi -2q 1 q 1 , -2q. 2 q 2 '-2q 2 q 3 ' H =p l p. 2 p z +2q l q 2 q s -p 1 q 1 2 -p 2 q 2 2 -p. 3 q^ M c ) He) 174 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. -ftft'ft' - ft'ft ft' "ft'ft'ft +Pi ft' 2 + ft ft' 2 + ft ft' 2 + 2 (ft.' ft ft' + ft'ft ft' + ft' ft ft') K=pMp& + ^^'^Is-Pi^-P^-Pil^ - ft'ft ft - ftft'ft - Pi P2P3 - 2 (fc'ft ft + ft ft' ft + ft ft ^s') + Kft 2 + iVft 2 +Kft 2 + 2 (/ ? lftft' + ^ftft' + ^3ftft')- Also, let *! = (^ - E) q x - p{q x ' - ftft + ft'ft' *2 = (ft - E ) ft " ft'ft' - ft ft + ft'ft' *3 = (ft ~ E ) ft - ft'ft' ~ ft ft + ft'ft' %l = (^ - E) jfc' + ^'ft - q 2 2s - ft'ft X 2 ' = (p 2 - E) q 2 ' + ^ 2 '<7 2 - ft ?/ - ? 3 'ft *s = (ft - E ) ft' + ft'ft - ft ft' - ft'ft J Then the equations of condition relative to the coefficients become the following : — = E 8 -flE2+ |jE-h, = g'E 2 - b'E + k', • lt x + 1'X{= m% 2 + m'X 2 ' = nx z + w%' "| lx{- l'X x = mXo- m'%2 =n% & ' - n'x z . J (8.) (9.) (10.) The three original 'equations of condition are transformed into the following six, to which (8), (9), (10) are equivalent : — = I (p x — E) -r I'pi + mq s + m'q 3 ' + nq 2 + n'q. 2 = l'(Pi — E) — lp x + m'q z — mq s ' + n'q 2 — nq 2 = I q z + l'q z ' + m (p 2 — E) + m'p 2 + nq x + n'q x = l'q s - lq z ' + «i'(ft ~ E) - mp z ' + n'q x - nq x = lq 2 + l'q 2 + mq x + m'q-l + n{p 3 - E) + rip z = l'q 2 - lq 2 + m'q x — mq x + n'(p s — E) — np z y (10A.) To give an intelligible result, the terms of the series in equation (7) must be taken in pairs, with the imaginary exponents in each pair of equal arithmetical value and opposite signs. VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. Hence equations (7) are equivalent to the following : — — {a'x + b'y + cz) f 2 tt . e x -J I. cos-— (^Z e . i — ax — by — cz) 2-rr ") "1 + T . sin — (^/e .t — ax — by — cz)r r\ = S {terms in w, m'}, £= 2 {terms in w, ft'} 175 ^ (H-) The above equations (11), together with the equations of condition (8), (9), (10), or their equivalent (10A), and the equations of notation (c), contain the complete representation of the laws of small molecular oscillations in a homogeneous body of any dimensions and figure ; it being understood that in the symbol of summation 2 are included as many definite integrations as the problem may require with respect to inde- pendent variables of which the coefficients A, ^ z,a,b,c,a' ,b' ,c ,l,m,n,l' ,m ,n , are functions. As there are fourteen coefficients, connected by seven equations — viz., a?-\-b 2 -{-c 2 =l, and the six equations of condition, the greatest number of independent variables is limited to seven ; therefore, in the most general case, the symbol 2 {...} in equations (11) may be replaced by ^ffflffl^v 0» %, e v dp e 6 , e 7 ) { . . } do v ae v ae 9 de„ ae„ ae & ae 7 , (i 2.) V &c, being variables of which the coefficients are functions, and F an arbitrary function.* 10. Let us consider the physical meaning of a single set of terms of the sums in equations (11), containing but one set of values of the coefficients. It represents a system of plane waves, the wave surfaces, or planes of equal phase, in which are normal to the line whose direction- cosines are a, b, c. A is the length of a wave measured along that line. 2 7T I -7T- (*Je .t — ax — by — cz)-\- tan -1 -y ~t- (s/s .t — ax — by — cz) + tan -1 — > A ' m — (*/s .t — ax — by — cz) + tan -1 -7 are the phases of vibration for %, *To make the functions in equations (11) satisfy the conditions of equilibrium, instead of those of oscillation, it is only necessary to make s = 0, and to substitute h=0,k' = 0, for equations (8) and (9). Some additional functions, however, are necessary in order to complete the values of £ , n, £. 176 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. ^/e = J(—^- ) is the normal velocity of propagation, —j- is the periodic time of an oscillation of a particle and the corresponding expressions in m and n are the semi-amplitudes of vibration parallel to x, y, z, respectively; a, b\ c, are proportional to the direction-cosines of a normal to a series of planes of equal amplitude of vibration. The trajectory of each particle affected by a single series of plane waves is in general an ellipse, the position and magnitude of which are 2 7T found as follows. Let (p denote the value of (y/e . t — ax — by — cz), which makes the total displacement ^/{^ + tj 2 + £ 2 ) a maximum or minimum. It is easily seen that where J2 + m 2 + n 2 _ p _ m >2 _ n t 1 1' + in m + /i n' The values of £, tj, ^, calculated from <f> by equations (11), are the co-ordinates of the extremities of the axes of the elliptic trajectory, referred to the natural position of the particle as origin. The processes of summation and definite integration denote the repre- sentation of an arbitrary manner of oscillation by the combination of a definite or indefinite number of systems of plane waves. Case of an Indefinitely Extended Medium. 11. Let the medium, in the first place, be supposed to be indefinitely extended in all directions. This case having been thoroughly investigated by MM. Poisson, Cauchy, Green, MacCullagh, Haughton, Stokes, and others, I shall give merely an outline of the general results. The condition that the motion shall consist of small oscillations, here makes it necessary that the exponential factor in the displacements should in all cases be equal to unity, and therefore that a' = ; V = ; e = j VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 177 and, consequently", each of the accented symbols in equations (c) = 0. Equation (9) vanishes, and the normal velocity of propagation for each set of direction-cosines a, b, c, has, generally speaking, three values, corre- sponding to the three values of E, roots of equation (8). Equations (10) become , ,, , , 1 1 1 I : m : n : : I : m : n : : - : - : — . r, X., r. • (14.) consequently, the phases of £, rj, £, are simultaneous; so that >v /(^ 2 + Z /2 ), x /(m 2 + m' 2 ), *J(n 2 + «' 2 ), are proportional to the direction-cosines of a rectilinear vibratory movement of the semi-amplitude +J(l 2 -\-l' 2 -\-m 2 + m' 2 -\-n 2 -\-n' 2 ), which cosines have in general three sets of values corresponding to the three values of E. It is easily shown that those three directions are at right angles to each other. The number of coefficients being in this case reduced to eleven, connected by six equations — viz., a 2 + 6 2 +c 2 = 1, equation (8), and the proportional equation (14), which is equivalent to four, the greatest number of definite integrations in the operation (12) is restricted to five. Thus it appears that the velocity of transmission of vibratory movement through an indefinitely extended mass, has a set of definite values, not exceeding three, for each position of plane waves. When the direction of propagation coincides with an axis of elasticity, we find those values to be : For vibrations parallel to Velocity of propagation along X . X m % m V ■ ■ m m m z . . m m V(¥) (15.) When the substance is equally elastic in all directions, we have simply, Velocities of propagation in any direction for longitudinal vibrations V(t& 178 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. For transverse vibrations, in any direction perpendicular to that of propagation, m- Hence, experiments on the velocity of sound in an indefinitely extended mass, or one so large as to be practically such, afford the means of directly calculating the coefficients of elasticity. General Case of a Body of Limited Dimensions. 12. It is not so, however, in a body of limited dimensions; for the coefficients a, V , c, in the exponents of the exponential factors, are no longer necessarily null, but have values which must depend on the mole- cular condition of the external surface of the body, and on the forces applied to it. The velocity of propagation is no longer a function of the direction-cosines a, b, c, alone, but also of the coefficients a, b', c. It has in general but one value, corresponding to the common root of the equations (8) and (9). By substituting successively the two roots of equation (9) — viz., 6' + \(\r- r for E in equation (8), the latter is converted into two alternative equations between the six quantities a, b, c, a', b', c\ showing the relations they must have in order that equations (8) and (9) may have a common root. In the only particular problems, however, of which I shall here give the solutions, those relations are obvious without going through that process, for they belong to a class of cases in which the three quantities g', ])', f{', have a common factor; which being made = 0, the necessary conditions are fulfilled. It is obvious that in all cases the effect of the coefficients a', b', c, is to diminish the velocity of propagation. 1 3. The following are the values of the three components of the velocity of a particle : 2* 1 d£ JT2ir , y {a'z + b'y+e'z) C ^ ^7= 2 I -y V E - C { -lsm—(^/£.t-ax-by-cz) + V cos ~ (J t . t- ax - by - cz) } ] f ( 16 -) -t.—^ {terms in m, m) ; y- = 2 {terms in n, n). VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 179 The strains, or coefficients of relative molecular displacement, are as follows : Along T2 tr —{a'x + b'y + c'z) Longitudinal Strains. ■j {la —I' a) cos— (y/s .t — ax— by— cz) -\- (la + 1' a') sin —(^e.t — ax— by — cz) r (1 11 -p = 2 {terms in (in V — m b) } (m b + m' V) } 7 Y -5- = 2 {terms in (nc'—n'c), (nc-\-n'c)} (17.) In the plane yz xy z^dy- Z L Distortions. 2tt — (a'x + b'y + c'z) dz di 1 C / 27T 1 (mc'—m'c -\-nV — n'b) cos — (^A . t— ax— by— cz) + (wc +m'c'-f-7i5 +?''^ / )sin — (^/s.t— ax— by— cz) r dZ d£ -j- + -7- =2 {terms in (na' — na -\-lc—l'c), (na -{-n'a -\-lc-\-l'c')} jy 7 j- -+— = 2 {terms in (Ib' — l'b -\-ma —rn'a), Lb if Cv JO (lb -\-l'b'-\-ma +mV)} J, (17 A.) The pressures on the co-ordinate planes, arising from those strains (using the notation of my paper on the Laws of Elasticity, Cambridge and Dublin Mathematical Journal, February, 1851), are the following: — On yz Normal. n *> — **f.-*- — - — B„ - dy 2 d JZ 180 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. L2tt —(a'x + b'y + c'z) (/.,,, r , ~e x | [A^la -la) \ 2 7T + B 3 (m V - m'6) + B 2 (« c' - n'en . cos -zr-^e.t — ax — by-c .:) + (A x (/a + Va) + B 3 (m6 + m V) + B 2 (» c + »' c')) sin - - (Je .t — a x — ly — e .: A )}] (18.) The other normal pressures are found by substituting symbols accord- ing to the following table : — Plane. Pressure. Coefficients. yz . . . P x . . . A lt B 3 , B 2 , zx . . . P 2 . . . B 3 , A 2 , B v -■' y B 2 , B p A 3 . Tangential I'r- Plane of Distortion Along On the plane "- xy fa y*1 y*J *=-*®+fi Case of an Uncrystallised Medium. 14. I shall now take the particular case of an uncrystallised medium, in which the coefficients of elasticity are the same for all axes, and may be represented thus : — rigidity = C ; fluid elasticity = J ; longitudinal elasticity A = 3 C + J, lateral elasticity B = C + J = A— 2 C. VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 181 The position of the axes being in this case arbitrary, I shall take the direction of propagation as the axis of x, so as to make o=l, b = 0, c = 0. To fulfil the condition that equations (8) and (9) shall have common roots, we must make a = 0, being in this case a common factor of g', |j', fe'. The equations of notation (c) now become i 7 2 = -A6' 2 + C(l-c' 2 ) ^ 3 = -Ac' 2 + C(l -V 1 ) 2l = -(A-C)6V; q 2 = 0; q, = g = (A + 2C)(l-Z/ 2 -c' 2 ) ft = (2AC + C 2 )(1 -b' 2 -c'- 2 ) 2 fe =AC 2 (1 -6' 2 -c' 2 ) 3 g' =0; jj'=0; fe'=0 '1 = (Pi — E )?i + fe'&'i r 2 = °; r s = °; Xi=0; x:=(p 2 - e)? 2 '- &'&; r s ,= (#$- E )&'- &&'• ^ Hence it appears, that for an uncrystallised medium, equation (8) has three roots, viz., — r- (d.) one root . . E = A two equal roots, each E = C i(i - & ,a - n, \ ) (1 _ tf* _ C '2). J (19.) So that the velocity of propagation is less than that in an unlimited mass, in the ratio ^/(l — &' 2 — c' 2 ) : 1. Equation (9) disappears. Equations (10) become or lx 1 = m'x% = w'r 3 ' — Vx 1 = ror 2 ' = nx 3 ' I : m : n' : : — I' : m : n (20.) i.i.I ! ti f a r' 3 J (20A.) 182 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. Equations (10A) become = I {p x — E) + m'jg + riq 2 ' = r(jpj - E) - m j g ' - n j,' = Z'g s ' + m (jp 2 — E) + n q x = — lq^ + m {p, — E) + »'& = l'q 2 ' + m q 1 + n (p 3 - E) = - lq 2 ' + m'q 1 + n'(jp 3 - E) ^ 15. It may be shown that the vibrations corresponding to the roots C (1_Z/2_ C '2) canno t take place in a body of which the surface is free, unless V = 0, c = 0, in which case they are reduced to ordinary transverse vibrations. (See Appendix, No. II.) Nearly-longitudinal Vibrations in an Uncrystallised Medium. 16. For the present, therefore, I shall confine the investigation to the root E = A (1 - V- - c' 2 ), corresponding to the velocity of propagation V—V{^< 1 -'* -*">}• • • (21 -> The vibrations to which this root is applicable may be called nearly-longi- tudinal; because in them the longitudinal component predominates, and their velocity of transmission is a function of the longitudinal elasticity A. This value being substituted for E in the expressions for X v &c, gives ' = — V I ; n = — c I } . = V X ; ri = c X. J . (22.) Which values being substituted in equations (11), (16), (17), (18), give the following results : — For brevity's sake, let also, let $ = 2 { — e^ (Z' cos $ - I sin 0) J- . VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 183 Then the displacements are £ = 2 {e* (Z cos (f> + V sin <p)} = n = 2 {6' e^ (I' cos — Z sin <£)} = £ = 2 {c' e^ (/' cos - J sin 0)} = da <Zs (23.) The velocities of the particles are : ^5 = 2 | ^ ^ • «* (*' cos - Z sin 0) } ^=-2J^ r /v /^ & '^G cos + rsin 0)} [" (24.) dZ = - ? \ ^ +/e . c e* (I cos $ + I' sin 0) J- The longitudinal strains : dj dx = - 2 j ^e^ (Z' cos - Z sin 0) J- ^ = 2 | ~6'2 e * (Z' cos - Z sin 0) j ^ = 2 j ^ c" 2 ^ (r cos - Z sin 0) } a z (. A J The total change of volume : .xdvdz IX j y (25.) The distortions : ^ + ^ = 22 j ^-h'c'e* (V cos rf> - Z sin 0) } <Z,s tfy l A J ^ + ^ = 22J^c'^(Zcosri + Z'sin 9 f>)} V (25.) dx dz I A J ^ + ^ = 22 ( ~ b'e+{l cos + r sin 0) } dy dx IX J J > (26.) 184 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. The pressures due to the displacements are as follows : Normal Pressures. Pj = 2r^ r ^{A(l-P-c /2 ) + 2C(&' 2 + c' 2 )}(r cos 0-Z sin tf>)J P 2 = 2 r^e^{A(l-&' 2 -r' 2 )-2C(l-c' 2 )}(rcos</>-?sin0)] P3 = 2r~ 7r e^{A(l-i /2 -c' 2 )-2C(l-^ 2 )}(rcos«/,-^sin^ The tangential pressures % Q 2 , Q 3 , are found by multiplying the distortions by — C. Let Rj, R 2 , R 3 , be the three components of the pressure exerted by the particles of the body, in consequence of the molecular displacements, at any part of its external surface, the normal to which makes with the axes the angles a, /3, y Then R x = P x cos a + Q 3 cos /3 + Q 2 cos 7 R 2 = Q 3 cos a + P 2 cos /3 + Q x cos y R 3 = Q 2 cos a + Qi cos /3 + P 3 cos y ^ Should there be any surface along which the particles are constrained to slide, it is obvious that at that surface the following condition must be fulfilled : = £ cos o + i? cos /3 + Z cos y ; or if z x = / (x, y) be the equation of the surface, f (27/) (28.) J2. d z x dx dy Were we acquainted with the laws which determine the superficial pressures in vibrating bodies, equations (27) would enable us to determine the values which b' and c must have, in virtue of those laws, during the transmission of sound in a limited mass of an uncrystallised material, and thence the ratio ^/(l — Z>' 2 — c' 2 ) : 1, in which the velocity of sound in such a body is less than in an unlimited mass of the same material. Those laws, however, are as yet a matter of conjecture only. Transmission of a Definite Musical Tone. 17. When the body transmits one or more definite musical tones (which is the case in all experiments capable of yielding useful results), the VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 18/ 50 velocity of propagation must be the same for all the elementary vibrations into which the motion may be resolved: that is to say, 1 — b" 2 — c' 2 must have the same value in all the terms of the sums 2. This affords the means of simplifying the equations. Let 6'2 + c '-2 = tf . i> - ] h cos q . c ' = hsin 6; h being the same for all the terms in the sums 2. Then the velocity of propagation is ' ^ = a/{tt (1 -* 2) } • • • (29 ° and this factor may be removed outside the sign of summation. 2 7T When but one musical tone is transmitted, the factor -y- also may be removed outside that sign, and for 2 { } may be substituted a definite integration, 2/F0 { . . . }d0, F 6 being arbitrary. We have also r e* = e x h (y cos 6 + z sin 6) (30.) 2tt in which : ~- h, y, and z, are independent of 9, and may be treated as A constants in the definite integration. Introducing these modifications into equations (23), &c, we find $ = A (l' cos <j> - I sin <f) 2/«* FQdO Displacements. l = {l cos <p +1' sin <j>) 2/^ F OdO r, = {V cos $ - I sin tfh.'S/cos 6e*F 9d0 I = (l f cos <j> - l sin 0) h. 2/ sin 0e*F0d0 > (31.) Velocities of the Particles. ~ = ^V £ - {V cos - I sin 0)2/^ F0d0 ttf A ^ = - ^ ^e. (Z cos + r sin 0) A. 2/cos 0^ F 0c7 (it A 1SG VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. j^= -^Js.Qcos <p + r sin 0) ft. 2/ sin Oc^FOdO dt dx Longitudinal Strain*. 27r (/' cos - I sin <j>) 2/e^ F6d0 p = ?£(T cos<j> -l8m$)h 2 .'2fcoB 2 0e*F6d0 ^? = ^ (7 cos - Z sin <f) h 2 . 2/ sin 2 Oe^FBdO. clz A CuJ/c Dilatation, d% dri dZ dx dy d : *T ,v {t cosf-l sin <£) ( 1 - /r) 2/e* F d 0. Distortions. 7 7X A -,-" + ', = J (I' cos d> - I sin <h) h- 2 /cos sin 0e*F0d0 dz dy A ^? + ^ = 1^(Z costf, + r sin 0)A2/ S in 0e*F0<Z0 J~ + P= y (' costf, + Z' sin 0)&2/cos de^FOdO. Which, being multiplied by — C, give the tangential pressures Qi» Q» Q 3 , on the co-ordinate planes. Normal Pressures on the Co-ordinate Planes, due to the Displacements. P x = -=- (/' cos $ - I sin <f) {A (1 - /j 2 ) + 2 C/r} 2/ e ^ F d p 2 = ^ (*' cos * - * sin 0) [{A (1 - !r) - 2 C} 2/e^ F 6 d + 2C/r2/sin 2 0^F0</0] 2tt P 3 = — (T cos<j>-l sin 0) [ {A (1 - F) - 2 C} 2/e^ F 0rf + 2C/?2/cos 2 0<^F0<2 0]. J (31.) VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 187 Let R x , R 2 , R 3 , be the components of the pressure exerted by the body, in consequence of the molecular displacements, at a point of its surface normal to the direction (a, /3, y). Also, let cos (3 — sin a cos X , cos y = sin a sin x , so as to make x the axis of polar co-ordinates, and xy the plane from which longitudes X are measured. Then, I> = ?£[cosa(rcos0-Zsin0){A(l -A 2 ) + 2CA 2 } Ife+VBdB 2 sin a (Z cos <f> + X sin <p) C /t 2/cos (0 - X ) e^ F d 0] 2tt, R 2 = — [ - 2 cos a (Z cos + Z' sin <p) C A 2/cos e^ F 0<Z + sina(rcos«/,-/sin^){cos x (A(l-/r)-2C)v^F0 ( /0 + 2 C /r 2/sin sin (0 - x ) e^ F d 0}] R = — [ - 2 cos a (/ cos + I' sin 0) C h 2/sin e^ F d A + sin a (r cos tf> - / sin j,) {sin X (A (1 -A 2 ) - 2 c) 2/«* F rf - 2 C /r 2/cos s i n (0 _ x ) c ^ F rf 0}] , [> (32.) Let P' represent the normal pressure at the given point of the surface due to molecular displacements : then P' = R x cos a + sin a (II., cos X + R 3 sin X ) = P x cos 2 a + P 2 sin 2 a cos 2 X + P 3 sin 2 a sin 2 X -f 2 Q x sin 2 a cos X sin x + 2 Q 2 cos a sin a sin X + 2 Q 3 cos a sin a cos x 2tt / = — [(? cosf—l sin 0) {(A (1 — lr) )-(32A) + 2 C (/i 2 cos 2 a - sin 2 a) ) 2/V* F d + 2 C K- sin 2 a 2/sin 2 (0 - X ) c^ F 6d 0} — 4(/cos0 + *' sin <p) C A cos a sin a 2/ C0S (0 - x) e * FedB 188 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. Propagation of Sound by Nearly-longitudinal Vibrations along a Horizontal Prism of Liquid contained in a Rectangular Trough, investigated according to a Peculiar Hypothesis. 18. I shall now suppose the vibrating body to be a rectangular horizontal prism of liquid contained in a trough of some substance so dense, hard, and smooth, that the particles at the sides and bottom of the trough are constrained to slide along those surfaces, and that the vertical ends of the trough are capable of perfectly reflecting a wave of sound travelling horizontally; so that the propagation of that wave may take place as if in a trough of indefinite length ; and I shall investigate the velocity of such a wave according to a peculiar hypothetical view of the molecular condition of the upper surface of the liquid. The axis of x being the horizontal axis of the trough, and parallel to the direction of propagation, let that of y be transverse, and that of 2 vertical. Let the middle of the bottom of the trough be the origin of co-ordinates, 2y 1 being its breadth, and z 1 the depth of liquid in it. The conditions to be fulfilled at the bottom are, when z = 0, a = ^ ir, and x = ~ i ~- Let , h y cos 6 „ . , „ 2/sin0^F0(Z0 = 2/sin0c A * F6dd = at the sides, when y = ± y v a = l ■ ir, and y_ = or 7r. Let 2-r t „ — ^(±2/1 cos0 + 2Sin4) 2/cos0^F0 f Z0 = 2y"cos0e* F0<Z0 = O; which conditions are fulfilled by making cos = 0, sin = ± 1, and putting for 2/F 6 d 6 a summation of two terms in which the signs of the exponent are respectively positive and negative. Thus we obtain ~^1, £ = (I cos $ + V sin (p) \e A + e 7/ = Z, = (I' cos — I sin (f)h\c x — e x / > y (33.) VELOCITY OF SOUND IX LIQUID AND SOLID BODIES. 189 The trajectory of each particle is an ellipse in a vertical longitudinal plane ; the motion being direct in the upper part of the ellipse, because the sien of — is the same with that of £. The axes are vertical and ° at horizontal respectively, and have the following values : — - V , -hz - —hz Horizontal axis = 2 J (I 2 + I" 2 ) . \e x Vertical axis = 2 J (I 2 + l' 2 ) . h. \e * * - e * 7 ; so that the motion is analogous to that of waves propagated by gravitation, being entirely horizontal at the bottom of the trough, and elliptical else- where, the ellipse being larger and less eccentric as the height above the bottom increases. The ratio of the axes, however, instead of approaching equality as the depth of the trough increases (which is the case with waves of gravitation), approaches 1 : h. 19. To determine this ratio, upon which the velocity of sound along such a mass of liquid must depend, I shall assume the following hypo- thetical principles respecting the state of the particles at the upper surface : First, That (as laid down in a previous paper, Cambridge and Dublin Mathematical Journal, February, 1851) the elasticity of bodies is due partly to the mutual actions of atomic centres producing elasticity both of volume and figure, and partly to a mere fluid elasticity resisting change of volume only, and exerted by atmospheres surrounding those centres ; and that the effect of the mutual actions of the atomic centres in producing pressure is very small in liquids, and absolutely inappreciable in gases and vapours. Secondly, That every liquid maintains at its surface, by molecular attraction, an atmosphere of its own vapour, under these conditions — that the total pressures of the liquid and vapour, and also their fluid pressures, shall be equal at the bounding surface. (From this hypothesis I have already deduced the form of an approximate equation between the pressure and temperature of vapour at saturation.) The total pressure of the vapour on the liquid is sensibly equal to its fluid pressure ; the total pressure of the liquid on the vapour consists of its fluid pressure, and a pressure due to atomic centres ; the latter quantity must, therefore, be null. Thirdly, That the pressure of the vapour follows that of the liquid throughout its variations during the propagation of sound; so that the portion of the pressure of the liquid on the vapour, due to atomic centres, must continue null throughout these variations. Let & be the mutual pressure of the liquid and its vapour in a state 190 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. of rest, then a» + P' is their momentary mutual pressure during the passage of a wave of sound horizontally along the trough. The portion of P' depending on the coefficient of rigidity C being made = 0, we shall obtain an equation from which the value of h may be deduced. Making the proper substitutions in equation (3 2 A), viz. — cos a — 0, sin a = 1, cos ^ = 0, sin ^ = 1, \L = ± -st- h .. cos 6 = 0, sin = ± 1, F 6 = 1, z = z v &c, we find /2r. _2» \ w + r'=w+~(rcos^-/sin0){A(l-/t 2 )-2C}Vc x Zl +e x 7- A The part of this depending on mere fluid elasticity, in which the liquid is followed by the vapour, is \dx dz / 2 t 2 * 2 7T ( h ~l T-nZi = u+ T (/' cos (p — I sin 0) . J (1 — Ir)\c x + e * A which, being subtracted, there remains for the part depending on atomic centres, / 2*- It \ A Consequently, = ^f (T cos - Z sin <j>) C (1 - 3 h°-)\e * + e 1 - 3/i 2 = 0, or h = Jl . . . (33.) is the equation of condition sought, arising from the state of the free surface ; and this equation is independent of the amount of rigidity of the liquid, requiring only that it shall be something, however small, while that of the vapour is null. It follows from this equation, that the velocity of propagation of sound along a trough of liquid of the density D, and longitudinal elasticity A, is += J {&-»}=&¥)• ■ <*> or less than the velocity in an unlimited mass in the ratio of *J2 to y/3. 20. This is precisely the result arrived at by M. Wertheim from a comparison of his numerous experiments on the propagation of sound in water at various temperatures, from 15° to 60° Centigrade, in solutions of various salts, in alcohol, turpentine, and ether (Ann. de Chim,, Ser. Ill, VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 191 torn, xxiii.), with those of M. Grassi on the compressibility of the same substances (Comptes Rendus, XIX., p. 153), and with the experiments of MM. Colladon and Sturm on the velocity of sound in an expanse of water. M. Wertheim having given this comparison in detail, I shall quote one example only. The velocity of sound in an unlimited mass of water, at the temperature of 16° Centigrade, as ascertained by MM. Colladon and Sturm, was 1435 metres per second. That of sound in water contained in a trough, the vibrations of which were regulated by an organ-pipe, was found by M. Wertheim, at 15° Centigrade, to be 11 734 metres per second. The ratio of the squares of those quantities is 0-6G8G : 1, differing from f by 0-0009 only. REMARKS ON THE PROPAGATION OF SOUND ALONG SOLID RODS. 21. I refrain from giving, in the body of this paper, detailed investigations of particular problems respecting the propagation of sound along a solid prism or cylinder; for, in the present state of our knowledge of the condition of the superficial particles of such bodies, the conclusions would be almost entirely speculative and conjectural. I may mention briefly, however, the following general results. If we adopt for solids the same hypothesis as for liquids, then the ratio of the velocity of sound in a rod of an uncrystallised material to that in an unlimited mass has the following values : For a rectangular prismatic rod, the lateral vibrations of the particles of which are confined to planes parallel to one pair of faces of the prism, but are perfectly free in other respects, the ratio is ^2:^/3, being the same as for a liquid. For a cylindrical rod, the surface being perfectly free, the ratio has various values, from ^Jl : ^/f , approaching the less value as the diameter of the rod diminishes, and the greater as it increases ad infinitum. (See Appendix, No. I.) 22. These conditions, however, cannot be realised in practice; and the lateral vibrations being more or less confined by the means used in fixing the rods, we find that the ratio generally exceeds ,J'2 : ^3, and sometimes approaches equality. The following table illustrates this fact. I have selected, in the first place, the experiments of M. Wertheim on tubes of crystal (Ann. de Chim., Ser. III., torn, xxiii.), because in them the coefficients of elasticity and the velocities of sound were ascertained by experiments on the same 192 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. pieces of the material. To these I have added a calculation, founded on a comparison of the experiments of M. Wertheim on the elasticity of brass, with those of M. Savart on the velocity of sound in it, as being the only other data of the kind now existing from which a satisfactory conclusion can be drawn. The coefficients of longitudinal elasticity, calculated by myself from M. Wertheim's experiments, are extracted from my paper on elasticity in the Cambridge and Dublin Mathematical Journal for February, 1851. The quantities - - for crystal are given as calculated by M. Wertheim. For brass I have used the following data : tj e — velocity of sound in brass rods ; mean of many experiments by M. Savart =3560 metres per second. D = density, in kilogrammes per cubic metre, 8395. TABLE. Crystal. Longitudinal Elasticity A Kilogrammes per square millimetre. jD .'/ Kilogrammes per square millimetre. llatio 1 - /** Tube No. L, . 5514-2 5354-0 0-970, „ „ HI, . 5611-0 5476-7 0-976, „ „ IV., . 6183-1 5597-3 0-905, v . 6659-9 5489-8 0-824, Brass, . 15625 10847 0-694. Concluding Remarks. 23. The chief positive results arrived at in this paper may be summed up as follows: — (i.) In liquid and solid bodies of limited dimensions, the freedom of lateral motion possessed by the particles causes vibrations to be propagated less rapidly than in an unlimited mass. (u.) The symbolical expressions for vibrations in limited bodies are distinguished by containing exponential functions of the co-ordinates as factors; and the retardation referred to depends on the coefficients of the co-ordinates in the exponents of those functions, which coefficients depend on the molecular condition of the body's surface — a condition yet imperfectly understood. (in.) If we adopt the hypothetical principle, that at the free surface of a vibrating mass of liquid the normal pressure depending on the actions of atomic VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 193 centres is always null, then we deduce from theory that the ratio of the velocity of sound along a mass of liquid contained in a rectangular trough to that in an unlimited mass is ^f'2 : ^3, that ratio being independent of the specific rigidity of the liquid — a conclusion agreeing with our present experimental knowledge. 24. I do not put forward the hypothetical part of these researches as more than a probable conjecture; nor should I be justified in so doing in the present state of our knowledge of molecular forces. I think, however, that these investigations are sufficient to prove that we are not warranted in concluding from M. Wertheim's experiments (as he is disposed to do) that liquids possess a momentary rigidity as great as that of solids, seeing that any amount of rigidity, howsoever small, will account for the phenomena if we adopt certain suppositions as to mole- cular forces; and to show that our knowledge of those forces is not yet sufficiently advanced to enable us to use experiments on sound as a means of determining the coefficients of elasticity of solids. Appendix. — No. I. Propagation of Sound by Nearly-Longitudinal Vibrations along a Cylindrical Uncrystallised Eod. Let the vibrating body be cylindrical round the axis of x, and let the vibrations of all particles in a given circle round that axis be assumed to be equal and simultaneous. Let r represent the distance of any particle A from the axis of x, and ^, the angle y r. Then To make the results of the definite integrations ^fFQdd independent of the angle ^, we must have F $ = constant, and the limits of integra- tion and 2 tt. The following are the definite integrals which enter into the solution of this problem. N 194 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES, Let 2 7T — — hr = k, /■2t fja-.fccos^ , p -kcosS e= <f«»iao = i\- = y * to Jo Jo A -^L + i la*(r(»+i)) , JJ 9': cosOe 7ccoa tdd > l36.) ""■2| 22n _! pi-i ] !».!>+ 1) i O' cos 2 Oe kcos <>dO ~ d& ~Jo l r v j (2w+ i)* 2n ) "1 _ 2,r L* + 1 2 2n+1 i> + i)i> + 2) J J the values of n comprehending all integers from 1 inclusive. Those series have the following properties : (i.) The term (n) of = term {n — 1) X — - , ; therefore, this series always becomes convergent at the term for which n > \ k. k 2 (n.) Term (n) of 0' = term (n — 1) X series becomes convergent when n 2 — n > i W, 4 (n — l)w ; therefore, this (in.) Term (w) of 0" = term (w — 1) X (2n+ \)k 2 4(2»- !).».(»+ 1)' w 2 n - 1 > 4" therefore, it begins to converge when n 2 — (iv.) Q'=k(e-G"). (v.) Term (n) of 0"= term (n) of x —77 ; a ratio which is \ for w ' 2 n + 2 the first term (w = 0), and approaches equality as n increases ; therefore, 0" when ^ k- is an inappreciably small fraction, — - = \ sensibl}'. 6' And the larger k is, the more nearly is — =1. VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 195 The following table of a few numerical results illustrates this : — ¥ e G" e" 4 2<r 2* e 1-0000 0-5000 0-5000 1 1-2661 0-7010 0-5537 i 3 1-3622 0-7741 0-5683 1 1-5661 09302 0-5490 1 2-2796 1-4843 0-6511 2 4-2523 3-0550 0-7160 3 • 7-1590 5-4238 0-7576 4 11-3019 9-3620 0-8284 The displacements in this case are as follows : — £ = (I cos <p + X sin 0) G rj = (I' cos (j) — I sin <f) h G' cos ^ £ = (7 cos — Z sin ^)/i6' sin x'> (37.) whence it appears that the two transverse displacements ij and £ compose a radial displacement, p = (I' cos <j> — I sin 0)A 6'. . (37A.) Therefore, the trajectory of each particle is an ellipse, in a plane passing through the axis of the cylinder; and the axes of the ellipse are longi- tudinal and radial, and have the following values: Longitudinal axis, = 2 A /(7 2 + P) . ") L . (38.) Eadial axis, . = 2 J{p + P) . h Q' j If we now adopt the same hypotheses with respect to the outer surface of the cylinder that have been used in the problem respecting liquids, we shall have for the mutual pressure of the solid and its atmosphere of vapour »+!?=» +^(l' cos <f>-ltmfi{(A-2V)(l-h*)Q 1 -2Ch*e i "}> A 196 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. © 1? ©j" being the values of those integrals corresponding to the radius of the cylinder. The portion of this pressure depending on mere fluid elasticity is w _j(^+^ + ^ =w + ^ _ Zs \dx ay dz/ A * which, being subtracted, leaves = ~(r cos - J sin 0)C {©! - h 2 (S 1 + 2 ©/')} ; therefore, according to the hypothesis adopted, h* = 1 -ttt,:. ■ • . (39.) 1 + 2 e x and the velocity of sound along the cylinder is Now, the limits of the ratios in the above formulae are the following : • 1, T . ., . 2 7r A r, , Lrmits of — — l = \ . . . A ©/ J • A V2 • • */i» „ */(!-]£) Jl . . Jl That is to say, if the hypothesis already explained with reference to liquids is applicable to a solid cylinder of an uncrystallised material, the velocity of sound along such a cylinder, when its surface is perfectly free, will be less than that in an unlimited mass in some ratio between ^ and J? 6 . VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 197 Appendix. — No. II. General Equations of Nearly-Transverse Vibrations. The two equal roots of equation (8) in uncrystallised bodies, viz. E = C (1 - b' 2 - c'% correspond to what may be called nearly-transverse vibrations, propagated with the velocity x/ £ = V{^ (1 ~ & ' 2 " C ' 2) }- ' * (41>) Equations (20) in this case give no result; but equations (20A) are reducible to the following two : I = — m'b' — n' c' . . . . ) f • • (42); I' — mb' + nc' ) the ratios m : n and m' : n' are arbitrary. Equations (11), (16), (17), (18), become the following: Displacements. £ — 2 [e+ {- (m! b' + ri c') cos + (m b' + n c) sin }] t) = 2 {e^ (m cos + m sin <£) } £ = 2 {e^ (n cos + n' sin cj>)} Velocities of the Particles. ^ = 2^^Je.e* {(m'b' + n'c)sm<p + (mb' + nc')cos<p}J y ^ — = 2 -j ^— ^/ g . e^ (— m sin + m' cos 0) >• dt dj dx = 2 -j ^— a/ e . e^ ( — w sin + %' cos $>) r Longitudinal Strains. = _ 2 I^V {(»» &' + n cos + (m' b' + n' c') sin <p} \ 198 VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. ^ = 2 \~e* (mV cos <h + m'V sin d>)\ ay (A ) dz 2 \ -t— e^ (w c' cos $ -\- n' c sin </j) r Cufo'c Dilatation. dj, , <*JJ , dt __ Q d re rf y rZ « Distortions. > (43.) g+f!= 2 [x^ { -( n/ ( i+ ^ +m ' &v ) co ^ + (n (1 + 0" + rub' A sin 0} | + (m (1 + Z>' 2 ) + nb' A sin 0} | Which, being multiplied by — C, give the tangential pressures %> ^2' **3' Normal Pressures on the Co-ordinate Planes due to the Displacements. P x = 2 C . 2 1 ~ e* {(m V + n c) cos + {m V + w' c) sin </>} | P2 — — 2 C . 2 1 -j—e^ {m V cos <f> -\- m b' sin 0} I P3 — — 2 C . 2j ~y-e^ {71 c' cos <j> -\- n c sin 0} P x + P 2 + P 3 = 0. The normal pressure due to the displacements at any point of the surface of a prism or cylinder described round x is VELOCITY OF SOUND IN LIQUID AND SOLID BODIES. 199 P' = — 2 C . 2 I --- e$ {(m b' cos'-' ^ + n c sin 2 ^ + (m <:' + n b') cos ^ sin ^J cos <p + ( wi' 6' cos" x + w' c' sin 2 y_ + (m' c' + n b') cos ^ sin y) sin 0} I (44.) If this pressure is to be null at all points of the surface, we must have b' = 0, c'=0, and, consequently, Z=0, l' = 0; and the motion is restricted to common exactly-transverse vibrations, for which E = Cand^a = >/ Cd)' Nearly-transverse vibrations, therefore, cannot be transmitted along a cylindrical or prismatic uncry stall ised body whose surface is absolutely free. PART II. PAPERS RELATING TO ENERGY AND ITS TRANSFORMATIONS, THERMODYNAMICS, MECHANICAL ACTION OF HEAT IN THE STEAM ENGINE, &a PART IT. PAPERS RELATING TO ENERGY AND ITS TRANSFORMATIONS, THERMODYNAMICS, MECHANICAL ACTION OF HEAT IN THE STEAM ENGINE, &a X.— ON THE ^CONCENTRATION OF THE MECHANICAL ENERGY OF THE UNIVERSE.* THE following remarks have been suggested by a paper by Professor William Thomson f of Glasgow, on the tendency which exists in nature to the dissipation or indefinite diffusion of mechanical energy originally collected in stores of power. The experimental evidence is every day accumulating, of a law which has long been conjectured to exist, — that all the different kinds of physical energy in the universe are mutually convertible; that the total amount of physical energy, whether in the form of visible motion and mechanical power, or of heat, light, magnetism, electricity, or chemical agency, or in other forms not yet understood, is unchangeable ; the transformations of its different portions from one of those forms of power into another, and their transference from one portion of matter to another, constituting the phenomena which are the objects of experimental physics. Professor William Thomson has pointed out the fact, that there exists (at least in the present state of the known world), a predominating tendency to the conversion of all the other forms of physical energy into heat, and to the uniform diffusion of all heat throughout all matter. The form in which we generally find energy originally collected, is that of a store of chemical power, consisting of uncombined elements. The com- bination of these elements produces energy in the form known by the name of electric currents, part only of which can be employed in analysing * Read before the British Association at Belfast, on September 2, 1S52, and published in the Philosophical Magazine for November, 1852. f ISIow Sir William Thomson. MECHANICAL ENERGY OF THE UNIVERSE. 201 compounds, and thus reconverted into a store of chemical power; the remainder is necessarily converted into heat : a part only of this heat can be employed in analysing compounds, or in reproducing electric currents. If the remainder of the heat be employed in expanding an elastic substance, it may be entirely converted into visible motion, or into a store of visible mechanical power (by raising weights, for example), provided the elastic substance is enabled to expand until its temperature falls to the point which corresponds to absolute privation of heat ; but unless this condition be fulfilled, a certain proportion only of the heat, depending upon the range of temperature through which the elastic body works, can be con- verted, the rest remaining in the state of heat. On the other hand, all visible motion is of necessity ultimately converted, entirely into heat by the agency of friction. There is thus, in the present state of the known world, a tendency towards the conversion of all physical energy into the sole form of heat. Heat, moreover, tends to diffuse itself uniformly by conduction and radiation, until all matter shall have acquired the same temperature. There is, consequently, Professor Thomson concludes, so far as we understand the present condition of the universe, a tendency towards a state in which all physical energy will be in the state of heat, and that heat so diffused that all matter will be at the same temperature ; so that there will be an end of all physical phenomena. ' Vast as this speculation may seem, it appears to be soundly based on experimental data, and to represent truly the present condition of the universe, so far as we know it. My object now is to point out how it is conceivable that, at some indefinitely distant period, an opposite condition of the world may take place, in which the energy which is now being diffused may be recon- centrated into foci, and stores of chemical power again produced from the inert compounds which are now being continually formed. There must exist between the atmospheres of the heavenly bodies a material medium capable of transmitting light and heat ; and it may be regarded as almost certain that this interstellar medium is perfectly transparent and diathermanous ; that is to say, that it is incapable of converting heat, or light (which is a species of heat), from the radiant into the fixed or conductible form. If this be the case, the interstellar medium must' be incapable of acquiring any temperature whatsoever ; and all heat which arrives in the conductible form at the limits of the atmosphere of a star or planet, will there be totally converted, partly into ordinary motion, by the expansion of the atmosphere, and partly into the radiant form. The ordinary motion will again be converted into heat, so that radiant heat is the ultimate form to which all physical energy tends ; and in this form it is, 202 MECHANICAL ENERGY OF THE UNIVERSE. in the present condition of the world, diffusing itself from the heavenly bodies through the interstellar medium. Let it now be supposed, that, in all directions round the visible world, the interstellar medium has bounds beyond which there is empty space. If this conjecture be true, then on reaching those bounds the radiant heat of the world will be totally reflected, and will ultimately be recon- centrated into foci. At each of these foci the intensity of heat may be expected to be such, that should a star (being at that period an extinct mass of inert compounds) in the course of its motions arrive at that part of space, it will be vaporised and resolved into its elements ; a store of chemical power being thus reproduced at the expense of a corresponding amount of radiant heat. Thus it appears, that although, from what we can see of the known world, its condition seems to tend continually towards the equable dif- fusion, in the form of radiant heat, of all physical energy, the extinction of the stars, and the cessation of all phenomena ; yet the world, as now created, may possibly be provided within itself with the means of recon- centrating its physical energies, and renewing its activity and life. For aught we know, these opposite processes may go on together ; and some of the luminous objects which we see in distant regions of space may be, not stars, but foci in the interstellar ether. GENERAL LAW OF THE TRANSFORMATION OF ENERGY. 203 XI.— ON THE GENERAL LAW OF THE TRANSFORMATION OF ENERGY.* Actual, or Sensible Energy, is a measurable, transmissible, and trans- formable condition, whose presence causes a substance to tend to change its state in one or more respects. By the occurrence of such changes, actual energy disappears, and is replaced by Potential or Latent Energy; which is measured by the product of a change of state into the resistance against which that change is made. (The vis viva of matter in motion, thermometric heat, radiant heat, light, chemical action, and electric currents, are forms of actual energy; amongst those of potential energy are the mechanical powers of gravitation, elasticity, chemical affinity, statical electricity, and magnetism). The law of the Conservation of Energy is already known — viz., that the sum of all the energies of the universe, actual and potential, is unchange- able. The object of the present paper is to investigate the law according to which all transformations of energy, between the actual and potential forms, take place. Let V be the magnitude of a measurable state of a substance; U, the species of potential energy which is developed when the state V increases; P, the common magnitude of the tendency of the state V to increase, and of the equal and opposite resistance against which it increases ; so that— dU = FdV; andP = ^. . . . (A.) Let Q be the quantity which the substance possesses, of a species of actual energy whose presence produces a tendency of the state V to increase. It is required to find how much energy is transformed from the actual form Q to the potential form U, during the increment d V; that is to say, the magnitude of the portion of d U, the potential energy developed, which is due to the disappearance of an equivalent portion of actual energy of the species Q. The development of this portion of potential energy is the immediate * Read before the Philosophical Society of Glasgow, on January 5, 1S53, and published in the Proceedings of that Society, Vol. III., No. V. 204 GENERAL LAW OF THE TRANSFORMATION OF ENERGY. effect of the presence in the substance of the total quantity Q of actual energy. Let this quantity be conceived to be divided into indefinitely small equal parts d Q. As those parts are not only equal, but altogether alike in nature and similarly circumstanced, their effects must be equal ; there- fore, the effect of the total energy Q must be equal simply to the effect of Q one of its small parts d Q, multiplied by the ratio -y-y But the effect of the indefinitely small part d Q in causing development of potential energy of the species U, during the increment of state d V, is represented by — whence it follows, that the effect of the presence of the total actual energy Q, in causing transformation of energy from the actual form Q to the potential form U, is expressed by the following formula : — «-^ v w which is the solution required, and is the symbolical expression of the General Law of the Transformation of Energy : — The effect of the whole actual energy present in a substance, in causing transformation of energy, is the sum of the effects of all its parts. The difference between this quantity and the potential energy developed, viz : — ( p -«-;nD« v < dW represents a portion of potential energy, clue to causes different from the actual energy Q. This difference is null, when the resistance ( P = -p\j) against which the state V increases, is simply proportional to the total actual energy Q. It is next proposed to find the quantity of actual energy of the form Q, which must be transmitted to the substance from without, in order that its total actual energy may receive the increment d Q, and its state V at the same time the increment d V. This quantity is composed of three parts — viz., actual energy, which preserves its form, dQ,; actual energy which transforms itself to some unknown form, in consequence of the resistance which is offered to the increase of the total actual energy, L d Q ; actual energy, already deter- GENERAL LAW OF THE TRANSFORMATION OF ENERGY. 205 mined, which transforms itself into potential energy of the form U, p Q . -7-^ • d V ; the sum of these parts being — d.Q = (l+L)dQ + Q.^|.dV, . . (2.) in which nothing remains to be determined except the function L. If Ave subtract from the above formula the total potential energy developed during the increment d Y, viz : — ■we obtain the algebraical sum of the energies, actual and potential, received and developed by the substance during the changes d Q, d V; which is thus expressed : — d¥ = d.Q-d.TJ = (l+L)dQ + (Q^-l)-p.dY. (B.) This quantity must be the exact differential of a function of Q and V; for otherwise it would be possible, by varying the order of the increments d Q, d V, to change the sum of the energies of the universe. It follows that — *J±-A-(qA. -\?-q JLy- dV-dQK^dq ) r - H 'dQ? r ' and, consequently, that l=/(Q) + Q.^/.wv, where /' (Q) is a function of Q and constants, the first derivative of/' (Q). We find at length the following equation — dV=d.Q-d.\J = (l+f®) + Q.^fvdv)dQ+(Q±-l) = ^{q+/(Q) + (q^-i)/p*v} • (3.) which represents the algebraical sum of the energy, actual and potential, received and developed by a substance, when the total actual energy of the species Q, and the state V, receive respectively the increments d Q, d V. It is to be observed, that in the last equation, the symbol J"P . d V denotes a partial integral, taken in treating the particular value of Q, to which it corresponds as a constant quantity ; while d . U represents the real magnitude of the potential energy developed. 206 GENERAL LAW OF THE TRANSFORMATION OF ENERGY. The application of the general law of the transformation of energy may be extended to any number of kinds of energy, actual and potential, by means of the following equation : = 2 {(l +/(«) + Q.S^i/p<T>q{ + 2 {(SQ^- 1>*TJ = rf jsQ + 2/(Q) + 2(2.Q^-l)/p<Zv| . (4.) This equation is the complete expression of the general law of the transformation of energy of all possible kinds, known and unknown. It affords the means, so soon as the necessary experimental data have been obtained, of analysing every development of potential energy, and referring its several portions to the species of actual energy from which they have been produced. Amongst the consequences of this law, the author deduces that which may be called the general principle of the maximum effect of engines. An engine consists essentially in a substance whose changes of state, and of actual energy, between given limits, are so regulated as to produce a permanent transformation of energy. Let Q x be the given superior limit of actual energy ; Q 2 , the inferior limit. To produce the maximum permanent transformation of energy from the actual to the potential form, the substance must undergo a cycle of four operations, viz : — First Operation. The substance, preserving the constant quantity Q l of actual energy, passes from the state V A to the state V B , receiving from without the following quantity of actual energy, which is converted into potential energy: — H -«.-/o/v: p -' n - Second Operation. The substance passes from the superior limit of actual energy Q v to the inferior limit Q 2 . Let V c be the value of the state V at the end of this operation. GENERAL LAW OF THE TRANSFORMATION OF ENERGY. 207 Third Operation. The substance, preserving the constant quantity Q 2 of actual energy,, passes from the state V c to the state V D , transmitting to external sub- stances the following quantity of actual energy, produced by the disap- pearance of potential energy: — h *= q 4/v: p <* v - Fourth Operation. The substance is brought back to its original actual energy Q ls and state V A , thus completing the cycle of operations. In order that the second and fourth operations may be performed without expenditure of energy, the following condition must be fulfilled : — hi k VllY (for Q = Qa) = m /£ p d v (fOT Q = *«■ This being the case, the total expenditure of energy during a cycle of operations will be H v being the quantity converted from the actual to the potential form during the first operation; the energy lost will be H 2 , the quantity reconverted to the actual form, and transmitted to external substances, during the third operation ; and the quantity of energy per- manently transformed from the actual to the potential form, that is to say, the work done by the engine, will be — H 1 -H 2 =(Q 1 -Q 2 )^|^P^V(forQ = Q 1 ) . (6.) Q The ratio of this work to the total expenditure of energy is H t — H 9 _ Q 2 — Q (7.) This principle is applicable to all possible engines, known and unknown. In the sequel of the paper, the author gives some examples of the application of the general principles of the transformation of energy to the theory of heat, and to that of electro-magnetism ; and deduces from them, as particular cases, several laws already known through specific researches. The details of the application of these principles to the theory of 208 GENERAL LAW OF THE TRANSFORMATION OF ENERGY. heat are contained in the sixth section of a memoir read before the Royal Society of Edinburgh, " On the Mechanical Action of Heat." The actual energy produced by an electric pile in unity of timo is expressed by — where M is the electro-motive force, and u, the strength of the current. The actual energy of an electric circuit is expressed by — ~Ru 2 , where R is the resistance of the circuit. This energy is immediately and totally transformed into sensible heat. The proportion of the actual energy produced in the pile, which is transformed into mechanical work by an electro-dynamic machine, is represented by — Q t -Q 2 _ M-Rtt Q 2 M ' The strength of the current is known to be found by means of the equation — M-N where N is the negative or inverse electro-motive force of the apparatus by means of which electricity is transformed into mechanical work. Hence, Qi - Q 2 _ n Qi ~M" The above particular forms of the general equation agree with formulae already deduced from special researches by Mr. Joule and Professor William Thomson. OUTLINES OF THE SCIENCE OF ENERGETICS. 209 XII— OUTLINES OF THE SCIENCE OF ENEKGETICS.* Section I. — What constitutes a Physical Theory. An essential distinction exists between two stages in the process of advancing our knowledge of the laws of physical phenomena; the first stage consists in observing the relations of phenomena, whether of such as occur in the ordinary course of nature, or of such as are artificially produced in experimental investigations, and in expressing the relations so observed by propositions called formal laws. The second stage consists in reducing the formal laws of an entire class of phenomena to the form of a science ; that is to say, in discovering the most simple system of principles, from which all the formal laws of the class of phenomena can be deduced as consequences. Such a system of principles, with its consequences methodically deduced, constitutes the physical theory of a class of phenomena. A physical theory, like an abstract science, consists of definitions and axioms as first principles, and of propositions, their consequences ; but with these differences: — First, That in an abstract science, a definition assigns a name to a class of notions derived originally from observation, but not necessarily corresponding to any existing objects of real pheno- mena; and an axiom states a mutual relation amongst such notions, or the names denoting them : while in a physical science, a definition states properties common to a class of existing objects, or real phenomena ; and a physical axiom states a general law as to the relations of phenomena. And, secondly, That in an abstract science, the propositions first discov- ered are the most simple ; whilst in a physical theory, the propositions first discovered are in general numerous and complex, being formal laws, the immediate results of observation and experiment, from which the definitions and axioms are subsequently arrived at by a process of reason- ing differing from that whereby one proposition is deduced from another in an abstract science, partly in being more complex and difficult, and partly in being, to a certain extent, tentative — that is to say, involving the trial of conjectural principles, and their acceptance or rejection, according as their consequences are found to agree or disagree with the formal laws deduced immediately from observation and experi- ment. * Read before the Philosophical Society of Glasgow on May 2, 1855, and published in the Proceedings of that Society, Vol. III., No. VL O 210 OUTLINES OF THE SCIENCE OF ENERGETICS. Section II. — The Abstractive Method of forming a Physical Theory distinguished from the Hypothetical Method. Two methods of framing a physical theory may be distinguished, characterised chiefly by the manner in which classes of phenomena are defined. They may be termed, respectively, the abstractive and the hypothetical methods. According to the abstractive method, a class of objects or phenomena is defined by describing, or otherwise making to be understood, and assigning a name or symbol to, that assemblage of properties which is common to all the objects or phenomena composing the class, as perceived by the senses, without introducing anything hypothetical. According to the hypothetical method, a class of objects or pheno- mena is defined, according to a conjectural conception of their nature, as being constituted, in a manner not apparent to the senses, by a modifica- tion of some other class of objects or phenomena whose laws are already known. Should the consequences of such a hypothetical definition be found to be in accordance with the results of observation and experiment, it serves as the means of deducing the laws of one class of objects or phenomena from those of another. The conjectural conceptions involved in the hypothetical method may be distinguished into two classes, according as they are adopted as a pro- bable representation of a state of things which may really exist, though imperceptible to the senses, or merely as a convenient means of expressing the laws of phenomena ; two kinds of hypotheses, of which the former may be called objective, and the latter subjective. As examples of objec- tive hypotheses may be taken, that of vibrations or oscillations in the theory of light, and that of atoms in chemistry; as an example of a subjective hypothesis, that of magnetic fluids. Section III. — The Science of Mechanics considered as an Illustration of the Abstractive Method. The principles of the science of mechanics, the only example yet exist- ing of a complete physical theory, are altogether formed from the data of experience by the abstractive method. The class of objects to which the science of mechanics relates — viz., material bodies — are defined by means of those sensible properties which they all possess — viz., the pro- perty of occupying space, and that of resisting change of motion. The two classes of phenomena to which the science of mechanics relates are distinguished by two words, motion and force — motion being a word OUTLINES OF THE SCIENCE OF ENERGETICS. 211 denoting that which is common to the fall of heavy bodies, the flow of streams, the tides, the winds, the vibrations of sonorous bodies, the revolutions of the stars, and, generally, to all phenomena involving change of the portions of space occupied by bodies ; and force, a word denoting that which is common to the mutual attractions and repulsions of bodies, distant or near, and of the parts of bodies, the mutual pressure or stress of bodies in contact, and of the parts of bodies, the muscular exertions of animals, and, generally, to all phenomena tending to produce or to prevent motion. The laws of the composition and resolution of motions, and of the composition and resolution of forces, are expressed by propositions which are the consequences of the definitions of motion and force respectively. The laws of the relations between motion and force are the consequences of certain axioms, being the most simple and general expressions for all that has been ascertained by experience respecting those relations. Section IV. — Mechanical Hypotheses in Various Branches of Physics. The fact that the theory of motions and motive forces is the only complete physical theory, has naturally led to the adoption of mechanical hypotheses in the theories of other branches of physics; that is to*say, hypothetical definitions, in which classes of phenomena are defined con- jecturally as being constituted by some kind of motion or motive force not obvious to the senses (called molecular motion or force), as when light and radiant heat are defined as consisting in molecular vibrations, thermo- metric heat in molecular vortices, and the rigidity of solids in molecular attractions and repulsions. The hypothetical motions and forces are sometimes ascribed to hypo- thetical bodies, such as the luminiferous ether; sometimes to hypothetical jparts, whereof tangible bodies are conjecturally defined to consist, such as atoms, atomic nuclei with elastic atmospheres, and the like. A mechanical hypothesis is held to have fulfilled its object, when, by applying the known axioms of mechanics to the hypothetical motions and forces, results are obtained agreeing with the observed laws of the classes of phenomena under consideration; and when, by the aid of such a hypo- thesis, phenomena previously unobserved are predicted, and laws antici- pated, it attains a high degree of probability. A mechanical hypothesis is the better the more extensive the range of phenomena whose laws it serves to deduce from the axioms of mechanics ; and the perfection of such a hypothesis would be, if it could, 212 OUTLINES OF THE SCIENCE OF ENERGETICS. by means of one connected system of suppositions, be made to form a basis for all branches of molecular physics. Section V. — Advantages and Disadvantages of Hypothetical Theories. It is well known that certain hypothetical theories, such as the wave theory of light, have proved extremely useful, by reducing the laws of a various and complicated class of phenomena to a few simple principles, and by anticipating laws afterwards verified by observation. Such are the results to be expected from well-framed hypotheses in every branch of physics, when used with judgment, and especially with that caution which arises from the consideration, that even those hypo- theses whose consequences are most fully confirmed by experiment never can, by any amount of evidence, attain that degree of certainty which belongs to observed facts. Of mechanical hypotheses in particular, it is to be observed, that their tendency is to combine all branches of physics into one system, by making the axioms of mechanics the first principles of the laws of all phenomena — an object for the attainment of which an earnest wish was expressed by Newton.* In the mechanical theories of elasticity, light, heat, and electricity, considerable progress has been made towards that end. The neglect of the caution already referred to, however, has caused some hypotheses to assume, in the minds of the public generally, as well as in those of many scientific men, that authority which belongs to facts alone ; and a tendency has, consequently, often evinced itself to explain away, or set aside, facts inconsistent with these hypotheses, which facts, rightly appreciated, would have formed the basis of true theories. Thus, the fact of the production of heat by friction, the basis of the true theory of heat, was long neglected, because inconsistent with the hypothesis of caloric ; and the fact of the production of cold by electric currents, at certain metallic junctions, the key (as Professor William Thomson recently showed) to the true theory of the phenomena of thermo-electricity, was, from inconsistency with prevalent assumptions respecting the so-called " electric fluid," by some regarded as a thing to be explained away, and by others as a delusion. Such are the evils which arise from the misuse of hypotheses. Utinam caetera naturae phenomena ex principiis mechanicis eodem argumentandi genere derivare liceret.— {Phil. Nat. Prin. Math.; Protf.) OUTLINES OF THE SCIENCE OF ENERGETICS. 213 Section VI. — Advantages of an Extension of the Abstractive Method of framing Theories. Besides the perfecting of mechanical hypotheses, another and an entirely distinct method presents itself for combining the physical sciences into one system; and that is, by an extension of the Abstractive Process in framing theories. The abstractive method has already been partially applied, and with success, to special branches of molecular physics, such as heat, electricity, and magnetism. We are now to consider in what manner it is to be applied to physics generally, considered as one science. Instead of supposing the various classes of physical phenomena to be constituted, in an occult way, of modifications of motion and force, let us distinguish the properties which those classes possess in common with each other, and so define more extensive classes denoted by suitable terms. For axioms, to express the laws of those more extensive classes of phenomena, let us frame propositions comprehending as particular cases the laws of the particular classes of phenomena comprehended under the more extensive classes. So shall we arrive at a body of principles, applicable to physical phenomena in general, and which, being framed by induction from facts alone, will be free from the uncertainty which must always attach, even to those mechanical hypotheses whose consequences are most fully confirmed by experiment. This extension of the abstractive process is not proposed in order to supersede the hypothetical method of theorising; for in almost every branch of molecular physics it may be held, that a hypothetical theory is necessary, as a preliminary step, to reduce the expression of the phenomena to simplicity and order, before it is possible to make any progress in framing an abstractive theory. Section VII. — Nature of the Science of Energetics. « Energy, or the capacity to effect changes, is the common characteristic of the various states of matter to which the several branches of physics relate; if, then, there be general laws respecting energy, such laws must be applicable, mutatis mutandis, to every branch of physics, and must express a body of principles as to physical phenomena in general. In a paper read before the Philosophical Society of Glasgow, on the 5 th of January, 1853 (seep. £08), a first attempt was made to investigate such principles by defining actual energy and potential energy, and by demonstrat- ing a general law of the mutual transformations of those kinds of energy, 214 OUTLINES OF THE SCIENCE OF ENERGETICS. of which one particular case is a previously known law of the mechanical action of heat in elastic bodies, and another, a subsequently demonstrated law which forms the basis of Professor William Thomson's theory of thermo-electricity. The object of the present ' paper is to present, in a more systematic form, both these and some other principles, forming part of a science whose subjects are, material bodies and physical phenomena in general, and which it is proposed to call the Science of Energetics. Section VIII. — Definitions of Certain Terms. The peculiar terms which will be used in treating of the Science of Energetics are purely abstract ; that is to say, they are not the names of any particular object, nor of any particular phenomena, nor of any particular notions of the mind, but are names of very comprehensive classes of objects and phenomena. About such classes it is impossible to think or to reason, except by the aid of examples or of symbols. General terms are symbols employed for this purpose. Substance. The term "substance" will be applied to all bodies, parts of bodies, and systems of bodies. The parts of a substance may be spoken of as distinct substances, and a system of substances related to each other may be spoken of as one complex substance. Strictly speaking, the term should be "material substance;" but it is easily borne in mind, that in this essay none but material substances are referred to. Properly. The term "property" will be restricted to invariable properties; whether such as always belong to all material substances, or such as constitute the invariable distinctions between one kind of substance and another. Mass. Mass means " quantity of substance." Masses of one kind of substance may be compared together by ascertaining the numbers of equal parts which they contain ; masses of substances of different kinds are compared by means to be afterwards referred to. Accident. The term "accident" will be applied to every variable state of substances, whether consisting in a condition of each part of a substance, how small soever, (which may be called an absolute accident), or in a physical relation OUTLINES OF THE SCIENCE OF ENERGETICS. 215 between parts of substances, (which may be called a relative accident). Accidents to be the subject of scientific inquiry, must be capable of being measured and expressed by means of quantities. The quantity, even of an absolute accident, can only be expressed by means of a mentally- conceived relation. The whole condition or state of a substance, so far as it is variable, is a complex accident; the independent quantities which are at once necessary and sufficient to express completely this complex accident, are independent accidents. To express the same complex accident, different systems of independent accidents may be employed; but the number of independent -accidents in each system will be the same. Examples. — The variable thermic condition of an elastic fluid is a complex accident, capable of being completely expressed by two independent accidents, which may be any two out of these three quantities — the temperature, the densihj, the pressure — or any two independent functions of these quantities. The condition of strain at a point in an elastic solid, is a complex acci- dent, capable of being completely expressed by six independent accidents, which may be the three elongations of the dimensions and the three distortions of the faces of a molecule originally cubical, or the lengths and directions of the axes of the ellipsoidal figure assumed by a molecule originally spherical ; or any six independent functions of either of those systems of quantities. The distinction of accidents into absolute and relative is, to a certain extent, arbitrary; thus, the figure and dimensions of a molecule may be regarded as absolute accidents when it is considered as a whole, or as relative accidents when it is considered as made up of parts. Most kinds of accidents are necessarily relative; but some kinds can only be considered as relative accidents when some hypothesis is adopted as to the occult condition of the substances which they affect, as when heat is ascribed hypothetically to molecular motions; and such suppositions are excluded from the present inquiry. Accidents may be said to be homogeneous when the quantities expressing them are capable of being put together, so that the result of the com- bination of the different accidents shall be expressed by one quantity. The number of heterogeneous kinds of accidents is evidently indefinite. Effort, or Active Accident. The term "effort" will be applied to every cause which .varies, or tends to vary, an accident. This term, therefore, comprehends not merely forces or pressures, to which it is usually applied, but all causes of variation in the condition of substances. Efforts may be homogeneous or heterogeneous. 21 G OUTLINES OF THE SCIENCE OF ENERGETICS. Homogeneous efforts are compared by balancing them against each other. An effort being a condition of the parts of a substance, or a relation between substances, is itself an accident, and may be distinguished as an " active accident." With reference to a given limited substance, internal efforts are those which consist in actions amongst its parts; external efforts those which consist in actions between the given substance and other substances. Passive Accident. The condition which an effort tends to vary may be called a "passive accident" and when the word "accident" is not otherwise qualified, ''passive accident" may be understood. Radical Accident. If there be a quantity such that it expresses at once the magnitude of the passive accident caused by a given effort, and the magnitude of the active accident or effort itself, let the condition denoted by that quantity be called a " radical accident." [The velocity of a given mass is an example of a radical accident, for it is itself a passive accident, and also the measure of the kind of effort called accelerative force, which, acting for unity of time, is capable of producing that passive accident.] [The strength of an electric current is also a radical accident.] Effort as a Measure of Mass. Masses, whether homogeneous or heterogeneous, may be compared by means of the efforts required to produce in them variations of some particular accident. The accident conventionally employed for this pur- pose is velocity. Work u Work " is the variation of an accident by an effort, and is a term comprehending all phenomena in which physical change takes place. Quantity of work is measured by the product of the variation of the passive accident by the magnitude of the effort, when this is constant; or by the integral of the effort, with respect to the passive accident, when the effort is variable. Let x denote a passive accident ; X an effort tending to vary it ; W the work performed in increasing x from x to x x : then, OUTLINES OF THE SCIENCE OF ENERGETICS. 217 W = f X Xdx, and W = X (x x —£(,), if X is constant. (10 Work is represented geometrically by the area of a curve, whereof the abscissa represents the passive accident, and the ordinate, the effort. Energy, Actual and Potential. The term "energy" comprehends every state of a substance which constitutes a capacity for performing work. Quantities of energy are measured by the quantities of work which they constitute the means of performing. " Actual energy" comprehends those kinds of capacity for performing work which consist in particular states of each part of a substance, how small soever ; that is, in an absolute accident, such as heat, light, electric current, vis viva. Actual energy is essentially positive. "Potential energy" comprehends those kinds of capacity for performing work which consist in relations between substances, or parts of substances; that is, in relative accidents. To constitute potential energy there must be a passive accident capable of variation, and an effort tending to produce such variation; the integral of this effort, with respect to the possible variation of the passive accident, is potential energy, which differs in work from this — that in work the change has been effected, which, in potential energy, is capable of being effected. Let x denote an accident; x v its actual value; X, an effort tending to vary it; x Q , the value to which the effort tends to bring the accident ; then X d x = U, denotes potential energy. r J x, Examples of potential energy are, the chemical affinity of uncombined elements ; the energy of gravitation, of magnetism, of electrical attraction and repulsion, of electro-motive force, of that part of elasticity which arises from actions between the parts of a body, and, generally, of all mutual actions of bodies, and parts of bodies. Potential energy may be passive or negative, according as the effort in question is of the same sign with the variation of the passive accident, or of the opposite sign ; that is, according as X is of the same sign with dx, or of the opposite sign. 218 OUTLINES OF THE SCIENCE OF ENERGETICS. It is to be observed, that the states of substances comprehended under the term actual energy, may possess the characteristics of potential energy also ; that is to say, may be accompanied by a tendency or effort to vary relative accidents ; as heat, in an elastic fluid, is accompanied by a ten- dency to expand; that is, an effort to increase the volume of the receptacle containing the fluid. The states to which the term potential energy is especially applied, aro those which are solely due to mutual actions. To put a substance into a state of energy, or to increase its energy, is obviously a hind of work. Section IX. — First Axiom. All kinds of Work and Energy are Homogeneous. This axiom means, that any kind of energy may be made the means of performing any kind of work. It is a fact arrived at by induction from experiment and observation, and its establishment is more especially due to the experiments of Mr. Joule. This axiom leads, in many respects, to the same consequences with the hypothesis that all those kinds of energy which are not sensibly the results of motion and motive force are the results of occult modifications of motion and motive force. But the axiom differs from the hypothesis in this, that the axiom is simply the generalised allegation of the facts proved by experience, while the hypothesis involves conjectures as to objects and phenomena which never can be subjected to observation. It is the truth of this axiom which renders a science of energetics possible. The efforts and passive accidents to which the branches of physics relate are varied and heterogeneous ; but they are all connected with energy, a uniform species of quantity which pervades every branch of physics. This axiom is also equivalent to saying, that energy is transformable and transferable (an allegation which, in the previous paper referred to, was included in the definition of energy) ; for, to transform energy, means to employ energy depending on accidents of one kind in putting a substance into a state of energy depending on accidents of another kind ; and to transfer energy, means to employ the energy of one substance in putting another substance into a state of energy, both of which are kinds of work, and may, according to the axiom, be performed by means of any kind of energy. OUTLINES OF THE SCIENCE OF ENERGETICS. 219 Section X. — Second Axiom. TJie Total Energy of a Substance cannot he altered by the Mutual Actions of its Parts. Of the truth of this axiom there can be no doubt; but some difference of opinion may exist as to the evidence on which it rests. There is ample experimental evidence from which it might be proved; but independently of such evidence, there is the argument, that the law expressed by this axiom is essential to the stability of the universe, such as it exists. The special application of this law to mechanics is expressed in two ways, which are virtually equivalent to each other, the principle of vis viva, and that of the equality of action and reaction. The latter principle is demonstrated by Newton, from considerations connected with the stability of the universe (Principia, Scholium to the Laws of Motion) ; for he shows, that but for the equality of action and reaction, the earth, with a continually accelerated velocity, would fly away through infinite space. It follows, from the second axiom, that all work consists in the transfer and transformation of energy alone; . for otherwise the total amount of energy would be altered. Also, that the energy of a substance can be varied by external efforts alone. Section XL — External Potential Equilibrium. The entire condition of a substance, so far as it is variable, as explained in Sect. VIII., under the head of accident, is a complex accident, which may be expressed in various ways by means of different systems of quantities denoting independent accidents ; but the number of independent accidents in each system must be the same. The quantity of work required to produce any change in the condition of the substance, that is to say, the potential energy received by it from without during that change, may in like manner be expressed in different ways by the sums of different systems of integrals of external efforts, each integrated with respect to the independent accident which it tends to augment ; but the number of integrals in each system, and the number of efforts, like the number of independent accidents, must be the same ; and so also must the suras of the integrals, each sum representing the same quantity of work in a different way. The different systems of efforts which correspond to different systems of independent accidents, each expressing the same complex accident, may 220 OUTLINES OF THE SCIENCE OF ENELIGETICS. be called equivalent systems of efforts; and the finding of a system of efforts equivalent to another may be called conversion of efforts. * When the law of variation of potential energy, by a change of condition of a substance, is known, the system of external efforts corresponding to any system of independent accidents is found by means of this principle : Each effort is equal to the rate of variation of the potential energy with respect to the independent accident which that effort tends to vary; or, symbolically, X-£ .... (,) External Potential Equilibrium of a substance takes place, when the external effort to vary each of the independent accidents is null ; that is to say, when the rate of variation of the potential energy of the substance with the variation of each independent a null. For a given substance there are as many conditions of equilibrium, of the form - = 0, .... (3.) as there are independent accidents in the expression of its condition. The special application of this law to motion and motive force consti- tutes the principle of virtual velocities, from which the whole science of statics is deducible. Section XII. — Internal Potential Equilibrium. The internal potential equilibrium of a substance consists in the equili- brium of each of its parts, considered separately ; that is to say, in the nullity of the rate of variation of the potential energy of each part with respect to each of the independent accidents on which the condition of such part depends. Examples of particular cases of this principle are, the laws of the equilibrium of elastic solids, and of the distribution of statical electricity. Section XIII. — Third Axiom. The Effort to Perform Work of a Given Kind, caused by a Given Quantity of Actual Energy, is the Sum of the Efforts caused by the Parts, of that Quantity. A law equivalent to this axiom, under the name of the "General * The conversion of efforts in physics is connected with the theory of lineal trans- formations in aliiehra. OUTLINES OF THE SCIENCE OF ENERGETICS. 221 Law of the Transformation of Energy," formed the principal subject of the previous paper already referred to. (See p. £03.) This axiom appears to be a consequence of the definition of actual energy, as a capacity for performing woi'k possessed by each part of a substance independently of its relations to other parts, rather than an independent proposition. Its applicability to natural phenomena arises from the fact, that there are states of substances corresponding to the definition of actual energy. The mode of applying this third axiom is as follows : — Let a homogeneous substance possess a quantity Q, of a particular kind of actual energy, uniformly distributed, and let it be required to determine the amount of the effort arising from the actual energy, which tends to perform a particular kind of work "W, by the variation of a particular passive accident x. The total effort to perform this kind of work is represented by the rate of its increase relatively to the passive accident, viz., — d X ' Divide the quantity of actual energy Q into an indefinite number of indefinitely small parts SQ; the portion of the effort X due to each of those parts will be and adding these partial efforts together, the effort caused by the whole quantity of actual energy will be „dX _ d*W , x If this be equal to the effective effort X, then that effort is simply proportional to, and wholly caused by, the actual energy Q. This is the case of the pressure of a perfect gas. and the centrifugal force of a moving- body. If the effort caused by the actual energy differs from the effective effort, their difference represents, when the former is the less, an additional effort, and when the former is the greater, a counter effort }■ (5.) due to some other cause or causes. 222 OUTLINES OF THE SCIENCE OF ENERGETICS. Section XIV. — Rate of Transformation; Metamorphic Function. The effort to augment a given accident x, caused by actual energy of a given kind Q, may also be called the "rate of transformation" of tho given kind of actual energy, with increase of the given accident ; for the limit of the amount of actual energy which disappears in performing work by an indefinitely small augmentation dx, of the accident, is dII = Q d ~dx .... (6.) a v^j n <?w , aw — QjTTj - dx = Qd- dQ,dx aQ The last form of the above expression is obviously applicable when the work W is the result of the variation of any number of independent accidents, each by the corresponding effort. For example, let x, y, z, &c, be any number of independent accidents, and X, Y, Z, &c, the efforts to augment them ; so that dV? = Xdx + Ydy + Zdz + &c. Then, 7TT ._ f d X 7 dY , dZ , . ) /n . '*-*il$*' + i$'t+i$" + *- J ' (7 -> = Q d -777 , as before. d*4 The function of actual energy, efforts, and passive accidents, denoted by S-i?-'.- • • ■« whose variation, multiplied by the actual energy, gives the amount of actual energy transformed in performing the work d W, may be called the " Metamorphic Function " of the kind of actual energy Q, relatively to the kind of work W. When this metamorphic function is known for a given homogeneous substance, the quantity H of actual energy of the kind Q transformed to the kind of work W, during a given operation, is found by taking the integral H JQdF (9.) The transformation of actual energy into work by the variation of passive accidents is a reversible operation; that is to say, if the passive OUTLINES OF THE SCIENCE OF ENEEGETICS. 223 accidents be made to vary to an equal extent in an opposite direction, potential energy will be exerted upon the substance, and transformed into actual energy: a case represented by the expression (9) becoming negative. The metamorphic function of heat relatively to expansive power, was first employed in a paper on the Economy of Heat in Expansive Machines, read before the Royal Society of Edinburgh in April, 1851. (Trans. Boy. Soc. Edin., Vol. XXI.) The metamorphic function of heat relatively to electricity was employed by Professor William Thomson, in a paper on Thermo-Electricity, read before the Eoyal Society of Edinburgh in May, 1854 (Trans. Boy. Soc. Edin., Vol. XXL), and was the means of anticipating some most remarkable laws, afterwards confirmed by experiment. Section XV. — Equilibrium of Actual Energy ; Metabatic Function. It is known by experiment, that a state of actual energy is directly transferable ; that is to say, the actual energy of a particular kind (such as heat), in one substance, may be diminished, the sole work performed being an equal augmentation of the same kind of actual energy in another substance. Equilibrium of actual energy of a particular kind Q between substances A and B, takes place when the tendency of B to transfer this kind of energy to B is equal to the tendency of B to transfer the same kind of energy to A. Laws respecting the equilibrium of particular kinds of actual energy have been ascertained by experiment, and in some cases anticipated by means of mechanical hypotheses, according to which all actual energy con- sists in the vis viva of motion. The following law will now be proved, respecting the equilibrium of actual energy of all possible kinds : — Theorem. — If equilibrium of actual energy of a given kind take PLACE BETWEEN A GIVEN PAIR OF SUBSTANCES, POSSESSING RESPECTIVELY QUANTITIES OF ACTUAL ENERGY OF THAT KIND IN A GIVEN RATIO, THEN THAT EQUILIBRIUM WILL SUBSIST FOR EVERY PAIR OF QUANTITIES OF ACTUAL ENERGY BEARING TO EACH OTHER THE SAME RATIO. Demonstration. — The tendency of one substance to transfer actual energy of the kind Q to another, must depend on some sort of effort, whose nature and laws may be known or unknown. Let Y A be this effort for 224 OUTLINES OF THE SCIENCE OF ENERGETICS. the substance A, Y B the corresponding effort for the substance B. Then a condition of equilibrium of actual energy is Y A = Y B (10.) The effort Y may or may not be proportionate to the actual energy Q multiplied by a quantity independent of Q, Case first. — If it is so proportional, let K being independent of Q ; then the condition of equilibrium becomes or Qb _ Kj, a ratio independent of the absolute amounts of actual energy. Case second. — If the effort Y is not simply proportional to the actual energy Q, the portion of it caused by that actual energy, according to the principle of Sect, XIII., deduced from the third axiom, is, for each substance, o' /v and a second condition of equilibrium of actual energy is furnished by the equation *& = *& ' • ' (11 ' ) In order that this condition may be fulfilled simultaneously with the con- dition (10), it is necessary that <ZQ A _ dQ B that is to say, that the ratio of the quantities of actual energy in the two substances should be independent of those quantities themselves ; a con- dition expressed, as before, by ?i = ?? (11.) Q.E.D. This ratio is a quantity to be ascertained by experiment, and may be OUTLINES OF THE SCIENCE OF ENERGETICS. 225 called the ratio of the specific actual energies of the substances A and B, for the kind of energy under consideration. The function K - K -e, .... (i2 v whose identity for the two substances expresses the condition of equili brium of the actual energy Q between them, may be called the " meta- batic FUNCTION " for that kind of energy. In the science of thermo-dynamics, the metabatic function is absolute temperature; and the factor K is real specific heat. The theorem stated above, when applied to heat, amounts to this : that the real specific heat of a substance is independent of its temperature. Section XVI. — Use of the Metabatic Function; Transformation of Energy in an aggregate. From the mutual proportionality of the actual energy Q, and the meta- batic function 9, it follows that the operations are equivalent ; and that the latter may be substituted for the former in all the equations expressing the laws of the transformation of energy. "We have therefore dX_ dX_ d?W , q d-q- e Te- deTx- ' ■ (W for the effort to transform actual energy of the kind Q into work of the land W, when expressed in terms of the metabatic function ; and dn = 9d~, .... (H.) for the limit of the indefinitely small transformation produced bj» an indefinitely small variation of the accidents on which the kind of work W depends. There is also a form of metamorphic function, ♦=£=/" = "• ■ : w 22G OUTLINES OF THE SCIENCE OF ENERGETICS. suited for employment along with the metabatic function, in order to find, by the integration B. = fed<t>, .... (1G.) the quantity of actual energy of a given kind Q transformed to the kind of work W during any finite variation of accidents. The advantage of the above expressions is-, that they are applicable not merely to a homogeneous substance, but to any heterogeneous substance or aggregate, which is internally in a state of equilibrium of actual and potential energy; for throughout all the parts of an aggregate in that condition, the metabatic function 6 is the same, and each of the efforts X, &c, is the same, and consequently the metamorphic function <p is the same. " Carnot's function " in thermo-dynamics is proportional to the reciprocal of the metabatic function of heat. Section XVII. — Efficiency of Engines. An engine is a contrivance for transforming energy, by means of the periodical repetition of a cycle of variations of the accidents of a sub- stance. The efficiency of an engine is the proportion which the energy perma- nently transformed to a useful form by it, bears to the whole energy com- municated to the working substance. In a perfect engine the cycle of variations is thus : — I. The metabatic function is increased, say from 6 to 9 V II. The metamorphic function is increased by the amount A <f>. III. The metabatic function is diminished from Q x back to 6 . IV. The metamorphic function is diminished by the amount A (p. During the second operation, the energy received by the working sub- stance, and transformed from the actual to the potential form is B x A 0. During the fourth operation energy is transformed back, to the amount O A 0. So that the energy permanently transformed during each cycle a a is (9 1 — ) A <f> ; and the efficiency of the engine -— — °. "i Section XVIII. — Diffusion of Actual Energy; Irreversible or Frictional Operations. There is a tendency in every substance, or system of substances, to the equable diffusion of actual energy; that is to say, to its transfer between the OUTLINES OF THE SCIENCE OF ENERGETICS. 227 parts of the substance or system, until the value of the metabatic function becomes uniform. This process is not directly reversible; that is to say, there is no such operation as a direct concentration of actual energy through a tendency of the metabatic function to become unequal in different parts of a substance or system. Hence arises the impossibility of using the energy reconverted to the actual form at the lower limit" of the metabatic function in an engine. There is an analogy in respect of this property of irreversibility, between the diffusion of one kind of actual energy and certain irreversible trans- formations of one kind of actual energy to another, called by Professor William Thomson, " Frictional Phenomena " — viz., the production of heat by rubbing, and agitation, and by electric currents in a homogeneous sub- stance at a uniform temperature. In fact, a conjecture may be hazarded, that immediate diffusion of the actual energy produced in frictional phenomena, is the circumstance which renders them irreversible ; for, suppose a small part of a substance to have its actual energy increased by the exertion of some kind of work upon it, then, if the increase of actual energy so produced be immediately diffused amongst other parts, so as to restore the uniformity of the metabatic function, the whole process will be irreversible. This speculation, how- ever, is, for the present, partly hypothetical ; and, therefore, does not, strictly speaking, form part of the science of energetics. Section XIX. — Measurement of Time. The general relations between energy and time must form an important branch of the science of energetics; but for the present, all that I am prepared to state on this subject is the following DEFINITION OF EQUAL times : — Equal times are the times in which equal quantities of the same kind of work are performed by equal and similar substances, under loholly similar circumstances. Section XX. — Concluding Eemarks. It is to be observed, that the preceding articles are not the results of a new and hitherto untried speculation, but are the generalised expression of a method of reasoning which has already been applied with success to special branches of physics. £28 OUTLINES OF THE SCIENCE OF ENERGETICS. In this brief essay, it has not been attempted to do more than to give an outline of some of the more obvious principles of the science of ener- getics, or the abstract theory of physical phenomena in general ; a science to which the maxim, true of all science, is specially applicable — that its subjects are boundless, and that they never can, by human labours, be exhausted, nor the science brought to perfection. THE PHRASE "POTENTIAL ENERGY." 229 XIII.— ON THE PHEASE "POTENTIAL ENERGY," AND ON THE DEFINITIONS OF PHYSICAL QUANTITIES* 1. In the course of an essay by Sir John Herschel "On the Origin of Force," which appeared some time ago in the Fortnightly Review, and has lately been republished in a volume, entitled Familiar Lectures on Scientific Subjects, the opinion is expressed that the phrase " Potential Energy " is " unfortunate, inasmuch as it goes to substitute a truism for the announce- ment of a great dynamical fact" (Familiar Lectures, page 469). 2. There is here no question as to the reality of the class of relations amongst bodies to which that phrase is applied, nor as to any matter of fact concerning those relations, but as to the convenient and appropriate use of language. This is a sort of question in the discussion of which authority has much weight ; and when an objection to the appropriateness of a term is made by an author who is not less eminent as a philosopher than as a man of science, and whose skill in the art of expressing scientific truth in clear language is almost unparalleled, it becomes the duty of those who use that term to examine carefully their grounds for doing so. 3. As the phrase "Potential Energy," now so generally used by writers on physical subjects, was first proposed by myself in a paper " On the General Law of the Transformation of Energy," f read before the Philosophical Society of Glasgow, on the 5th of January, 1853 (seep. 203), I feel that the remark of Sir John Herschel makes it incumbent upon me to explain the reasons which led me, after much consideration, to adopt that phrase for the purpose of denoting all those relations amongst bodies, or the parts of bodies, which consist in a power of doing work dependent on mutual configurations. 4. The kind of quantity now in question forms part of the subject of the thirty -ninth proposition of Newton's Principia; but it is there repre- sented by the area of a figure, or by symbols only, and not designated by a name ; and such is also the case in many subsequent mathematical writings. 5. The application of the word "force" to that kind of quantity is open * Pvead before the Philosophical Society of Glasgow on Jan. 23, 1867, and published in the Proceedings of that Society, Vol. VI., No. III. t Viz.,— that the effect of the presence of a quantity of actual energy, in causing transformation of energy between the actual and the potential forms, is the sum of the effects of all the parts of that quantity. 230 THE PHRASE "POTENTIAL ENERGY." to the objection, that when "force" is taken in the sense in which Newton defines "vis motrix," the power of performing work is not simply force, but force multiplied by space. To make such an application of the word "force," therefore, would have been to designate a product by the name properly belonging to one of its factors, and would have added to the confusion which has already arisen from the ambiguous employ- ment of that word. 6. The word " power," though at first sight it might seem very appro- priate, was already used in mechanics in at least three different senses : — viz., first, the power of an engine, meaning the rate at which it performs work, and being the product of force and space divided by time; secondly, the power, in the sense of effort or pressure, which drives a machine ; and thirdly, "mechanical powers," meaning certain elementary machines. Thus "'power" was open to the same sort of objection with "force." 7. About the beginning of the present century, the word "energy" had been substituted by Dr. Thomas Young for "vis viva," to denote the capacity for performing work due to velocity ; and the application of the same word had at a more recent time been extended by Sir William Thomson to capacity of any sort for performing work. There can be no doubt that the word "energy" is specially suited for that purpose; for not only does the meaning to be expressed harmonise perfectly with the etymology of avipyua, but the word "energy" has never been used in precise scientific writings in a different sense; and thus the risk of ambiguity is avoided. 8. It appeared to me, therefore, that what remained to be done, was to qualify the noun "energy" by appropriate adjectives, so as to distinguish between energy of activity and energy of configuration. The well-known pair of antithetical adjectives, " actual " and " potential," seemed exactly suited for that purpose ; and I accordingly proposed the phrases " actual energy " and " potential energy," in the paper to which I have referred. 9. I was encouraged to persevere in the use of those phrases, by the fact of their being immediately approved of and adopted by Sir William Thomson ; a fact to which I am disposed to ascribe, in a great measure, the rapid extension of their use in the course of a period so short in the history of science as fourteen years.* I had also the satisfaction of receiving a very strong expression of approval from the late Professor Baden Powell. 10. Until some years afterwards I was not aware of the fact, that the idea of a phrase equivalent to " potential energy," in its purely mechanical sense, had been anticipated by Carnot, who, in an essay on machines in general, employed the term "force vive virtuelle," of which "potential * Sir William Thomson and Professor Tait have lately substituted the word "kinetic " for "actual." THE PHRASE "POTENTIAL ENERGY." 231 energy" might be supposed to be almost a literal translation. That coin- cidence shows how naturally the phrase " potential energy," or something equivalent, occurs to one in search of words appropriate to denote that power of performing work which is due to configuration, and not to activity. 11. Having explained the reasons which led me to propose the use of the phrase " potential energy," I have next to make some observations on the objection made by Sir John Herschel to that phrase, that " it goes to substitute a truism for a great dynamical fact." 12. It must be admitted that the use of the term "potential energy" tends to make the statement of the law of the conservation of energy wear, to a certain extent, the appearance of a truism. It seems to me, however, that such must always be the effect of denoting physical relations by words that are specially adapted to express the properties of those relations ; or, what amounts virtually to the same thing, of drawing up precise and com- plete definitions of physical terms. Let A and B denote certain conceivable relations, and let them be precisely and completely defined ; then, from the definitions follows the proposition, that A and B are related to each other in a certain way; and that proposition wears the appearance of a truism, and is virtually comprehended in the definitions. But it is not a bare truism ; for when with the definitions are conjoined the two facts, ascer- tained by experiment and observation, that there are relations amongst real bodies corresponding to the definition of A, and that there are also relations amongst real bodies corresponding to the definition of B, the pro- position as to relation between A and B becomes not a bare truism, but a physical fact. In the present case, for example, "actual energy" and "potential energy" are defined in such a way as to make the proposition : That what a body or a system of bodies gains in one form of energy through mutual actions, it loses in the other form — in other words, that the sum of actual and potential energies is "conserved" — follow from the definitions, so as to sound like a truism ; but when it is proved by experiment and observation that there are relations amongst real bodies agreeing with the definitions of " actual energy" and " potential energy," that which otherwise would be a truism becomes a fact. 1 3. A definition cannot be true or false ; for it makes no assertion, but says, " let such a word or phrase be used in such a sense;" but it may be real or fantastic, according as the description contained in it corresponds, or not, to real objects and phenomena; and when, by the aid of experi- ment and observation, a set of definitions have been framed which possess reality, precision, and completeness, the investing of a physical fact with the appearance of a truism is often an unavoidable consequence of the use of the term so defined. 14. In the case of physical quantities in particular, the definition involves a rule for measuring the quantity ; and the proof of the reality of the 232 THE PHRASE "POTENTIAL ENERGY." definition is the fact, that the application of the rule to the same quantity under different circumstances gives consistent results, which it would not do if the definition were fantastic ; and hence the definitions of a set of physical quantities necessarily involve mathematical relations amongst those quantities, which, when expressed as propositions and compared with the definitions, wear the appearance of truisms, and are at the same time statements of fact. 15. In illustration of the foregoing principles, it may be pointed out that there is a certain set of definitions of the measurement of time, force, and mass, which reduce the laws of motion to the form of truisms, thus — I. Let " equal times " mean the times in which a moving body, under the influence of no force, describes equal spaces. This definition is proved to be real by the fact, that times which are equal when compared by means of the free motion of one body, are equal when compared by means of the free motion of any other body. If the definition were fantastic, times might be equal as measured by the free motion of one body, and unequal as measured by that of another. II. Let "fwce" mean a relation between a pair of bodies such that their relative velocity changes, or tends to change, in magnitude or direction, or both ; and let " equal forces " mean those which act when equal changes of the relative velocity of a given pair of bodies occur in equal times. This definition is proved to be real by the fact, that the comparative measurements of forces made in different intervals of time are consistent with each other, which would not be the case if the definition were fantastic. III. Let the " mass " of a body mean a quantity inversely proportional to the change of velocity impressed on that body in a given time by a given force. This definition is proved to be real by the fact, that the ratio of the masses of two given bodies is found experimentally to be always the same, when those masses are compared by means of the velocities impressed on them by different forces, and in different times ; and is also the same, whether each of the masses is measured as a whole or as the sum of a set of parts. Assuming those definitions as merely verbal, without reference to their reality, the laws of motion take the form of verbal truisms; but when experiment and observation inform us that permanent relations exist amongst real bodies and real events corresponding to the definitions, those apparent truisms become statements of fact. 16. One of the chief objects of mathematical physics is to ascertain, by the help of experiment and observation, what physical quantities or functions are "conserved." Such quantities or functions are, for example — I. The mass of every particle of matter, conserved at all times and under all circumstances. THE PHRASE " POTENTIAL ENERGY." 233 II. The resultant momentum of a body, or system of bodies, conserved so long as internal forces act alone. III. The resultant angular momentum of a body or system of bodies, conserved so long as internal forces act alone. IV. The total energy of a body, or system of bodies, conserved so long as internal forces act alone. V. The tliermo-dynamic function, conserved in a body while it neither receives nor gives out heat. In defining such physical quantities as those, it is almost, if not quite, impossible to avoid making the definition imply the property of con- servation ; so that when the fact of conservation is stated, it has the form of a truism. 1 7. In conclusion, it appears to me that the making of a physical law wear the appearance of a truism, so far from being a ground of objection to the definition of a physical term, is rather a proof that such definition has been framed in strict accordance with reality. 234 THE MECHANICAL ACTION OF HEAT. XIV.— ON THE MECHANICAL ACTION OF HEAT, ESPECIALLY IN GASES AND VAPOURS. * Introduction — Summary of the Principles of the Hypothesis of Molecular Vortices, and its Application to the Theory of Temperature, Elasticity, and Real Specific Heat. The ensuing paper forms part of a series of researches respecting the consequences of an hypothesis called that of Molecular Vortices, the object of which is, to deduce the laws of elasticity, and of heat as connected with elasticity, by means of the principles of mechanics, from a physical supposition consistent and connected with the theory which deduces the laws of radiant light and heat from the hypothesis of undulations. Those researches were commenced in 1842, and after having been laid aside for nearly seven years, from the want of experimental data, were resumed in consequence of the appearance of the experiments of M. Regnault on gases and vapours. The investigation which I have now to describe, relates to the mutual conversion of heat and mechanical power by means of the expansion and contraction of gases and vapours. In the introduction, which I here prefix to it, I purpose to give such a summary of the principles of the hypothesis as is necessary to render the subsequent investigation intelligible. The fundamental suppositions are the following : — First, That each atom of matter consists of a nucleus, or central physical point, enveloped by an elastic atmosphere, which is retained in its position by forces attractive towards the nucleus or centre. Suppositions similar to this have been brought forward by Franklin, iEpinus, Mossotti, and others. They have in general, however, conceived the atmosphere of each nucleus to be of variable mass. I have treated it, on the contrary, as an essential part of the atom. I have left the question indeterminate, whether the nucleus is a small body of a character distinct from that of the atmosphere, or merely a portion of the atmosphere in a highly condensed state, owing to the mutual attraction of its parts. According to this first supposition, the boundary between two con- * Read before the Royal Society of Edinburgh on Feb. 4, 1850, and published in the Transactions of that Society, Vol. XX., Part I. (See also p. 16.) THE MECHANICAL ACTION OF HEAT. 235 tiguous atoms of a body is an imaginary surface at which the attractions of all the atomic centres of the body balance each other; and the elasticity of the body is made up of two parts : First, the elasticity of the atomic atmospheres at the imaginary boundaries of the atoms, which I shall call the superficial-atomic elasticity; and, secondly, the force resulting from the mutual actions of distinct atoms. If the atmospheres are so much con- densed round their nuclei or centres, that the superficial-atomic elasticity is insensible, and that the resultants of the mutual actions of all parts of the distinct atoms are forces acting along the lines joining the nuclei or centres, then the body is a perfect solid, having a tendency to preserve not only a certain bulk, but a certain figure ; and the elasticity of figure, or rigidity, bears certain definite relations to the elasticity of volume. If the atmospheres are less condensed about their centres, so that the mutual actions of distinct atoms are not reducible to a system of forces acting along the lines joining the atomic centres, but produce merely a cohesive force sufficient to balance the superficial-atomic elasticity, then the condition is that of a perfect liquid ; and the intermediate conditions between this and perfect solidity constitute the gelatinous, plastic, and viscous states. When the mutual actions of distinct atoms are very small as compared with the superficial-atomic elasticity, the condition is that of gets or vapour; and when the substance is so far rarefied that the influence of the atomic nuclei or centres in modifying the superficial elasticity of their atmos- pheres is insensible, it is then in the state of perfect gas. So far as our experimental knowledge goes, the elasticity of a perfect gas, at a given temperature, varies simply in proportion to its density. I have therefore assumed this to be the law of the elasticity of the atomic atmospheres, ascribing a specific coefficient of elasticity to each substance. The second supposition, being that from which the hypothesis of mole- cular vortices derives its name, is the following : — That the elasticity due to heat arises from the centrifugal force of revolutions or oscillations among the particles of the atomic atmospheres ; so that quantity of heat is the vis viva of those revolutions or oscillations. This supposition appears to have been first definitely stated by Sir Humphry Davy. It has since been supported by Mr. Joule, whose valuable experiments to establish the convertibility of heat and mechanical power are well known. So far as I am aware, however, its consequences have not hitherto been mathematically developed. To connect this hypothesis with the undulatory theory of radiation, I have introduced a third supposition : That the medium which transmits light and radiant heat consists of the nuclei of the atoms, vibrating independently, or almost independently, of their atmospheres; so that the absorption of 236 THE MECHANICAL ACTION OF HEAT. light and of radiant heat, is the transference of motion from the nuclei to their atmospheres; and the emission of light and of radiant heat, the transference of motion from the atmospheres to their nuclei. Although in all undulations of sensible length and amplitude, such as those of sound, the nuclei must carry their atmospheres along with them, and vibrating thus loaded, produce a comparatively slow velocity of propagation; yet, in all probability, the minute vibrations of light and radiant heat may be performed by the atomic nuclei in transparent and diathermanous bodies, without moving the atmospheres more than by that amount which constitutes absorption; and those vibrations will therefore be transmitted according to the laws of the elasticity of perfect solids, and with a rapidity corresponding to the extreme smallness of the masses set in motion, as compared with the mutual forces exerted by them. This supposition is peculiar to my own view of the hypothesis, and is, in fact, the converse of the idea hitherto adopted, of an ether surrounding ponderable particles. The second and third suppositions involve the assumption, that motion can be communicated between the nuclei and their atmospheres, and between the different parts of the atmospheres; so that there is a tendency to produce some permanent condition of motion, which constitutes equili- brium of heat. It is now to be considered what kind of motion is capable of producing increase of elasticity, and what are the conditions of perma- nency of that motion. It is obvious, that the parts of the atomic atmospheres may have motions of alternate expansion and contraction, or of rectilinear oscillation about a position of equilibrium, without affecting the superficial atomic elasticity, except by small periodical changes. Should they have motions, however, of revolution about centres, so as to form a group of vortices, the centrifugal force will have the effect of increasing the density of the atmosphere at what I have called the bounding surfaces of the atoms, and thus of augmenting the elasticity of the body. In this summary, I shall not enter into the details of mathematical analysis, but shall state results only. The following, then, are the con- ditions which must be fulfilled, in order that a group of vortices, of small size as compared with the bulk of an atom, and of various diameters, may permanently co-exist, whether side by side, or end to end, in the atomic atmospheres of one substance, or of various substances mixed. First, The mean elasticity must vary continuously, which involves the condition, that at the surface of contact of two vortices of different substances, side by side, or end to end, the respective densities at each point of contact must be inversely proportional to the coefficients of elasticity. Hence, the specific gravities of the atmospheric parts of all substances, under -precisely similar circumstances as to heat and molecular forces THE MECHANICAL ACTION OF HEAT. 237 (a condition realised in perfect gases at the same pressure and temperature), are inversely proportional to the coefficients of atmospheric elasticity. Therefore, let juL represent the mass of the atmosphere of one atom of any substance, b its coefficient of elasticity, and n the number of atoms which, in the state of perfect gas, occupy unity of volume under unity of pressure at the temperature of melting ice; — then n fib (I.) is a constant quantity for all substances. Secondly, The superficial elasticity of a vortex must not be a function of its diameter : to fulfil which condition, the linear velocity of revolution must be equal throughout all parts of each individual vortex. Thirdly, In all contiguous vortices of the same substance, the velocities of revolution must be equal; and in contiguous vortices of different substances, the squares of the velocities must be proportional to the coefficients of elasticity of the molecular atmospheres. The second and third conditions are those of equilibrium of heat, and are equivalent to this law : — Temperature is a function of the square of the velocity of revolution in the molecular vortices, divided by the coefficient of elasticity of the atomic atmos- pheres; or Temperature = ( —J, . . . (II.) where w represents that velocity. The mean elasticity which a vortex exerts endways is not affected by its motion, being equal to b P , (HI.) where p is its mean density. The superficial elasticity at its lateral surfaces, however, is expressed by % + *p (iv.) The additional elasticity ■—-*-, being that which is due to the motion, is independent of the diameter. The divisor #,(the force of gravity) is introduced, on the supposition of the density p being measured by weight. Supposing the atmosphere of an atom to be divided into concentric spherical layers, it may be shown that the effect of the co-existence of a great number of small vortices in one of those layers whose radius is r, and mean density p, is to give it a centrifugal force, expressed by ^ (V.) gr v 238 THE MECHANICAL ACTION OF HEAT. which tends to increase the density and elasticity of the atmosphere at the surface, which I have called the boundary of the atom. The layer is also acted upon by the difference between the mean elasticities at its two surfaces, and by the attraction towards the atomic centre; and these three forces must balance each other. I have integrated the differential equation which results from this condition, for substances in the gaseous state, in which the forces that interfere with the centrifugal force and atmospheric elasticity are com- paratively small ; and the result is p = J s D (5 +1 )( 1 - F ) + /( D )- • < VL > P is the entire elasticity of the gas, and D its mean density. M repre- sents the total mass of an atom, measured by weight, and /.i that of its atmospheric part; so that D is the mean density of the atomic atmospheres. / (D) denotes the effect of the mutual actions of separate atoms. The first term represents the superficial-atomic elasticity. F denotes the effect of the attraction of the nucleus in modifying that elasticity, and can be represented approximately by a converging series, in terms of ID 2 the negative powers of „ r + 1, commencing with the inverse square, the coefficients being functions of the density D. By using the first term of such a series, and determining its coefficient and the quantity /(D) empirically, I have obtained formula? agreeing closely with the results of M. Eegnault's experiments on the expansion of atmospheric air, carbonic acid, and hydrogen. In a perfect gas, the above expression is reduced to p=i H D W> + 1 > < VIL > Let n, as before, denote the number of atoms of a substance which, in the state of perfect gas, occupy unity of volume under unity of pressure, at the temperature of melting ice, so that wM is its specific gravity in that state : then, p = ^ J (sT* + 1 )- • ( VIIL > The factor by which -^-= is here multiplied, fulfils the condition of THE MECHANICAL ACTION OF HEAT. 239 being a function of - --, and of constants which are the same for all substances, and is, therefore, fitted for a measure of temperature. It obviously varies proportionally to the pressure of a perfect gas of a given density, or its volume under a given pressure. Let r, therefore, denote temperature, as measured from an imaginary zero, C degrees of the scale adopted, below the temperature of melting ice, at which 3gb Then, for all substances and in perfect gases • O^*) »lff r may be termed absolute temperature, and the point from which it is measured, the absolute zero of temperature. This, as I have observed, is an imaginary point, being lower than the absolute zero of heat by the quantity Ciifib, which is the same for all substances. The value of C, or the absolute temperature of melting ice, as determined from M. Eegnault's experiments, is 274°-6 Centigrade, being the reciprocal of 0-00364166 per Centigrade degree, the value to which the coefficients of dilatation of gases at the temperature of melting ice approximate as they are rarefied. For Fahrenheit's scale C = 494°-28. In the sequel I shall represent temperatures measured from that of melting ice by T = r - C. We have now to consider the absolute quantity of heat, or of mole- cular vis viva which corresponds to a given temperature in a given substance. It is obvious that /AW 2 represents, in terms of gravity, the portion of vis viva, in one atom, due 240 THE MECHANICAL ACTION OF HEAT. to the molecular vortices ; but besides the vortical motion, there may be oscillations of expansion and contraction, or of rectilinear vibration about a position of equilibrium. The velocity with which these additional motions are performed will be in a permanent condition, when the mean value of its square, independent of small periodic changes, is equal throughout the atomic atmosphere. We may therefore represent by fX V L _ fX w* 27 "27' (X.) the total vis viva of the atomic atmosphere. To this we have to add that of the nucleus, raising the quantity of heat in one atom to Mr 2 2<7 = <Z> while the quantity of heat in unity of weight is y . (XI.) 2? H The coefficient k (which enters into the value of specific heat) being the ratio of the vis viva of the entire motion impressed on the atomic atmospheres by the action of their nuclei, to the vis viva of a peculiar kind of motion, may be conjectured to have a specific value for each substance depending, in a manner as yet unknown, on some circumstance in the constitution of its atoms. Although it varies in some cases for the same substance in the solid, liquid, and gaseous states, there is no experimental evidence that it varies for the same substance in the same condition. In the investigation which follows, therefore, I have treated it as sensibly constant. The following, then, are the expressions for quantity of heat in terms of temperature : in one atom, — v 1 __ 3fcM , n .. In unity of weight, 3k Q = 2^ = 2^ r - Cw ^- 1 Y (xii.) J Real specific heat is, consequently, expressed by the following equa- tions : — THE MECHANICAL ACTION OF HEAT. 241 For one atom, — dq dr~ 3kM 2Crc// For unity of weight, — dQ 'dr ~ 3k 2Cwu y (xiii.) For so much of a perfect gas as occupies unity of volume under unity of pressure at the temperature of melting ice, — dq 3kM d< 2C fi The laws established experimentally by Dulong, that the specific heats of simple atoms, and of certain groups of compound atoms, bear to each other simple ratios, generally that of equality, and that the specific heats of equal volumes of all simple gases are equal, show that the specific factor depends on the chemical constitution of the atom, and thus M confirm the conjecture I have stated respecting the coefficient k. As I shall have occasion, in the investigation which follows, to refer to and to use the equation for the elasticity of vapours in contact with their liquids, which I published in the Edinburgh New Philosophical Journal for July, 1849, I shall here state generally the nature of the reasoning from which it was deduced. The equilibrium of a vapour in contact with its liquid depends on three conditions : First, The total elasticity of the substance in the two states must be the same. Secondly, The superficial atomic elasticity must vary continuously; so that if at the surface which reflects light there is an abrupt change of density (which seems almost certain), there must there be two densities corresponding to the same superficial-atomic elasticity. Thirdly, The two forces which act on each stratum of vapour parallel to the surface of the liquid — namely, the preponderance of molecular attraction towards the liquid, and the difference of the superficial-atomic elasticities at the two sides of the stratum — must be in equilibrio. Close to the surface of the liquid, therefore, the vapour is highly condensed. The density diminishes rapidly as the distance from the liquid increases, and at all appreciable distances has a sensibly uniform value, which is a function of the temperature and of certain unknown molecular forces. Q 242 THE MECHANICAL ACTION OF HEAT. The integration of a differential equation representing the third condition of equilibrium, indicates the form of the approximate equation, P v Log P = a - ^ - T T" (XIV.) the coefficients of which have been determined empirically by three experimental data for each fluid. For proofs of the extreme closeness with which the formula? thus obtained agree with experiment, I refer to the Journal in which they first appeared. I annex a table of the coefficients for water, alcohol, ether, turpentine, petroleum, and mercury, in the direct equation, and also in the inverse formula, J- • A 1 y 4 r ay (XV.) by which the temperature of vapour at saturation may be calculated from the pressure. In the ninth and tenth columns are stated the limits between which the formulae have been compared with experiment. For turpentine, petroleum, and mercury, the formula consists of two terms only. LokP /3 a — — . (XVI.) the small range of the experiments rendering the determination of y impossible. The following are some additional values of the constant a for steam, corresponding to various units of pressure used in practice. Units of Pressure. Atmospheres of 7 GO millimetres of mercury, = 29*922 inches of mercury, = 14" 7 lbs. on the square inch, = 1 , 0333 kilogrammes on the square centimetre, Atmospheres of 30 inches of mercury, = 761 - 99 millimetres, = 14 - 74 lbs. on the square inch, = T036 kilogrammes on the sqv Kilogrammes on the square centimetre, Kilogrammes on the circular centimetre, Pounds avoirdupois on the square inch, Pounds avoirdupois on the circular inch, Pounds avoirdupois on the square foot, Values of a. 4-950433 are centimetre, 4-949300 4-964658 4-859748 6-117662 6-012752 8-276025 All the numerical values of the constants are for common logarithms. THE MECHANICAL ACTION OF HEAT. 243 H a A o O m W 5 o PL, < o m W »— i H a S o ;«5 2 9 33 o S + •s + OO fe(M W a W H « O P a O PR W W H 02 H H O O a W H O PQ -H CO !>. —I °3 "3 ^ rd r^ ,£3 Cm Cm &H fe CM -d A Eq .O .S . c3 . c3 .3 a tJD bD p ^.3 c.g "3 O --H « ."^ Oh £ is 3 -d'o -d'o K' a 1 H Cm 244 THE MECHANICAL ACTION OF HEAT. Section I. — Of the Mutual Conversion of Heat and Expansive Power. 1. The quantity of heat in a given mass of matter, according to the hypothesis of molecular vortices, as well as every other hypothesis which ascribes the phenomena of heat to motion, is measured by the mechanical power to which that motion is equivalent, that being a quantity the total amount of which, in a given system of bodies, cannot be altered by their mutual actions, although its distribution and form may be altered. This is expressed in equation XII. of the introduction, where the quantity v 2 of heat in unity of weight, Q, is represented by the height - , from which a body must fall in order to acquire the velocity of the molecular oscilla- tions. This height, being multiplied by the weight of a body, gives the mechanical power to which the oscillations constituting its heat are equivalent. The real specific heat of unity of weight, as given in equa- tion (XIII.) of the introduction, d Q 3 h dr 2Cn /•- represents the dijdli of fall, which is equivalent to one degree of rise of temperature in any given weight of the substance under consideration. We know, to a greater or less degree of precision, the ratios of the specific heats of many substances to each other, and they are commonly expressed by taking that of water at the temperature of melting ice as unity ; but their actual mechanical values have as yet been very imper- fectly ascertained, and, in fact, the data necessary for their determination are incomplete. 2. Mr. Joule, indeed, has made several very interesting series of experiments, in order to ascertain the quantity of heat developed in various substances by mechanical power employed in different ways — viz., by electric currents excited by the rotation of a magnet, by the forcing of water through narrow tubes, by the agitation of water and oil with a paddle, by the compression of air, and by the friction of air rushing through a narrow orifice. The value of the depth of fall equivalent to a rise of one degree of Fahrenheit's scale in the temperature of a mass of water, as determined by that gentleman, varies, in the different series of experiments, between the limits of 760 feet and 890 feet, the value in which Mr. Joule appears to place the greatest confidence being about 780 feet. Although the smallness of the differences of temperature measured in those experiments renders the numerical results somewhat uncertain, THE MECHANICAL ACTION OF HEAT. 245 it appears to me that, as evidence of the convertibility of heat and mechanical power, they are unexceptionable. Nevertheless, there is reason to believe that the true mechanical equivalent of heat is consider- ably less than any of the values deduced from Mr. Joule's experiments; for in all of them there are causes of loss of power the effect of which it is impossible to calculate. In all machinery, a portion of the power which disappears is carried off by waves of condensation and expansion, along the supports of the machine, and through the surrounding air : this portion cannot be estimated, and is, of course, not operative in producing heat within the machine. It is also impossible to calculate, where friction is employed to produce heat, what amount of it has been lost in the production of electricity, a power which is, no doubt, conver- tible into heat, but which, in such experiments, probably escapes without undergoing that conversion. To make the determination of the mechanical equivalent of heat by electro-magnetic experiments correct, it is necessary that the whole of the mechanical power should be converted into magnetic power, the whole of the magnetic power into what are called electric currents, and the whole of the power of the electric currents into heat, not one of Avhich conditions is likely to be exactly fulfilled. Even in producing heat by the compression of air, it must not be assumed that the whole of the mechanical power is expended in raising the tempera- ture. 3. The best means of determining the mechanical equivalent of heat are furnished by those experiments in which no machinery is employed. Of this kind are experiments on the velocity of sound in air and other gases, which, according to the received and well-known theory of Laplace, is accelerated by the heat developed by the compression of the medium. The accuracy of this theory has lately been called in question. There can be no doubt that it deviates from absolute exactness, in so far that the magnitude of the displacements of the particles of air is neglected in comparison with the length of a wave. It appears to me, however, that the Astronomer-Eoyal, in his remarks on the subject in the London and Edinburgh Philosophical Magazine for July, 1849, has shown, in a satisfactory manner, that although the effect of the appreciable magnitude of those displacements, as compared with the length of a wave of sound, is to alter slowly the form of the function representing the wave, still that effect is not sufficiently great to make Laplace's theory practically erroneous. I have, therefore, in the sequel, adhered to the experiments of Dulong, and to those emoted by Poisson, on the velocity of sound, as the best data for determining the mechanical equivalent of heat. 4. The expression already given for the real specific heat of unity of weight of a given substance may be resolved into two factors, thus : — 246 THE MECHANICAL ACTION OF HEAT. dQ, 1_ 3&M (l) dr C«M 2/i ' ' k '' The first factor, ~ — r>, may be considered in general as a known quantity; CwM for C represents, as already stated, 274"6 Centigrade degrees, the absolute temperature of melting ice, and n M the theoretical weight, in the per- fectly gaseous state, of unity of volume of the substance, under unity of pressure, at that temperature; or what is the same thing, — ^ is the height of an imaginary column of the substance, of uniform density, and at the temperature of melting ice, whose pressure by weight upon a given area of base is equal to its pressure by elasticity, supposing it to be . 3&M . perfectly gaseous. The determination of the ratio — — is necessary to complete the solution of the problem. 5. The relation now to be investigated between heat and mechanical power, is that which exists between the power expended in compressing a body into a smaller volume, and the increase of heat in consequence of such a compression ; and, conversely, between the heat which disappears, or, as it is said, becomes latent during the expansion of a body to a greater volume, and the mechanical power gained or developed by that expansion. Those phenomena, according to the hypothesis now under consideration, as well as every hypothesis which ascribes heat to motion, are simply the transformation of mechanical power from one shape into another. It is obvious, in the first place, without the aid of algebraical symbols, that the general effect of the compression of an oscillating atomic atmos- phere, or molecular vortex, must be to accelerate its motion, and of its dilatation, to retard its motion; for every portion of such an atmosphere is urged toward the nucleus or atomic centre by a centripetal force equal to the centrifugal force arising from the oscillation; so that when, by compression, each portion of the atmosphere is made to approach the centre by a given distance, the vis viva of its motion will be increased by the amount corresponding to the centripetal force acting through that dis- tance ; and, conversely, when by expansion each portion of the atmosphere is made to retreat from the centre, the vis viva of its motion will be diminished by a similar amount. It is not, however, to be taken for granted, that all the power expended in compressing a body appears in the form of heat. More or less power may be consumed or developed by changes of molecular arrangement, or of the internal distribution of the density of the atomic atmospheres; and changes of molecular arrangement or distribution may develop or consume heat, independently of changes of volume. 6. We shall now investigate, according to the hypothesis of mole- THE MECHANICAL ACTION OF HEAT. 247 cular vortices, the amount of heat produced by an indefinitely small compression of one atom of a body in that state of perfect fluidity which admits of the bounding surface of the atom being treated as if it were spherical : its radius being denoted by R, and the radius of any internal spherical layer of the atmosphere by multiplying R by a fraction u. I shall denote by the ordinary symbol of differentiation d, such changes as depend on the various positions of portions of the atomic atmosphere relatively to each other, when changes of volume and temperature are not taken into consideration; while by the symbol 8 of the calculus of variations, I shall represent such changes as arise from the variations of volume and temperature. Let us consider the case of an indefinitely thin spherical layer of the atomic atmosphere, whose distance from the nucleus is Rw, its thickness R d u, its area 4 tt R 2 u 2 , and its density ^ : D \p (u, D, t). The weight, then, of this layer is 4 ttR 3 ^ Dm 2 ?/,(«, D, T )du. Its velocity of oscillation is v, and having, in virtue of that velocity, a mean centrifugal force, as explained in the introduction (Equation V.), equal to / t' 2 2Q \ its weight X ( ? p = y~— ) ° \ g k R u kliu/ it is kept in equilibrio by an equal and opposite centripetal force, arising from attraction and elastic pressure, which is consequently represented by 4ttR 2 £^-Dmi/,(w, D > T)du M gk = 8 7t R 2 r^ Q D u $ («, D, r) d u. Let the mean density of the atom now be increased by the indefinitely small quantity § D. Then the layer will approach the nucleus through the distance — S(Rw)=— wSR — RSm, and being acted upon through that distance by the centripetal force already stated, the vis viva of oscillation will be increased by a quantity corresponding to the mechanical power (that is to say, the heat), represented by the product of that distance by that force, or by -8ttR 2 t^|QDm^ («, J),T)duxd(Ru) 248 THE MECHANICAL ACTION OF HEAT. = -8»^QD^D,r)^ + ^. which, because SR 1 SD , 47rE 3 D __ . — = — - . -jp and = M, is equal to + QM.^frD,r)tf(^-8^)*iL "We must suppose that the velocity of oscillation is equalised throughout the atomic atmosphere, by a propagation of motion so rapid as to be practically instantaneous. Then, if the above expression be integrated with respect to du, from u = to u=l, the result will give the whole increase of heat in the atom arising from the condensation SD; and dividing that integral by the atomic weight M, we shall obtain the corresponding development of heat in unity of weight. This is expressed by the following equation : — 8Q' = *<*&{irfo du • ** ( * D ' r) -3 ( l du.vBu\l,(u t 'D, r)| . (2.) The letter Q' is here introduced to denote, when negative, that heat which is consumed in producing changes of volume and of molecular arrangement ; and when positive, as in the above equation, the heat which is produced by such changes. The following substitutions have to be made in equation (1) of this Section : For Q is to be substituted its value, according to equation XII. of the introduction; or abbreviating Cn/nb into k, — «=rab<'— > • • • < 3) The value of the first integral in equation (2) of this Section is /•i 1 I du . u 2 \p (u, D, t) = x. JO "J The value of the second integral, — 31 du . uBu-ip (u, D, t) J o remains to be investigated. The first step in this inquiry is given by THE MECHANICAL ACTION OF HEAT. 2-19 the condition, that whatsoever changes of magnitude a given spherical layer undegoes, the portion of atmosphere between it and the nucleus is invariable. This condition is expressed by the equation „ A + 8 r 4- + § D ^)fd u . M 2 + (n, D, r), (4.) du ' dr from which it follows that d~D, Su = - 1 T — (8 r f + S D -A ) f\* » . ^ ( w , D, r), u, D, t) \ dr a D/J o u 2 \p (m, D, r) and, consequently, that — 3 I du . u$uip (u, D, r) = •Jo Hence, making 9 I 1 — I" du. u 2 xL (u, D, r) = U. . . (5.) J U J The second integral in equation (2) is transformed into By means of those substitutions we obtain for the mechanical value of the heat developed in unity of weight of a fluid by any indefinitely small change of volume or of molecular distribution — SQ' = r — k CnM (SD(1 + ^WC^ dW or taking V = ==r to denote the volume of unity of weight of the substance, C»M\ \V d V/ d t (6.) Of this expression, the portion -k SD T-.C SV CnM ' Y~ repre " CnM D sents the variation of heat arising from mere change of volume. t-k gyfg = JLZJLgr)45, denotes the variation of heat pro- CnM dY CnM dD duced by change of molecular distribution dependent on change of volume. 250 THE MECHANICAL ACTION OF HEAT. — — — - 8 t —, — expresses the variation of heat due to change of mole- (jnM dr cular distribution dependent on change of temperature. 7. The function U is one depending on molecular forces, the nature of which is as yet unknown. The only case in which it can be calculated directly is that of a perfect gas. Without giving the details of the integra- tion, it may be sufficient to state, that in this case 1 y . . . (7.) T and, therefore, that dU _ K clU _ dr ~ t 25 dV~ J In all other cases, however, the value of this function can be determined indirectly, by introducing into the investigation the principle of the conservation of vis viva. Suppose a portion of any substance, of the weight unity, to pass through a variety of changes of temperature and volume, and at length to be brought back to its primitive volume and temperature. Then the absolute quantity of heat in the substance, and the molecular arrangement and distribution, being the same as at first, the effect of their changes is eliminated ; and the algebraical sum of the vis viva expanded and produced, whether in the shape of expansion and compression, or in that of heat, must be equal to zero: that is to say, if, on the whole, any mechanical power has appeared, and been given out from the body, in the form of expansion, an equal amount must have been communicated to the body, and must have disappeared in the form of heat ; and if any mechanical power has appeared and been given out from the body in the form of heat, an equal amount must have been communicated to the body, and must have disappeared in the form of compression. This principle expressed symbolically is A n + A Q' = 0, . . . . (8.) where n, when positive, represents expansive power given out, when negative, compressive power absorbed; and Q' represents, when positive, heat given out, when negative, heat absorbed. To take the simplest case possible, let the changes of temperature and of volume be supposed to be indefinitely small, and to occur during distinct intervals of time, so that t and V are independent variables; let the initial absolute temperature be t, the initial volume V, and the initial total elasticity P ; and let the substance go through the following four changes : THE MECHANICAL ACTION OF HEAT. 251 First, Let its temperature be raised from r to r + 8 r, the volume remaining unchanged. Then the quantity of heat absorbed is - ( d ® T ~ K dJJ dT \dr CmM dr and there is no expansion nor compression. Secondly, Let the body expand, without change of temperature, from the volume V to the volume V + S V. Then the quantity of heat absorbed is + Sr-ic/l d nT , d\J while the power given out by expansion is SY(P + ^Sr). Thirdly, Let the temperature fall from r + 8 r, to its original value r, the volume V + S V continuing unchanged ; then the heat given out is + «'(£-wh £<" + £">> and there is no expansion nor compression. Fourthly, Let the body be compressed, without change of temperature, to its original volume V ; then the heat given out is + 6V C7iMVV dYJ' while the power absorbed in compression is — 8 V . P. The body being now restored in all respects to its primitive state, the sum of the two portions of power connected with change of volume, must, in virtue of the principle of vis viva, be equal to the sum of the four quantities of heat with their signs reversed. Those additions being made, and the sums divided by the common factor S V 8 t, the following equation is obtained : dV _ 1 (1 dU\ dr C»M\V dVJ' ' ' ' K ' J The integral of this partial differential equation is U = 0.r + fdV^-CnM^). . . (10.) 252 THE MECHANICAL ACTION OF HEAT. Now <f).r being the same for all densities, is the value of U for the perfectly gaseous state, or -; for in that state, the integral = 0. The values of the partial differential coefficients are accordingly dV dY ~ 1 V — c dr dJJ dr ~ - K — CnM fdY d 2 ~P dr* i J and they can, therefore, be determined in all cases in which the quantity ic = Cii{xb, and the law of variation of the total elasticity with the volume and temperature are known, so as to complete the data required in order to apply equation 6 of this section to the calculation of the mechanical value of the variations of heat, due to changes of volume and molecular arrangement. The total elasticity of an imperfect gas, according to equations VI. and XII. of the introduction, being p -uiavO- F M)+'<w its first and second partial differential coefficients with respect to the temperature arc dV dr d?V dr 2 i C n M V 1 0- 1+T a t- d t d r C n M V Consequently, for the imperfectly gaseous state, u du dY 'M^l )j<ivX T) >0 l A' n-r>fc y (I 2 -) — = --i a r r" *£+'£)' *)/< rt 8. It is to be observed that the process followed in ascertaining the nature of the function U is analogous to that employed by M. Carnot in his theory of the motive power of heat, although founded on contrary THE MECHANICAL ACTION OF HEAT. 253 principles, and leading to different results. Carnot, in fact, considers heat to be something of a peculiar kind, whether a condition or a substance, the total amount of which in nature is incapable of increase or of diminution. It is not, therefore, according to his theory, convertible into mechanical power ; but is capable, by its transmission through substances under parti cidar circumstances, of causing mechanical power to be developed. He supposes a body to go through certain changes of temperature and volume, and to return at last to its primitive volume and temperature, and conceives, in accordance with his view of the nature of heat, that it must have given out exactly the same quantity of heat that it has absorbed. The transmission of this heat he regards as the cause of the production of an amount of mechanical power depending on the quantity of heat transmitted, and on the temperature at which the transmission has taken place. According to these principles, a body, having received a certain quantity of heat, is capable of giving out, not only all the heat it has received, but also a quantity of mechanical power which did not before exist. According to the theory of this essay, on the contrary, and to every conceivable theory which regards heat as a modification of motion, no mechanical power can be given out in the shape of expansion, unless the quantity of heat emitted by the body in returning to its primitive tem- perature and volume is less than the quantity of heat originally received ; the excess of the latter quantity above the former disappearing as heat, to appear as expansive power, so that the sum of the vis viva in these two forms continues unchanged. Section II. — Of Eeal and Apparent Specific Heat, especially in the State of Perfect Gas. 9. The apparent specific heat of a given substance is found by adding to the real specific heat (or the heat which retains its form in producing an elevation of one degree of temperature in unity of weight) that additional heat which disappears in producing changes of volume and of molecular arrangement, and which is determined by reversing the sign of Q 1 in equation 6 of Sect. I (so as to transform it from heat evolved to heat absorbed), and taking its total differential coefficient with respect to the temperature. Hence, denoting total apparent specific heat by K, K = dQ_ d. Q 1 _dQ dQ 1 dQ 1 dV dr dr dr dr dV ' d r 1 (31M, , ,/tiV/l dV\ dV\) ,„„. 251 THE MECHANICAL ACTION OF HEAT. Another mode of expressing this coefficient is the following : Denote the ratio . M by N, and the real specific heat by ft = ClM' .... (H.) then K -i {1 +N(r - k) ( 7z7 ( y - <nr ) - 7/ J }. (15.) The value of ■=— is to be determined from the conditions of each par- ticular case, so that each substance may have a variety of apparent specific heats, according to the manner in which the volume varies with the temperature. dV If the volume is not permitted to vary, so that — - = 0, there is a t obtained the following result, being the apparent specific heat at constant volume : — *'-o^(i-fr-'»£)-*( l -*<'-")i7> ( 10 -» 10. Then the substance under consideration is a perfect gas, it has already been stated, equation (7), that d]J k dV _ dr == r 2 ' d V ~ ' and because the volume of unity of weight is directly as the absolute temperature and inversely as the pressure. 1 cTV _ 1 1 dP_ V dr ~ r P dr ' " " ( '^ Hence, the following are the values of the apparent specific heats of unity of weight of a theoretically perfect gas under different circum- stances : — General value of the total apparent specific heat : THE MECHANICAL ACTION OF HEAT. 255 K-- M1- + c»m In t *-*&+4h)} = chi { s + (r _ k) ( ? + r " ^7) 1 Apparent specific heat at constant volume : y (18.) *-jot{s+!-S-»0-+»(-:-s)) Apparent specific heat under constant pressure : ^ = o^m(s + 1 -?) = s { 1 + n ( 1 -?)} , The ratio of the apparent specific heat under constant pressure to the apparent specific heat at constant volume, is the following : — 1 +N (i-a 1 - l+N -- = 1 +N 1 +N -;-5) (19.) The value of k is unknown, and, as yet, no experimental data exist from which it can be determined. I have found, however, that practically, results of sufficient accuracy are obtained by regarding k as so small in comparison with that -, and ic fortiori —,, may be neglected in T T~ calcvdation. Thus are obtained the following approximate results for perfect gases, and gases which may without material error be treated as perfect. General value of the total apparent specific heat : — _ _L M r dY CnM\W + Y dr = fe+P dY 1 -1 _ r dV N + P dr CwM Apparent specific heat at constant volume 1 K, = % y (20.) CnMN being equal to the real specific heat. Apparent specific heat under constant pressure :- K„ = cin(-i + i)=ft(i+N). 256 THE MECHANICAL ACTION OF HEAT. Eatio of those two specific heats : — |- P =1 + N (21.) This ratio is the quantity called by Poisson y, in his researches on the propagation of sound. 11. It is unnecessary to do more than to refer to the researches of Poisson, and to those of Laplace, for the proof that the effect of the production of heat by the compression of air is the same as if the elasticity varied in proportion to that power of the density whose index is the ratio of the two specific heats ; so that the actual' velocity of sound is greater than that which it would have if there were no such development of heat, in the proportion of the square root of that ratio. The following is the value of the velocity of sound in a gas, as given by Poisson in the second volume of his TraiU de M6caniqu& : — B =^.y.(l+ET)^, . . . (22.) where a denotes the velocity of sound, g the velocity generated by gravity in unity of time, E the coefficient of increase of elasticity with temperature, at the freezing point of water, T the temperature measured from that point, m the specific gravity of mercury, A that of the gas at the tem- perature of melting ice, and pressure corresponding to a column of mercury of the height h. It follows that the ratio y is given by the formula Y = l + Nneari y = gm/(( ^ ET) . . . (23.) Calculations have been made to determine the ratio y from the velocity of sound ; but as many of them involve erroneous values of the coefficient of elasticity E, the experiments have to be reduced anew. The following calculation is founded on an experiment quoted by Poisson on the velocity of sound in atmospheric air, the values of E, w, and A being taken from the experiments of M. Eegnault. a = 340-89 metres per second. g = 9 m -S0896. h = m -76. T = 15 0< 9 Centigrade. in. E = 0-003665; ~ = 10513. A Consequently, for atmospheric air, y = 1-401. THE MECHANICAL ACTION OF HEAT. 257 The results of a reduction, according to correct data, of the experiments of Dulong upon the velocity of sound in atmospheric air, oxygen, and hydrogen, are as follows : — Atmospheric air, Oxygen, . Hydrogen, 7- 1-410 1-426 1-426 Thus it appears, that for the simple substances, oxygen and hydrogen, the ratio N is the same, while for atmospheric air it is somewhat smaller* 12. The ordinary mode of expressing the specific heats of gases is to state their ratios to that of an equal volume of atmospheric air at the same pressure and temperature. When - is a very small fraction, specific heats of unity of volume of a perfect gas are given by the equations — M K n = n M K t 1 CN 1 /l SGr + (24.) That is to say, the specific heat of unity of volume at constant volume is inversely proportional to the fraction by which the ratio of the two specific heats exceeds unity; a conclusion already deduced from experiment by Dulong. The following is a comparison of the ratios of the apparent specific heats under constant pressure, of unity of volume of oxygen and hydrogen * The following are some additional determinations of the value of y for atmospheric air, founded upon experiments on the velocity of sound : — Observers. Bravals and Martins : mean of several experi- ments at temperatures varying from 5° to 11° Centigrade, reduced to 0° (Comptes JRendus, xix.) ...... Moll and Van Beeh : reduced to Stampfer and Myrbach: reduced to 0° (not corrected for moisture) .... Academic des Sciences, 173S : (not corrected ) for moisture) . . . . . . S A variation of one metre per second in the velocity of sound at 0° corresponds to a variation of '0085 in the value of y. K T Centigrade. a Metres per second. y 0° 332-37 1-40955 0° 332-25 1-40853 1 0' 332-96 1-41456 G°l 337-10 1-41S 258 THE MECHANICAL ACTION OF HEAT. respectively, to that of atmospheric air, as deduced from equation (24), with those determined experimentally by De la Roche and Berard: — _ A . wMK p (Gas) Ratio ■ n M K r (Atmos. air) Gas. By Theory. By Experiment. Oxygen, . . . 0-973 0*9705 Hydrogen, . . . 0-973 0-9033 This comparison exhibits a much more close agreement between theory and experiment than has been hitherto supposed to exist, the errors in the constants employed having had the effect of making the ratio 1+N seem greater for atmospheric air than for oxygen and hydrogen, while in fact it is smaller. To treat the other substances on which both M. Dulong and MM. De la Roche and Berard made experiments as perfect gases, would lead to sensible errors. I have, therefore, confined my calculations for the present to oxygen, hydrogen, and atmospheric air. 13. The heat produced by compressing so much of a perfect gas as would occupy unity of volume under the pressure unity, at the temperature 0° Centigrade, from its actual volume nMY 1 = p-p> into a volume which is less in a given ratio ^ (when k is neglected as compared with r), is expressed by the following motion : — sV s «MQ' = -i( 1 dY.^ = -nM\ 1 t Vds, . (25.) v 1 being, in fact, equal to the mechanical power used in the compression. When the temperature is maintained constant, this becomes »MQ' (T) = £log e .i . . (26.) which is obviously independent of the nature of the gas. Hence, equal volumes of all substances in the state of perfect gas, at the same pressure, and at equal and constant temperatures, being compressed by the same amount, disengage equal quantities of heat; a law already deduced from experiment by Dulong. 14. The determination of the fraction N affords the means of calculating the mechanical or absolute value of specific heat, as defined by equation (1), section first. The data for atmospheric air being taken as follows : — N =: 0-4, C = 274°-6 Centigrade, THE MECHANICAL ACTION OF HEAT. 259 —=-= := height of an imaginary column of air of uniform density, at the n M temperature 0° Cent., whose pressure by weight on a given base is equal to its pressure by elasticity, = 7990 metres, =26214 feet :— the real specific heat of atmospheric air, or the depth of fall equivalent to 1 Centigrade degree of temperature in that gas, is found to be C»H = 72-74 metres = 238*60 feet. . (27.) The apparent specific heat of atmospheric air, under constant pressure, according to De la Eoche and Berard, is equal to that of liquid water at 0° Centigrade x 0-2669. The ratio of its real specific heat to the apparent specific heat of water at 0° Centigrade is, therefore, •2669 x ~= -1906. 1"4 And, consequently, the mechanical value of the apparent specific heat of liquid water, at the temperature of melting ice, is ; n '7 = 38T64 metres = 1252 feet per Centigrade degree, ( . x •1906 >- [to.) or 695*6 feet per degree of Fahrenheit's scale. ) This quantity we shall denote by K w . It is the mechanical equivalent of the ordinary thermal unit I have already pointed out (in article 2 of the first section) the causes which tend to make the apparent value of the mechanical equivalent of heat, in Mr. Joule's experiments, greater than the true value. The differences between the result I have just stated, and those at which he has arrived, do not seem greater than those causes are capable of producing when combined with the uncertainty of experiments, like those of Mr. Joule, on extremely small variations of temperature. 15. Besides the conditions of constant volume and constant pressure, there is a third condition in which it is of importance to know the apparent specific heat of an elastic fluid — namely, the condition of vapour at saturation, or in contact with its liquid. The apparent specific heat of a vapour at saturation is the quantity of heat which unity of weight of that vapour receives or gives out, while its temperature is increased by one degree, its volume being at the same time compressed so as to bring it to the maximum pressure corresponding to the increased temperature. 2G0 THE MECHANICAL ACTION OF HEAT. It has been usually taken for granted, that this quantity is the same, with the variation for one degree of temperature, of what is called the total heat of evaporation. Such is, indeed, the case according to the theory of Carnot; but I shall show that, according to the mechanical theory of heat, these two quantities are not only distinct, but in general of con- trary signs. I shall, for the present, consider such vapours only as may be treated in practice as perfect gases, so as to make the first of the equations (20) applicable. It has been shown that the logarithm of the maximum elasticity of a vapour in contact with its liquid may be represented by the expression Log. P = a - £ - -<. T T- The coefficients a, /3, y, being those adapted for calculating the common logarithm of the pressure, I shall use the accented letters a, /3', y, to denote those suited to calculate the hyperbolic logarithm, being equal respectively to the former coefficients X 2 , 3025851. Then for vapour at saturation, 11. _ £ + Ix (29 , Prfr _ r- + J*' ■ ■ ■ (->) Making this substitution in the general equation (21,) wo find the following value for the apparent specific heat of perfectly gaseous vapour at saturation : K s ^h + P ( - = h(l+N. ? -) . (30.) >{i+*(i-f£)} C n M \N ^ T r 2 / 16. For the vapours of which the properties are known, the negative terms t>f this expression exceed the positive at all ordinary temperatures, so that the kind of apparent specific heat now under consideration is a negative quantity : — that is to say, that if a given weight of vapour at saturation is increased in temperature, and at the same time maintained by compression at the maximum elasticity, the heat generated by the compression is greater than that which is required to produce the elevation of temperature, and a surplus of heat is given out; and on the other hand, if vapour at saturation is allowed to expand, and at the same time THE MECHANICAL ACTION OF HEAT. 261 maintained at the temperature of saturation, the heat which disappears in producing the expansion is greater than that set free by the fall of temperature ; and the deficiency of heat must be supplied from without, otherwise a portion of the vapour will be liquefied in order to supply the heat necessary for the expansion of the rest. This circumstance is obviously of great importance in meteorology, and in the theory of the steam-engine. There is as yet no experimental proof of it. It is true that, in the working of non-condensing engines, it has been found that the steam which escapes is always at the tem- perature of saturation corresponding to its pressure, and carries along with it a portion of water in the liquid state ; but it is impossible to distinguish 1 ictween the water which has been liquefied by the expansion of the steam, and that which has been carried over mechanically from the boiler. The calculation of the proportion of vapour liquefied by a given expansion, requires the knowledge of the latent heat of evaporation, which forms the subject of the next section. Section III— Of the Latent and Total Heat of Evaporation, ESPECIALLY FOR WATER. 1 7. The latent heat of evaporation of a given substance at a given temperature, is the amount of heat which disappears in transforming unity of weight of the substance from the liquid state, to that of vapour of the maximum density for the given temperature, being consumed in producing an increase of volume, and an unknown change of molecular arrangement. It is obvious, that if the vapour thus produced is reconverted into the liquid state at the same temperature, the heat given out during the lique- faction must be equal to that consumed during the evaporation; for as the sum of the expansive and compressive powers, and of those dependent on molecular arrangement during the whole process, is equal to zero, so must the sum of the quantities of heat absorbed and evolved. The heat of liquefaction, at a given temperature, is therefore equal to that of evaporation, with the sign reversed. 18. If to the latent heat of evaporation at a given temperature, is added the quantity of heat necessary to raise unity of weight of the liquid from a certain fixed temperature, (usually that of melting ice), to the temperature at which the evaporation takes place, the result is called the total heat of evaporation from the fixed temperature chosen. According to the theory of Carnot, this quantity is called the constituent heat of vapour ; and it is conceived, that if liquid at* the temperature oi melting ice be raised to any temperature and evaporated, and finally 262 THE MECHANICAL ACTION OF HEAT. brought in the state of vapour to a certain given temperature, the whole heat expended will be equal to the constituent heat corresponding to that given temperature, and will be the same, whatsoever may have been the intermediate changes of volume, or the temperature of actual evaporation. According to the mechanical theory of heat, on the other hand, the quantity of heat expended must vary with the intermediate circumstances; for otherwise no power could be gained by the alternate evaporation and liquefaction of a fluid at different temperatures. 19. The law of the latent and total heat of evaporation is immediately dcducible from the principle of the constancy of the total vis viva in the two forms of heat and expansive power, when the body has returned to its primitive density and temperature, as already laid down in article 7. That principle, when applied to evaporation and liquefaction, may be stated as follows : — Let a portion of fluid in the liquid state be raised from a certain temperature to a higher temperature : let it be evaporated at the higher temperature : let the vapour then be allowed to expand, being maintained always at the temperature of saturation for its density, until it is restored to the original temperature, at which temperature let it be liquefied : — then, the excess of the heat absorbed by the fluid above the heat given out, will he equal to the expansive powt v generated. To represent those operations algebraically, — let the lower absolute temperature be t ; the volume of unity of weight of liquid at that temperature r , and that of vapour at saturation V : let the pressure of that vapour be P ; the latent heat of evaporation of unity of weight L ; and let the corresponding quantities for the higher absolute temperature t v be v v V v P x , L r Let K L represent the mean apparent specific heat of the substance in the liquid form between the temperatures r and r r Then,— First, Unity of weight of liquid being raised from the temperature r to the temperature r v absorbs the heat, K l( t i - r o)> and produces the expansive power, J dv.V. Secondly, It is evaporated at the temperature r v absorbing the heat and producing the expansive power, THE MECHANICAL ACTION OF HEAT. 263 P x (V x - vj. Thirdly, The vapour expands, at saturation, until it is restored to the original temperature r. In this process it absorbs the heat, -r dr . K 3 , and produces the expansive power I dV.F. Fourthly, It is liquefied at the original temperature, giving out the heat \y and consuming the compressive power, Po(V -*'o)- The equation between the heat which has disappeared, and the expansive power which has been produced, is as follows : — L x - L + K t (r x -r ) - dr . K s = P x (V x - v x ) - P (Y - v ) + /\ * . P +/ ' ° <Z V . P. y (3i.) j ■V If the vapour be such that it can be regarded as a perfect gas without ^forK s ,andof^ d V r sensible error, the substitution of ft + P -r- for K s , and of n M = HNr for P V, transforms the above to L x - L + {K L - k (1 + N)} (r, - r ) ^j fix /- P o \ (32.) = -P 1 t; 1 + P » +J dv.? = -j dP.v I In almost all cases which occur in practice, v is so small as compared with V, that — [ d P . v may be considered as sensibly = ; and, there- fore, (sensibly) L 1 + K t (r 1 -r ) = L to + fc(l+N)(r 1 -r ). . (33.) 2G4 THE MECHANICAL ACTION OF HEAT. Now this quantity, which I shall denote by H, is the total heat required to raise unity of weight of liquid from r to rj of absolute temperature, and to evaporate it at the latter temperature. Therefore, the total heat of evaporation, where the vapour may be treated as a perfect gas, increases sensibly at an unifwm rate icith the temper attire of evaporation; and the coefficient of Us increase with temperature is equal to the apparent specific heat of the vapour at constant pressure, lt(l + X). 20. There have never been any experiments from which the apparent specific heat of steam under constant pressure can be deduced in the manner in which that of permanent gases has been ascertained. The experiments of M. Eegnault, however, prove that the total heat of evaporation of water increases uniformly with the temperature from 0° to 200° Centigrade, and thus far fully confirm the results of this theory. The coefficient of increase is equal to K w x 0-305. Its mechanical value is, consequently, 1 16*4 metres = 382 feet per Centigrade degree, or 212 feet per degree of Fahrenheit. (34.) Although the principle of the conservation of vis viva has thus enabled us to ascertain the law <>f increase of the total heat of evaporation, it does not enable us to calculate a priori the constant L of the formula, being the latent heat of evaporation at the fixed temperature from which the total heat is measured ; for the changes of molecular arrangement which constitute evaporation are unknown. "When the fixed temperature is that of melting ice, M. Eegnault's experiments give 606 5 Centigrade degrees, applied to liquid water as the value of this constant : so that H = K w (606°-5 + -30:)T°), for the Centigrade scale, H = K w (l091°-7 + -305 (T° - 32°)\ for Fahrenheit's scale, (35.) is the complete expression for the heat required to raise unity of weight of water from the temperature of melting ice to T° above the ordinary zero, and to evaporate it at the latter temperature. This formula has- been given by M. Eegnault as merely empirical ; but we have seen that it closely represents the physical law when quantities depending on the expansion of water are neglected. THE MECHANICAL ACTION OF HEAT. 2G5 It must be remarked, that the unit of heat in M. Regnault's tables is not precisely the specific heat of water at 0° Centigrade, but its mean specific heat between the initial and final temperatures of the water in the calorimeter. The utmost error, however, which can arise from this circumstance, is less than y^- of the total heat of evaporation, so that it may safely be neglected. The coefficient -305 K w = 382 feet per Centigrade degree, is the apparent specific heat of steam at constant pressure ; that is to say, for steam, — f{ + — — m = 382 feet per Centigrade degree, but ~ — ^ = 153 ft. 1 Therefore, the real specific heat of steam is K = 7. — ^r^p =229 1 C 11 JV1 -N feet per Centigrade degree = 127*4 feet per deg. of Fahrenheit 153 _ 2 229 ~ 3' > (36.) 153 ° = K W X -183,andX = — = The quantity — I d P . v has been neglected, as already explained, in these calculations, on account of its smallness. When r = C, or the fixed point is 0° Centigrade, this integral is nearly ecpial to which for steam, is equal to — T\ x -122 V - t v 1 For a pressure of eight atmospheres, ~= = -—7 nearly, r a = 445 0, 5 (T = 170°-9 Cent.); consequently, - v P 2 = - K w X 0°-22 Cent., a quantity much less than the limit of errors of observation in experiments on latent heat. This shows that in practice we are justified in overlook- ing the influence of the volume of the liquid water on the heat of evaporation. Section IV. — Or the Mechanical Action of Steam treated as a Perfect Gas, and the Power of the Steam Engine. 21. In the present limited state of our experimental knowledge of the density of steam at pressures differing much from that of the atmosphere. 200 THE MECHANICAL ACTION OF HEAT. it is desirable to ascertain whether any material error is likely to arise from treating it as a perfect gas. For this purpose the ratio of the volume of steam at 100° Centigrade under the pressure of one atmosphere, to that of the water which produces it at 4 0, 1 Centigrade, as calculated theoretically on the supposition of steam being a perfect gas, is to be compared with the actual ratio. The weight of one volume of water at 4 0- l Centigrade being taken as unity, that of half a volume of oxygen at 0° Centigrade, under the pressure of one atmosphere, according to the experiments of M. Eegnault, is 0*000714900 That of one volume of hydrogen, .... 0-000089578 The sum being, 0-000804478 The reciprocal of this sum being multiplied by -»,„ = 1'3 64 16 6, the ratio of dilatation of a perfect gas from 0° to 100° Centigrade, the result gives for the volume of steam of saturation at 100° Centigrade, as com- pared with that of water, At 4°-l 1095-72 And for its density, .... 0-00058972 The agreement of those results with the known volume and density of steam is sufficiently close to show, that at pressures less than one atmos- phere, it may be regarded as a gas sensibly perfect ; from which it may be concluded, that, in the absence of more precise data, the errors arising from treating it as a perfect gas at such higher pressures as occur in practice, will not be of much importance. Representing, then, by v the volume of unity of weight of water at 4 "1 Centigrade, that of unity of weight of steam at any pressure and temperature will be given by the formula _ 1696e<o r \ - — ^y- . p . • • • (38.) (o representing the number of units of weight per unit of area in the pressure of one atmosphere, and (r) the absolute temperature at which the pressure of saturation is one atmosphere ; being for the Centigrade scale 374°-6, and for Fahrenheit's scale 674 0> 28. The mechanical action of unity of weight of steam at the temperature r and pressure P, during its entrance into a cylinder, before it is permitted to expand, is represented by the product of its pressure and volume, or by THE MECHANICAL ACTION OF HEAT. 207 PV, 16 ^., . . . (39.) The coefficient — y-r — represents a certain depth of fall per degree of absolute temperature, and is the same with the coefficient ~ — ^., already referred to. By taking the following values of the factors: — v = 0*016 cubic foot per pound avoirdupois, w = 2117 pounds avoirdupois per square foot, we find this coefficient to be 153*35 feet = 46*74 metres per Centigrade degree, 85*19 feet per degree of Fahrenheit. (40.) This determination may be considered correct to about 12 1 0o part. When French measures are used in the calculation, the following is the result : — v = 1 cubic centimetre per gramme, to = 1033*3 grammes per square centimetre. 1 ,, = 46*78 metres per Centigrade degree, C»M 153*48 feet, or 85*27 feet per degree of Fahrenheit. (41.) The difference, which is of no practical importance in calculating the power of the steam-engine, arises in the estimation of the density of liquid water. 22. Unit of weight of steam at saturation, of the elasticity P x and volume V 19 corresponding to the absolute temperature r v being cut off from external sources of heat, it is now to be investigated what amount of power it will produce in expanding to a lower pressure P 2 and temperature r 2 . It has already been shown, at the end of the second section, that if vapour at saturation is allowed to expand, it requires a supply of heat from without to maintain it at the temperature of saturation, otherwise a portion of it must be liquefied to supply the heat required to expand the rest. Hence, when unity of weight of steam at saturation, at the pressure Pj and volume V v expands to a lower pressure P, being cut off from external sources of heat, it will not occupy the entire volume V corre- sponding to that pressure, according to equation (38), but a less volume, S = mV, where m represents the weight of water remaining in the gaseous state, the portion 1 — m having been liquefied during the expan- 2GS THE MECHANICAL ACTION OF HEAT. sion of the remainder. The expansive action of the steam will, therefore, be represented by S %ZS.P (42.) /: The law of variation of the fraction m flows from the following con- siderations : — Let d m represent the indefinitely small variation of m corresponding to the indefinitely small change of temperature S r ; L, the latent heat of evaporation of unity of weight ; K s , as in equation (30), the specific heat of vapour at saturation, which is a negative coefficient varying with the temperature ; then we must have — L o m = m EL o r, or — = — T - S t, in L in order that the heat produced by the liquefaction of $ m may be equal to the heat required to expand m. Hence making, according to equation (30) — K s Sr = 1i(Sr + N^Sv), and Sr = -y8V-^ 1 + 2 V we obtain m " IA ft' 27 - + -I- - 1 ) v' and denoting the coefficient of (43.) SV by - v, d log. m d log. V and because d log. V d log. P d log. m d log. P d log. S V cl log. S V ' d log. V P...I 2 V T + T 2 + V 1 - 1 /3' + 2y d log. P = -0-v) 1- 1 ^4 T T~ 2y ^ (44.) THE MECHANICAL ACTION OF HEAT. 269 As the mean temperature of the liquid thus produced, more or less exceeds that of the remaining vapour, a small fraction of it will be reconverted into vapour, if the expansion is carried on slowly enough ; but its amount is so small, that to take it into account would needlessly complicate the calculation, without making it to any material extent more accurate. 23. The extreme complexity of the exponent a, considered as a function of the pressure P, would render a general formula for the expansive action /PfZS very cumbrous in its application. For practical purposes, it is sufficient to consider the exponent o- as constant during the expansion which takes place in any given engine, assigning it an average value suitable to the part of the scale of pressures in which the expansion takes p>lace. For engines in which the steam is introduced at pressures not exceeding four atmospheres, I conceive that it will be sufficiently accurate to make tr = — - } while for engines in which the initial pressure lies 5 between four and eight atmospheres, the suitable value is a = -. The utmost error which can arise from using these exponents is about yijj of the whole power of the engine, and that only in extreme cases. Making, therefore, * = *>$ we obtain for the value of the expansive action of unity of weight of steam, V, 1 P,V y (45.) ^iU l -' l ' s ) s s being used to denote ==^, or the ratio of the volumes occupied by steam at the end and at the beginning of the expansion respectively. A table to facilitate the computation is given in the Appendix. The gross mechanical action of unity of weight of steam on one side of the piston is found by adding to the above quantity the action of the steam before it begins to expand, or P x V v and is therefore, (46.) 270 THE MECHANICAL ACTION OF HEAT. The values of the coefficients and exponent being 1 a !_1 1 - a 1 - <7 x — - . a For initial pressures between 1 and 4 atmospheres, . 7 6 1 6' 4 and 8 atmospheres, . 6 5 1 5' 24. The following deductions have to be made from the gross action, in order to obtain the action effective in overcoming resistance. First, For loss of power owing to a portion of the steam being employed in filling steam-passages, and the space called the clearance of the cylinder at one end. Let the bulk of steam so employed be the fraction c S 2 of the space filled by steam at the end of the expansion ; then the loss of power from this cause is PjCSo = csP 1 V 1 . Secondly, For the pressure on the opposite side of the piston, of the steam which escapes into the condenser, or into the atmosphere, as the case may be. Let P 3 be the pressure of this steam ; the deduction to be made for its action is P 3 S.,(1 -c)= P 8 V t (l - c)s. These deductions having been made, there is obtained for the effect of unity of weight of water evaporated, VijP, (y^ - ^-J ~ »- cs) - P,(l - c)s) (47.) 25. The effect of the engine in unity of time is found by multiplying the above quantity by the number of units of weight of water evaporated in unity of time. If this number be denoted by W, "W S 2 (1 - c) = W V 1 (1 - c) s = Au, . . (48.) will represent the cubical space traversed by the piston in unity of time, A denoting the area of the piston, and u its mean velocity. Now, let the whole resistance to be overcome by the engine be reduced, by the principles of statics, to a certain equivalent pressure per unit of area of piston, and let this pressure be denoted by P. Then, THE MECHANICAL ACTION OF HEAT. 271 E A u = E W Vj (1 - c) 5, . . . (49.) expresses the effect of the engine in terms of the gross resistance. We have now the means of calculating the circumstances attending the "working of a steam-engine, according to the principle of the conservation of vis viva, or, in other words, of the equality of power and effect, which regulates the action of all machines that move with an uniform or periodical velocity. This principle was first applied to the steam-engine by the Count de Pambour ; and, accordingly, the formulae which I am about to give only differ from those of his work in the expressions for the maximum pressure at a given temperature, and for the expansive action of the steam, which are results peculiar to the theory of this essay. In the first place, the effect as expressed in terms of the pressure, is to be equated to the effect as expressed in terms of the resistance, as follows: — ea^kwv^i -c)s = wVijp^j-^- yzt^ s a ~ cs ) -P 3 (l-c)s} . . . (50.) This is the fundamental equation of the action of the steam-engine, and corresponds with equation A. of M. de Pambour's theory. 26. Dividing both sides of equation (50) by the space traversed by the piston in unity of time, "W V x (1 — c) s, and transferring the pressure of the waste steam, P 3 , to the first side, we obtain this equation : — i-L cs R+p 3= p 1 1 ' ll-'). ■ ■ («•) which gives the means of determining the pressure P x at which the steam must enter the cylinder, in order to overcome a given resistance and counter-pressure with a given expansion ; or, supposing the expansion s to be variable at pleasure, and the initial pressure P x fixed, the equation gives the means of finding, by approximation, the expansion best adapted to overcome a given resistance and counter-pressure. The next step is to determine, from equations (XV.) of the introduction and (38) of this section, the volume V 1 of unity of weight of steam corresponding to the maximum pressure P r Then equation (48) gives the space traversed by the piston in unity of time, which, being multiplied by the resistance E per unit of area of piston, gives the gross effect of the engine. 272 THE MECHANICAL ACTION OF HEAT. 27. If, on the other hand, the space traversed by the piston in unity of time is fixed, equation (48) gives the means of determining, from the evaporating power of the boiler W, either the volume Y 1 of unity of weight of steam required to work the engine at a given velocity with a given expansion, or the expansion s proper to enable steam of a given initial density to work the engine at the given velocity. The initial pressure P x being then determined from the volume Y v the resistance which the engine is capable of overcoming with the given velocity is to be calculated by means of equation (51). 28. This calculation involves the determination of the pressure P 2 from the volume V 1 of unity of weight of steam at saturation, which can only be done by approximation. The following formula will be found useful for this purpose : — 12 F-i^Hv/ 1 • • • • ( 52 -) where to represents the pressure of one atmosphere, V the volume of steam of saturation at that pressure (being 1G96 times the volume of water at 4°'l Cent., or 27*130 cubic feet per pound avoirdupois), and V, the volume of steam of saturation at the pressure P r This formula is only applicable between the pressures of one and eight atmospheres : that is to say, when the volume of steam is not greater than 27 cubic feet per pound, nor less than 4, and the temperature not lower than 100° Centigrade, nor higher than 171° Centigrade (which correspond to 212° and 340° Fahrenheit). The greatest error in computing the pressure by means of this formula is about -Jg- of an atmosphere, and occurs at the pressure of four atmos- pheres, so that it is pj of the whole pressure. This is sufficiently accurate for practice, in calculating the power of steam-engines ; but should a more accurate result be required, the approximate value of the pressure may be used to calculate the temperature by means of equation (XV.) ; and the temperature thus determined, (which will be correct to ~ of a Centigrade degree), may then be used in conjunction with the volume to compute a corrected value of the pressure, according to equation (38). The pressure, as thus ascertained, will be correct to ^wo of its amount, which may be considered the greatest degree of accuracy attainable. The most convenient and expeditious mode, however, of computing the pressure from the volume, or vice versd, is by interpolation from the table given in the Appendix to this paper. 29. The resistance denoted by R may be divided into two parts ; that which arises from the useful work performed, and that which is independent of it, being, in fact, the resistance of the engine when unloaded. Now it is evident, that the maximum usrful effect of the steam has been attained, THE MECHANICAL ACTION OF HEAT. 273 as soon as it has expanded to a pressure which is in equilibrio with the pressure of the waste steam added to the resistance of the engine when unloaded ; for any further expansion, though increasing the total effect, diminishes the useful effect. Therefore, if we make R = K'+/, R' being the resistance arising from the useful work, and / the resistance of the engine when unloaded, both expressed in the form of pressure on the piston, the expansion corresponding to the maximum of useful effect will take place when P 2 = P 3 +/> the corresponding ratio of expansion being ! ,_ . \p.+// The maximum useful effect with a given pressure on the safety-valve has been so fully discussed by M. de Pambour, that it is unnecessary to do more than to state that it takes place when the initial pressure in the cylinder is equal to that at the safety-valve : that is to say, when it and the useful resistance are the greatest that the safety-valve will permit. 30. Annexed is a table of the values of some of the quantities which enter into the preceding equations in the notation of the Count de Pambour's works : — Expression in the Notation Equivalent Expression in of this Paper. M. tie Pambour's Notation. R = R'+/ (1 + 8)r+/ Aw av W . . . S x weight of one cubic foot of water. P 3 . . p I + c V + c I' + c 31. As an illustration, I shall calculate the maximum useful effect of one pound, and of one cubic foot of water, in a Cornish double-acting S 27-i THE MECHANICAL ACTION OF HEAT. engine, in the circumstances taken by M. de Pambour as an example for that kind of engine, that is to say, — Clearance one-twentieth of the stroke, or c = — Eesistance not depending on the useful load, / = 72 lbs. per sq. ft. Pressure of condensation, . . . . P 3 = 576 lbs. „ „ Consequently, to give the maximum useful effect, P 2 = P S +/ = 048 lbs. „ „ Total pressure of the steam when first admitted, V 1 — 7200 lbs. „ „ Volume of 1 lb. of steam V x = 8*7825 cubic feet. Therefore P x Vj = G3234 lbs. raised one foot. P 7200 — 1 = — — ; and, consequently, P 2 04o Expansion to produce the maximum useful effect s = (rr ) ' — 7'877. Space traversed by the piston during the action of one pound of steam, = V x (1 - c) s = G5-8SG cubic feet. Gross effect of one pound of steam, in pounds raised one foot high, 1 = PjV, (7 -6s' 6 - £) - P 3 V 1 (1 - c)s = 112001 Deduct for resistance of engine when unloaded f\ 1 (1 — c)s = 4744 Effect of one pound of steam in overcoming resistance depend-"! io7°60 ing on useful load, . . . . . . -J This being multiplied by G2£, gives for the effect of one cubic foot of water evaporated, in pounds raised one foot, . G,703,750 It is here necessary to observe, that M. de Pambour distinguishes the useful resistance into two parts, the resistance of the useful load indepen- dently of the engine, and the increase in the resistance of the engine arising from the former resistance, and found by multiplying it by a constant fraction, which he calls $. In calculating the net useful effect, he takes into account the former portion of the resistance only ; consequently : Net useful effect as defined by M. de Pambour THE MECHANICAL ACTION OF HEAT. 275 Gross effect — f Y, (1 — c) s /t , . N = i + g • • <"•> The value of S, for double-acting steam-engines generally, is considered by M. de Pambour to be y ; consequently, to reduce the effect of one cubic foot of water, as calculated above, to that which corresponds with his definition, we must deduct £, which leaves, 5,865,781 lbs. raised one foot. M. de Pambour's own calculation gives, 6,277,560, being too large by about one-fifteenth. 32. In order to show the limit of the effect which may be expected from the expenditure of a given quantity of heat in evaporating water, and also to verify the approximate method employed in calculating the expansive action of the steam, I shall now investigate the maximum gross cjfed, including resistance of all kinds, producible by evaporating unity of weight of Avater at a higher temperature and liquefying it at a lower, and compare, in two examples, the power produced with the heat which disappears during the action of the steam, as calculated directly. To obtain the maximum gross effect, the steam must continue to act expansively until it reaches the pressure of condensation, so that P 2 = P 3 . The clearance must also be null, or c = 0. Making those substitutions in the formula (47), we find, for the maximum gross effect of unity of weight of water, evaporated under the pressure P x , and liquefied under the pressure P OJ , ( -M ^ST" p^r-V 1 - 8 '' - P ' Y ' i - . (55) In order to calculate directly the heat which is converted into power in this operation, let r v r 2 , respectively represent the absolute temperatures of evaporation and liquefaction, and L 2 the latent heat of evaporation at the lower temperature r 2 ; then the total heat of evaporation at t v starting from t as the fixed point, by equation (33), is H 2 , , = L 2 + -305 K w (r, - r 2 ). This is the heat communicated to the water in raising it from r 2 , to t x and evaporating it. Now a weight 1 — m of the steam is liquefied during the expansion at temperatures varying from r a to r 2 , so that it may be looked upon as forming a mass of liquid water approximately at the mean 276 THE MECHANICAL ACTION OF HEAT. temperature ^-— — -, and from which a quantity of heat, approximately a represented by K w (i-m)^p, must be abstracted, to reduce it to the primitive temperature r 2 . Finally, the weight of steam remaining, m, has to be liquefied at the temperature r 2 , by the abstraction of the heat mL 2 . The difference between the heat given to the water, and the heat abstracted from it, or Hjs,, - K w (l - m) Tl T * - mL 2 = (1 - m)U + K w (-305 - 1 - J =-^)(r 1 - r 2 ) > ( 56 is the heat which has disappeared, and ought to agree with the expression (55) for the power produced, if the calculation has been conducted correctly. As a first example, I shall suppose unity of weight of water to be evaporated under the pressure of four atmospheres, and liquefied under that of half an atmosphere ; so that the proper values of the coefficients and exponent are 1 7 1 l = 7, 1 — a = 1 -a ' T The data, in this case, for calculating the power are, P x = 84G8 lbs. per square foot. Vj = 7 - 58 4 cubic feet for 1 lb. of steam. P x V x = 64221 lbs. raised one foot. P 1 " — = -, whence s = 8 7 = 5 '9 44. J-i b Maximum possible effect of one pound of water, = P 1 V 1 x7H- Qm = 115600 lbs. raised one foot. Being the mechanical equivalent of 9 2° '3 Centigrade applied to one pound of liquid water at 0° C; or, THE MECHANICAL ACTION OF HEAT. 277 92°-3 K w . Maximum possible effect of one cubic foot of water, 7,225,000 lbs. raised one foot. In order to calculate directly the heat converted into power, we have, Tj = C + 144°-1 Cent. r 2 = C + 81°'7. L 2 = 549°-7 K w . H 2n = 5 6 8 0- 7 K w = heat expended in the boiler. 1 — m = '14 nearly = proportion of steam liquefied during the expansion. The heat converted into mechanical power, as calculated from these data, is found to be, 01°'6K W , differing by only o, 7 from the amount as calculated from the power produced. The direct method, however, is much less precise than the other, and is to be regarded as only a verification of the general principle of calculation. 92'3 The heat rendered effective, in the above example, is , or less than one-sixth of that expended in the boiler. As a second example, I shall suppose the steam to be produced at a pressure of eight atmospheres, and to expand to that of one atmosphere. In this case, Pj = 16936 lbs. per square foot. Vj = 4'03 cubic feet per lb. of steam. V l Y 1 = 68252 lbs. raised one foot. PI 6 £?= * -.s = 5-657 = 8*. Maximum possible effect of one pound of water, = P 1 V 1 xen- (q)0 = 119 ' 942 lbs - raised one foot ' Being the equivalent of 95 0- 8 K w (Centigrade). Maximum possible effect of one cubic foot of water = 7,496,375 lbs. raised one foot. The data for calculating directly the heat rendered effective are, 278 THE MECHANICAL ACTION OF HEAT. Tl = C + 170 o, 9 Cent, r, = C + 100°. L 2 = 537° K w . Hojj = 558 0, 6 K w = heat expended in the boiler. 1 — m = '148 nearly = steam liquefied during the expansion. Whence, the heat converted in power, as calculated directly, is 95°-S K w , agreeing with the calculation from the power produced. 95"8 In this example, the heat rendered effective is - , or somewhat more than one-sixth of that expended in the boiler. 33. The results of the calculations of maximum possible effect, of which examples have just been given, are limits which may be approached in practice by Cornish and similar engines, but which cannot be fully realised ; and yet it has been shown, that in those theoretical cases only about one-sixth of the heat expended in the boiler is rendered effective. In practice, of course, the proportion of heat rendered effective must be still smaller; and, in fact, in some unexpansivc engines, it amounts to only onc-t ve a f {/-fourth, or even less. Dr. Lyon Playfair, in a memoir on the Evaporating Power of Fuel, has taken notice of the great disproportion between the heat expended in the steam-engine and the work performed. It has now been shown that this waste of heat is, to a great extent, a necessary consequence of the nature of the machine. It can only be reduced by increasing the initial pressure of the steam, and the extent of the expansive action; and to both of those resources there are practical limits, which have already, in some instances, been nearly attained. APPENDIX TO THE FOURTH SECTION. Containing Tables to be used in Calculating the Pressure, Volume, and Mechanical Action of Steam, Treated as a Perfect Gas. The object of the First of the annexed Tables is to facilitate the calcula- tion of the volume of steam of saturation at a given pressure, of the pressure of steam of saturation at a given volume, and of its mechanical action at full pressure. THE MECHANICAL ACTION OF HEAT. 279 The pressures are expressed in pounds avoirdupois per square foot, and the volumes by the number of cubic feet occupied by one pound avoirdu- pois of steam, when considered as a perfect gas ; those denominations being the most convenient for mechanical calculations in this country. The columns to be used in determining the pressure from the volume, and vice versd, are the third, fourth, sixth, and seventh. The third column contains the common logarithms of the pressures of steam of saturation for every fifth degree of the Centigrade thermometer, from — 30° to + 260°: that is to say, for every ninth degree of Fahrenheit's thermometer, from — 22° to + 500°. The fourth column gives the differences of the successive terms of the third column. The sixth column contains the common logarithms of the volume of one pound of steam of saturation corresponding to the same temperatures. The seventh column contains the differences of the successive terms of the sixth column, which are negative ; for the volumes diminish as the pressures increase. By the ordinary method of taking proportional parts of the differences, the logarithms of the volumes corresponding to intermediate pressures, or the logarithms of the pressures corresponding to intermediate volumes, can be calculated with great precision. Thus, let X + h be the logarithm of a pressure not found in the table, X being the next less logarithm which is found in the table ; let Y be the logarithm of the volume cor- responding to X, and Y — h the logarithm of the volume corresponding to X + h; let H be the difference between X and the next greater logarithm in the table, as given in the fourth column, and K the corresponding difference in the seventh column ; then by the proportion H : K : : h : i either Y — Jc may be found from X -f h, or X + h from Y - k In the fifth and eighth columns respectively, are given the actual pressures and volumes corresponding to the logarithms in the third and sixth columns, to five places of figures. In the ninth column are given the values of the quantity denoted by P Vj in the formulae, which represents the mechanical action of unity of weight of steam at full pressure, or before it has begun to expand, in raising an equal weight. Those values are expressed in feet, being the products of the pressures in the fifth column by the volumes in the eighth, and have been found by multiplying the absolute temperature in Centigrade degrees by 153*48 feet. Intermediate terms in this column, for a given pressure or a given volume, may be approximated to by the method of differences, the constant difference for 5° Centigrade being 767'4 280 THE MECHANICAL ACTION OF HEAT. feet ; but it is more accurate to calculate them by taking the product of the pressure and volume. When the pressure is given in other denominations, the following logarithms are to be added to its logarithm, in order to reduce it to pounds avoirdupois per square foot : — For Millimetres of mercury, . . . 0*44477 „ Inches of mercury, .... 1 '84960 „ Atmospheres of 7 GO millimetres, . . 3*32559 „ Atmospheres of 30 inches, . . . 33 2 G 72 . Kilogrammes on the square centimetre, . 3-31136 „ Kilogrammes on the circular centimetre, 3*41 G 27 „ Kilogrammes on the square metre, . 1*31136 „ Pounds avoirdupois on the square inch, . 2*15836 „ Pounds avoirdupois on the circular inch, 2*26327 To reduce the logarithm of the number of cubic metres occupied by one kilogramme to that of the number of cubic feet occupied by one pound avoirdupois, add 1*20463. The logarithms are given to five places of decimals only, as a greater degree of precision is not attainable in calculations of this kind. The Second Table is for the purpose of calculating the mechanical action of steam in expansive engines. The first column contains values of the fraction of the entire capacity of the cylinder which is filled with steam before the expansion commences (being the quantity - of the formulas), for every hundredth part, from 1*00, or the whole cylinder, down to 0*10, or one-tenth. If I be the entire length of stroke, V the portion performed at full pressure, and c the fraction of the entire capacity of the cylinder allowed for clearance, then 1 _ I' s C , 1 „ , V i=Tzre and-=(i -,)- + ,. The entire capacity of the cylinder is to be understood to include clearance at one end only. The second column gives the reciprocals of the quantities in the first, or the values of the ratio of expansion s. THE MECHANICAL ACTION OF HEAT. 281 The third and fourth columns, headed Z, give the values of the quantity 1 C7 1 ~i 1 - a- 1 - a of article 23, which represents the ratio of the entire gross action of the steam to its action at full pressure, without allowing for clearance. The third column is to be used for initial pressures of from one to four atmospheres ; and the fourth for initial pressures of from four to eight atmospheres. The deduction to be made from the quantity Z for clearance is c s, or the product of the fraction of the cylinder allowed for clearance by the ratio of expansion. Hence, to calculate from the tables the net mechanical action of unity of weight of steam, allowing for the counter-pressure of the waste steam P 3 , as well as for clearance, we have the formula p 1 y 1 (z- cs )-p 3 v 1 (i- c ),, being equivalent to the formula (47) of this paper. 282 THE MECHANICAL ACTION OF HEAT. ° 2 .2 t? - £ cm O to 1 " a <DfM N 5 's .g a &* CO w S Ph .j H O »— * H O < w H ft o p ►J o > Q « £> CO W Ph Ph PP o _ £ >■ o CI OS tD CO -rH ph in H -i CC CO 0) rt IO (M -*< O H< CO o^^op^oooipp<»0305c<icpgsec«ioj>> -jO^w-iHOoi^'tioeHHi — t<icrc-ticcci-HC©mio i - -o i- — i- >~. ~ o cc — re o re gc — -t ~ it ci os x o io t -h CI i - -t 3 X :c -r t- im a [^ li ■* n CI « h n a>'«inot>^cQNHH COW i-l i-H i - ( s -i i- /■ :r;oicoi - ;-i-KC. i-cMcsnO <r. » i - 1 - -o i c -? -f e e - ~ c i -i -— — ~ o c. 05 ci ao go go go ooooooooooooooooooobooo ■ o 5 = 3 3, g 5 nor " t- us >~ '-r •- oo m ci — - ~i uo o ei ci © so >o o -# -n i - go -r r. • -r i - oi t ~ ■ i a cc — i - EC — os cc — ■ — • cc re — i - e i — — i - x / •- : - r. c i - 'i i: r. x : i - x c i o o -f o O CO CO' Ci — -+ X : c t --. ~ I - — C I CG OS Ci co GO Ci Ci — ■ CI »* O io«" ; /■ ■-: ifi cc "i — — x i - cc -r re ci — i o Ci os go t^ so "* -i"# -* cc co co re re re ci ci ci ci ci ci ci ci ci . o — i Ci re r~- -t :c 3! r. otOHHCOON ie ics r- co i- t- — is re re e i — c. re re in ic -f ei — < -* re re co os r^ r. ic ■jji go i_- ip rfi x — z. o 1 ip cr. i - :c -t c. 9 m o ^ -t o «) o r^-< ci re ih go ci 1 - ic w k >e 1'- co re cs o go >b ^ o ■* go Os « 1-1 ci n -t a co «o cc ic ci - ci ic: o o o 1— 1 1—1 1-1 ci ce •"# o o co Ci ci m ci c -t o co h ci -f a -t c t^ 51 c ei •* t^ h. o o eo co o cc « o co -t o - o w « x o -t r- o -1 -t cc a m 10 10 ic cc 1 - gc ~ re cc — -r co ci 1 - ci x -t cr ■ cc ci co ie 01 o co o o co 1- 1- a ic io t m k ci :i « 1- h c o o o o co co P ( T < 'TC- i TC- , 7 l 7 l 7 l 7 l rc- , 7 1 7 l C H 7 l 7 H - l 99P9? OOOOOOOOOOOOOOOOOOOOOO co — 1 re re t^ ci o -^ re »■- re r^ r- -** co o co ci o co ci ci 10 ro ,. — 1< -t ic — — o — ci ci co -* ci ci co 1 - 1- c. re re o >o o o ci go -r — o <-> -t — 1 — < 10 re co ■* t- ic o co m co t^ re m 10 -h o o o x a cc cc 1 s c -t cc m -r a x cc o h - h h c co o rt n o 1- cs o ci -t iq a co k s h ci ■* n o t~>» o a o l-< O O OO ©1 1 CI CI CI CI CI CI CI CI CI CI CO rl © O id o o o o O LC O >C OiCOiOOOO i-O O 10 OiOOiC cecicirtrt rtr-icicirere'*Tt<ioLe)cooi^t--coco I I I I I I + a 9 si cire-rcHuC)-i'reci--iooGoi>'Co»ci-f<roci^ococot~~oio ci^h rtCiroTtiioi-ocot-'COO©^cico-*Ttiiciot^co I , I + rH rt rl ^H rt rH rH -I p-I -H THE MECHANICAL ACTION OF HEAT. 283 1~~ CO CI toint3ooHi>.o(M!OMoooHcoHb.ioioHia^o5io»i>Hi^iono rt< o co (M««(»n(MQ'<)ijio'MH-jioooH©cio-Hati(ooo*tKinio i~- o ■* (NHHOl^QOOGO-f ^lOOOHOHOMONrtKMOCOt'-lOTllCOIMHO O O CO o o o CO CO O tO T* CO CO ">* lO 00 -H •># C5 -+ 05 lO 0) Ci !•- "O -* CO 71 CI Ol Ol CO -+ CO !>■ Ci CM •** t^ Ci t-IO(NOl»tO'i<(MO'JOh-lfl«CIOO)OOOWiieCIN'HOO(»N0 1STi(MM Ol ^H O M>Nt- O 'J a C C I" 1.1 O IS O « 1"* -t -f 4 i< i"* i" K K K M ft :^ « W CO CO CO 99900099909999900009999000000099 9 9 9 00000000006000000000000000060000 6 6 6 OCOMC5Clt>.CO?li.'5'^COi.OIMOi-ic)T)('OOM'HO-*HNCOTt(tO'#«Ow-^ t- t~- CO CO Ci O CO Ci lO 01 CO 01 -t" CO CO 3i lO lO OS K3 -O CI :<: Q O « H 35 O IM Nrt M ■* tl t^ O O Ci » i* n I* » ■* h rt »i o ci o « ci c< <* n -1 1> m h a o O ci o 00 « M tji -n 00 t>. co OOOW'BOClCO'COOClO'HOrtOHOClt-nOOHh.nocCIBi.'JCI CO O Ol ipip^nciciHiHooocpcp^^ooinip^^nMMWNrtrtrtpop 9 9 9 HHHHrtrtHHHooboobooooooboooooooooolHlHlH oci^^oiocococioo-^o Miobcococib-HNint^bo' o o h w o 05 •# m -^ ■* 10 o n 1 — iiNHLcaiSHtoooicsaMciooonociciooaccf cinoooo o t~- ci rtH«CINM'*^OOl-00QHCl<*t0C0OnaoCIOO«!»:i5»C0Ou') ci r-~ riHrtrti- 1 ci ci ci o co co co Tf rf o m o o i> co co 03 «rtcotO'*-HQr.LC'*cioaNOT(imr-ooK-'C't::'M^oaacoN to 10 10 00 00 N nn n o o o 'O o a 11 o 10 ic o o a •* t -t "t -f -t -t •* ■# m w n w co co co 99999999999999999999999999990990 o 9 9 6666666666666666666 6 666666666666 6 6 6 HO©o^ocotoeoi>i.o^Hcicooa)cicoocico-i<cii>-ii>cioc)^-i ,-h co o IOMiOOHHCiMCOC»COOi-<oo-H?jCU^-Hfl-OCOin«00!COOiHifl H Ci o lOtOiOHOCOfLOHlOOOOOOIOHOOHHMQ-OCrCOKiCOOiiMO I>» CI CO tS'^CIOI^'*rtOOin-<|>MO>IOHt9-HNCIt^-lOrtlOO-) , CCCI0 3-t(CO 1—1 lO CO HNcp^^LOoot^xcpffiooHH^NMeo^^inoofflo^L^cpcpcp 9 9 fOCOCOCOCOCOCOCOCOCOCOCOCO^^^^^T^^^^rttTji^^^^^-^TtHTj* ^ -^1 ^JH oooiooooooooioooooooouoiooiooooicoioon o »o o OOOOHHCIClMni<'*10in(OONNCO»OOC2' J 'HCIC!-CnTlT)( 10> 10 to HHHHHrtHi-IHrtrtHrtHrtHHrtrlrtCINNNClClCICICIC) CI CI d ■"fieONp-iOQoONtOLOTfiMClHOQcOlNOiOTfiMClHOOMNtJiaiiM CI — t O Ci o —1 oi co cc -* 10 to r- co OS o — 1 ci ci co -+ 1 o to t— co o O 1— 1 — ' ci co tj< 1.0 to r-~ co ci o r^c^cicicicicicicicicacicococococococococococO'*-*-*'*'*'^ | -i*'#-* tjh -^ m 284 THE MECHANICAL ACTION OF HEAT. TABLE II. — Expansive Action of Steam. (1.) (2.) (3.) (4.) (1.) (2.) (3.) (4.) Fraction of Coefficient of Gross Fraction of Coefficient of Cros3 Cylinder Action = Z. i rider Action = Z. filled with Steam at full Pressuro Patio of Expansion Cnitj il Pres- sure One Initial Pres- sure Four filled with Steam at full Pressure Ratio of Expansion Initial Pres- sure One Initial Pros- sure Four 1 ' to Four to Eight 1 = •'• to Four to Eight Atmos- Atmos- Atmos- — s' pheres. ph( l s pheres. pheros. 1-00 1-000 1-000 1-000 •54 1-852 1-586 1 -5S0 ■99 1010 1010 1010 •53 1-887 1G02 1-596 •98 1-020 1020 1 020 •52 1 -923 1 -620 1-613 •97 1031 1030 1 -030 •51 1-961 1-637 1-G30 •9G 1 -042 1041 1041 •50 2 000 1 -655 1 -647 •95 1 053 1051 1-051 •49 2 041 1-G73 L-665 •94 1-064 1-0G2 1-0G2 •48 2-083 1-691 1-683 •93 1075 1072 L-072 •47 2-12S 1-709 1-701 •92 1-087 1-083 10S3 •46 2174 1-728 1-719 •91 1 -099 1-094 1 093 •45 2*222 1-748 1-738 •90 1-111 1-104 1-104 •44 , 2-273 1-767 1-757 •89 1-124 1115 1-115 •43 2-326 1-787 1777 •88 1-136 1-126 1-126 •42 2 381 1-803 1 -79G •87 1-149 1-138 1-137 •11 2-439 1-S29 1-817 •86 1-1G3 1-149 1-149 •40 2-500 1-850 1-837 •85 1-176 1-1G0 1-1G0 •39 2-564 1-871 1-858 •84 1-190 1-172 1-171 •38 2-632 1-894 1-880 •83 1-205 1183 1-1S3 •37 2-703 1-916 1-902 •82 1 -220 1-195 1-195 •36 2-778 1-939 1-924 •81 1 -235 1-207 1 206 •35 2-857 1-963 1-947 •80 1-250 1-219 1-218 •34 2-941 1-987 1-970 •79 1-266 1-231 1-230 •33 3 030 2012 1-994 •78 1-282 1-243 1-242 •32 3125 2-038 2 019 •77 1 -209 1 -256 1 -255 •31 3-225 2 064 2 044 •76 1-316 1-268 1-2G7 •30 3-333 2091 2 070 •75 1 -333 1-281 1-280 •29 3-44S 2119 2 097 •74 1-351 1-294 1 -292 •28 3 571 2-147 2-124 •73 1-370 1 -307 1-305 •27 3-704 2176 2152 •72 1-389 1-320 1-318 •26 3-S46 2-207 2-181 •71 1-408 1-333 1-331 •25 4-000 2-238 2-211 •70 1-429 1-346 1-344 •24 4-167 2-270 2-242 •69 1-449 1-3G0 1-358 •23 4-34S 2 304 2-273 •6S 1-471 1 -374 1-371 •22 4-545 2-338 2-306 •67 1-493 1-387 1-385 •21 4-762 2-374 2-341 •66 1-515 1-401 1-399 •20 5 000 2412 2-376 •65 1-538 1-416 1-413 •19 5-203 2-451 2-413 •64 1-563 1-430 1-427 •18 5-556 2-492 2-452 •63 1-587 1-445 1-441 •17 5-8S2 2-534 2-492 •62 1-613 1-459 1 -456 •16 6-250 2-579 2-434 •61 1-640 1-474 1-471 •15 6-667 2-626 2-579 •60 1-667 1-490 1-486 •14 7143 2-676 2-626 •59 1-695 1-505 1-501 •13 7-692 2-730 2-675 •58 1-724 1-521 1-516 •12 8-333 2-786 2-728 •57 1-754 1-537 1-532 •11 9 091 2-847 2-784 •56 1-786 1-553 1-547 •10 10 000 2-912 2-845 •55 1-818 1-569 1-563 THE MECHANICAL ACTION OF HEAT. 285 XV.— NOTE AS TO THE DYNAMICAL EQUIVALENT OF TEMPERATURE IN LIQUID WATER, AND THE SPECIFIC HEAT OF ATMOSPHERIC AIR AND STEAM: Being a Supplement to a Paper On the Mechanical Action of Heat.* 33. In my paper on the Mechanical Action of Heat (see p. *23Jt), published in the first part of the twentieth volume of the Transactions of Hie Royal Society of Edinburgh, some of the numerical results depend upon the dynamical equivalent of a degree of temperature in liquid -water. The value of that quantity which I then used, was calculated from the experi- ments of De la Roche and Be>ard on the apparent specific heat of atmospheric air under constant pressure, as compared with liquid water. The experiments of Mr. Joule on the production of heat by friction give, for the specific heat of liquid water, an equivalent about one-ninth part greater than that which is determined from those of De la Roche and B6rard. I was formerly disposed to ascribe this discrepancy, in a great measure, to the smallness of the differences of temperature measured by Mr. Joule, and to unknown causes of loss of power in his apparatus, such as the production of sound and of electricity; but, subsequently to the publication of my paper, I have seen the detailed account of Mr. Joule's last experiments in the Philosophical Transactions for 1850, which has convinced me, that the uncertainty arising from the smallness of the elevations of temperature, is removed by the multitude of experi- ments (being forty on water, fifty on mercury, and twenty on cast iron) ; that the agreement amongst the results from substances so different, shows that the error by unknown losses of power is insensible, or nearly so; and that the necessary conclusion is, that the dynamical value assigned by Mr. Joule to the specific heat of liquid water — viz., 772 feet per degree of Fahrenheit — does not err by more than two, or, at the utmost, three feet; and therefore, that the discrepancy originates chiefly in the experiments of De la Roche and Berard. I therefore take the earliest opportunity of correcting such of my calculations as require it, so as to correspond with Mr. Joule's equivalent. * Eead before the Eoyal Society of Edinburgh on December 2, 1850, and published in the Transactions of that Society, Vol. XX., Part II. 286 THE MECHANICAL ACTION OF HEAT. They relate to the specific heat of atmospheric air as compared with liquid water, and to that of steam, and are contained in the second and third sections of my paper, Articles 14 and 20 ; equations 28, 34, and 36. Specific Heat of Atmospheric Air as Compared with liquid Water. — (Section II., Article 14.) The dynamical values of the specific heat of atmospheric air are calculated independently from the velocity of sound, without reference to the specific heat of liquid water; and from the closeness of the agree- ment of the experiments of MM. Bravais and Martins, Moll and Van Beek, Stampfcr and Myrbach, Wertheim and others, it is clear that the limits of error are about ^ for the velocity of sound, j^ for the ratio, and from - 4 T - to - 5 V for the dynamical values of the specific heat of air, at constant volume and constant pressure. Those values, as given by equation (27), are — Real specific heat, — ft = 23S'GG feet = 7274 metres per Centigrade degree. = 132-G feet per degree of Fahrenheit. Apparent specific heat under constant pressure, — K p = 334 fcet= 101-8 metres per Centigrade degree, = 185*6 feet per degree of Fahrenheit. The ratio of these two quantities being taken as ^- p =l +N= 1-4. R The dynamical equivalent of the specific heat of liquid water, as determined by Mr. Joule, is K w = 1389-6 feet = 423-54 metres per Centigrade degree, = 772 feet per degree of Fahrenheit. The specific heat of air, that of liquid water being taken as unity, has therefore the following values : — Real specific heat, — It 1326 IF-— r-nn = 1< 1 I. K w 772 THE MECHANICAL ACTION OF HEAT. 287 Apparent specific heat under constant pressure, — K P . 185-6 nnlnt — - = = ,o 404 K w 772 V ~ Wi - This last quantity, according to De la Eoche and Berard, is 0-2669 The discrepancy being . . . . 0'0265 or one-ninth of the value, according to Mr. Joule's equivalent. Specific Heat of Steam. — (Section III, Art. 20.) The apparent specific heat of steam (equations 34 and 36) as a gas under constant pressure, is equal to that of liquid water x0'305. Its dynamical value is, therefore, K P = fc + ?1 Vr = 1389-6 x 0-30.3 C n M = 422-83 feet = 129-18 metres per Cent, degree. But — — — r = 153*48 feet = 46 - 78 metres per Cent, decree. Therefore, the real specific heat is ft = 269*35 feet = 82 - 40 metres per Cent, degree. Or, that of liquid water being taken as unity, A = 2^ = 0-194 K w 1389-6 U1J *- The ratio of these two values of the specific heat of steam is l+N=l-57. Their dynamical equivalents for Fahrenheit's scale are, ft = 149-64 feet, . . . K P = 235'46 feet. Neither the formula? in the fourth Section, respecting the working of the steam-engine, nor the tables at the end of the paper, require any alteration ; for the action of steam at full pressure being calculated from data independent of its specific heat, is not at all affected by the discrepancy I have mentioned ; and the expansive action is not affected to an extent appreciable in practice. 2SS THE MECHANICAL ACTION OF HEAT. XVI.— ON THE POWER AND ECONOMY OF SINGLE-ACTING EXPANSIVE STEAM-ENGINES : Being a Supplement to the Fourth Section of a Paper On the Mechanical Action of Heat.* 34. The objects of this" paper are twofold : First, To compare the results of the formula? and tables relative to the power of the steam- engine, which have been deduced from the dynamical theory of heat, with those of experiments on the actual duty of a large Cornish engine at various rates of expansion ; and, Second!;/, To investigate and explain the method of determining the rate of expansion, and, consequently, the dimensions and proportions of a Cornish engine, which, with a given maximum pressure of steam in the cylinder, at a given velocity, shall perform a given amount of work at the least possible pecuniary cost, taking into account the expense of fuel, and the interest of the capital required for the construction of the engine. This problem is solved with the aid of the tables already printed, by drawing two straight lines on a diagram annexed to this paper. The merit of first proposing the question of the economy of expansive engines in this definite shape belongs, I believe, to the Artizan Club, who have offered premiums for its solution ; having done so (to use their own words) ''with a view to enable those who, from their position, cannot take part in the discussions of the various scientific societies, to give the profession the benefit of their studies and experience." The 5th of April is the latest day fixed by them for receiving papers; and as this communication cannot possibly be read to a meeting before the 7th April, nor published until some months afterwards, I trust I may feel confident that it will not be considered as interfering with their design. Formula Applicable to the Cornish Engine. 35. The equations of motion of the steam-engine, in this and the original paper, are the same in their general form with those of M. de * Eead before the Royal Society of Edinburgh on April 21, 1851, and published in the Transactions of that Society, Vol. XX., Part II. SINGLE-ACTING EXPANSIVE STEAM-ENGINES. 289 Pambour. The differences consist in the expressions for the pressure .and volume of steam, and for the mechanical effect of its expansion ; the former of which were deduced from a formula suggested by peculiar hypothetical views, and the latter from the dynamical theory of heat. Those equations are Nos. (50) and (51) of the original paper. (Seep. 971.) I shall now express them in a form more convenient for practical use, the notation being as follows : — Let A be the area of the piston ; /, the length of stroke ; v, the number of double strokes in unity of time ; c, the fraction of the total bulk of steam above the piston when down, allowed for clearance, and for filling steam-passages; so that the total bulk of steam at the end of the effective stroke is ... . . . (a.) 1 — c I', the length of the portion of the stroke performed when the steam is cut off. s, the ratio of expansion of the steam, so that (b.) Let W be the weight of steam expended in unity of time. Pj, the pressure at which it enters the cylinder. A" , the volume of unity of weight of steam at saturation at the pressure P x j which may be found from Table I. of the Appendix to the original paper. (See p. 282) F, the sum of all the resistances not depending on the useful load, reduced to a pressure per unit of area of piston; whether arising from imperfect vacuum in the condenser, resistance of the air-pump, feed-pump, and cold-water pump, friction, or any other cause. E, the resistance arising from the useful load, reduced to a pressure per unit of area of piston. Z, the ratio of the total action of steam Avorking at the expansion s, to its action without expansion. Values of this ratio are given in the second table of the Appendix to the original paper. Then the following are the two fundamental equations of the motion of the steam-engine as comprehended in equation (50) of the original paper. . First, Equality of power and effect, — T 1 J' -\ s = (!■ -;h + ; 1 - c f V s I = T^ - c J 290 SINGLE-ACTING EXPANSIVE STEAM-ENGINES. R A I a = W V, {P 2 (Z - c s) - F (1 - c) *}. . (c.) Secondly, Equality of two expressions for the weight of steam expended in unity of time, — w = v ,f* n . (d.) \j(l - c)s From these two equations is deduced the following, expressing the ratio of the mean load on the piston to the initial pressure of the steam : — li + F _ Z — cs . . being equivalent to equation (51). In computing the effect of Cornish engines these formulae require to be modified, owing to the following circumstances : — The terms depending on the clearance c have been introduced into equations (c), (//), on the supposition that the steam employed in filling the space above the piston at the top of its stroke is lost, being allowed to escape into the condenser, without having effected any work; so that a weight of steam Wcs is wasted, and an amount of power WVj (Pj — F)ox lost, in unity of time. But in Cornish engines this is not the case; for by closing the equilibrium-valve at the proper point of the up or out-door stroke, nearly the whole quantity of steam necessary to fill the clearance and valve-boxes may be kept imprisoned above the piston, so as to make the loss of power depending on it insensible in practice. This portion of steam is called a cushion, from its preventing a shock at the end of the upstroke; and, as Mr. Pole in his valuable work on the Cornish engine has observed, its alternate compression and expansion compensate each other, and have no effect on the duty of the engine. The proper moment of closing the equilibrium-valve is fixed by trial, which is, perhaps, the best way; but if it is to be fixed by theory, the following is the proper formula : let I" be the length of the portion of the upstroke remaining to be performed after the equilibrium-valve has been closed : then — 1" _ C(8-1) m I ' I - c ' ' ' * u ' ; A slight deviation from this adjustment will produce little effect in practice, if the fraction c is small. In forming the equations of motion, therefore, of the Cornish engine, we may, without material error, in practice omit the terms denoting a waste of steam and loss of power due to clearance and filling of steam- passages ; and the results are the following : — Equation of effect and power in unity of time : — SINGLE-ACTING EXPANSIVE STEAM-ENGINES. 291 Useful effect E = E AZ» = W V, {P x Z - F}. . (57.) "Weight of steam expended in unity of time : — w = 4^ ( 58 o From those two fundamental equations the following are deduced: — Ratio of mean load on piston to maximum pressure, — R + F _ Z Pi = » (59.) Duty of unity of weight of steam, — W-V^Z-F), • • (6Q.) which, being multiplied by the number of units of weight of steam produced by a given weight of fuel, gives the duty of that weight of fuel. Weight of steam expended per stroke, — ^ = ^ (61.) In fact, it is clear that if any five quantities out of the following seven be given, the other two may be determined by means of the equations : R + F, the mean load on unit of area of piston. P p the maximum pressure of steam in the cylinder. s, the ratio of expansion. W, the weight of steam produced in unity of time. A, the area of the piston. /, the length of stroke. n, the number of strokes in unity of time. The other quantities, E, Y v Z, are functions of those seven. Comparison of the Theory with Mr. Wicksteed's Experiments. 36. In order to test the practical value of this theory, I shall compare its results with those of the experiments which were made by Mr. Wicksteed on the large Cornish pumping engine, built under the direction of that eminent engineer, by Messrs. Harvey and West, for the East London 292 SINGLE-ACTING EXPANSIVE STEAM-ENGINES. Water-Works- at Old Ford, and which were published in 1841. The dimensions and structure of the engine, and the details of the experiments, are stated with such minuteness and precision, that there is none of that uncertainty respecting the circumstances of particular cases, which is the most frecpient cause of failure in the attempt to apply theoretical principles to practice. The eno-ine was worked under a uniform load at five different rates of expansion successively. The number of strokes, and the consumption of steam during each trial, having been accurately registered, Mr. ^Wieksteed gives a tabic showing the weight of steam consumed per stroke for each of the five rates of expansion. I shall now compute the weight of steam per stroke theoretically, and compare the results. Throughout these calculations I shall uniformly use the foot as the unit of length, the avoirdupois pound as that of weight, and the hour as that of time. Pressures are consequently expressed in pounds per square foot for the purpose of calculation; although in the table of experimeuts I have reduced them to pounds per square inch, as being the more familiar denomination. The data respecting the dimensions and load of the engine, which are constant throughout the experiments, are the following : — Area of piston, . .... A =: 34'854 square feet. Stroke, . ... 7 = 10 feet. Cubic space traversed by piston during one down stroke, . . . . = A I — 3 48 '5 4 cubic feet Clearance and valve-boxes, .... 18 - 00 „ Sum, . 300-54 Therefore, c = 0*05 E = useful load of piston, . . • = 1597-0 lbs. per sq.ft. F = additional resistance, . . • =200*0 „ R + F = total mean pressure on piston, . . = 1803-0 „ The mode of calculation is the following : — Mr. Wicksteed states the fraction - of the stroke performed at full pressure in each experiment. From this the ratio of expansion s is computed by equation (b), giving in this case - = 0-95 - 7 + 0-05. s I SINGLE-ACTING EXPANSIVE STEAM-ENGINES. 293 The value of Z corresponding to s is then found by means of the third column of table second; that column being selected because the initial pressures were all below four atmospheres. This affords the means of determining the initial pressure of the steam by equation (59), viz. — P x = |(E + F) = 1863-6 |. , By using table first according to the directions prefixed to it, the volume of one pound of steam at the pressure P p in cubic feet, is calculated, and thence, by equation (60), the weight of steam per stroke, according to theory, which is compared with the weight as ascertained by experiment. Further, to illustrate the subject, the useful effect, or duty of a pound of steam, is computed according to the theory and the experiments respectively, and the results compared. The following table (See j>. ,!UJ t ) exhibits the results. This comparison sufficiently proves that the results of the theory are practically correct. It is remarkable, that in every instance except one (experiment E), the experimental results show a somewhat less expenditure of steam per stroke, and a greater duty per pound of steam, than theory indicates. This is to be ascribed to the fact, that although the action of the steam is computed theoretically, on the assumption that during the expansion it is cut off from external sources of heat, yet it is not exactly so in practice; for the cylinder is surrounded with a jacket or casing communi- cating with the boiler, in which the temperature is much higher than the highest temperature in the cylinder, the pressure in the boiler being more than double the maximum pressure of the steam when working, as columns (2) and (5) show. There is, therefore, a portion of steam, of whose amount no computation can be made, which circulates between the boiler and the jacket, serving to convey heat to the cylinder, and thus augment by a small quantity the action of the steam expended ; and hence the formulae almost always err on the safe side. Supposing one pound of the best Welsh coals to be capable (as found by Mr. Wicksteed) of evaporating 9'493 lbs. of water at the pressure in the boiler during the experiment F, then the duty of a Cornish bushel, or 94 lbs. of such coals, in the circumstances of that experiment would be — By theory, 88,288,000 ft. lbs. By experiment, 90,801,000 „ Difference, . . + 2,513,000 „ 294 SINGLE-ACTING EXPANSIVE STEAM-ENGINES. 12! « O © « O m H W w C-i X W Q W W H OQ M o Q PS O h <=> O w w w H O o Oh o © © O i-~ § CO CO • : CO ".0 CN CO o ■71 o CI ^ + + + 1 + 3 a SO 01 a -o CI l~ O >>.2 - V5 1^ 1 - c; ■- -co to C5 ct a " © - 1- CO CO P< o . M o p-g H Eh O C3 • o >, so eo IO CO 3 l>>5 ca © £12 CO OS c7 OS 10 CI -f * CO • '- 1~ CO o fa a lO o 10 — o CO CO L0> 01 p p 1 — ' CC^ «2 © b b b b *j i i 1 + 1 a ■a , s -3 o • !C 10 >■ pa CO P CO 1 ~ Si '-. 7.- 71 - . y t © © So 1 » b b '"• 10 c3 is 2«2 X u. c © o a fj M '-2 10 >> o 00 O CO 01 •a a I - C5 01 0) te 'r a o H t - ■j: b 10 b - 6-S.2 »"§ S © £ ~ = 1 - — j C5 -* -•1 2 CI ■a Si b CO 1- OS b S - 5 rt ,,; - a ss CO 01 'O 01 © CO -r CO p OS CO CO -f £ 5 ~ '-' CI CI 01 3 £ g s ° ! -.1 t^ 1^ CJ CO o l- O 10 p b <* b CO b CO CO b CO • *> a - ^ -a ■"fcjd tt — — ~ IO aMa £ © CI CO CI b C) 1- 10 cT S a> = 5 CO cc -* -* ■-.a o 3 — ~Ph * 73 °a t-i © «' 0" Q H & C- w SINGLE-ACTING EXPANSIVE STEAM-ENGINES. 295 Economy of Single-Acting Expansive Engines. 37. By increasing the ratio of expansion in a Cornish engine, the quantity of steam required to perform a given duty is diminished ; and the cost of fuel, and of the boilers, is lowered. But at the same time, as the cylinders and every part of the engine must be made larger, to admit of a greater expansion, the cost of the engine is increased. It thus becomes a problem of maxima and minima to determine what ratio of expansion ought to be adopted under given circumstances, in order that the sum of the annual cost of fuel, and the interest of the capital employed in construction, may be the least possible, as compared with the work done. That this problem may admit of a definite solution, the following five quantities must be given : — Pj, the initial pressure in the cylinder. F, the resistance not depending on the useful load. I n, the amount of the length of the effective strokes made in unity of time. k, the annual cost of producing unity of weight of steam in unity of time, which consists of two parts — the price of fuel, and the interest of the cost of the boilers. k, the interest of the cost of the engine, per unit of area of piston. Hence the annual expenditure to be taken into consideration, reduced to unity of weight of steam, is , And the useful effect of unity of weight of steam being V^Z-Fs), the problem is to determine the ratio of expansion s, so that V^Z-Fs) V s In shall be a maximum. Dividing the numerator of this fraction by V t P 1? and the denominator by -r- 1 , both of which are constants in this problem, we find that it will 296 SINGLE-ACTING EXPANSIVE STEAM-ENGINES. be solved by making the ratio 7 F Z — =r-S Py .... (G2.) a maximum. The algebraical solution would be extremely complicated and tedious. The graphic solution, on the other hand, is very simple and rapid, and sufficiently accurate for all practical purposes ; and 1 have therefore adopted it. In the diagram (See Piatt II., Fig. I), the axis of abscissa; — XO + X, is graduated from towards + X into divisions representing ratios of expansion, or values of & The divisions of the axis of ordinates OY represent values of Z. The curve marked "locus of Z," is laid down from the third column of Table II. of the Appendix to the original paper, being applicable to initial pressures not exceeding four atmospheres. Through the origin draw a straight line BOA, at such an inclination F to — X + X that its ordinates are represented by s. Then the -*■ i ordinates measured from this inclined line to the locus of Z represent the F value of the numerator Z — s, of the ratio (62), corresponding to the -*- i various values of .*. Take a point at C on the line B A, whose abscissa, measured along — X, represents — 7 . Then the ordinates, measured from BOA, k V j of any straight line drawn through C, vary proportionally to the denomi- nator — — + s of the ratio (62). /,' V , Through the point C, therefore, draw a straight line C T, touching the locus of Z : then the ratio (62) is a maximum at the point of contact T, and the abscissa at that point represents the ratio of expansion required. Example. i 38. To exemplify this method, let us take the following data: — Greatest pressure in the cylinder P x = 20 lbs. per square inch, = 2880 lbs. per square foot. The corresponding value of V x is 20 - 248 cubic feet per pound of steam. To obtain this initial pressure in the cylinder, it will be necessary to have a pressure of about 50 lbs. per square inch in the boiler. F, resistance not depending on the useful load = 2 lbs. per square inch, = 288 lbs. per square foot, = T V P r SINGLE-ACTING EXPANSIVE STEAM-ENGINES. 21)7 In, amount of clown strokes, = 4800 feet per hour; being the average speed found to answer best in practice. To estimate h, the annual cost of producing one pound of steam per hour, I shall suppose that the engine works 6000 hours per annum; that the cost of fuel is one penny per 100 lbs. of steam;'" that the cost of boiler for each pound of steam per hour is 0*016 ton, at £27, = £0*432; and that the interest of capital is five per cent, per annum. Hence h is thus made up — Fuel for 6000 lbs. of steam at 0*01d., . . . £0*2500 Interest on £0*432, at 5 per cent,, . . . 0*0216 h = £0*2716 Estimating the cost of the engine at £250 per square foot of piston, we find /; = 5 per cent, per annum on £250 = £12*5. , h „ , ^-. ~ h 1 n _ , . . and =0*0217; =-=■ = 5*144. I; k V j The line E O A, then, is to be drawn so that its ordinates are F 1 p; 5 = To 5 - The point C is taken on this line, at £=- = 5*144 divisions of the axis of abscissa? to the left of O Y. The tangent C T being drawn, is found to touch the locus of Z at 2*800 divisions to the right of O Y. Then s = 2*800 is the ratio of expansion sought, corresponding to the greatest economy. If we make c=r0'05, as in Mr. Wicksteed's engine, then the fraction of the stroke to be performed at full pressure is - = 0*323, I being nearly the same as in experiment F. The mean resistance of the useful load per square foot of piston is E = — R - F -=1713*6 lbs. s 1 * This estimate is maile on the supposition that coals capable of producing nine times their weight of steam are worth about 16s. 9d. per ton. 298 SINGLE-ACTING EXPANSIVE STEAM-ENGINES. The duty of one square foot of piston per liour — B,ln, = 8,225,300 foot-lbs. And one horse-power being 1,980,000 foot-lbs. per hour, the real horse- power of the engine is 4 - 154 per square foot of piston. The duty of one pound of steam is RV 1 s = 97,154 foot-lbs. To give an example of a special case, let the duty to be performed be 198,000,000 foot-pounds per hour, being equal to 100 real horse-power, for G,000 hours per annum. This being called E, we find from the above data that the area of piston required is i; A = _ . = 21-072 square feet, li / n The consumption of steam per hour is W =,,!': = 2038 lbs., Jtt V x .S' which requires 2038 x 0-01 G = 32-608 tons of boilers. The expenditure of steam per annum is 2038 x G000 = 12,228,000 lbs. Hence we have the following estimate : — Cost of engine, 24*072 square feet of piston at £250, £6018-000 Cost of boilers, 32-G08 tons at £27, . . . 880*416 Total capital expended, . . . £6898-416 Interest at five per cent, per annum, . . . 344-921 Cost of fuel per annum, 12,228,000 lbs. of steam at 0-01d., 509-500 Annual cost for interest and fuel, . . £854*421 I wish it to be understood that the rates I have adopted in the fore- SINGLE-ACTING EXPANSIVE STEAM-ENGINES. 299 going calculations, for interest, cost of fuel, and cost of construction, are not intended as estimates of their average amount, nor of their amount in any particular case ; but are merely assumed in order to illustrate, by a numerical example, the rules laid down in the preceding article. It is, of course, the business of the engineer to ascertain those data with reference to the special situation and circumstances of the proposed work; and having clone so, the method explained in this paper will enable him to determine the dimensions and ratio of expansion which ought to be adopted for the engine, in order that it may effect its duty with the greatest possible economy. 300 ECONOMY OF HEAT IN EXPANSIVE MACHINES. XVII.— ON THE ECONOMY OF HEAT IN EXPANSIVE MACHINES :* Forming the Fifth Section of a Paper On the Mechanical Action of Heat. 39. A MACHINE working by expansive power consists essentially of a portion of some substance to which heat is communicated, so as to expand it, at a higher temperature, being abstracted from it, so as to condense it to its original volume, at a lower temperature. The quantity of heat given out by the substance is less than the quantity received; the difference disappearing as heat, to appear in the form of expansive power. The heat originally received by the working body may act in two ways: to raise its temperature, and t<< expand it. The heal given out may also act in two ways: to lower the temperature, and to contract the body. Now, as the conversion of heat into expansive power arises from changes of volume only, and not from changes of temperature, it is obvious, that the proportion of the heat received which is converted into expansive power will lie the greatest possible, when the reception of heat, and its emission, each take place at a constant temperature. 40. Carnot was the first to assert the law, that the ratio of the maximum mechanical effect to the vial heat expended in an expansive machine, is a function solely of tin fir,, temperatures at which the loot is respectively received ami emitted, and is independent of (he nature of the working substance, But his investigations not being based on the principle of the dynamical convertibility of heat, involve the fallacy that power can be produced out of nothing. •41. The merit of combining Carndt's Law, as it is termed, with that of the convertibility of heat and power, belongs to Mr. Clausius and Professor William Thomson; and in the shape into which they have brought it, it may be stated thus : — The maximum proportion of heat converted in la expansive power by any machine, is a function solely of the temperatures at which heat is received and emitted by the working substance; which function for each pair of temperatures is the same for all substances in nature. * Bead before the Eoyal Society of Edinburgh on April 21, 1851, and published in the Transactions of that Society, Vol. XX., Part II. in which the negative sign denotes aDsorpuou, ana cue jjustvttg emission. Bn.l. ECONOMY OF HEAT IN EXPANSIVE MACHINES. 301 This law is laid down by Mr. Clausius. as it originally had been by Carnot, as an independent axiom; and I had at first doubts as to the soundness of the reasoning by which he maintained it. Having stated those doubts to Professor Thomson, I am indebted to him for having induced me to investigate the subject thoroughly; for although I have not yet seen his paper, nor become acquainted with the method by which he proves Carnot's law, I have received from him a statement of some of his more important results. 42. I have now come to the conclusions, — First, That Carndt's Law is not an independent principle in the theory of heat; hut is deducible, as a consequence, from the equations of the mutual conversion of heat and expansive power, as given in the first section of this paper. Secondly, That the function of the temperatures of reception and emission, which expresses the maximum ratio of the heat converted into power to the total heat received by the working body, is the ratio of the difference of those temper- atures to the absolute temperature of reception diminished by the constant, which I have called tc = Cn/j.b ; and which must, as I have shown in the intro- duction, be the same for all substances, in order that molecular equilibrium may be possible. 43. Let abscissa^ parallel to X in the diagram, Plate II, Fig. 2, denote the volumes successively assumed by the working body, and ordinates, parallel to Y, the corresponding pressures. Let r x be the constant absolute temperature at which the reception of heat by the body takes place : t , the constant absolute temperature at which the emission of heat takes place. Let A B be a curve such that its ordinates denote the pressures, at the temperature of reception t v corresponding to the volumes denoted by abscissa\ Let D C be a similar curve for the temper- ature of emission r . Let A D and B C be two curves, expressing by their co-ordinates how the pressure and volume must vary, in order that the body may change its temperature, without receiving or emitting heat ; the former corresponding to the most condensed and the latter to the most expanded state of the body, during the working of the machine. The quantity of heat received or emitted during an operation on the body involving indefinitely small variations of volume and temperature, is exj>ressed by adding to equation (G) of section fourth the heat due to change of temperature only, in virtue of the real specific heat. "We thus obtain the differential equation — fe £ r, in which the negative sign denotes absorption, and the positive emission. 302 ECONOMY OF HEAT IN EXPANSIVE MACHINES. If wo now put for } v , , , their values according to equation (11), d V (It we find 8Q'_SQ= _( T _«)^?.gy _ jfc + 1 (« _ «*) + (T _ .) '' j™ ,, V \» T . (63.) (. C«M\t rv d tJ (It ) The first term represents the variation of heat due to variation of volume only; the second, that due to variation of temperature. Let us now apply this equation to the cycle of operations undergone by the working body in an expansive machine, as denoted by the diagram. First operation. — The body, being at first at the volume V A and pressure P A , is made to expand, by the communication of heat at the constant temperature t v until it reaches the volume V B and pressure P B , A B being the locus of the pressures. Here S r = ; therefore, the total heat received is H 1 = -Q' 1 = (r 1 - K )J v v 1 dV d r = 0"!- *){<P(V*>tJ- *(V*r,)}.J (a.) Set: lion. — The body, being prevented from receiving or emitting heat, exj lands until it falls to the temperature r , the locus of the pressures being the curve B C. During this operation the following condition must be fulfilled, — = 8 q - 8 < >. which, attending to the fact that V is now a function of r, and trans- forming the integrals as before, gives the equation = ,! + cTm \r ~ ?) + < r - K) KTr + ,77 • 2 v) * <*■ T > This equation shows that (V. r x ) - 96 (V c , r ) = ^ (r r r ). . . (b.) Third operation. — The body, by the abstraction of heat, is made to contract, at the constant temperature r , to the volume V D and pressure P Dr which are such as to satisfy conditions depending on the fourth operation. C D is the locus of the pressures. The heat emitted is evidently H = Q' = (r - k) {cp (V c , r ) - d> (V w r )}. . (&) ECONOMY OF HEAT IN EXPANSIVE MACHINES. 303 Fourth operation. — The body, being prevented from receiving or emitting heat, is compressed until it recovers its original temperature t v volume V A , , and pressure P A ; the locus of the pressures being DA. During this operation, the same conditions must be fulfilled as in the second operation ; therefore, <P ( Y a, O - </> (V D , r ) = xP (t v r ). . . (d.) ib being the same function as in equation (b). By comparing equations (b) and (d), we obtain the relation which must subsist between the four volumes to which the body is successively brought, in order that the maximum effect may be obtained from the heat. It is expressed by the equation tiVr,^) - c/>(Y x ,T 1 ) = ci>(Y c ,T )- <P(\\,T ). • (64.) From this and equations (a) and (c), it appears that Ho _ To — K H l T l - * (65.) That is to say : ivhen no heat is employed in producing variations of tempera- ture, the ratio of the heat received to the heat emitted by the working body of an ivi machine, is equal to that of the absolute temperatures of reception and emission, each diminished by the constant k, which is the same for all substances. Hence, let n = - q\ - Q' = h x - h denote the maximum amount of power which can be obtained out of the total heat H^ in an expansive machine working between the tempera- tures t x and r . Then £ = ^=A .... (66.) H i T i ~ K being the law which has been enunciated in Article 42, and which is deduced entirely from the principles already laid down in the introduction and first section of this paper. The value of the constant K is unknown; and the nearest approximation to accuracy which we can at present make, is to neglect it in calculation, as being very small as compared with r. 44. This approximation having been adopted, I believe it will be found that the formula (66), although very different in appearance from that arrived at by Professor Thomson, gives nearly the same numerical results. For example : let the machine work between the temperatures 140° and 30° Centigrade : then 304 ECONOMY OF HEAT IX EXPANSIVE MACHINES. r, = 414°-G, r = 304°'6, and " = 0-2653. Trofessor Thomson has informed me, that for the same temperatures he finds this ratio to be 0*271 3.* 45. To make a steam-engine work according to the conditions of maximum effect here laid down, the steam must enter the cylinder frorn^ the boiler without diminishing in pressure, and must be worked expan- sively down to the pressure and temperature of condensation. It must then be so far liquefied by conduction alone, that on the liquefaction being completed by compression, it may be restored to the temperature of the boiler by means of that compression alone. These conditions are unattainable in steam-engines as at present constructed, and different from those which form the basis of the formulas and tables in the fourth section of this paper; hence it is found, both by experiment and by calculation from those formulas, that the proportion of the total heat converted into power in any possible steam-engine is less than that indicated by equation (GG). The annexed table illustrates this. The heat transformed into power, as given in the fifth column, has been reduced to Centigrade degrees in liquid water, by dividing the duty of a pound of steam by Mr. Joule's equivalent, 1389-G feet per Centigrade degree. Hence, the first two numbers in that column are less than those given in Art. 32, which were computed from too small an equivalent. The first two cases fulfil the conditions required by Canmt's law in every respect except one : — viz., that the steam remaining at the end of the stroke, instead of being partially liquefied by refrigeration, and then reduced to water at the temperature of the boiler by compression, is supposed to be entirely liquefied by refrigeration. This occasions the loss of the heat necessary to raise the water from the temperature of the condenser to that of the boiler; but at the same time, there is a gain of the power which would be required to liquefy part of the steam by compression, and those two quantities partially compensate for each other's effects on the ratio of the power to the heat expended, so that although it is below the maximum, the difference is small. * From information which I have received from Professor Thomson subsequently to the completion of this paper, it appears that his formula becomes identical with the approximate formula here proposed, on making the function called bj' him fc = — , J being Joule's equivalent. Mr. Joule also, some time since, arrived at this approximate formula in the particular case of a perfect gas. ECONOMY OF HEAT IN EXPANSIVE MACHINES. 305 ~ ° * .H o5 - n »§28g <*6 .o , is »— 3 ° £ fl © O c f- tu p d es -" -*H IC t^ © o.y c? M M a s II I! !l P. u «p ep ep 6 Sll ■>* -r O Eh S ct CN ti 2s + J. + 7 ':- r~ O © o CO IQ OD CO i~ O "* o -* «* "* gM *n II 3 3 co © o go o tH c ^ S o £ I! « '3 <N C) CM — J + + + — ^H ,—t C5 CI g M Z*, o Ifi — s~ c~. — ' ^_/ , • s ■- r-i C5 .5 w a cc p 1 cs o S £ ' « -e c3 X o o o -a £ M 3 '-3 o 1— 1 o 2 § -2 .2 § O fc. 02 ^ a ' 306 ECONOMY OF HEAT IN EXPANSIVE MACHINES. In the third and fourth examples, founded on the calculated and observed duty of Mr. Wicksteed's engine during experiment F, the actual ratio is less than half the maximum. This waste of heat is to he ascribed to the following causes : — First, The mode of liquefaction, which has already been referred to. Secondly, The initial pressure in the cylinder is but 18 "9 3 lbs. on the square inch, while that in the boiler is 45-7; so that although the steam is produced at 135°-2 Centigrade, it only begins to work at 107°'2G. This great fall of pressure is accounted for by the fact, that the steam for each stroke, which is produced in the boiler in about seven or eight seconds, escapes suddenly into the cylinder in a fraction of a second. Thirdly, The expansive working of the steam, instead of being continued down to 30° Centigrade, the temperature of the condenser, stops at a much higher temperature, 74°-GG. This is the most important cause of loss of power. If we now take for r x and r the absolute temperatures at the beginning and end of the expansive working, and calculate the maximum duty of one pound of steam by Carn&t's Law between those temperatures, we find, — t x = 107°-2G + 274°-G = 381°-8G Tq = 74°-GG + 274°-G = 349 '26 ~ = 0-08f) ! 2 K 1 =5U '5 i .:Il= 48°'22 To this has to be added the duty, at full pressure, of steam at r , diminished by one-third for back- pressure and friction, and by OHe-fifteenth for liquefaction in the cylinder, = . . . . 23 0, 14 The whole amounting to . . . 71° # 3G Which agrees very nearly with 73°'23, the observed duty, and almost exactly with 71°*2, the duty as calculated by the formulae and tables of section fourth. These examples show clearly the nature and causes of the waste of heat in the steam-engine. ABSOLUTE ZERO OF THE PERFECT GAS THERMOMETER. 307 XVIIL— ON THE ABSOLUTE ZERO OF THE PERFECT GAS THERMOMETER : * Being a Note to a Paper On the Mechanical Action of Heat. {Seep. 23 If.) Temperature being measured by the pressure of a perfect gas at constant density, the absolute zero of temperature is that point on the thermometric scale at which, if it were possible to maintain a perfect gas at so low a temperature, the pressure would be null. The position of this point is of great importance, both theoretically and practically; for by reckoning temperatures from it, the laws of phenomena depending on heat are reduced to a more simple form than they are when any other zero is adopted. As we cannot obtain any substance in the perfectly gaseous condition (that is to say, entirely devoid of cohesion), Ave cannot determine the position of the absolute thermometric zero by direct experiment, which furnishes us with approximate positions only. Those approximate posi- tions are always too high ; because the effect of cohesion is to make the pressure of a gas diminish more rapidly with a diminution of temperature, than if it were devoid of cohesion. As a gas is rarefied, the cohesion of its particles diminishes, not only in absolute amount, but also in the proportion which it bears to the pressure due to heat. The gas, therefore, approaches more and more nearly to the state of a perfect gas as its density diminishes ; and from a series of experiments on the rate of increase of its elasticity with temperature, at progressively diminishing densities, may be calculated the positions of a series of points on the thermometric scale, approaching more and more nearly to the true absolute zero. By observing the law which those successive approximations follow, the true position of the absolute zero can be determined. Having performed this operation by means of a graphic process, soon after the publication of the experiments of M. Regnault on the elasticity and expansion of gases, I stated the result in a paper on the Elasticity of Vapours (Edinburgh New Philosophical Journal, July, 1849), {See p. 1). and * Read before the Royal Society of Edinburgh on January 4, 1333, and pnblis! ed in the Transactions of that Society, Vol. XX. Part IV. 80S ABSOLUTE ZERO OF THE TERFECT GAS THERMOMETER. also in a paper on the Mechanical Action of Heat (Trans. Royal Sue. Edin., Vol. XX., Part I.), (See p. ..'■'.'/' — viz., that the absolute zero is — 274°-G Centigrade, .) below ^ tc rature of me lting ice ; or 494°-28 Fahrenheit, J or 4G2 0- 28 below the ordinary zero of Fahrenheit's scale. To enable others to judge of the accuracy of this result, 1 shall now explain the method by which it was obtained. Let E denote the mean rate of increase, per degree, between the freezing and boiling points, of the pressure of a gas whose volume is maintained constant. Then the reciprocal of this coefficient, , is an approximation to the number of degrees below the freezing point, at which the absolute zero is situated. The experimental data in the following table were copied from the memoirs of M. Regnault on the Expansion of Gases. The numbers in the first column designate the series of experiments. The second column contains the pressures of the gases at the freezing point. The third column contains the mean coefficients of increase of pressure per Centi- grade degree, between 0° sod 100° Centigrade. The fourth column contains the reciprocals of those coefficients, with the negative Bign, being approximate positions of the absolute zero, in Centigrade degrees, below the temperature of melting ice. The gases employed were atmospheric air and carbonic acid. The approximate positions of the absolute zero contained in this table wen* laid down on ;i diagram, in which they were marked by crosses. The longitudinal divisions representing Centigrade degrees divided into tenths; the transverse divisions, atmospheres of pressure a1 (l Centigrade, also divided into tenths. The positions of the crosses indicating at once the pressures in the second column of the table, and the approximate zeros in the fourth ; and the numbers affixed to them corresponding with those in the first column. As the effect of cohesion is greater and more easily eliminated in car- bonic acid gas than in atmospheric air, the determination of the true absolute zero was made from the experiments on the former gas. The approximate positions of the absolute zero for carbonic acid lie nearly in a straight line. A straight line having been drawn so that it should, as nearly as possible, traverse them, was found to intersect the line corre- sponding to the zero of pressure, that is, to the state of perfect gas, at a point on the scale of temperatures 274 - 6 Centigrade degrees below the temperature of melting ice, which point was accordingly taken as the true absolute zero of the perfect gas thermometer. ABSOLUTE ZERO OF THE PERFECT GAS THERMOMETER. 309 No. Pressure at 0° Centigrade in Atmospheres. Coefficient of Increase of Elasticity with Temperature = E. Approximate Positions of the Absolute Zero in Centigrade Degrees 1 ~ E" CARBONIC ACID. 1. 0-99S0 0-0036S5G -271-33 o 1-1857 0036943 -270-63 3. 22931 0-0037523 -266-50 4, 4-7225 0-003S59S -259 -OS • ATMOSPHERIC AIR. 1. 0-1444 00036482 -274-11 o 0-2294 0-003G513 -273-88 3. 0-3501 0-0036542 -273-66 4. 0-4930 00365S7 -27332 5. 0-4937 0036572 -273-43 6. 1-0000 0-003G650 -272-85 7. 2-2084 0-003G700 -272-03 8. 2-2270 0-0036800 -27174 9. 2-8213 0036S94 -27105 10. 4-S100 0037091 -269 61 So far as their irregularity permits, the experiments on atmospheric air confirm this result, for the approximate positions of the absolute zero deduced from them evidently tend towards the very same point on the diagram with those deduced from the experiments on carbonic acid. The values of the coefficient of dilatation and of increase of pressure of a perfect gas, per degree, in fractions of its volume and pressure, at the temperature of melting ice, are, accordingly,— For the Centigrade Scale 7 —— = 0-003641 CO ° 274-0 For Fahrenheit's Scale 1 494-L'S 0-00202314. 810 THE MECHANICAL ACTION OF HEAT. XIX.— OX THE MECHANICAL ACTION OF HEAT.* Section VI. — A Review of the Fundamental Principles of the Mechanical Theory of Heat; with Remarks on the Thermic Phenomena of Currents of Elastic Fluids, as illustrating those Principles. 4G. I have been induced to write this section in continuation of a paper on the Mechanical Action of Heat, by the publication (in the Philosophical Magazine for December, 1852, Supplementary Number) of a series of experiments by Mr. Joule and Professor "William Thomson, On the Thermal Effects experienced by Air in rushing through Small Apertures. Although those authors express an intention to continue the experiments on a large scale, so as to obtain more precise results; yet the results already obtained are sufficient to constitute the first step towards the experimental determination of that most important function in the theory of the mechanical action of heat, which has received the name of Carnot's function. By the theoretical investigations of Messrs. Clausius and Thomson, — which are based simply on the fact of the convertibility of heat and mechanical power, the determination of their relative value by Mr. Joule, and the properties of the function called temperature, without any definite supposition as to the nature of heat, — Carnot's function is left wholly indeterminate. By the investigations contained in the previous sections of this paper, and in a paper on the Centrifugal Theory of Elasticity (See p. ^9), — in which the supposition is made, that heat consists in the revolutions of what are called molecular vortices, so that the elasticity arising from heat is in fact centrifugal force, — a form is assigned to Carnot's function; but its numerical values are left to be ascertained by experiment. The recent experiments of Messrs. Joule and Thomson serve (so far as the degree of precision of their results permits) at once to determine numerical values of Carnot's function for use in practice, and to test the * Read before the Eoyal Society of Edinburgh on January 17, 1853, and pub- lished in the Transactions of that Society, Vol. XX., Part IV. THE MECHANICAL ACTION OF HEAT. oil accuracy with which the phenomena of heat are represented by the consequences of the hypothesis of molecular vortices, from which the investigation in this paper sets out. Sub-Section 1. — Properties of Expansive Heat. 47. To show more clearly the nature of the questions, towards the decision of which these experiments are a step, I shall now briefly review the fundamental principles of the theory of heat, and the reasoning on which they are based; and the object of this being illustration rather than research, I shall use algebraical symbols no further than is absolutely necessary to brevity and clearness, and shall follow an order of investigation which, though the same in its results with that pursued in the previous sections of this paper, is different in arrangement. By a mind which admits as an axiom that, in the present order of things, physical power cannot be annihilated, nor produced out of nothing, the law of the mutual convertibility of heat and motive power must be viewed as a necessary corollary from this axiom, and Mr. Joule's experi- ments, as the means of determining the relative numerical value of those two forms of power. By a mind which does not admit the necessity of the axiom, these experiments must be viewed also as the proof of the law. This law was virtually, though not expressly, admitted by those who introduced the term latent heat into scientific language; for Avhen divested of ideas connected with the hypothesis of a subtle fluid of caloric, and regarded simply as the expression of a fact, this term denotes heat which has disappeared during the appearance of expansive power in a mass of matter, and which may be made to reappear by the expenditure of an equal amount of compressive power. 48. Without for the present framing any mechanical hypothesis as to the nature of heat, let us conceive that unity of weight of any substance occupying the bulk V under the pressure P, and possessing the absolute quantity of thermometric heat whose mechanical equivalent is Q, undergoes the indefinitely small increase of volume d V; and let us investigate how much heat becomes latent, or is converted into expansive power, during this process; the thermometric heat being maintained constant, so that the heat which disappears must be supplied from some external source. During the expansion d V, the body, by its elastic pressure P, exerts the mechanical power P d V. Part of this power is produced by mole- cular attractions and repulsions; and although this part may be modified by the influence of heat upon the distribution of the particles of the body, it is not the direct effect of heat. The remainder must be considered as directly caused by the heat possessed by the body, of which the pressure 312 THE MECHANICAL ACTION OF HEAT. P is a function; and to this portion of the power developed, the heat which disappears during the expansion must be equivalent. To determine the portion of the mechanical power P<ZV which is the effect of heat, let the total heat of the body, Q, be now supposed to vary by an indefinitely small quantity d Q. Then the mechanical power of expansion VdY will vary by the indefinitely small quantity This is the development of power for the expansion dV, caused by each indefinitely small portion rf'Q of the total heat possessed by the body; and, consequently, the whole mechanical power for the expansion dY due to the whole heat possessed by the body Q, is expressed as follows: — «ii" (67 -» and this is the equivalent of the heat transformed into mechanical power, or the latent heat of expansion of unity of weight, fur the small increment of volume d V, at the volume V and total heat Q. Now, a part only of this power, viz. — P d Y, is visible mechanical energy, expended in producing velocity in the expanding body itself, or in overcoming the resistance of the bodies which inclose it. The remainder („;:;;- .•)..- v. . w r/O is, therefore, expended in overcoming molecular attraction. Molecular attraction depends on the density and distribution of the particles of the body; and is, consequently, a function of the volume and total heat of unity of weight. It is, therefore, possible to find a potential S, being a function of V and Q, of such a nature, that the difference between its two values S 2 — 9j, corresponding respectively to two sets of values of the volume and total heat (V 15 Q, and V„ Q 2 ), shall represent the power which is the equivalent of the heat consumed in overcoming molecular attraction, during the passage of the body from the volume Y 1 and heat Q x to the volume V 2 THE MECHANICAL ACTION OF HEAT. 313 and heat Q 2 . The form of the expression (68) shows that this potential has the following property : — The integration of which partial differential equation gives the following value for the potential of molecular action: — ■ S = /(Q^!-P)eZV + tf>(Q), • • (70.) <j) (Q) being some unknoAvn function of the heat only, and the integral being taken as if the heat Q were constant. The heat which disappears in overcoming molecular action, during a small increase of total heat d Q, while the volume remains constant, is expressed as follows: — the heat Q being treated as a constant in the integration. If we now investigate the entire quantity of heat, both sensible and latent, which is consumed by a body during a simultaneous small change of total heat d Q and volume d V, we find the following results : — Sensible heat (which retains its condition), . = d Q Latent heat, or heat which disappears in overcoming molecular action, . . yf\ ^ ^ • Tv " Latent heat equivalent to the visible me- chanical effect, P d V The amount being ^ + ^S + P.V=(l + ^)iQ+(^ + p).V = (72.) This formula expresses completely the relations between heat, mole- cular action, and expansion, in all those cases in which the expansive power developed, P d V, is entirely communicated to the bodies inclosing the substance which expands. 49. The following coefficients are contained in, or deducible from it, 314 THE MECHANICAL ACTION OF HEAT. The ratio of the specific heat at constant volume to the real specific heat : — T = 1+ s| = 1 + Q /^- dV + ^ (Q) ' • (73) The coefficient of latent heat of expansion at constant heat :— dS d\ + P = Q dV (74.) The ratio of the specific heat at constant pressure to the real specific heat is found as follows. To have the pressure constant we must have dV lrt dF ITT , dV a (^ d\ d Q dV dQ dl y dX consequently, the ratio in question is, dV dX dVY- \d ( ) ^ (75.) 50. In order to investigate the laws according to which heat is con- verted into mechanical power, in a machine working by the expansion of an elastic body, it will be convenient to use a function, F= j^dY(Q = const.), of such a nature that the difference between two of its values, correspond- ing to different volumes of the body at the same total heat, represents the ratio of the heat converted into power by expansion between those volumes, to the given constant total heat. I shall call this function a heat-potential. Introducing this function into equation (72), we find for the total heat consumed by a body during the increase of total heat d Q, and the expan- sion d V, dQ, + d . S + P<7V = (l + f . (Q) W + Qd . F (7G.) THE MECHANICAL ACTION OF HEAT. 315 ( d F dF observing that d . F = T — d Q + T, r d V « y a V "OS"- )'« + £«•> Let us now suppose that the body changes its volume without either losing or gaining heat by conduction. This condition is expressed by the equation = (1 +$' .Q)dQ + Qd. F, from which we deduce the following, -i.r-l+^W.iq. . . (77.) which expresses the following theorem : — When the quantity of heat in a body is varied by variation of volume only, the variation of the heat-potential depends on the heat only, and is independent of the volume. In order that a machine working by the expansive power of heat may produce its greatest effect, all the heat communicated from external bodies should be employed in producing expansive power, and none in producing variations of the quantity of heat in the body; for heat employed for the latter purpose would be wasted, so far as the production of visible motion is concerned. To effect this the body must receive heat by conduction, and convert it into expansive power, while containing a certain constant quantity of heat Q : ; give out by conduction heat produced by compression, while containing a smaller constant quantity of heat Q 9 ; and change between those two quantities of thermometric heat by means of changes of volume only, without conduction. For this purpose a cycle of opera- tions must be performed similar to that described by Carnot, as follows : — (I.) Let F A be the initial value of the heat-potential ; let the body expand at the constant heat Q l till the heat-potential becomes F B . Then the heat received and converted into expansive power is H a = Q, (F B - F A ). (II.) Let the body further expand without receiving or emitting heat till the quantity of heat in it falls to Q 2 ; the heat-potential varying according to equation (77), and becoming at length F c . The heat con- verted into expansive power in this operation is Q, - Q,- 31G THE MECHANICAL ACTION OF HEAT. (III.) Let the body be compressed, at the constant heat Q„ till the heat-potential becomes F D ; a quantity differing from the initial heat- potential F v by as much as F c differs from F B . In this operation the following amount <>;' power is reconverted into heat, and given out by conduction : — IL = Q, (F c - F„). (IV.) Let the body be further compressed, till the heat-potential returns to F A , its original value. Then, by the power expended in this com- pression alone, without the aid of conduction, the total heat of the body will be restored to its original amount, exactly reversing the operation II. At the end of this cycle of operations, the following quantity of heat will have been converted into mechanical power: — H x - H 2 = Q, (F B - F A ) - Q 2 (F c - F D ), but it is obvious that the difference between the heat-potentials is the same in the first and third operations ; therefore, the useful effect is simply 11,-11,, = ^,- t>,KF u -F A ), 1 while the whole heat expended is, ' . . (78.) H^Q^F,- V J Hence, the ratio of the heat convert* J into mechanical effect, in cm expansive machine, working to the greatest advantage, to the whole heat expended, is the same with that which the difference between the quantities of heat possessed by the expansive body during the i to the quantity of heat possessed by it during the operation of receiving . and is independent of the nature and condition of the body. This theorem is thus expressed symbolically, — H, - H., Effect _ Q, - Q., \l i i teat expended Q x (79.) 51. When a body expands without meeting with resistance, so that all its expansive power is expended in giving velocity to its own particles, and when that velocity is ultimately extinguished by friction, then a quantity of heat equivalent to the expansive power is reproduced. The heat consumed is expressed by taking away the term representing the expansive power, ~P dY, from the expression (72), the remainder of which consists merely of the variation of actual heat, and the heat expended in overcoming molecular attraction, viz. : — / d S\ , „ d S , ,, THE MECHANICAL ACTION OF HEAT. 317 This expression is a complete differential, and may be written thus : — d (Q + S) = d { Q + <p (Q) + (Q ~ - l) j P rf v|- (80.) (Q being treated as a constant in performing the integration I P d X ). Its integral, Q + 8, the sum of the heat of the body, and of the potential of its molecular actions, is the same quantity which I have denoted by the symbol ^F in the tenth article of a paper on the Centrifugal Theory of Elasticity (See p. G2), and whose differences are there stated to repre- sent the total amount of power which must be exercised on a body, whether in the form of expansive or compressive power, or in that of heat, to make it pass from one volume and temperature to another. This integral corresponds also to the function treated of by Professor William Thomson in the fifth part of his paper on the Dynamical Theory of Heat, under the name of " Total Mechanical Energy." 52. We have now obtained a system of formula 3 , expressing all the relations between heat and expansive power, analogous to those deduced from a consideration of the properties of temperature, by Messrs. Clausius and Thomson, and from the hypothesis of molecular vortices in the previous sections of this paper; but, in the present section, both the theorems and the investigations are distinguished from former researches by this circumstance — that they are independent, not only of any hypo- thesis respecting the constitution of matter, but of the properties, and even of the existence, of such a function as temperature; being, in fact, simply the necessary consequences of the following " Definition of Expansive Heat. Let the term Expansive Heat he used to denote a hind of physical energy •convertible with, and measurable by, equivalent quantities of mechanical power, and augmenting the expansive elasticity of matter in which it is -present. 52a. It is further to be remarked, that the theorems and formulae in the preceding articles of this section are applicable, not only to heat and expansive power, but to any two directly convertible forms of physical energy, one of which is actual, and the other potential. They are, in fact, the principles of the conversion of energy in the abstract, when interpreted according to the following definitions of the symbols : — 318 THE MECHANICAL ACTION OF HEAT. Let Q denote the quantity of a form of actual physical energy present in a given body ; V, a measurable state, condition, or mode of existence of the body, whose tendency to increase is represented by P, a force, depending on the condition V, the energy Q, and permanent properties of the body, so that P d V is the increment of a form of potential energy, corresponding to a small increment d V of the condition V. Let d& be the quantity Avhcrcby the increment of potential energy P(7V falls short of the quantity of actual energy of the form Q, which is converted into the potential form by the change of condition d V. Then, as in equation (G9), dX ~ H dQ ' an equation from which all those in the previous articles are deducible, and which comprehends the whole theory of the mutual conversion of the actual form of energy Q, and the potential form I "PdV, whatsoever those forms may be, when no other form of energy interferes. The application of these principles to any form, or any number of forms, of actual and potential energy, is the subject of a paper read before the Philosophical ►Society of Glasgow, on the 5th January, 18.33, and published in the Philosophical Magazine for February, 1853. (See p. 203.) Si B-SECTION 2. — PROPERTIES OF TEMPERATURE. 53. Still abstaining from the assumption of any mechanical hypothesis, let us proceed a step beyond the investigation of the foregoing articles, and introduce the consideration of temperature — that is to say, of an arbitrary function increasing with heat, and having the following pro- perties : — Definition of Equ.d Temperatures. Two portion; of matter are said to have equal temperatures ivhen neither tends to communicate heat to the other. Corollary. All bodies absolutely destitute of heat have equal temperatures. The ratio of the real specific heats of two substances is that of the THE MECHANICAL ACTION OF HEAT. 319 quantities of heat which equal weights of them possess at the same temperature. Theorem. The ratio of the real specific heats of any pair of substances is the same at all temperatures. For, suppose equal weights of a pair of homogeneous substances to be in contact, containing heat in such proportions as to be in equilibrio. Then, let additional portions of each substance, of equal weight, and destitute of heat, be added to the original masses ; so that the quantities of heat in unity of weight may be diminished in each substance, but may continue to be in the same ratio. Then, if the equality of temperature do not continue, portions of heat which were in equilibrio must have lost that equilibrium, merely by being transferred to other particles of a pair of homogeneous substances, which is absurd. Therefore, the temperatures continue equal. It follows, that the quantity of heat in unity of weight of a substance at a given temperature, may be expressed by the product of a quantity depending on the nature of the substance, and independent of the temperature, multiplied by a function of the temperature, which is the same for all substances. Let r denote the temperature of a body according to the scale adopted ; k, the position, on the same scale, of the temperature corresponding to absolute privation of heat ; ft, a quantity depending on the nature of the substance, and independent of temperature. Then the quantity of heat in unity of weight may be expressed as follows : — Q = feO/,.r-^. K ). . . . (81.) 54. If we introduce this notation into the formula (79), which expresses the proportion of the total heat expended which is converted into useful power by an expansive machine working to the best advantage, the quantity ft, peculiar to the substance employed, disappears, and we obtain Carnot's theorem, as modified by Messrs. Clausius and Thomson — viz., that this ratio is a function solely of the temperatures at which heat is received and emitted respectively, and is independent of the nature of the substance ; or symbolically, Effect _ Qi^Q. 2 _ \p . Tj - }p . r, , g2 . Heat expended Q x \f; . r x — \p . k 55. Let us now apply the same notation to the formula (G7) for the latent heat of a small expansion, d V, at constant heat, viz : — 320 THE MECHANICAL ACTION OF HEAT. we have evident lv d? 1 dV 1 dv dQ ~ dQ dr ~ \l \jj . T ll 7 dr and. consequently, the heat which disappears by the expansion dX is dV dr T d V, dQ \f/ from which formula the specific quantity ft has disappeared. Now, in the notation of Professor Thomson we have \p . T — Tp . K _ «J where J is Joule's equivalent, and ju. a function of the temperature, the same for all substances, to be determined empirically ; and, consequently, hyp. log. (i// . t — i// . k) = 1 J ' fx d -. if /J- ip . t — ifj . k = e and (84.) sr ^ j Q = fe (i/, . t - i£ . k) = ft These expressions will be re by those who have studied Professor Thomson's papers on the dynamical theory of heat. By introducing the value given above of the quantity of heat in unity of weight, into the formula' of the preceding articles of this section, they are at once trans- formed to those of Professor Thomson, and in particular, the formulae (79) and (82) become the following: — 1 "I 7 1 ~, j) " dr J, * Effect of Machine Heat expended M = 1 - £ J k d ,"- (85.) * It is to be observed, that in Professor Thomson's notation, heat is supposed to be measured by an arbitrary unit, whose ratio to a unit of mechanical power is denoted by J ; while in this paper, the same unit is employed in expressing quantities of heat and of mechanical power. THE MECHANICAL ACTION OF HEAT. 321 Sub-Section 3. — On the Hypothesis of Molecular Vortices. 56. The use of a mechanical hypothesis in the theory of heat, as in other branches of physics, is to render it a branch of mechanics, the only "complete physical science ; and to deduce its principles from the laws of force and motion, Avhich are better understood than those of any other phenomena. The results of the investigations in the £>receding part of this section are consistent alike with all conceivable hypotheses which ascribe the phenomena of heat to invisible motions amongst the particles of bodies. Those investigations, however, leave undetermined the relation between temperature and quantity of heat, except in so far as they show that it must follow the same law of variation in all substances. By adopting a definite hypothesis, we are conducted to a definite relation between temperature and quantity of heat; which, being intro- duced into the formulae, leads to specific results respecting the phenomena of the mutual transformation of heat and visible mechanical power ; and those results, being compared with experiment, furnish a test of the soundness of the hypothesis. Thus, the hypothesis of molecular vortices, which forms the basis of the investigations in the first five sections of this paper, and in a paper on the centrifugal theory of elasticity, leads to the conclusion, that, if tempera- ture be measured by the expansion of a perfect gas, the total quantity of heat in a body is simply proportional to the elevation of its temperature above the temperature of absolute privation of heat ; or, in the notation of the preceding article, xp . r = r, \f/ . r = 1, and Q = fc (r - k), . . . . (86.) ft being the real specific heat of the body. If this value be substituted for the quantity of heat Q, in all the formulae, from (67) to (80) inclusive, which are founded simply on the definition of expansive heat, it reproduces all the formula? which, in this and the other paper referred to, have been deduced directly from the hypothesis. In the sequel, I shall apply one of these formulae to the calculation, from the experiments of Professor Thomson and Mr. Joule on the heating of currents of air by friction, of approximate values of the absolute temperature corresponding to total privation of heat, that the mutual consistency of those values may serve as a test of the soundness of the hypothesis, and the accuracy of the formulae deduced from it. 57. Before proceeding further, it may be desirable to point out how far x 322 THE MECHANICAL ACTION OF HEAT. this hypothesis agrees with, and how far it differs from, that proposed by Mr. Herapath and Mr. Waterstom, which supposes bodies to consist of extremely small and perfectly clastic particles, which fly about in all directions with a velocity whose half-square is the mechanical equivalent of the heat possessed by unity of weight, and are prevented from dispersing by their collisions with each other and with the particles of surrounding bodies. Let v be the velocity of motion, then 9 represents the heat possessed by unity of weight, expressed in terms of the force of gravity. The expansive pressure due to such motions is found by conceiving a hard, perfectly elastic plane of the area unity to be opposed to the collision of the particles, and calculating the pressure which would be required to maintain its position against them. If all the particles were to strike and rebound from such a plane at right angles, the pressure would bi represented thus: 1_ 9 ' T » where V is the volume which contains so many particles as amount to unity of weight. But the particles are supposed to tly in equal numbers in all directions. Then, if 9 denote the angle of incidence on the plane / sin <1 9 . „ , » = sin 9 I' 9 — sin 6 d represents the proportion of the whole particles which fly in those direc- tions which make the angle with the normal to the plane. Of this proportion, again, the fraction cos 9 only strikes the plane; while the force of the blow also is less than that of a normal blow in the ratio cos 0:1. Hence, the mean force of collision is / .7T 2 1 cos 2 9 sin 9 (I 9 = ^ 6 of the force of a perpendicular collision ; so that the expansive pressure i& represented by 1 *2 ^_2 Q 3 ' cj ' V ~ 3 ' V THE MECHANICAL ACTION OF HEAT. 323 Hence, according to this hypothesis, we should have for a perfect gas P V = | Q, o or the product of the pressure and volume of a mass of a perfect gas equal to two-thirds of the mechanical equivalent of its total heat. It is known, however, that the product of the pressure and volume of a mass of sensibly perfect gas is only about four-tenths of the equivalent of its total heat. The hypothesis, therefore, requires modification. By supposing the particles to attract each other, or to be of appreciable bulk compared with the distances between them, the ratio in question is diminished; but either of these suppositions is inconsistent with the perfectly gaseous condition. * It appears to me, that, besides this difficulty connected with the gaseous condition, there exists also great difficulty in conceiving how the hypothesis can be applied to the solid condition, in which the particles preserve definite arrangements. The limited amount of time and attention, however, which I have hitherto bestowed on this hypothesis, is not sufficient to entitle me to pronounce whether these difficulties admit of a solution. 58. The idea of ascribing expansive elasticity to the centrifugal force of vortices or eddies in elastic atmospheres surrounding nuclei of atoms, originated with Sir Humphry Davy. The peculiarity of the view of the hypothesis taken in this paper consists in the function ascribed to the nuclei or central physical points of the atoms, which, besides retaining the atmospheres round them by their attraction, are supposed, by their actions on each other, to constitute the medium which transmits radiant heat and light ; so that heat is radiant or thermometric, according as it affects the nuclei or their atmospheres. In this form the hypothesis of molecular vortices is not a mere special supposition to elucidate the theory of expansive heat, but becomes connected with the theory of the elasticity of matter in all conditions, from solid to gaseous, and with that of the transmission of radiations. I have already investigated mathematically the consequences of this hypothesis by two different processes, which are necessarily somewhat complicated. When the question, however, is confined to the relations between tem- peratures and quantities of heat, a more simple process may be followed, analogous to that which has been applied in the preceding article to the hypothesis of molecular collisions. If a mass of elastic fluid, so much rarefied that the effect of molecular attraction is insensible, be entirely filled with vortices, eddies, or circu- lating currents of any size and figure, so that every particle moves with 324 THE MECHANICAL ACTION OF HEAT. the common velocity iv, then, if the planes of revolution of these eddies be uniformly distributed in all possible positions, it follows, from reasoning precisely similar to that employed in the preceding article, that the pressure exerted by the fluid against a plane, in consequence of the cen- trifugal force of the eddies, has the following value in terms of gravity : — \-'?-l (870 3 g V or two-thirds of the hydrostatic pressure due to the velocity of the eddies w j V being, as before, the volume occupied by unity of weight. It is, however, reasonable to suppose, that the motion of the particles of atomic atmospheres does not consist merely in circulating currents ; but that those currents afe accompanied with a certain proportionate amount of vibration, — a kind of motion which docs not produce centrifugal force. To these Ave have to add the oscillations of the atomic nuclei, in order to obtain the mechanical equivalent of the whole molecular motions ; which is thus found to be expressed for unity of weight by I f- • = Q, • • ■ • (88.) 2g 2 k h being a specific coefficient. Hence it follows (denoting — by N), that the expansive pressure due to molecular motions in a perfect gas is equal to the mechanical equivalent of those motions in unity of volume multi- plied by a specific constant N • | (89.) The coefficient N has to be determined by experiment ; its value for atmospheric air is known to be between 0'4 and 0*41. In order to account for the transmission of pressure throughout the molecular atmospheres, it is necessary to suppose them possessed of a certain amount of inherent elasticity, however small, varying proportionally to density, and independent of heat. Let this be represented by h V' then P=(NQ + A)i . • (90.) is the total pressure of a perfect gas. Equilibrium of heat and pressure between portions of two different perfect gases in contact requires that the pressures independent of heat THE MECHANICAL ACTION OF HEAT. 325 and the pressures caused by heat, shall separately be in equilibria Let the suffixes a and b be used to distinguish quantities relative to two different substances in the perfectly gaseous condition. Then the first condition of equilibrium is expressed as follows : — h\ , N (h (4) w =(£)<»» • (9L) that is to say, the densities of two perfect gases in equilibrio are inversely proportional to the coefficients of elasticity of their atomic atmospheres. The second condition is expressed as follows : — ("*)« = &*)» which, being taken in connection with the first condition, gives (!«)«=(!«)» • < 93 -> Now, by equation (90), we have Hence the condition of equilibrium of heat between two perfect gases is (^)(°> = (i>>> • • • o*> consequently, temperature may be measured by the product of the pressure and volume of a perfect gas, divided by a coefficient, which is proportioned to the volume of the gas at a standard pressure and temperature. Temperatures thus measured are reckoned from the point known as the zero of gaseous tension, or absolute zero of a perfect gas thermometer, 274°*6 Centigrade below the temperature of melting ice. Let V denote the volume of unity of weight of a perfect gas, at a standard pressure P , and absolute temperature t ; then any other absolute temperature has the following value : — PV t — r, -^-(NQ + A), • • (94.) op v ~p v x v x v while the absolute temperature of total privation of heat is K — T o p v " r v (94a.) 320 THE MECHANICAL ACTION OF HEAT. Hence it appears that quantity of heat in unity of weight bears the following relation to temperature, — Q = i(PT-*) = ^.(r-.4 • • (95.) in which, if we substitute the symbol of real specific heat, » p V ft =iv • • • • ^ we obtain the formula already given (8*6) for the relation between heat and temperature.* 59. The introduction of the value given above of the quantity of heat in terms of temperature, into the formula (67), gives for the latent heat of a .small expansion d V at constant temperature (r- K )7 > -<>V. . . . (97 .) The formula' (79) and (82), for the proportion of heart rendered available by an expansive engine working to the greatest advantage, becomes rj - r 2 (98.) or the ratio of the difference between the temperatures of receiving and emitting heat, to the elevation of the former temperature above that of total privation of heat. This is the law already arrived at by a different process in Section V of this paper. When the same substitution is made in equation (80), which represents the total energy, whether as heat or as compressive power, which must be applied to unity of weight of a substance to produce given changes of heat and volume, the following result is obtained : — d . ¥ = dQ + d . S = I ft +/'(t) + (r - k)J~ d V I dr = d . { ft r +/(r) + ((t - k) f~ T - l) / P d V } . (99.) As it cannot be simplified, it is unnecessary here to recapitulate the investigation, which leads to the conclusion that the functions / (r) and / (t) have the following values : — * See Appendix, Note A, p. 33G. THE MECHANICAL ACTION OF HEAT. 327 f{r) = * N (k hyp. log. r + £) ; /' (r) = ft N (^ - J). (99 A.) We have thus reproduced equation (26) of the paper formerly referred to, on the Centrifugal Theory of Elasticity. (See p. 49.) The coefficient of the variation of temperature in the first form of equation (99) is the specific heat of the suhstance at constant volume. Denoting this by K v , the formula becomes d . * = K v . d t + | (r - k) ^f - P | d V. . (100.) Sub-Section 4. — Thermic Phenomena of Currents of Elastic Fluids. 60. When a gas previously compressed is allowed to escape through small apertures, as in the experiments of Mr. Joule and Professor Thomson, and has its velocity destroyed entirely by the mutual friction of its particles, without impediment from any other substance, and without conduction of heat to or from any other substance ; then its condition is expressed by making d ¥ = 0, that is to say, 1 f (d? P\ r/P) If we assume (as is really the case in the experiments) that the specific heat of the gas at constant volume does not sensibly vary within the limits of the experiments as to temperature and volume, so that K T is sensibly constant, and also that the variation of temperature is very small as compared with the absolute temperatures, then we have the following approximate integral : i i which represents the cooling effect of an expansion from the volume V t to the volume V 2 . If it were possible to obtain any substance in the state of perfect gas to be used in experiments of this kind, the first integral in the above expres- sion would disappear, because for a perfect gas dr t 328 THE MECHANICAL ACTION OF HEAT. and as the other terra is negative, the result would be a slight heating dP , , P , effect. As no gas, however, is perfect, and as — always exceeds — , the (I T T mode of reducing the experimental data is to calculate the value of the first term, which represents the effect of cohesion, from the known pro- perties of the gas, to subtract from it the actual cooling, and from the remainder to compute values of k, the temperature of absolute privation of heat, according to the following formula : — v 2 riP (103.) r i T- dY K T J v dr When the gas is nearly perfect, as in the case of atmospheric air, it is unnecessary to take into consideration its deviation from the perfect condition in computing the integral in the denominator, whose approxi- mate value is found to be P V V P - ° — - . hyp. log. ~ = N . hyp. log. - nearly (t being nearly constant), K v r Vj 1 2 and K v nearly = ft. The value of the integral in the numerator is found as follows : — The centrifugal theory of elasticity indicates that the pressure of an imperfect gas may be represented by the following formula : — P = P »v{r + A »- A ;-r'- &C -}' ' (1M0 where V is the volume in the perfectly gaseous state, at a standard pressure P , and absolute temperature r , and A , A 1} &c, are a series of functions of the density, to be determined empirically. From this formula it is easily seen that dP P_ p V f A 2A ) so that the first term in the numerator of the expression (103) has the following value : — i ii p v in which — ^— ° = N r nearly. THE MECHANICAL ACTION OF HEAT. 329 In order to represent correctly the result of M. Regnault's experiments on the elasticity and expansion of gases, it was found sufficient to use, in the formula for the pressure (104), the first three terms; and the functions of the density which occur in these terms, as determined empirically from the experiments, were found to have the following values, in which the unit of volume is the theoretical volume of unity of weight of air under the pressure of one atmosphere, at the temperature of melting ice, * and the values of the constants are given for the Centigrade scale. v =t (v) ? ; v 1 = "G) f • • < 107 -> Com. log. b — 3-8181545 ; Com. log. a = 0-317616S. Hence it appears that the integrals in the formula (106) have the following values : — />y=«.A.(^ ! !/>T=¥.t. ? .A.(^)*a M i.) i i in which the common logarithms of the constants are Com. log. 2 b = 2-1101845 ; log. — . - = 2-4017950 ; 3 r and these values suit any scale of temperatures. In calculating, for use in these formulae, the densities — from the observed pressures, it is sufficiently near the truth, in the case of air, to use the approximate equation 1 T — ° . P (in atmospheres). V T The common logarithm of r , the absolute temperature of melting ice, for the Centigrade scale, is 2 - 4387005. The constant N for atmospheric air is 0"4 nearly ; therefore Com. log. (N X hyp. log. 10) = 1-9642757. The following, therefore, is the approximate value of the formula (103) to be used (with the numerical constants already given) in reducing the experiments of Mr. Joule and Professor Thomson on atmospheric air, so as to obtain approximate values of the absolute temperature of total privation of heat : — * This unit of volume is greater than the actual volume of air, under the circum- stances described, in the ratio of 1 00085 to 1. 330 THE MECHANICAL ACTION OF HEAT. -={^(jg.0)»A.lP*)-«^A.CP»))-X-A^ -4- N hyp. log. 10 x A . com. log. ,,. . (106.) In using this formula, the mean absolute temperature should be taken as the value of r. The following table shows the values of the quantity k, computed from ten mean experimental data, taken respectively from the first ten series of experiments described in the recent paper of Messrs. Joule and Thomson, in the supplementary number of the Philosophical Magazine for December, L852. The temperatures in the table, for the sake of convenience, are reduced to the Centigrade scale, because that scale has been used through- out the previous sections of this paper. The final pressure in each case was thai of the atmosphere. Professor Thomson and Mr. Joule have expressed the opinion, which is undoubtedly correct, that those experiments in which the largest quantities of air Avere used were the least liable to error from disturbing causes, such as the conduction of heat. Now, it may 1 bserved in the table, that the calculated values of k are generally greatest, and the discrepancies amongst them least, for the experiments in which most air was used. To illustrate this, the results of the last eight series are arranged below in the order of the quantities of air i mployed. Cubic inches 1 1>4 ^ ^ g . fi &l ^ n . 2 {] .., per second, ) Values of ic, 1-683 1'762 2'09 2-228 151 2-087 2-345 2'14 It is further to be remarked, that the discrepancy between the highest and the lowest of the values of k is 2°-345 - l°-08 = r-265 Centigrade: a quantity which corresponds to a difference of less than one three-hundredth part in computing the proportion of heat converted into mechanical power by any ordinary expansive engine, according to the formula (98), which has been deduced from the hypothesis of molecular vortices. The experiments, therefore, may 1 »e considered as tending to prove, that the formulas deduced from this hypothesis are sufficiently correct for practical purposes; and also as affording a strong probability that the principles to which it leads are theoretically exact, and that the tempera- ture of absolute privation of heat is a real fixed point on the scale, THE MECHANICAL ACTION OF HEAT. 331 ° ' £ SpSo':? s rt ,Otn O ,2 t^- >0 CO Cl CO 'zz t ~ o o 2 % 1 ri M CO o CO -* CO o Cl O - 3 O ^^ 1 ^3 oco p p p CI p CI P CI p 1^ Cl Cl Cl <P3;S SS O © o a tH Cl CO CO CO CI CO C) L01 g C C5 ■* CO p F^H p -T ■■C p Cl *2 b b 6 b b b ■CS P>> . O <d.q a ts 1-0) ta 6C.2 g -r: 3 03 fcn o o >o OS CO CO CO CO t- CI ■<* CO p p OS ooli 'oho °^ b ~ ^ CI c. b b ■- 1 ■- 1 jO u O -Mi's c3 a S 'O CO t^. o 3 CI CO CO Cl Actu Cooli entigi i- b o p b p b - p b Cl b p b b b O .3 o Tf< 73 © 2 , i ■* CO -* i-H o OJ ^— Cl i-H •-: p CI p 7^ "7?* p Tf p T -1 sSo CO Cl 4* QQ T* tH rH 01 Cl ■* hH g g ^ 2 * 3 ID gs "? p p r^ p V CO p p p <l cO lO ^_, ,Ih CO — Cl Cl Cl Cl OS -* in CO i^ 05 C5 OJ S p. Cl CO CO CI Cl Cl Cl Cl «* S 03 3 k— i C3 «H 6 h © 3 -3 §«1 1 2 ,_, ,_! ^H OJ o _- t— -+ __ t- ep p p p p Cl co Init mpei eutig c b .1, 1~ t- CO ■ o r~- CO CO CO t^ l~ £ ° "o a O, o ,_; T3 03 0) ^■.•cg S o o tH p Tl* CI •jH 00 p 01 fi3 * rt rt b b CO ^H i-H Cl b ^H g:nJ-K 3 o o o •"* 1—1 =?«*P, £ £ 5 • • • a 03 EC Ul W in . 03 CQ ~ z > s •2 2 M 3 — o 03 Q. 03 '2 '2 ° k— 1 03 H- 1 — to "g .2 S I— ' (5 a 03 S-" •r* 2 05 5 > % W p ■g § h^ 3 <o 2 1 ^ -1 03 t, 03 x 00 pa ^rJ S x ^^ s 10 1T5 !>• co CI Tf o LO rH CO t5 ^— . t*H t|H =« «H en tfH t<-i <+-! o O O o o o O o o o a r< fj rt c s d s c rt 03 ri ri rt c3 rt rt rt cS 03 03 03 a> o (U 03 03 03 03 » s £ <H S 3 3 g 3 332 THE MECHANICAL ACTION OF HEAT. somewhat more than two Centigrade degrees above the absolute zero of a perfect gas-thermometer (which is, of course, an imaginary point) ; that is to say, about 272h Centigrade degrees, or 490| degrees of Fahrenheit, below the temperature of melting ice. If these conclusions be correct, it follows, that when the temperatures T t and T.„ between which an expansive engine works, are measured from the ordinary zero points of the Centigrade and of Fahrenheit's scales respectively, the following are the utmost proportions of the total heat expended which it can be made to convert into mechanical power : — For the Centigrade scale, For Fahrenheit's scale, T, - T 2 Tj + 2721 T x + 458£ J (109.) In the fifth section of this paper, where a comparison is made between the actual duty of the Cornish engine at Old Ford, as determined by Mr. Wicksteed, and the greatest possible duty which could be obtained from a given quantity of heat by a theoretically perfect engine working between the same temperatures, the constant k is treated as being so small that it may be neglected in practice. If the value of k is really 2 0, 1 Centigrade, as computed above, the calculated maximum theoretical duty in Section \ is too small by about one one-hundred-and-ninetieth part of its amount,, — a quantity of no practical importance in such calculations. 61. It may be anticipated, that when Mr. Joule and Professor Thom- son shall have performed experiments on the thermic phenomena exhibited by air in more copious currents, and by gases of more definite composition, and more simple laws of elasticity, much more precise results will be obtained. When a gas deviating considerably from the perfectly gaseous condition, or a vapour near the point of saturation, is employed, it will no longer be sufficiently accurate to treat the specific heat at constant volume as a con- stant quantity, nor the cooling effect as very small. It will, therefore, be necessary to employ, for the reduction of the experiments, the integral form of equation (99) — that is to say, = A ^ = A - h t + fe X k- (hyp. log. r + -J + (('-^/ r -0f p '' v } THE MECHANICAL ACTION OF HEAT. 335 - k { A (-^ rf V - k N (A . 5 + A hj T . log. t) } . (110.) 62. Preliminary to the application of this equation, it is necessary to determine the mechanical value of the real specific heat fe. Supposing the law which connects the pressure, density, and temperature of the gas to be known, it is sufficient for this purpose to have an accurate experimental determination, either of the apparent specific heat at constant pressure for a given temperature, or the velocity of sound in the gas under given circumstances. First, let us suppose that the apparent specific heat at constant pressure is known. The value of this coefficient (Centrifugal Theory of Elasticity, Art. 1 2) is ( dV \ dr v dv ; In order that the lower limit of the integral may correspond with the condition of perfect gas, it is convenient to transform it into one in terms of the density. Let D be the weight of unity of volume, then /£"--#•£<*■ < 1I1A -> If, then, we have the pressure of the gas undeu consideration expressed by the following approximate formula : — B -^U+v*} The following will be the values of the functions of the pressure which enter into the above equation : — :p P o V f 1 A t 1 ^P P V A, _ , pv A 1 ] 334 THE MECHANICAL ACTION OF HEAT. ) dr> ~ i D 2 " dr** + " r- i D ( dF . S \dr, — = p v d p ° ° ,/ v 1 + AA» rf. A D 1 < Z . A 1 D To illustrate the application of these formula?, let us calculate the difference between the real specific heat and the apparent specific heat, at constant pressure, of carbonic acid gas. at the temperature of melting ice, and at the density which, it' the gas were perfect, would correspond to a pressure of one atmosphere al the temperature of melting ice. Let this density be denoted by D , and its reciprocal by V . As the constants have been deduced from M. Regnault's experiments, the calculations will be made in French measures and for the latitude of Paris The actual density of carbonic acid at 0° Centigrade, and under one atmosphere of pressure, exceeds the theoretical density, in the perfectly gaseous state, in the ratio of 1'0065 to 1 nearly. Hence, the height of a homogeneous atmosphere of actual carbonic acid at 0° Centigrade being ......... 5225*5 metres, the corresponding height in the state of perfect gas is P V =5259 - 5 P V and " - = 19*53 metres per Centigrade i --- 02*84 !■ r The functions which express the influence of density on thejdeviation of carbonic acid gas from the perfectly gaseous state, have the following values : — b D A T) A 1 = «.- when > Com. log. h = 3*1083932; Com. log. a = 03344538 6 = 0-00128349 a =2*16: Mine.) B n d.D D d J n dD-J A 1]D . ^ -«- DflJ rfD'-^o^ = 2b D d D ' rfD' A,D i> For the purposes of a first approximation, we may assume that the THE MECHANICAL ACTION OF HEAT. 335 value of k already found is sufficiently near the truth — viz., 2 0, 1 Centi- grade, so that, in the present instance, t — k = 2 7 2° 5 Centigrade. Then we find the following results when r = r , and D=D : Metres. Feet. ( T - K ) P ° - V ° . - 7 = per Centigrade degree, . . . 0*145 0-48 /' <T- P ( T — K \\ — - d V =■ per Centigrade degree, . . . 0*150 - 49 J tl t Sum = K v — ll = excess of apparent specific heat at con- stant volume above real specific heat 0-295 0-97 ,/r r (j _ K ) T/ = difference between apparent specific heats at constant volume and at constant pressure, . . . 19'565 64 - 19 K r — ft = excess of apparent specific heat at con- stant pressure above real specific heat 19-8G0 65-16 4 of the above quantities are of course the corresponding quantities for Fahrenheit's scale. Secondly, If the velocity of sound in the gas is given, let this = u. Then Ave know that v? = q - P K '' • • • (112.) 9 dD K v V ; in which d? d P p y f r , d.A D ld^AjD) . A) So that from the velocity of sound we can calculate the ratio of the specific- heats at constant pressure and at constant volume. Let this ratio be denoted by y, and let K v = fc+ c; K p = ft + c'; then fc + c ; ana k = - 7 = l±^; andft = ^- C . . (H2B.) B+C y — 1 i.3G THE MECHANICAL ACTION OF HEAT. in which c and c are to be calculated as above.* G3. In using the formula (110) for a gas whose pressure is represented by the formula the integrals may be transformed so as to be taken, with respect to the density, as in the preceding article. Thus we obtain *J(^*)"--4*v-S-*)»= - fl.^Dn: -PoV A(^hy P .log.I)+^/; V ;^D) (113.) For carbonic acid, the first of these formula; becomes simply + ^gQ-^KOw. J and the second, )■ (113 A.) n „ fl, . D, a /D t DA ) + P Y u -|. ) hvp.log. I) '- Do (^-^)^ APPENDIX. Note A. (to Article 58). — Since this section was read, the theoretical views relative to the relation between heat and temperature contained in it and the previous sections of this paper, have received a strong confir- mation by the publication by M. Eegnault of the fact, that he has found the specific heat of air to be sensibly constant at all temperatures from — 30° Centigrade to + 225°, and at all pressures from one to ten atmospheres (Comptes Rcnclus, April 18, 1853); so that equal lengths on the scale of the air thermometer represent equal quantities of heat. Note B. (to Article 62). — Until very recently, there existed no exact * See Appendix, Note B. THE MECHANICAL ACTION OF HEAT. 337 <! o i— i W o fd w a H H W P3 O H w w o W 172 «i H Szi 1—1 CO a ■J3 <i> p < o i^ a H H D o '3 rt bb t5] to bO o o o o -< (5 § ft fi n P o i 6 CO CO F-i gip ^ CO ^ p CN O a! £ uo b ci o t~- 00 •°.2 £C5 CS Th o o LO Si O-S p.© o o OO H t>> *SH r-( l-H ^ f— < o o e fc i- w tA O LO K* t< £"* -? CO op ip o hO b eo b t^ SO CO o o CO H P.© o OO >> >* a £ TJH tH o r- -# M o lO -* .i. o o o I-- i ~? ■JK ■* "5* 1 tN H M ~ ^H - 1 i— i 1—1 1-1 LO « CO -? p O) >o >-i > «■* b CO b b <4 M o CO -f rH CO feci ci CO C4 CN H CO a p !» lO H M .© qo p 01 01 %--0 © 00 b tJ( 1 <£> C5 co 00 OS CO Ph fe CO M CO 01 p LO r- t. oO cV\ _ -* b m a CO o CO -f o fcco CO l^ ■* CO CO «« -« W| os <M o c^ os j I— 00 -* t- CO < CO o M< •/■* CI t>' "? Ol <N a o b CO b b E pH v. b^ -* "* p op i>- o"5 -J - * b os I— tH W «8 OS Tf< if a <dO! I>- OO CS 60 fee© CO 00 t~ CI Cl 1- co CM 1-1 P4 of • W rH rs S3 • 'S "3 o CD o << o o • n" DO '3 '3 <o o o tT Q0 ^2 < o w o O 338 THE MECHANICAL ACTION OF HEAT. experimental determination of the specific heat of any gas. The specific heat of air at constant pressure, as compared with that of water, was calculated theoretically in the previous part of this paper, from Joule's equivalent and the velocity of sound, and found to be 0'24. This value has since been confirmed very closely by Mr. Joule's experiments, whose mean result was 0'23, and still more exactly by M. Kegnault's experiments, already referred to, which give the value 0-2379. The table {See p. 337), shows the results of the application of the formula? of this paper to the specific heats of five different gases at constant pressure, selected from M. Kegnault's table (Comptes Rendus, April 1 8), as being those in which the velocity of sound can be computed, and has been determined experi- mentally. The table shows also a comparison of the calculated and observed velocities of sound. This table appeared originally, in French measures, in the Philosophical Magazine for June, 1853: the metres are here reduced to feet. K p , K v , and K w , are expressed in feet of fall per Centigrade degree. K w (Joule's equivalent) = 1389'G The real specific heat of carbonic acid gas is 235*5 feet of fall per Centigrade degree. That of the other gases does not differ from the apparent specific boat at constant volume by an amount appreciable in practice. ON THERMODYNAMICS. 339 XX.— ON THE GEOMETRICAL REPRESENTATION OF THE EXPANSIVE ACTION OF HEAT, AND THE THEORY OF THERMODYNAMIC ENGINES * Section I. — Introduction and General Theorems. 1. The first application of a geometrical diagram to represent the expan- sive action of heat was made by James Watt, when he contrived the well-known steam-engine indicator, subsequently altered and improved by others in various ways. As the diagram described by Watt's Indicator is the type of all diagrams representing the expansive action of heat, its general nature is exhibited in Fig. 1. Let abscissas, measured along, or par- allel to, the axis X represent the vol- umes successively assumed by a given mass of an elastic substance, by whose alternate expansion and contraction heat is made to produce motive power ; V A and V B being the least and greatest volumes which the substance is made to assume, and OV any intermediate vol- ume. For brevity's sake, these cmantities will be denoted by V A , Y B , and V, respec- tively. Then V B — V A may represent the space traversed by the piston of an engine during a single stroke. Let ordinates, measured parallel to the axis Y, and at right angles to X, denote the expansive pressures successively exerted by the substance at the volumes denoted by the abscissae. During the increase of volume from V A to V B , the pressure, in order that motive power may be produced, must be, on the whole, greater than during the diminution of volume from V B to V A ; so that, for instance, the ordinates V P x and V P 2 , or the symbols * Read before the Royal Society of London on January 19, 1834, and published in the Philosophical Transactions for 1854. 340 ON THERMODYNAMICS. P, and P , may represent the pressures corresponding to a given volume V during the expansion and contraction of the substance respectively. Then the area of the curvilinear figure, or Indicator-diagram, A P x B P 2 A, will represent the motive power, or " potential energy," developed or given out during a complete stroke, or cycle of changes of volume of the elastic substance. The algebraical expression for this area is P,) i V (l.) The practical use of such diagrams, in ascertaining the power and the mode of action of the steam in steam-engines, where the curve A P t B P 2 A is described by a pencil attached to a pressure-gauge on a curd whose motion corresponds with that of the piston, is sufficiently well known. 2. It appears that the earliest application of diagrams of energy (as they may be called) to prove and illustrate the theoretical principles of the mechanical action of heat, was made either by Carnot, or by M. Clapeyron in his account of Carnot's theory ; but the conclusions of those authors were in a great measure vitiated by the assumption of the substantiality of heat. In the fifth section of a paper on the Mechanical Action of Heat, published in the Transactions of the Royal Society of Edinburgh (Seep. 300), a diagram of energy is employed to demonstrate the general law of the economy of heat in thermodynamic engines according to the correct principle of the action of such machines — viz., that the area of the diagram represents at once the potential energy or motive power which is de- veloped at each stroke and the mechanical equivalent of the actual energy, or heat, Avhich permanently disappears. As the principles of the expansive action of heat are capable of being presented to the mind more clearly by the aid of diagrams of energy than by means of words and algebraical symbols alone, I purpose, in the present paper, to apply those diagrams partly to the illustration and demonstration of propositions already proved by other means, but chiefly to the solution of new questions, especially those relating to the action of heat in all classes of engines, whether worked by air, or by steam, or by any other material ; so as to present, in a systematic form, those theoretical principles which are applicable to all methods of transforming heat to motive power by means of the changes of volume of an elastic substance. Throughout the whole of this investigation, quantities of heat, and coefficients of specific heat, are expressed, not by units of temperature in a unit of weight of water, but by equivalent quantities of mechanical power, stated in foot-pounds, according to the ratio established by Mr. Joule's experiments on friction (Phil. Trans., 1850); that is to say, ON THERMODYNAMICS. 341 772 foot-pounds per degree of Fahr., or 1389 - 6 foot-pounds per Centigrade degree, applied to one pound of liquid water at atmospheric temperatures. 3. Of Isothermal Curves, and Curves of No Transmission of Heat. A curve described on a diagram of energy, such that its ordinates represent the pressures of a homogeneous substance corresponding to various volumes, while the total sensible or actual heat present in the body- is maintained at a constant value, denoted, for example, by Q, may be called the isothermal curve of Q for the given substance (See Fig. 2). Suppose, for instance, that the co-ordinates of the point A, V A and P A , represent respectively a volume and a pressure of a given substance, at which the actual heat is Q ; and the co-ordinates of the point B — viz., V B and P B , another volume and pressure ' at which the actual heat is the same ; then are the points A and B situated on the same isothermal curve Q Q. On the other hand, let the substance be allowed to expand from the volume and pressure V A , P A , without receiving or emitting heat; and when it reaches a certain volume, V c , let the pressure be represented by P c , which is less than the pressure would have been had the actual heat been maintained constant, because by expansion heat is made to disappear. Then C will be a point on a certain curve N N passing through A, which may be called a Curve of No Transmission. It is to be understood, that "during the process last described, the potential energy developed during the expansion, and which is represented by the area A C V c V A , is entirely communicated to external substances ; for if any part of it were expended in agitating the particles of the expanding substance, a portion of heat would be reproduced by friction. If o o o be a curve whose ordinates represent the pressures corresponding to various volumes when the substance is absolutely destitute of heat, then this curve, which may be called the Curve of Absolute Cold, is at once an isothermal curve and a curve of no transmission. So far as we yet know, the curve of absolute cold is, for all substances, an asymptote to all the other isothermal curves and curves of no trans- Fig. 2. 342 ON THERMODYNAMICS. mission, which approach it and each other indefinitely as the volume of the substance increases without limit. Note. — The following remarks are intended to render more clear the precise meaning of the term Total Actual Heat. The Total Actual Heat of a given mass of a given substance at a given temperature, is the cpiantity of physical energy present in the mass in the form of heat under the given circumstances. If, for the purpose of illustrating this definition, we assume the hypo- thesis that heat consists in molecular revolutions of a particular kind, then the Total Actual Heat of a mass is measured by the mechanical power corresponding to the vis viva of those revolutions, and is repre- sented by - 2 . m v 2 , m being the mass of any circulating molecule, and i> 2 the mean square of its velocity. But the meaning of the term Total Actual Heat may also be illustrated without the aid of any hypothesis. For this purpose, let us take the ascertained fact of the production of heat by the expenditure of mechanical power in friction, according to the numerical proportion determined by Mr. Joule; and let E denote the quantity of mechanical power which must be expended in friction, in order to raise the temperature of unity of weight of a given substance from that of absolute privation of heat to a given temperature r. During this operation, let the several elements of the external surface of the mass undergo changes of relative position expressed by the variations of quantities denoted generally by^, and let the increase of each such quantity as p be resisted by an externally-applied force such as P. Then, during the elevation of temperature from absolute cold to r, the energy converted to the potential form in overcoming the external pressures P will be fvdp. Also, let the internal particles of the mass undergo changes of relative position, expressed by the variations of quantities denoted generally by r, and let the increase of each such quantity as r be resisted by an internal molecular force such as R : Then the energy converted to the potential form in overcoming internal molecular forces will be 2 . I E d r. ON THERMODYNAMICS. 343 Subtracting these quantities of energy converted to the potential form by means of external pressures and internal forces, from the whole power converted into heat by friction in order to raise the temperature of the mass from that of absolute privation of heat to the given temperature t, we find the following result : — Q = E-2.[p<Zp-2.( Hdr and this remainder is the quantity of energy which retains the form of heat in unity of weight of the given substance at the given temperature ; that is to say, the Total Actual Heat. It is obvious that Total Actual Heat cannot be ascertained directly ; first, because the temperature of total privation of heat is unattainable ; and, secondly, because the molecular forces R are unknown. It can, however, be determined indirectly from the latent heat of expansion of the substance. For the heat which disappears during the expansion of unity of weight of an elastic substance at constant actual heat from the volume V A to the volume V B , under the constant or variable pressure P, is expressed (as will be shown in the sequel) by q -/q/,> V; so that from a sufficient number of experiments on the amount of heat transformed to potential energy by the expansion of a given substance, the relations, for that substance, between pressure, volume, and total actual heat, may be determined. 4. Proposition I. — Theorem. The mechanical equivalent of the heat absorbed or given out by a substance in passing from one given state as to pressure and volume to another given state, through a series of states represented by the co-ordinates of a given curve on a diagram of energy, is represented by the area included between the given curve and two curves of no transmission of heat drawn from its extremities, and indefinitely prolonged in the direction representing increase of volume. (Demonstration : see Fig. 3.) Let the co-ordinates of any two points, A and B, represent respectively the volumes and pressures of the substance in any two conditions ; and let a curve of any figure, A C B, represent, by the co-ordinates of its points, an arbitrary succession of volumes and pressures through which the substance is made to pass, in changing from the condition A to the condition B. From the points A and B respectively let two curves of no transmission AM, BN, extend indefinitely towards X; then the area referred to in the enunciation is that contained between 344 ON THERMODYNAMICS. the given arbitrary curve A C B and the two indefinitely prolonged curves of no transmission; areas above the curve AM bciii^- considered as representing heat absorbed by the substance, and those below heat given out. To fix the ideas, let us, in the first place, suppose the area MACBN Re. 3. to be situated above A M. After the substance has reached the state B, let it be expanded according to the curve of no transmission BN, until its volume and pressure are represented by the co-ordinates of the point D'. .Next, let the volume V„ be maintained constant, while heat is abstracted, until the pressure falls so as to be represented by the ordinate of the point D, situated on the curve of no transmission A M. Finally, let the substance be compressed, according to this curve of no transmission, until it recovers its primitive condition A. Then the area AcBD'DA, which represents the whole potential energy developed by the substance during one cycle of operations, represents also the heat which disappears, that is. the difference between the heat absorbed by the substance during the change from A to 1!. and emitted during the change from D' to D; for if this were not so, the cycle of operations would alter the amount of energy in the universe, which is impossible. The farther the ordinate V r ,PD' is removed in the direction of X. the smaller does the heat emitted during the change from D' to D become; and, consequently, the more nearly does the area ACBD'DA approximate to the equivalent of the heat absorbed during the change from A to B; to which, therefore, the area of the indefinitely prolonged diagram M A C B N is exactly equal. Q. E. D. It is easy to see how a similar demonstration could have been applied, ■mutatis mutandis, had the area lain below the curve AM. It is evident also, that when this area lies, part above and part below the line A M. the difference between these two parts represents the difference between ON THERMODYNAMICS. 345 the heat absorbed and the heat emitted during- different parts of the operation. 5. First Corollary. — Theorem. The difference between the whole, heat absorbed, and the whole expansive power developed, during the operation repre- sented by any curve, such as A C B, on a diagram of energy, depends on the initial and final conditions of the substance alone, and not on the intermediate process. (Demonstration.) In Fig. 3, draw the ordinates AV A , BV B parallel to Y. Then the area V A A C B V„ represents the expansive power developed during the operation ACB; and it is evident that the difference between this area and the indefinitely-prolonged area MAC B N, which represents the heat received by the substance, depends simply on the positions of the points A and B, which denote the initial and final conditions of the substance as to volume and pressure, and not on the form of the curve ACB, which represents the intermediate process. Q.E.D. To express this result symbolically, it is to be considered, that the excess of the heat or actual energy received by the .substance above the expansive power or potential energy given out and exerted on external bodies, in passing from the condition A to the condition B, is equal to the whole energy stored "jp in the substance during this operation, which consists of two parts, viz. — Actual energy; being the increase of the actual or sensible heat of the substance in passing from the condition A to the condition B, which is to be represented by this expression, A.Q = Q B -Q A ; Potential energy; 1 icing the power which is stored up in producing changes of molecular arrangement during this process; and which it appears from the theorem just proved, must be represented, like the actual energy, by the difference between a function of the volume and pressure corre- sponding to A, and the analogous function of the volume and pressure corresponding to B ; that is to say, by an expression of the form, Let AS = S B -S A . H A>B = areaMACBX represent the heat received by the substance during the operation A C B, and P d V = area V A A C B V B the power or potential energy given out. 346 ON THERMODYNAMICS. Then, the theorem of this article is expressed as follows : — ,V„ H AlB - I PrfV = Q B -Q A + S B -S A = AQ + A.S, (2.) being a form of the general equation of the expansive action of heat, in which the potential of molecular action, S, remains to be determined. 6. Second Corollary (see Fig. 4). — The Latent Heat of Expansion of a substance from one given volume V A to another V B , for a given amount of actual heat Q; Fig. 4. that is to say, the heat which must be absorbed by the substance in expand- ing from the volume V A to the volume V B , in order that the actual heat Q may be maintained constant, is represented geometrically as follows: Let Q Q be the isothermal curve of the given actual heat Q on the diagram of energy; A, B two points on this curve, whose co-ordinates represent the two given volumes and the corresponding pressures. Through A and B draw the two curves of no transmission A M, B N, produced indefinitely in the direction of X. Then the area contained between the portion of isothermal curve AB, and the indefinitely-produced curves AM, B N, represents the mechanical equivalent of the latent heat sought, Avhose symbolical expression is formed from equation (2) by making Q,. — Q A = 0, and is as follows: — H A , B (for Q = const.) = J ^ P tf V + S B - S A . . (3.) Section II. — Propositions Belative to Homogeneous Substances. 7. Proposition II.— Theorem. In Fig. 5, let A x A 2 M, B x B 2 N he any two curves of no transmission, indefinitely extended in the direction of X, ON THERMODYNAMICS. 347 intersected in the points A v B v A 2 , B 2 , by two isothermal curves Q x A x B x Q x , Q A B Q , which are indefinitely near to each other; that is to say, which correspond to two quantities of actual heat, Q x and Q 2 , differing by an indefinitely small quantity Q x — Q 2 = SQ. Then the elementary quadrilateral area, A 1 B x B 2 A 2 , bears to the whole indefinitely-prolonged area MA 1 B 1 N, the same proportion which the indefi- nitely small difference of actual heat SQ bears to the ivhole actual heat Q : ; or area A x B a B 2 A 2 S Q area M A x B 1 N ~~ Q x ' (Demonstration.) Draw the ordinates A 1 V Al , A 2 V A2 , B^^, B 2 V B2 . Suppose, in the first place, that 8 Q is an aliquot part of Q 1? obtained by dividing the latter quantity by a very large integer n, which we are at liberty to increase without limit. The entire indefinitely-prolonged area M A 1 B 1 N represents a quantity of heat which is converted into potential energy during the expansion of the substance from V Al to V Bl , in consequence of the continued presence of the total actual heat Q x ; for if no heat were present no such conversion would take place. Mutatis mutandis, a similar statement may be made respecting the area MA.,B,K By increasing without limit the number n and diminishing 8 Q, we may make the expansion from V A2 to V B2 as nearly as we please an identical phenomenon with the expansion from V Al to V Bl . The quadrilateral A x B l B 2 A 2 represents the diminution of conversion of heat to potential energy, which results from the abstraction of any one whatsoever of the n small equal parts § Q into which the actual heat Q : is supposed to be divided, and it therefore represents the effect, in conversion of heat to potential energy, of the presence of any one of those small portions of actual heat. And as all those portions 8 Q, are similar and similarly circumstanced, the effect of the presence of the whole actual heat Q x in causing conversion of heat to potential 348 ON THERMODYNAMICS. energy, will be simply the sum of the effects of all its small portions, and will bear the same ratio to the effect of one of those small portions which the whole actual heat bears to the small portion. Thus, by virtue of the general law enunciated below and assumed as an axiom, the theorem is proved when SQ is an aliquot part of Q l ; but SQ is either an aliquot part, or a sum of aliquot parts, or may be indefinitely approximated to by a series of aliquot parts; so that the theorem is universally true. Q. E. D. The symbolical expression of this theorem is as follows : when the actual heat Q l5 at any given volume, is varied by the indefinitely small quantity 8 Q, let the pressure vary by the indefinitely small quantity (IP y-r- B Q ; then the area of the quadrilateral A x ~B 1 B 2 A 2 will be rcpre- (l v^ sented by v V A, 1 and, consequently, that of the whole figure MAjBjN", or the latent heat of expansion from V Al to V Bl , at Q u by V ' "' clV ■V; . . . (4.) ^ A 1 a result identical with that expressed in the sixth section of a paper published in the Transactions of th Royal Society of Edinburgh. (Seep. 310) The demonstration of this theorem is an example of a special application of the following General Law of the Transformation of Energy. The effect of the presence, in a substance, of a quantify of actual energy, in causing transformation of energy, is the sum of the effects of all its parts: — a law first enunciated in a paper read by me to the Philosophical Society of Glasgow on the 5th of January, 1853. (See p. 203.) 8. General Equation of the Expansive Action of Heat. The two expressions for the latent heat of expansion at constant actual heat, given in equations (3) and (4) respectively, being equated, furnish the means of determining the potential energy of molecular action S, so far as it depends on volume, and thus of giving a definite form to the general equation (2). The two expressions referred to may be thus stated in words: — ON THERMODYNAMICS. 349 I. The heat which disappears in producing a given expansion, while the actual heat present in the substance is maintained constant, is equiva- lent to the sum of the potential energy given out in the form of expansive power, and the potential energy stored up by means of molecular attractions. II. It is also equivalent to the potential energy due to the .action during d P the expansion of a pressure Q-ttt, at each instant equal to what the pressure would be, if its actual rate of variation with heat at the instant in question were a constant coefficient, expressing the ratio of the whole pressure to the whole actual heat present. The combination of these principles, expressed symbolically, gives the following result : V V H A , B (for Q = const.) = Q f ^d\ = f ' P dV + S B - 8 A ; * A ' A whence Ave deduce the following general value for the potential of mole- cular action : — S = /( Q ff- P )<* V + *- Q ' * • < 5 -> in which <£.Q denotes some function of the total actual heat not depending on the density of the substance. This value being introduced into equation (2), produces the following : — H A ,J PiV = Q.-Q 4 + S,-S A = Qb - Qa + <j> . Q B - <j> . Q A + / "(q ~- p)tf v = ¥ B - * A . (6.) ' A The symbol ¥" = Q + S is used to denote the sum of the actual energy of heat, and the potential energy of molecular action, present in the substance in any given condition. The above is the General Equation of the Expansive Action of Heat in a Homogeneous Substance, and is the symbolical expression of the Geometrical Theorems I. and II. combined. When the variations of actual heat and of volume become indefinitely small, this equation takes the following differential form : — d.¥ = d.K-FdV = dQ + d.S = (l + '.Q + Q^/ P(J v)iQ + (Qf|-P>T ) otherwise y (<) *.H-£.*Q + Qg.iV. 350 ON THERMODYNAMICS. The coefficient of d Q in the above expressions, viz. — (8.) is the ratio of the apparent specific heat of the substance at constant volume to its real specific heat ; that is, the ratio of the whole heat consumed in producing an indefinitely small increase of actual heat, to the increase of actual heat produced. These general equations are here deduced independently of any special molecular hypothesis, as they also have been, by a method somewhat different, in the sixth section of a paper previously referred to. Equations equivalent to the above have also been deduced from the hypothesis of molecular vortices, in the paper already mentioned, and in a paper on the Centrifugal Theory of Elasticity. (See p. 4$.) 9. First Corollary from Proposition II. — Theorem. If a succession of isothermal curves corresponding to quantities of heal diminishing by equal small differences $ Q, be drawn across any pair of curves of no transmission, they ivill cut off a series of equal small quadrilaterals. Second Corollary. — Theorem. In Fig. G, let ADM,BCN be any two curves of no transmission, indefinitely prolonged in the direction of X, and let Fig. 6. any two isothermal curves Q t Q 15 Q 2 Q 2 , corresponding respectively to any two quantities of actual heat Q x , Q 2 , be drawn across them. Then will the indefinitely-prolonged areas MABN, M DCN, bear to each other the simple ratio of the quantities of actual heat, Q v Q 2 . Or, denoting those areas respectively by ~H_ V H 2 — H Hi ~ Q/ | .... (9 , This corollary is the geometrical expression of the law of the maximum ON THERMODYNAMICS. 351 efficiency of a perfect thermodynamic engine, already investigated by other methods. In fact, the area MABN represents the whole heat expended, or the latent heat of expansion, the actual heat at which heat is received being Q t ; M D C N, the heat lost, or the latent heat of com- pression, which is carried off by conduction at the actual heat Q 2 ; and ABCD (being the indicator-diagram of such an engine), the motive power produced by the permanent disappearance of an equivalent quantity of heat ; and the efficiency of the engine is expressed by the ratio of the heat converted into motive power to the whole heat expended, viz. : — ABCD Ht-Hg, Qt-Qg (w) MABN H x Q x " ' K '' 1 0. Third Corollary (of Thermodynamic Functions). If the two curves of no transmission in Fig. 6, ADM, BCN, be indefinitely close together, the ratio of the heat consumed in passing from one of those curves to the other to the actual heat present, will be the same, whatever may be the form and position of the curve indicating the mode of variation of pressure and volume, provided it intersects the two curves of no transmission at a finite angle; because the area contained between this connecting curve and the two indefinitely-prolonged curves of no transmission will differ from an area whose upper boundary is an isothermal curve, by an indefinitely small area of the second order. To express this symbolically, let f = 8F, be the ratio in question, for a given indefinitely-close pair of curves of no transmission. Let the change from one of these curves to the other be made by means of any indefinitely-small changes of actual heat and of volume, SQ, SV. Then by the general equation (7), the following quantity — + m- SY = m-^ + '^ v ' ' ' (11) is constant for a given pair of indefinitely-close curves of no transmission, and is, therefore, the complete variation of a function, having a peculiar constant value for each curve of no transmission, represented by the following equation : — 352 ON THERMODYNAMICS. This function, which I shall call a thermodynamic function, has the following properties : — H=(QdF, .... (13.) is equivalent to the general equation (6) ; <1F = 0, . . . . (14.) is the equation common to all curves of no transmission; and F = a given constant, . . . (14A.) is the equation of a particular curve of no transmission. 11. Proposition 111. — Problem. Let it be supposed find for a given substance, the forms of all pc \ermal cm . hut of only one curve of no tra ; it is required /■ determination of points, another curve of no transmi sing through a given point, situated anywhere out of the known curve. (Solution : see Fig. 7.) Let L M be the known curve of no trans- mission ; B the given point. Through B draw an isothermal curve Q 1 ABQ 1 , cutting LM in A. o, being the quantity of heat to which this curve corresponds, draw, indefinitely near to it, the isothermal curve q l q v corresponding to the quantity of heat Q, — 2 Q, where c Q is an indefinitely small quantity. Draw any other pair of indefinitely close isothermal Kg. 7. curves Q„ Q 2 , q 2 q 2 , corresponding to the quantities of heat Q 2 Q 2 — $ Q; 2 Q being the same as before. Let D be the point where the isothermal curve Q 2 Q 2 cuts the known curve of no transmission. Draw the ordinates A V A , B V B parallel to Y, enclosing, with the isothermal curves of Q x and Qj — S Q, the small quadrilateral A B b a. Draw the ordinate D V D ON THERMODYNAMICS. 353 parallel to Y, intersecting the isothermal curve of Q 2 — § Q in d. Lastly, draw the ordinate C V c in such a position as to cut off from the space between the isothermal curves of Q 2 and Q 2 — 8 Q a quadrilateral D C c d, of area equal to the quadrilateral ABba. Then will C, where the last ordinate intersects the isothermal curve of Q 2 , approximate indefinitely to the position of a point in the curve of no transmission passing through the given point B, when the variation of actual heat S Q is diminished without limit. And thus may be determined to as close an approximation as we please, any number of points in the curve of no transmission NBE which passes through any given point B, when any one curve of no transmission L M is known. (Demonstration.) For when the variation § Q diminishes indefinitely, the curves q x q v q. 2 q 2 , approach indefinitely towards the curves Q x Q 15 Q 2 Q 2 respectively; and the small quadrilaterals bounded endways by the ordinates approximate indefinitely to the small quadrilaterals bounded endways by the curves of no transmission ; which latter pair of quadri- laterals are equal, by the first corollary of Proposition II. The symbolical expression of this proposition is as follows : — Let V A , V B , V c , V D , be the volumes corresponding to the four points of intersection of a pair of isothermal curves with a pair of curves of no transmission; A and B being on the isothermal curve of Q p C and D on that of Q 2 , A and D on one of the curves of no transmission, B and C on the other ; then C § d v (for Q = Ql) = -C* 8 '' v (for Q = Qi> ' 1 ' a v i> y (is.) or F — F = F — F J -B- L A- 1 -C- I -D- The second form of this equation is in the present case identical, because F = F • F = F 12. Proposition IV. — Problem (see Fig. 8). The forms of all isothermal curves for a given substance being given, let E F be a curve of any form, representing an arbitrarily assumed succession of pressures and volumes. It is required to find, by the determination of points, a corresponding curve passing through a given point B, such that the quantity of heat absorbed or emitted by the substance, in passing from any given isothermal curve to any other, shall be the same, ichether the pressures and volumes be regulated according to the original curve E F, or according to the curve passing through the point B. (Solution.) The process by which the latter curve is to be deduced z 354 ON THERMODYNAMICS. from the former is precisely the same with that by Avhich one curve of no transmission is deduced from another in the last problem. «*\ <Di^ \"~~"~~~><1L S ^~- H a p — -^£__ ^^\ ^c 0. H ^^^- **" ~^5j-m ^Kl L Fig. 8. (Demonstration.) Let GBH be the required curve. This curve, and the curve EF, in their relation to each other, may be called Curves of Equal Transmission. Through B draw the isothermal curve Q x Q 19 inter- secting the curve E F in A. Draw also any other isothermal curve Q 2 Q 2 , intersecting E F in D and G H in C. Through A, B, C, D, respectively, draw the four indefinitely-prolonged curves of no transmission, AK, intersecting Q 2 Q 2 in d, B L, intersecting Q 2 Q 2 in c, C M, and D N. Conceive the whole space between the isothermal curves Q, Q t , Q 2 Q 2 , to be divided by other isothermal curves, into a series of indefinitely-narrow stripes, corresponding to equal indefinitely-small variations of actual heat. Then, by the construction of the solution, the quadrilaterals cut from those stripes by the pair of curves E F, G H are all equal ; and so also are the quadrilaterals cut from the stripes by the pair of curves of no transmission, A K, B L. Therefore the area A B C D is equal to the area A B c d. The indefinitely-prolonged areas, M C D N, L c d K, are evidently equal ; therefore, adding this pair of equal areas to the preceding, the pair of indefinitely-prolonged areas LEAK. MCBADN are equal. Subtracting from each of these areas the part common to both, ABR, and adding to each the indefinitely -prolonged area K K C M, we find, finally, that the indefinitely-prolonged areas KADN, LBCM are equal. But the former of those areas (by Prop. I.) represents the mechanical equivalent of the heat absorbed by the substance in passing from the actual heat Q 2 to the actual heat Q x through a series of pressures and volumes represented by the co-ordinates of the curve E F ; and the latter, the corresponding quantity for the curve G H ; therefore those curves are, with respect to each other, curves of equal transmission, which was required. The algebraical expression of this result is that the equation (15) holds ON THERMODYNAMICS. >oa for any pair of curves of equal transmission, as well as for a pair of curves of no transmission ; or, in other terms, let F A , F B , F c , F D be the thermo- dynamic functions for the curves of no transmission passing through the four points where a pair of isothermal curves cut a pair of curves of equal transmission : A, B being on the upper isothermal curve ; C, D on the lower; A, D on one curve of equal transmission, B, C on the other: then, F. = K - F r (16.) 13. Proposition V. — Theorem. The difference between the quantities of heat absorbed by a substance in passing from one given amount of actual heat to another, at two different constant volumes, is equal to the difference between the two latent heats of expansion in passing from one of those volumes to the other at the two different amounts of actual heat respectively, diminished by the corresponding difference between the quantities of expansive power given out. (Demonstration: see Fig. 9.) Let Q t Q a be the isothermal curve of the higher amount of actual heat; Q 2 Q 2 that of the lower. Let V A , V B be Fig. 9. the two given volumes. Draw the two ordinates V A a A, V B b B, and the four indefinitely-prolonged curves of no transmission A M, a m, B 1ST, b n. the quantities of heat absorbed, in passing from the actual heat Q 2 to the actual heat Q v at the volumes V A and V B , are represented respectively by the indefinitely-prolonged areas MAam, NBbn. Then adding to each of those areas the indefinitely-prolonged area n iBAM (observing that the space below the intersection R is to be treated as negative), we find for their difference NBJn- MAam = NBAM- ubBAam =(NBAM -nbam) -(V B BAV A -V B 5«V A ); but NBAM and n b a m represent the latent heats of expansion from V A to V B , at the actual heats Q x and Q 2 respectively ; and V B B A V A and 856 ON THERMODYNAMICS. V B b a V A represent the power given out by expansion from V A to V,, at the actual heats Q x and Q 2 respectively : therefore, the proposition is proved. Q. E. D. This proposition, expressed symbolically, is as follows : A Q being the difference of actual heat, Q x — Q 2 , let A (Q + S A ) be the heat absorbed in passing from Q 2 to Q t at the volume V A , and A (Q + S„) the corre- sponding quantity ;tt the volume V B ; A S A and A S B representing quantities of potential energy stored up in altering molecular arrangement. Then »H-«-i(^-l)/'MT. . (17.) A 14. Of Curves of Free Expansion. In all the preceding propositions, the whole motive power developed by an elastic substance in expanding is supposed to be communicated to external bodies; to a piston, for example, which the substance causes to move and to overcome the resistance of a machine. Let us now suppose that as much as possible of the motive power developed by the expansion is expended in agitating the particles of the expanding substance itself, by whose mutual friction it is finally recon- verted into heat (as when compressed air escapes freely from a small orifice); and let us examine the properties of the curves which, on a diagram of energy, represent the law of -expansion of the substance under these circumstances, and which may be called Curves of free Expansion. 15. Proposition VI. — Theorem. If from two points on a curve of free expansion there be drawn tiro straight lines perpendicular to and terminating at the axis of ordinates, and also tv:o curves of no transmission, indefinitely prolonged away from the origin of co-ordinates ; then the area contained between the curve of free expansion, the two straight lines, and the axis of ordinates, will be erpial to the area contained between the curve of free expansion and the two indefinitely-prolonged curves of no trans- mission. (Demonstration.) Let FF (Fig. 10) be a curve of free expansion ; G H any two points in it ; G V G , H V H ordinates ; G P G , H P n lines per- pendicular to O Y ; G M, HN ■* curves of no transmission, indefinitely Fig. 10. prolonged in the direction of X. Then the indefinitely-prolonged area MGHN represents the heat which would have to be communicated to the substance, if the motive power ON THERMODYNAMICS. 357 developed were entirely transferred to external bodies, while the area Y,. G H V H represents that motive power. The excess of the rectangular area P H H V H above the area P G G V G 0, is the power necessarily given out by the elastic fluid in passing from a vessel in which the pressure is P G and volume V G , to a vessel in which the pressure is P H and volume V H . The remainder of the expansive power, represented by the area P G H P H , by the mutual friction of the particles of the expanding substance, is entirely reconverted into heat, and is exactly sufficient (by the definition of the curve of free expansion) to render the communication of heat to the substance unnecessary; from which it follows, that this area is equal totheareaMGHX. Q. E. D. The equation of a curve of free expansion is d (¥ + P V) = 0. (17 a.) 16. Corollary. — In Fig. 11, the same letters being retained as in the last figure, through G draw an isothermal curve Q L Q x , which the line P H H produced cuts in h; and from h draw the indefinitely-prolonged curve of no transmission, hn. Then because, by the proposition just proved, the areas P G G H P H and M G H N are equal, it follows that the indefinitely-prolonged area, MGhn, which represents the latent heat of expansion at the constant actual heat Q 1? from the volume V G to the volume Y h , exceeds P G G h P H , by the indefinitely-prolonged area NH/j n, which represents the heat which the substance would give out, in falling, at the pressure P H , from the actual heat Q t to the actual heat corresponding to the point H on the curve of free expansion passing through G. Subtracting from this area the excess of the rectangle P H Y h above the rectangle P G V G , we obtain the excess of the area M G h n above the area V G G h Y h . This conclusion may be thus expressed : — Let Q 2 be the actual heat for 358 ON THERMODYNAMICS. K, the point H ; -~ the ratio of specific heat at the constant pressure P H to real specific heat ; then / Qa ■rfQ-P H V A + PaV fl (^- 1 )L ApdV ( f0rQ = Ql); r- • (18.) otherwise : — Qi r P o Equation (18) may he used, either to find points in the curve of free expansion which passes through G, when the isothermal curves and the curves of no transmission are known; or to deduce theoretical results from experiments on the form of curves of free expansion, such as those which have been for some time carried on by Mr. Joule and Professor "William Thomson. Considered geometrically, these experiments give values of the area XH/ni, The area ,P <; P fl G/»P H = VdT ' I\, is known in each case from previous experiments on the properties of the gas employed ; and this area, by Proposition VI., is equal to the area MG/iHN; to which, adding the area N H h n, ascertained by experiment, we obtain the area MG/j n, that is, the latent heat of expansion from the volume V to the volume Y h , at the constant actual heat Q 1} denoted symbolically by * G Now the problem to be solved is of this kind. "We know the differences of actual heat corresponding to a certain series of isothermal curves for the substance employed ; and we have to ascertain the absolute quantities of actual heat corresponding to those curves. Of the above expression for the area M G h n, therefore, the factor Q t is to be determined, while the other factor, being the difference between two thermodynamic ON THERMODYNAMICS. 359 functions, is known ; and the experiments of Messrs. Thomson and Joule, by giving the value of the product, enable us to calculate that of the unknown factor, and thence to determine the point on the thermometric scale corresponding to absolute privation of heat. 1 7. Proposition VII. — Problem. To determine the ratio of the apparent specific heats of a substance at constant volume and at constant pressure, for a given pressure and volume; the isothermal curves and the curves of no transmission being known. (Solution.) In Fig. 12, let A be the point whose co-ordinates represent the given volume V A and pressure P A ; Q A Q the isothermal curve passing Fis. 12. through A; qq another isothermal curve, very near to Q Q. Through A draw the ordinate V A A a parallel to Y, cutting qq in a; draw also A B parallel to X, cutting q q in B. From A, a, B, draw the three indefi- nitely-prolonged curves of no transmission A M, a in, B N. Then the heat absorbed in passing from the actual heat Q to the actual heat q, at the constant volume V A , is represented by the indefinitely- prolonged area MAaro, while at the constant pressure P A it is represented by the area MABN. Let the curve qq be supposed to approximate indefinitely to QQ. Then will the three-sided area A«B diminish indefinitely as compared with the areas between the curves of no trans- mission A M, a m, B N ; and, consequently, the area MABN will approxi- mate indefinitely to the sum of the areas MA am and m«BN; the ultimate ratio of which sum to the area M A a m is therefore the required ratio of the specific heats. Now in a B N, as qq approaches Q Q, approxi- mates indefinitely to the latent heat of the small expansion V B — V A at the actual heat Q, and this small expansion bears ultimately to the increment of pressure P a — P A , the ratio of the subtangent of the isothermal curve Q Q to its ordinate at the point A. The symbolical expression of this proposition is as follows : — Let o Q denote the indefinitely-small difference of actual heat between the 360 ON THERMODYNAMICS. isothermal curves Q Q, qq; B V the indefinitely-small variation of volume V B — V A ; B P the indefinitely small variation of pressure P a — P A ; the quantities of heat required to produce the variation BQ, at the constant volume V A , and at the constant pressure P A respectively. Then d? BY = SP d? dV dQ d P " dY ■ BQ; and ^.8Q=^.8Q + Qi£.SV consequently. dQ ,/p r~ (19.) _ ' ; ? J equations agreeing with equation (31) of a paper on the Centrifugal Theory of Elasticity before referred to. 18. First Corollary. — As the curves AM, am, BN approximate indefi- nitely towards parallelism, and the point a towards C, where am intersects A B, the ratio of the areas MABN: MAaih, approximates indefinitely to that of the lines AB:AC, which are ultimately proportional, respectively, to the subtangents of the isothermal curve and the curve of no transmission passing through A. Therefore, Kv Subtangent of Isothermal Curve Subtangent of Curve of No Transmission" (20.) 19. Second Corollary.— Velocity of sound. The subtangents of different curves at a given point on a diagram of energy being inversely proportional to the increase of pressure produced by a given diminution of volume according to the respective curves, are inversely proportional to the squares of the respective velocities with which waves of condensation and rare- faction will travel when the relations of pressure to volume are expressed by the different curves. Therefore, if there be no sensible transmission of heat between the particles of a fluid during the passage of sound, the- ON THERMODYNAMICS. 3G1 square of the velocity of sound must be greater than it would have been had the transmission of heat been instantaneous in the ratio of the subtangent of an isothermal curve to that of a curve of no transmission at the same point, or of the specific heat at constant pressure to the specific heat at constant volume. This is a geometrical proof of Laplace's law for all possible fluids. The same law is deduced from the hypothesis of molecular vortices in the paper before referred to on the Centrifugal Theory of Elasticity. 20. Proposition VIII. — Problem. The isothermal curves for a given substance being knoum, and tin- quantities of heat required to produce all variations of actual heat at a given constant volume; it is required to find any number of points in a curve of no transmission passing through a given point in the ordinate corresponding to that volume. (Solution.) In Fig. 13, let V A A X be the given ordinate; Q 1 Q 1 , A 2 Q, isothermal curves meeting it in A lf A 2 , respectively; and let it be required. Fig. 13. for example, to find the point where the curve of no transmission passing through A 1 intersects the isothermal curve A 2 Q 2 . On the line V A A 2 A 1? as an axis of abscissa?, describe a curve C C, whose ordinates (such as A 2 C 2 , a A c A , &c.) are proportional to the specific heat of the substance at the constant volume V A , and at the degrees of actual heat corresponding to the points where they are erected, divided by the corresponding rate of increase of pressure with actual heat ; so that the area of this curve between any two ordinates (e.g., the area a 4 c 4 c 3 a 3 ) may represent the mechanical equivalent of the heat absorbed in augmenting the actual 362 ON THERMODYNAMICS. heat from the amount corresponding to the lower ordinate to that corre- sponding to the higher (e.g., from the amount corresponding to « 4 to that corresponding to a' 3 ). Very near to the isothermal curve A 2 Q 2 , draw another isothermal curve a 2 #2, and let the difference of actual heat corresponding to the interval between these curves be 3 Q. Draw a curve D D, such that the part cut off by it from each ordinate of the curve C C shall bear the same proportion to the whole ordinate which the difference 8 Q bears to the whole actual heat corresponding to the ordinate ; for example, let A^q : A7D x : : Qi = 8 Q ATO; : A^TI, : : Q 2 : 8 Q, &C. Then draw an ordinate V B B b, parallel to Y, cutting off from the space between the isothermal curves A, Q 2 , a 2 q 2 , a quadrilateral area A 2 B b a 2 equal to A { D 4 D 2 A 2 , the area of the curve D D between the ordinates at A, and A.. Then, if the difference 8 Q be indefinitely diminished, the point B will approximate indefinitely to the intersection required of the isothermal curve A 2 Q 2 with the curve of no transmission passing through A 4 ; and thus may any number of points in this curve of no transmission be found. (Demonstration.) Let A! M] be the curve of no transmission required. Let « 3 c 3 , « 4 c 4 be any two indefinitely-close ordinates of the curve C C, corresponding to the mean actual heat Q 3 4 . Let a 3 m 3 , a 4 w? 4 be curves of no transmission, cutting the curves a., q 2 , A, Q 2 , so as to enclose a small quadrilateral area e. Then, by the construction, and Proposition I., The area a 3 c 3 c 4 « 4 = the indefinitely-prolonged area m 3 a 3 « 4 m A ; and by the first corollary of the second proposition and the construction, the area e 8 Q area a 3 d 3 d 4 a 4 m 3 a 3 a 4 m i Q 3 4 area a 3 c s c 4 a^ Therefore, the area e = the area a 3 d 3 </ 4 a 4 ; but the area A x D x D 2 A 2 is entirely made up of such areas as a 3 d 3 d i a 4) to each of which there corre- sponds an equal area such as e; and when the difference 3 Q is indefinitely diminished, the area A 2 B b a 2 approximates indefinitely to the sum of all the areas such as e, that is, to equality with the area AjDjD^o. Q.E.D. The symbolical expression for this proposition is found as follows : — rQi K The area A x D x D 2 A<, ultimately = 8 Q . / ^ . d Q (for V = Y A ) ; J q KQ, ON THERMODYNAMICS. 363 V The area A 2 B b a. 2 ultimately = % Q . I TfidV (f ov Q = Q2) ; divide both sums by S Q and equate the results ; then, r*B f ;p A? 1 I" % " v (for Q - Qi> " L h " Q (for v - v - ) ' (21) which denotes the equality of two expressions for the difference, F x — F a between the thermodynamic functions for the curve of no transmission A x M, and for that passing through the point A 2 . When the relations between pressure, volume, and heat, for a given substance, are known, the equation (21) may be transformed into one giving the volume V B corresponding to the point at which the required curve of no transmission cuts the isothermal curve of Q.. Suppose, for instance, that for a perfect gas P V = N Q sensibly ; and \ Y = 1 sensibly; . (22.) N being a constant (whose value for simple gases and for atmospheric air and carbonic oxide is about - 41); then the thermodynamic function for a perfect gas is sensibly F = hyp. log. Q + N hyp. log. V; . . (22a.) and equation (21) gives, for the equation of a curve of no transmission, X, = a>". . . . (23.) whence P„ P ! = (r)"" 1_Sk - • • (24) Equations (23) and (24) are forms of the equation of a curve of no transmission for a perfect gas, according to the supposition of Mayer ; and are approximately true for a perfect or nearly perfect gas on any supposition. According to the hypothesis of molecular vortices, the relations between pressure, volume, and actual heat, for a perfect gas, are expressed by these equations : — PV = NQ + / i; I |=l + ^^ ; . (25.) T)G4 ON THERMODYNAMICS. where h is a very small constant, which is inversely proportional to the specific gravity of the gas, and whose value, in the notation of papers on the hypothesis in question, is h = Nlr, . . . . (25A.) k being the height, on the scale of a perfect gas-thermometer, of the point of absolute cold above the absolute zero of gaseous tension. Hence we find, for the thermodynamic function of a perfect gas, F = hyp. log. Q - y^r h + N hyp. log. V, . (26.) and for the equation of a curve of no transmission, 1 \ h h ) - B = (9AN . e t » Q* .+ h N Q x + A I _ /07) V . \\j.tJ For all practical purposes yet known, these equations may be treated as sensibly agreeing with equation (23), owing to the smallncss of h as compared with N Q, Section III. — Of the Efficiency of Thermodynamic Engines, Worked by the Expansion and Condensation of Permanent Gases. 21. The efficiency of a thermodynamic engine is the proportion of the whole heat expended which is converted into motive power ; that is to say, the ratio of the motive power developed to the mechanical equivalent of the whole heat consumed. To determine geometrically the efficiency of a thermodynamic engine, it is necessary to know its true indicator-diagram ; that is to say, the curve whose co-ordinates represent the successive volumes and pressures which the elastic substance working the engine assumes during a complete revolution. This true indicator-diagram is not necessarily identical in figure with the diagram described by the engine on the indicator-card ; for the abscissae representing volumes in the latter diagram include not only the volumes assumed by that portion of the elastic substance which really performs the work by alternately receiving heat while expanding, and emitting heat while contracting, in such a manner as permanently to transform heat into motive power, but also the volumes assumed by that portion of the elastic substance, if any, which acts merely as a cushion for transmitting pressure to the piston, undergoing, during each revolution, a ON THERMODYNAMICS. 3G5 scries of changes of pressure and volume, and then the same series in an order exactly the reverse of the former order, so as to transform no heat permanently to power. The thermodynamic engines to be considered in the present section, are those in which the elastic substance undergoes no change of condition. We shall, in the first place, investigate the efficiency of those which work without the aid of the contrivance called an ; ' economiser " or " regenerator," and afterwards, those which work with the aid of that piece of apparatus. 22. Lemma. — Problem. To determine the true from the apparent indicator-diagram of a thermodynamic engine ; the portion of the elastic substance which acts as a cushion being known, and the law of its changes of pressure and volume. -X Fig. 14 (Solution.) In Fig. 14, let abed be the apparent indicator-diagram. Parallel to X draw H a and L c, touching this diagram in a and c respectively ; then those lines will be the lines of maximum and minimum pressure. Let H E and L G be the volumes occupied by the cushion at the maximum and minimum pressures respectively : draw the curve E G, such that its co-ordinates shall represent the changes of volume and pressure undergone by the cushion during a revolution of the engine. Let K F d b be any line of equal pressure, intersecting this curve and the apparent indicator-diagram ; so that K b, K d shall represent the two volumes assumed by the whole elastic body at the pressure K, and K F the volume of the cushion at the same pressure. On this line take JB = Iff) = ~KF; then it is evident that B and D will be two points in the true indicator- diagram ; and in the same manner may any number of points be found. The area of the true diagram ABCD is obviously equal to that of the apparent diagram abed. 3G6 ON THERMODYNAMICS. 23. Proposition IX. — Problem. The true Indicator-diagram of a thermodynamic engine worked by the expansion and contraction of a substance which does not change its condition, and without a regenerator, being given, it is required to determine the efficiency of the engine. (Solution.) In Fig. 15, let A a a Bb' b A be the given true indicator- diagram. Draw two curves of no transmission, A M, B N, touching this figure at A and B respec- tively, and indefinitely pro- duced towards X. Then during the process denoted by the portion Aaa'B of the diagram the elastic sub- stance is receiving heat, and the mechanical equivalent of the total quantity received r is represented by the in- Firr. 15. definitely - prolonged area MAaa'BN; during the process denoted by the portion Bb'bA of the diagram, the substance is giving out heat, and the mechanical equivalent of the total heat given out is represented by the indefinitely-prolonged area M A b b' B N ; while the difference between those areas, that is, the area of the indicator-diagram itself, represents at once the heat which permanently disappears and the motive power given out. The EFFICIENCY of the engine is the ratio of this last quantity to the total heat received by the elastic substance during a revolution ; that is to say, it is denoted by the fraction, area A a a' B b' b A area M A a «' B N' To express this result symbolically, find the limiting points A and B by combining the equation of the indicator-diagram Avith the general equation of curves of no transmission, viz. : — d F = 0. Then draw two indefinitely-close and indefinitely-prolonged curves of no transmission, a b m, a V m, through any part of the diagram, cutting out of it a quadrilateral stripe, a b V a. Let Q x be the mean actual heat corre- sponding to the upper end ad of this quadrilateral stripe; Q 2 , that corresponding to the lower end, b V. The area of this indefinitely-narrow stripe, representing a portion of the heat converted into motive power, is found, according to the principles and notation of the third corollary to Proposition II. and of Proposition ON THERMODYNAMICS. 3G7 III., by multiplying the difference between the actual heats by the difference between the thermodynamic functions for the curves of no transmission that bound the stripe, thus : — SE = (Q 1 -Q 2 )SF: while the area of the indefinitely-prolonged stripe, mactfm', representing part of the total heat expended, is, according to the same principles, and that of the indefinitely-prolonged stripe m b V m', representing part of the heat given out, is o H 2 = Q 2 o F. Integrating these expressions we find the following results : — Whole heat expended, F B H 1 = f B Q^F; Heat given out. Motive power given out, -F, H„ = ( B Q,dF; F E = H 1 -H 2 = f "(Qi-Q^ZF; (28.) Efficiency, J V QtdF J formulae agreeing with equation (28) of a paper on the Centrifugal Theory of Elasticity (Seep. 63); it being observed that the symbol F in the last- mentioned paper denotes, not precisely the same quantity which is denoted by it in this paper, and called a thermodynamic function, but the pro- duct of the part of that function which depends on the volume by the real specific heat of the substance. 24. First Corollary. Maximum efficiency between given limits of actual heat. When the lushest and lowest limits of actual heat at which the engine 3GS ON THERMODYNAMICS. can work are fixed, it is evident that the greatest possible efficiency of an engine without a regenerator will be attained when the whole reception of heat takes place at the highest limit, and the whole emission at the lowest; so that the true indicator-diagram is such a quadrilateral as is shown in Fig. 6, and referred to in the second corollary of Proposition II.; bounded above and below by the isothermal curves denoting the limits of actual heat, and. laterally, by any pair of curves of no transmission. The •efficiency in this case, as has been already proved in various ways, is represented by E (29.) being the maximum efficiency possible between the limits of actual heat, Q x and Q,. 25. Second Corollary. — Problem. To drew the diagram of greatest efficiency of a thermodynamic engine without a regenerator, when the extent of variation of volume is limited, as well as that of the ra notion of actual heat. (Solution.) In Fig. 16, let Qi ( i i • ( I ■• Q-2 ^ e the- isothermal curves denoting the limits of actual heat; Y A , Y„ the limits of volume. Draw the ordinates \\ I > A. V B C B, intersecting the isothermal curves in the points A.IU',D. Through A and C respectively draw the curves of no transmission, A M cutting Q2Q2 in d, and ( ' \ cutting C^Q, in b. Then will AbCd be the diagram required An anal- ogous construction would give the diagram of greatest efficiency when the variations of pressure and of actual heat are limited ; as in the air-engine proposed by Mr. Joule. 26. Of the use of the Economizer or Regenerator in Thermodynamic Engines. As the actual heat of the elastic substance which works a thermo- dynamic engine requires to be alternately raised and lowered, it is obvious that unless these operations are performed entirely by compression and expansion, without reception or emission of heat (as in the case of maximum efficiency described in the first corollary of Proposition IX.), part, at least, of the heat emitted during the lowering of the actual heat may be stored up, by being communicated to some solid conducting substance, and used again by being communicated back to the elastic substance, when its actual heat is being raised. The apparatus used for ON THERMODYNAMICS. 3G9 this purpose is called an economiser or regenerator, and was first invented about 181 G, by the Rev. Robert Stirling. In the air-engine proposed by him, it consisted of a sheet-metal plunger surrounded by a wire grating or network; in that of Mr. James Stirling, it is composed of thin parallel plates of metal or glass through which the air passes longitudinally, and in the engine of Captain Ericsson of several sheets of wire gauze. A regenerator may be regarded as consisting of an indefinite number of strata with which the elastic substance is successively brought into contact; each stratum serving to store up and give out the heat required to produce one particular indefinitely-small variation of the actual heat of the working substance. A perfect regenerator is an ideal apparatus of this kind, in which the mass of material is so large, the surface exposed so extensive, and the conducting powers so great as to enable it to receive and emit heat instantaneously without there being any sensible difference of temperature between any part of the regenerator and the contiguous portion of the working substance; and from which no appreciable amount of heat is lost by conduction or radiation. In theoretical investigations it is convenient, in the first place, to determine the saving of heat effected by a perfect regenerator, and afterwards to make allowance for the losses arising from the non-fulfiment of the conditions of ideally perfect action; losses which, in the present imperfect state of our knowledge of the laws of the conduction of heat, can be ascertained by direct experiment only.* 27. Proposition X. — Problem. The true indicator-diagram of any thermodynamic engine being given, to determine the amount of heat saved by a perfect regenerator. (Solution.) Let ABCD (in Fig. 1 7) be the given indicator-diagram. Across it draw any two indefinitely-close isothermal curves, q 1 q x inter- secting it in a, b, and q 2 q 2 intersecting it in d, c To the stripe between those two curves, speaking generally, a certain layer or stratum of the regenerator corresponds, which receives heat from the working substance during the change from b to c, and restores the same amount of heat during the change from d to a. The amount of heat economised by the layer in question is thus found. Through the four points a, b, c, d, draw the indefinitely-prolonged curves of no transmission, ah, bl, cm, dn; then the smaller of the two indefinitely-prolonged areas, lb cm, Icadn, represents the heat saved by the layer of the regenerator corresponding to the indefinitely-narrow stripe between the isothermal curves q x q 1 and q 2 q 2 . * It is true that the problem of the waste of heat in the action of the regenerator is capable of a hypothetical solution by the methods of Fourier and Poisson; and I have by these methods obtained formulae which are curious in a mathematical point of view; but owing to our ignorance of the absolute values and laws of variation of the coefficients of conductivity contained in these formulae, they are incapable of being usefully applied ; and I therefore for the present refrain from stating them. 2 A 370 ON THERMODYNAMICS. Draw two curves of no transmission, BL, DN, touching the diagram; and through the points of contact, B and I), draw the isothermal curves, -X Fig. 17. Qj Q, cutting the diagram in A and B, and Q 2 Q 2 cutting it in C and D. Then because, during the whole of the change from D through A to B, the Avorking substance is receiving heat, and during the whole of the change from B through C to D, emitting heat, the regenerator can have no action above the isothermal curve QjQp nor below the isothermal curve Q 2 Q . The whole of the diagram between these curves is to be divided by indefinitely-close isothermal curves into stripes like abed; and the saving of heat effected by- the layer of the regenerator corresponding to each stripe ascertained in the manner described, when the whole saving may be found by summation or integration. The symbolical expression of this result is as follows : Let the points of contact, B, D, which limit the action of the regenerator, and the corresponding quantities of actual heat, Q 15 Q.„ be found, as in Proposition IX., by means of the equation dF = 0. Then, the saving of heat ■Qi Q, = L q w: dq = L It + %iqJ^ (3(X) Qs Q 2 .7 T? care being taken, when 'y- nas different values for the same value of Q, ON THERMODYNAMICS. 371 corresponding respectively to the two sides of the diagram, to choose the smaller in performing the integration. 28. Corollary. — It is evident that the regenerator acts most effectually when the outlines of the indicator-diagram from A to D, and from B to C, are portions of a pair of curves of equal transmission (determined as in Proposition IV.) ; for then, if the operation of the regenerator is perfect, the changes from B to C and from D to A will be effected without expenditure of heat; the. heat transmitted from the working substance to a given stratum of the regenerator, during any part, such as be, of the operation B C, being exactly sufficient for the corresponding part, da, of d . F the operation D A. In this case for each value of Q between Q, and d^ Q 2 , has the same value at either side of the diagram. In fact, the effect of a perfect regenerator is, to confer upon any pair of curves of equal transmission the properties of a pair of curves of no transmission. 29. Proposition XL — Theorem. The greatest efficiency of a thermo- dynamic engine, working between given limits of actual heat, with a perfect regenerator; is equal to the greatest efficiency of a thermodynamic engine, working between the same limits of actual heat without a regenerator. (Demonstration.) In Fig. 1 8, let Q x Q v Q 2 Q. 2 be the isothermal curves denoting the given limits of actual heat. Let AD, B C be a pair of curves of equal transmission of any form. Then by the aid of a perfect regenerator, the whole of the heat given out by the elastic substance during the operation B C may be stored up, and given out again to that substance in such a manner as to be exactly sufficient for the operation DA; so that the whole consumption of heat in one revolution by an engine whose Fig. 18. indicator-diagram is A B C D, may be reduced simply to the latent heat of expansion during the operation AB, which is represented by the 372 ON THERMODYNAMICS. indefinitely-prolonged area M A B N, A d M and B c N being curves of no transmission. The efficiency of such an engine is represented by the area A B C D the area MABY Now the maximum efficiency of an engine without a regenerator, working between the same limits of actual heat, is represented by the area AB cd Q, — Q 2 the area MABN Q, and from the mode of construction of curves of equal transmission, described in Proposition IV., it is evident that the area A B C D = the area ABn/; hence the maximum efficiencies, working between the given limits of actual heat, Qj and {}.» are equal, with or without a perfect regenerator. Q.E.D. 30. Advantage of a Regenerator. It appears from this theorem that the advantage of a regenerator is, not to increase the maximum efficiency of a thermodynamic engine between given limits of actual heat, but to enable that amount of efficiency to be attained with a less amount of expansion, and, consequently, with a smaller engine. Suppose, for instance, that to represent the isothermal curves, and the curves of no transmission, for a gaseous substance, Ave adopt the approxi- mate equations already given in Article 20, viz. : — For the isothermal curve of Q, P V = N Q ; ] - 1 I (31.) V /n \ - . /P V f I For a curve of no transmission ~ = ( - -- J ^ ' = ( ~ ) 1 + * ' ; j and let us compare the forms of the indicator-diagrams without and with a regenerator, for a perfect air-engine, working between given limits as to actual heat, defined by the isothermal curves Q 2 Q 1} ()., Q., in Fig. 19. The amount of expansion at the higher limit of heat being arbitrary, let us suppose it to be from the volume V A to the volume V B , corre- sponding respectively to the points A and B, and to be the same in all cases, whether with or without a regenerator. The engine being without a regenerator, the diagram corresponding to the maximum efficiency has but one form, viz., A B c d, where Be, A d are curves of no transmission. Hence, in this case, there must be an additional expansion, from the volume V B to the volume ON THERMODYNAMICS. 373 V. = V, Qi\n Q 2 / ' (32.) for the purpose merely of lowering the actual heat of the air without loss of heat ; and the engine must be made large enough to admit of this expansion, otherwise heat will lie wasted. Fig. 19. On the other hand, if the engine be provided with a perfect regenerator, any pair of curves of equal transmission passing through A and B will complete a diagram of maximum efficiency. The property of a pair of these curves being, as shown in Proposition IV., that the difference of their thermodynamic functions, AF ( = I -r-=- d V, when Q is constant j, is the same for every value of Q, it follows, that for a gas, according to the approximate equation (23), the property of a pair of curves of equal transmission is, that the volumes corresponding to the intersections of the two curves by the same isothermal curve, are in a ratio which is the same for every isothermal curve. Thus, let V a , Y b be such a pair of volumes, then this equation v. V.' (33.) defines a pair of curves of equal transmission. From this and from equation (31) it follows, that for such a pair of curves P P (34.) 374 OX THERMODYNAMICS. If one of the curves, or lines, of equal transmission is a straight line of equal volumes, that is, an ordinate A D parallel to Y, then the other is an ordinate B C, parallel to Y also. Then ABCI) is the diagram of maximum efficiency for an air-engine with a perfect regenerator, when the air traverses the regenerator without alteration of volume; and by adopting this diagram, the additional expansion from V B to V c is dispensed with. If one of the curves, or lines, of equal transmission is a straight line of equal pressures A D' parallel to X, then the other also is a straight line of equal pressures BC. The diagram thus formed, A BCD', is suitable, when the air, as in Ericsson's engine, has to traverse the regenerator without change of pressure. It must be observed, that no finite mass, or extent of conducting surface, will enable a regenerator to act with the ideal perfection assumed in Propositions X. and XL, and their corollaries. Owing to the want of a general investigation of the theory of the action of the regenerator based on true principles, those who have hitherto written respecting it have either exaggerated its advantages or unduly depreciated them. From this remark, however, must be excepted a calculation of the expenditure of heat in Captain Ericsson's engine, by Professor Barnard of the University of Alabama.* 31. General Hi urn 1 1,< on the preceding Propositions. The eleven preceding propositions, with their corollaries, are the geometrical representation of the theory of the mutual transformation of heat and motive power, by means of the changes of volume of a homo- geneous elastic substance which does not change its condition. All these propositions are virtually comprehended in the first two, of which, perhaps, the most simple enunciations are the following : — I. The mechanical equivalent of the heat absorbed or given out by a substance in passing from one given state as to pressure and volume to another given state, through a series of states represented by the co- ordinates of a given curve on a diagram of energy, is represented by the area included between the given curve and two curves of no transmission of heat drawn from its extremities, and indefinitely prolonged in the direction representing increase of volume. II. If across any pair of curves of no transmission on a diagram of energy there be drawn any series of isothermal curves at intervals corre- sponding to equal differences of actual heat, the series of quadrilateral areas thus cut off from the space between the curves of no transmission will be all equal to each other. These two propositions are the necessary consequences of the definitions of isothermal curves and curves of no transmission on a diagram of energy, and are the geometrical representation of the application to the particular * SUliman's Journal, September, 1853. ON THERMODYNAMICS. 375 case of heat and expansive power, of two axioms respecting Energy in the abstract, viz.: — I. The sum of Energy in the Universe is unalterable. II. The effect, in causing Transformation of Energy, of the whole quantity of Actual Energy present in a substance, is the sum of the effects of all its parts. The application of these axioms to Heat and Expansive Power virtually involves the following definition of expansive heat : — Expansive Heat is a species of Actual Energy, the presence of iddch in a substance affects, and in general increases, its tendency to expand. And this definition, arrived at by induction from experiment and observation, is the foundation of the theory of the expansive action of heat. Section IV. — Of Temperature, the Mechanical Hypothesis of Molecular Vortices, and the Numerical Computation of the Efficiency of Air-Engines. 32. In order to apply the propositions of the preceding articles to existing substances, besides experimental data sufficient for the determina- tion, direct or indirect, of the isothermal curves and curves of no transmission, it is necessary also to know the relation, for the substance in question, between the quantity of heat actually present in it under any circumstances, and its temperature ; a quantity measured by the product of the pressure, volume, and specific gravity of a mass of perfect gas, when in such a condition that it has no tendency to communicate heat to, or to abstract heat from, the substance whose temperature is ascertained. The nature of the relation between heat and temperature has been discussed in investigations already published, as a consequence deducible from a hypothesis respecting the molecular constitution of matter, with the aid of data supplied by the experiments of Messrs. Thomson and Joule and of M. Eegnault. Nevertheless, it seems to me desirable to add here a few words respecting the grounds, independent of direct experiment, for adopting the hypothesis of molecular vortices as a probable conjecture, the extent to which, by the aid of this hypothesis, the residts of experiment were anticipated, and its use, in conjunction with the results of experiment, as a means of arriving at a knowledge of the true law of the relation between temperatures and total quantities of heat. To introduce a hypothesis into the theory of a class of phenomena, is to suppose that class of phenomena to be, in some way not obvious to the senses, constituted of some other class of phenomena with whose laws we are more familiar. In thus framing a hypothesis, we are guided by some 376 ON THERMODYNAMICS. analogy between the laws of the two classes of phenomena : we conclude, from this analogy of laws, that the phenomena themselves are probably alike. This act of the mind is the converse of the process of ordinary physical reasoning; in which, perceiving that phenomena are alike, we conclude that their laws are analogous. The results, however, of the latter process of reasoning may be certainly true, while those of the former can never be more than probable ; for how complete soever the analogy between the laws of two classes of phenomena may be, there will always remain a possibility of the phenomena themselves being unlike. A hypothesis, therefore, is incapable of absolute proof; but the agreement of its results with those of experiment may give it a high degree of probability. The laws of the transmission of radiant heat are analogous to those of the propagation of a transverse oscillatory movement. The laws of thermometric heat are analogous to those of motion, inasmuch as both are convertible into mechanical effect ; and motion, especially that of eddies in liquids and gases, is directly convertible into heat by friction. If, guided by these analogies, we assume as a probable hypothesis that heat consists in some kind of molecular motion, we must suppose that thermometric heat is such a molecular motion as will cause bodies to tend to expand; that is to say, a motion productive of centrifugal force. Thus we are led to the hypothesis of Molecular Vortices. This hyp i :hesis, besides the principles already enunciated, of the mutual tram formation of heat and motive power in homogeneous sub- stances, leads to the following special conclusion respecting the Eelation between Temperature and Actual Heat: — When the temperature of a substance, as measured by a perfect gas ther- mometer, rises by equal increments, the actual heat present in the substance rises also by equal increments — a principle expressed symbolically by the equation Q = h (r - k), . . . . (35.) where Q is the actual heat in unity of weight of a substance, r its temperature, measured from the absolute zero of gaseous tension, k the temperature of absolute cold, measured from the same point, and f> the real specific heat of the substance, expressed in terms of motive power.* The enunciation of this law was originally an anticipation of the results of experiment; for when it appeared no experimental data existed by which its soundness could be tested. Since then, however, one confirmation of this law has been afforded * The hypothesis of Mayer amounts to supposing that » = 0, or that the zero of gaseous tension coincides with the point of absolute cold. ON THERMODYNAMICS. o77 by the experiments of M. Regnault, showing that the specific heat of atmospheric air is sensibly constant at all temperatures and at all densities throughout a very great range; and another, by the experiments of Messrs. Joule and Thomson, referred to in Proposition VI., on the thermic phenomena of gases rushing through small apertures, which not only verify the theoretical principle, but afford the means of computing approxi- mately the position k of the point of absolute cold on the thermometric scale. According to this relation between temperature and heat, every isothermal curve on a diagram of energy is also a curve of equal tempera- ture. The isothermal curve, for example, corresponding to a constant quantity of actual heat, Q, corresponds also to a constant absolute temperature, r = fH- K (36.) The curve of absolute cold is that of the absolute temperature k. Any series of isothermal curves at intervals corresponding to equal differences of heat, correspond to a series of equidistant temperatures. Hence we deduce Proposition XII. — Theorem. Everything that has been predicated, in the propositions of the preceding articles, of the mutual proportions of quantities of actual heat and their differences, may be predicated also of the mutual proportions of temperatures as measured from the point of absolute cold, and their differences. The symbolical expression of this theorem is, that in all the equations of the preceding sections, we may make the following substitutions: — Qa _ r 2 - k _ (A, g, or d) Q _ (A, g, or d) r ^ Ql T X — K ' Q r — K This theorem is not, like those which have preceded it, the consequence of a set of definitions. It is a law known by induction from experiment, aided by a hypothesis or conjecture, with the results of which those of experiment have been found to agree. It is true that the theorem itself might have been stated in the form of a definition of degrees of temperature; but then induction from experiment would still have been required, to prove that temperature, as measured in the usual way, agrees with the definition. By substituting symbols according to the above theorem, and making 378 ON THERMODYNAMICS. the general equation of the expansive action of heat is made to take the following form: — A . ¥ = A . H - IPdV = AQ+A.S = fc.Ar«f A/ . - which agrees with the equation deduced directly from the hypothesis of molecular vortices, if we admit that and, consequently, /.<r=ftNie (hyp. log. r + /\r = *N (37 a.) J The differential form of equation (37) is d .* = d .H - VdX = dQ + d .* = K v o? r (38.) where K v = k+/'.r + (r-K)| //r , rfV. The expression for the thermodynamic function denoted by F takes the form r l +£./'. r (39.) but a more convenient thermodynamic function, bearing the same relation to temperature as reckoned from the point of absolute cold, which the function F does to actual heat, is formed by multiplying the latter by the real specific heat fe, thus: — $ = kF= fii+/^, r+ (''l\,Y, . (40.) J r — K Jut which, being introduced into the general equation, transforms it to . (10 A.) A.¥= f(r-K)i$- fvdY ON THERMODYNAMICS. 370 33. Of the Numerical Computation of the Efficiency of Air-Engines, with or without a perfect Regenerator. The relation between temperature and heat being known, the preceding propositions can be applied to determine the efficiency, and other circum- stances relative to the working of thermodynamic engines. To exemplify this application of the theory, let the substance working the engine be atmospheric air, and let the real indicator-diagram be such as to develop the maximum efficiency between two given absolute temperatures r x and r 2 , being a quadrilateral, as in Fig. 1 9, of which two sides are portions of the isothermal curves of those temperatures, and the other two portions of a pair of curves of equal transmission, of such a form as may be best suited to the easy working of the engine. Should these curves be curves of no transmission, a regenerator may be dispensed with. In every other case a regenerator is necessary, to prevent waste of heat; and for the present, its action will be assumed to be perfect, as the loss which occurs from its imperfect action cannot be ascertained except by direct experiment. In this investigation it is unnecessary to use formula? of minute accuracy ; and for practical purposes those will be found sufficient which treat air as a perfect gas, whose thermometric zero of pressure coincides with the point of absolute cold, viz. — 272 J° Centigrade, or. ! below melting; ice ; * 490i° Fahrenheit, 3 whose real specific heat is equal to its specific heat at constant volume, being , 2'M'G feet of fall per Centigrade degree, Gi- ft = K v = I 130 - 3 feet of fall per degree of Fahrenheit; whose specific heat at constant pressure (as determined by M. Regnault) is - 23S x the specific heat of liquid water; or c 3 30 - 8 feet of fall per Centigrade degree, or K P = \ i. 183-8 feet of fall per degree of Fahrenheit; the ratio of these two quantities being |?= 1+X=1-41, as calculated from the velocity of sound. * This estimate of the position of the point of absolute cold is to be considered as merely approximate, recent experiments and calculations having shown that it may possibly be too high by about 1^° Centigrade. It is, however, sufficiently correct for all practical purposes. 380 ON THERMODYNAMICS. The volume occupied by an avoirdupois pound of air, at the temperature of melting ice, under the pressure of one pound on the square foot, as calculated from the experiments of M. Regnault, is P V = 2G214-4 cubic feet. This represents also the length in feet of a column of air of uniform density and sectional area, whose weight is equal to its elastic pressure on the area of its section at the temperature of melting ice. Tt will be found convenient, in expressing the temperature, as measured from the point of absolute cold, to make the following substitution : — k = T +T fl (41.) where T represents the temperature as measured on the ordinary scale from the temperature of melting ice, and T the height of the temperature of melting ice above the point of absolute cold, as already stated. Then we have P V IN B - T • (41 A.) According to these data, the equation of the isothermal curve of air for any temperature T is PV = P V .^l° = Nfe(T + T ). . (42.) The thermodynamic functions are — For quantities of actual heat, F = hyp. log. Q + N hyp. log. V; For temperatures, $ = hF + constant = K v {hyp. log. (T + T ) + N hyp. log. V] = K v hyp. log. (T + T^ + P „0 ° • hyp. log. V; 1 o consequently, the equation of any curve of no transmission is $ = constant ; otherwise N 1 + N (T + T ) . V = const.; orP . V = const.; K (43.) y (42a.) or (T + T ) . PI + N - constant ; ON THERMODYNAMICS. 381 111 W liicli N = 0-41, 1 + N = 1-41, N 1 + N 0-2908. The maximum possible efficiency between any two temperatures T x and T., is given by the universal formula, H t - H, T, - T, H x T x + T (44.) The latent heat of expansion of unity of weight of air at a given constant temperature T v from the volume V A to the volume V B , is sensibly equivalent simply to the expansive power developed, being given by the following formula : — H x = (T, + T ) (*,-*a) = P V„ ,Y V Tx + T hyp. log. c Pd v. (45.) Let Y a and V & be the volumes corresponding to the points at which any isothermal curve intersects a given pair of curves of no transmission, or of equal transmission ; then the ratio of these volumes, V' (46.) is constant for every such pair of points on the given pair of curves ; because the difference of the thermodynamic functions, which is pro- portional to the logarithm of this ratio, is constant. Hence, if in Fig. 19 a, two isothermal curves, T^, T 2 T 2 , be the upper and lower boundaries of an y indicator-diagram of maximum energy for an air-engine, AaD an arbitrary curve bounding the diagram at one side, and B the other limit of the expan- sion at the higher temperature ; the fourth boundary, being a curve of equal transmission to AftD, may be described by this construction; draw any isothermal curve 1 1 cutting A a D in a, and make V A :V B ::V ■ V,. . (47.) Fis- 19 A. 382 ON THERMODYNAMICS. then will b be a point in the curve sought, B b C. Suppose, for example, that the form assumed for AaD is a hyperbola, concave towards Y, and haviug the following equation : — . . . (47 A.) a /s - v; v in which a and (3 are two arbitrary constants; and let the ratio v " = r. Then must the curve B b C be another hyperbola concave towards Y, having for its equation p. = r ^. ■ • • («») The total expenditure of heat, per pound of air per stroke, in a perfect air-engine, is the latent heat of expansion from V A to V B , given by equation (45). The heat to be abstracted by refrigeration is the latent heat of com- pression from V c to V D j and is found by substituting in the same equation the lower temperature T 2 for the higher temperature T r The indicated work per pound of air per stroke, being the difference between those two quantities, is found by multiplying the range of temperature by the difference of the thermodynamic functions l l> for the curves AD, BC, or by multiplying the latent heat of expansion by the efficiency, and has the following value : — E^^-H^^-T,).^-^) -Po^o.^V^-^P-log.^. • • (48.) o ' o • nn • 'vr —&■ y The heat alternately stored up and given out by the regenerator (supposing it to work perfectly), is to be computed as follows : — Let the arbitrary manner in which volume is made to vary with temperature, on either of the curves D a A, C b B, be expressed by an equation then the thermodynamic function $ takes the form $ = K v hyp. log. (T + T ) + P ^-° hyp. log. V . T : and the total heat stored up and given out per pound of air per stroke, is ON THERMODYNAMICS. 383 T / 1 (T + T )^^T = K V (T 1 -T 2 ) T f - ' rp For example, if, as before, + gAJ H? + t )V .? dT (49) ft - V a be the equation of the curve D A, then P (T + T ) V„ = T + T o(l+pV)' x x v and the heat stored up per pound of air per stroke, is K v (T, - T 2 ) + « . hyp. log. <{ ) ^ V (49 a.) L T 3 + T (l + ^)j 33A. Numerical Examples?. To illustrate the use of these formula*, let us take the following; example : — Temperature of receiving heat, T x = 343°-3 Centigrade. T l + T = 615°-8 Centigrade. Temperature of emitting heat, T 2 = 35°-4 Centigrade. T 2 + T = 307°-9 Centigrade. Ratio of Effective Expansion, Y. _ lo _ Pa = ?d = 3 V A V D P B P c 2' From these data are computed the following results : — 3S4 OX THERMODYNAMICS. Maximum Efficiency. — 307°'9 1 G15°-8 2 Heat expended, or latent heat of expansion, — G15-S 3 H, = P \ x ^^ x hyp. log. - = 24020 foot-pounds per pound of working air per stroke. Heat abstracted by refrigeration, — H 2 = P V X y^ X hyp. log. - = 12010 foot-pounds per pound of working air per stroke. Work performed, — H 1 -H, = P V o x |^x h yP-log| = 12010 foot-pounds per pound of working air per stroke. To exemplify the computation of the heat stored by a perfect regenerator, let it be supposed, in the first place, that the indicator-diagram resembles ABC'D' in Fig. 1 9, where the curves of equal transmission are represented by a pair of lines of constant pressure. Then the heat to be stored is K p (T, — l\)) = 101,800 foot-pounds per pound of working air per stroke. (Secondly, let the diagram resemble ABCD in Fig. 19, where the curves of equal transmission are represented by a pair of lines of constant volume. Then the heat to be stored is K V (T 2 — T 2 ) = 72,233 foot-pounds per pound of working air per stroke. Thirdly, let the curves of equal transmission, as in a recent example, be hyperbolas, concave towards O Y, and let the arbitrary constant a have the following value, — a — . P V = 26214*4 foot-pounds; then the heat to be stored, according to equation (49 a), is 72,233 + 26214-4 x hyp. log.||^| = 72,233 +11,157 = 83,390 foot-pounds per pound of working air per stroke. ON THERMODYNAMICS. 385 The large proportions borne by these quantities to the whole heat expended, show the importance of efficient action in the regenerator to economy of fuel. The quantity of heat to be stored, however, becomes smaller, as the curves of equal transmission approach those of no trans- mission, for which it is null. The additional expansion requisite in this last case is found by the following computation : — /% + T \s " VT 2 + Tj l 2 0-41 = 5-423, the result of which shows the great additional bulk of engine required, in order to obtain the maximum efficiency without a regenerator. Supposing one pound of coal, by its combustion, to be capable of com- municating heat to the air working in an engine corresponding with the above example, to an amount equivalent to 6,000,000 foot-pounds (an amount which would evaporate about 7 lbs. of water), the maximum theoretical duty of one pound of such coal in such an engine, without waste of heat or power, would be 3,000,000 foot-pounds, corresponding to 3,000,000 = 249 strokes of a pound of working air, with the effective 12,010 . 3 expansion -. u The deductions to be made from this result in practice must, of course, be determined by experience. Section V. — Propositions Relative to a Heterogeneous Mass, or Aggregate, especially in Vapour-Engines. 34. The heterogeneous mass to which the present investigation refers, is to be understood to mean an aggregate of portions of different in- gredients, in which each ingredient occupies a space, or a number of spaces, of sensible magnitude. The results arrived at are not applicable to mixtures in which there is a complete mutual diffusion of the molecules of the ingredients, so that every space of appreciable magnitude contains every ingredient in a fixed pro- portion. A mixture of this kind, when the relations between its pressure, 2b 38G ON THERMODYNAMICS. volume, heat, and temperature are known, may be treated, so far as regards the expansive action of heat, as a homogeneous substance. The ingredients of an aggregate are heterogeneous with respect to the expansive action of heat, when either their specific heats, or their volumes for unity of weight at a given pressure and temperature, or both these classes of quantities, arc different. Hence a portion of a liquid, and a portion of its vapour, enclosed in the same vessel, though chemically identical and mutually transformable, are heterogeneous, and are to be treated as an aggregate, with respect to the expansive action of heat. M. Clausius and Professor William Thomson have applied their formulae to the aggregate composed of a liquid and its vapour, and have pointed out certain relations which must exist between the pressure and density of a liquid and its vapour, and the latent heat of evaporation. I shall now apply the geometrical method of this paper to the theory of the expansive action of heat in an aggregate, especially that consisting of a liquid and its vapour. The total volumes are, for the present, supposed not to be large enough to exhibit any appreciable differences of pressure due to gravitation. 35. Proposition XIII. — Theorem. In an aggregate in equilibria, the pressure of each ingredient must be the same; and the quantity of heat in unity of weight of each ingredient must fa inversely proportional to its real specific heat; that is to say, the temperature must be equal. The following is the symbolical expression of this theorem, with certain conclusions to which it leads : — Let t — k be the common temperature of the ingredients, as measured from the point of absolute cold ; P, their common pressure ; n v w 2 , ?? 3 , &c.j their proportions by weight, in unity of weight of tin- aggregate ; v v v 2 , v z , &c, the respective volumes of unity of weight of the several ingredients. V, the volume of unity of weight of the aggregate ; q v q 2 , q 3 , Sec, the respective quantities of actual heat in unity of weight of the several ingredients; fci> fc-2> ^3> c ^' c -' their respective real specific heats ; 0, the quantity of heat in unity of weight of the aggregate ; «1>, a thermodynamic function for the aggregate. ON THERMODYNAMICS. 387 Then these quantities are connected by the following equations: — 2 . n = 1 (50.) V-=.2..»«. (51.) T - I. - - — - cVC. . . . {0^.) **1 "2 ^3 Q = S . n J = (r - k) . 2 . n Is. . . . (53.) t. p''» +/, ' ,) .i,'+fg.<T. . (54.) ' It is evident that all these equations hold, whether the proportions of the ingredients n v &c, are constant, as in an aggregate of chemically distinct substances, or variable, as in the aggregate of a liquid and its vapour. Let 3H be the heat which disappears in consequence of a small expansion of aggregate at constant temperature, represented by SV = 2.S«, .... (55.) $ u representing any one of the parts arising from the changes undergone by the different ingredients, of which the whole expansion of the aggregate S V is made up. Then gH = 2-[(r-K)^.S«}; . • (56.) the same for every ingredient, as well as the tei perature ; therefore, the factor (r - k) -j- is the same for every ingredient. but the pressure P is the same for every ingredient, as well as the tem- (V? (It and, consequently, for the whole aggregate; that is to say, SH = (r- K )^.SV=(r- K )S$. . (57.) (It This equation shows that the relation of temperature to the mutual transformation of heat and expansive power is the same in an aggregate as in a homogeneous substance. Consequently, if we define isothermal curves for an aggregate to be cm of constant temperature, we arrive at the following conclusion: — Proposition XIV. — Theorem. Isothermal curves on the diagram of energy of an aggregate have the same properties, with reference to the mutual transformation of heat and expansive power, with those on the diagram of energy of a homogeneous substance. It is unnecessary to enunciate separately a similar proposition for 388 ON THERMODYNAMICS. curves of no transmission; for the demonstration of Proposition I., on which all their properties depend, is evidently appli cable to an aggregate, constituted in any manner. Hence it appears, that if the isothermal curves for an aggregate be drawn according to the above definition, all the propositions proved in this paper respecting homogeneous substances become true of the aggregate. 36. Proposition XV. — Theorem. Every isothermal line for an aggre- gate of a liquid and its vapour, is a straight line of equal pressure, from the volume corresponding to complete liquefaction to the volume corresponding to complete evaporation. This is a fact known by experiment. The theorem is equivalent to a statement, that the pressure of a liquid and its vapour in contact with each other is a function of the temperature only. Corollary. — Theorem. At any given temperature, the volume of an aggre- gate of liquid and vapour is arbitrary between and up to the limits of total liquefaction and total evaporation. To express this symbolically, let P be the pressure of an aggregate of liquid and vapour corresponding to the absolute temperature r ; and unity of weight being the quantity of the aggregate under consideration, let v be the volume corresponding to complete liquefaction, v that corresponding to complete evaporation, and V the actual volume at any time; let n be the proportion of liquid, and 1 — n that of vapour, corre- sponding to the aggregate volume V ; then V — nv + (1 - n)v', . . . (58.) and V may have any value not less than v nor greater than v, while P and r remain constant; the proportion of liquid, n, being regulated according to the foregoing equation. 37. Proposition XVI. — Problem. The density of a liquid and of Us vapour, when in contact at a given temperature, being given, and the isothermal lines of the aggregate, it is required to determine the latent heat of evaporation of unity of weight of the fluid. (Solution.) The densities of the liquid and of its vapour are respectively the reciprocals of the volumes of total liquefaction and total evaporation of unity of weight, above-mentioned. In Fig. 20, let the abscissae Ov, Ov represent these volumes, and the equal ordinates, vA, v'B, the pressure corresponding to the given temperature; so that AB, parallel to OX, is the isothermal line of the aggregate for that temperature. Suppose two curves of no transmission AM, BN, to be drawn from A and B respectively, and indefinitely prolonged towards X; then the indefinitely-prolonged area MABN represents the mechanical equivalent of the latent heat sought, and this area is to be computed in the following manner : — Draw a second isothermal line, a b, indefinitely near to A B, at an interval ON THERMODYNAMICS. 389 A 1 3 N M X Fv'. 20. corresponding to the indefinitely-small difference of temperature dr; then, ultimately, d t : t — k : : area A B b a : area MABN; or, symbolically, L d~P latent heat of evaporation = (r — k) -=- (v — v). (59.) This is simply the application of Propositions I. and II. to the aggregate of a liquid and its vapour, mutatis mutandis. (Remarks?) — The existence of a necessary relation between the density, pressure, and temperature of a vapour and its liquid in contact, and the latent heat of evaporation, was first shown by Carnot. If for r — k in the preceding equation be substituted, according to Professor Thomson's notation, -, J being "Joule's equivalent" and /x "Carnot's function," the ,u equation is transformed into that deduced by Messrs. Clausius and Thomson from the combination of Carnot's theory with the law of the mechanical convertibility of heat. 38. Corollary. — The volume occupied by unity of weight of vapour at saturation may be computed from its latent heat of evaporation by means of the inverse formula, — L (r - k) dj>' It (GO.) the latent heat, L, being, of course, always stated in units of motive power. The want of satisfactory experiments on the density of vapours of any kind, has hitherto prevented the use of the direct formula (59). 390 ON THERMODYNAMICS. It is otherwise, however, with the inverse formula (60), at all events in the case of steam; for, so far as we are yet able to judge, the experi- ments of M. Kegnault have determined the latent heat of evaporation of water with accuracy throughout a long range of temperature. M. Clausius, applying to those experimental data a formula founded on the supposition of Mayer (that is to say, similar to the above, with the exception that k is supposed = 0), has calculated the densities of steam at certain temperatures, so as to show how much they exceed the densities calculated from the pressures and temperatures on the supposition that steam is a perfect gas. From these calculations he concludes, that either the supposition of Mayer is erroneous, or steam deviates very much at high densities from the condition of a perfect gas. In the following table, the value of k is supposed to be 2 0, 1 Centigrade, and use has been made of the formula for calculating the pressure of steam and other vapours at saturation, first published in the Edinburgh Neiv Philosophical Journal for July, 1849 (Sec p. 1), viz. — log.P = a-£-^. • • • (61.) This table exhibits, side by side, the volume in cubic feet occupied by one pound avoirdupois of steam at every twentieth Centigrade degree, from - 20° to + 200° (that is, from -4° to + 500' Fahrenheit)— first, as extracted from a table for computing the power of steam-engines, in the Transactions of the Royal Society of Edinburgh, Vol XX. (See p. 2S2), which was calculated on the supposition that steam is a perfect gas; and, secondly, as computed by equation (GO) from the latent heat of steam as determined by M. Regnault The excess of the former quantity above the latter is also given in each case, with its ratio to the second value of the volume. For convenience sake, a column is added containing the pressures of steam corresponding to the temperatures in the table in pounds per scpiare foot. The fourth column of this table could easily be extended and filled up, so as to replace the column of volumes of steam for every fifth Centigrade degree in the table previously published; but it would be unadvisable to do so at present, for the following reasons : — First, the value of the constant k is still uncertain."" Secondly, the results of M. Regnault's direct experiments, on the density of steam and other vapours, may soon be expected to appear. * It is probable that x. may be found to be inappreciably small ; in which case the numbers in column (4) will have to be diminished to an extent varying from T l,-, to T J „ of their amount. ON THERMODYNAMICS. 391 Table of Computed Volumes of 1 lb. Avoirdupois of Steam. Temperature. Volume sup- posed a Per- fect Gas. Volume com- puted from Latent Heat. Difference. Ratio of Difference to lesser Value of Volume. Pressure. Fahrenheit. Centigrade. Deg. Deg. Cubic feet. Cubic feet. Cubic feet. lb. per square foot. - 4 -20 15757 15718 39 0-0025 2-4799 + 32 3390 4 3377-2 13-2 0-0039 12-431 68 + 20 936-81 934-50 2-31 0025 48-265 104 40 314-88 313 56 1-32 0042 153-34 140 60 123 65 122-63 1-02 0-0083 415-33 176 80 55 05 54-19 86 0-0158 988-67 212 100 27-166 26-47S 0-68S 0-0260 2116-4 248 120 14-596 14 076 0-520 0-0369 4149-3 284 140 8-420 8-004 0416 0-0502 7557-0 320 160 5-15S 4-838 0-320 0661 12931 356 180 3-326 3-071 0-255 0830 20979 392 200 2-241 2 033 0-20S 1023 32512 428 220 1-568 1-396 0-172 0T232 48425 464 240 1134 0-990 0144 0-1455 69680 500 260 0-843 722 121 0-1676 97275 Col. (1.) (2.) (3.) (4.) (5.) (6.) (7.) Thirdly, it is possible that the values of the latent heat of evaporation of water, as deduced from M. Eegnault's experiments, may still have to undergo some correction ; because, according to the theoretical definition of the latent heat of evaporation, the liquid is supposed to be under the pressure of an atmosphere of its own vapour, which atmosphere, as it increases in bulk, performs work of some kind, such as lifting a piston; whereas, in M. Regnault's experiments, the water is pressed by an atmo- sphere of mingled steam and air, whose united pressure is that corre- sponding to the tenrperature of internal ebullition of the water; so that the pressure of the steam alone on the surface of the water, which regulates the superficial evaporation, may be less than the maximum pressure corresponding to the temperature of ebullition; and this steam, moreover, has no mechanical work to perform, except to propel itself along the passage leading to the calorimeter, and to agitate the water in the latter vessel. Under these circumstances, it is possible, though by no means certain, that the latent heat of evaporation of water, as deduced from M. Eegnault's experiments, may be somewhat smaller than that which corre- sponds to the theoretical definition, especially at high pressures; and a doubt arises as to the precise applicability of the formulas (59) and (60) to those experimental results, which cannot be solved except by direct experiments on the density of steam. Notwithstanding this doubt, however, the preceding table must be regarded as adding a reason to those already known, for believing that 392 ON THERMODYNAMICS. saturated steam of high density deviates considerahly from the laws of the perfectly gaseous condition.* 39. Proposition XVII. — Problem. The isothermal lines for a liquid and Us vapour, and the apparent specific heat of the liquid at all temperatures hcing given, and the expansion of the liquid by heat being treated as inappreciably small: to determine a curve of no transmission for the aggregate, passing through a given point on the ordinate ichose distance from the origin approxi- mately represents the volume of the liquid, (Solution.) In Fig. 21, let Ov represent the volume of the liquid assumed to be approximately constant for all temperatures under con- sideration; let jjA be an ordinate parallel to Y, and let the heat Fig. 21. consumed by the liquid in passing from the temperature corresponding to any point on this ordinate to that corresponding to any other point, be known; let the isothermal lines for the aggregate of liquid and vapour, all of which are straight lines of equal pressure parallel to X, such as AT 1? «BT 2 , be known. Then to draw a curve of no transmission through any point A on the ordinate vA, the same process must be followed as in Proposition VIII. To apply to this case the symbolical representation of Proposition VIIL, viz., equation (21), let r x be the absolute temperature corresponding to the point A (that is, to the isothermal line A T x ) ; r 2 that corresponding to any lower isothermal line aBT 9 ; V B the volume of the aggregate of liquid and vapour corresponding to the point B, where the curve sought, AM, intersects the latter isothermal line; K L the apparent specific heat * Evidence in favour of this opinion is afforded by the experiments recorded by Mr. C. W. Siemens (Civil Engineer and Architect's Journal). A remarkable cause, however, of uncertainty in all such experiments has lately been investigated by Professor Magnus (Poggendorff's Annalen, 1853, No. 8), viz., a power which solid bodies have of condensing, by attraction on" their surfaces, appreciable quantities of gases. ON THERMODYNAMICS. 393 of the liquid; then, making the proper substitutions of the symbols of temperature for those of heat, and observing that the operation / dV -'v A is in this case equivalent to multiplication by V B — v, we have l^ = f T (Y B -v)(iovr = r. 1 )=f 1 ^ K dr, . (62.) being an equation between two expressions for the difference between the thermodynamic functions $ for the curve A B, and for that which passes through a. If the specific heat of the liquid is approximately constant, this equation becomes A * = ~ (V B - v) (for t = t.J = K t hyp. log. *LZ*> (63.) »T T 2 — K 40. Corollary. — Problem. The same data being given as in the preceding problem, and the expansion of the liquid by heat neglected, a mass of liquid, having been raised from the absolute temperature t 2 to the absolute temperature r l7 is supposed to be allowed to evaporate partially, under pressure, without receiving or emitting heat, until its temperature falls again to t 2 , at which temperature it is liquefied under constant pressure by refrigeration : it is required to find the power developed. (Solution.) The power developed is represented by the area of the three-sided diagram of energy in Fig. 21, A B a ; that is to say, by T 2 T 2 which, if K L is nearly constant, becomes K L j * hyp. log. ^^ .dr = K L { (r, - k) r 9 - (r 2 -k) . (l + hyp. log. J-^) }• • . (65.) 41. Numerical Example. Let one pound avoirdupois of water be raised, in the liquid state, from T 2 = 40° Centigrade to T\ = 140° Centigrade. Then 394 ON THERMODYNAMICS. Tl - K = T, + T = 140° + 272r = 4m° Centigrade. r 2 - k = T 2 + T = 40° + 272?,° = 3121° Centigrade. The mean apparent specific heat of liquid water between those tem- peratures is K L = K w (or Joule's equivalent) X l'OOG = 1398 feet per Cent, degree; consequently, the heat expended is equivalent to 139,800 foot-pounds. The other numerical data are, — dP -r— at 40° Centigrade = S'2075 lbs. per square foot per Cent, degree; (i - v = mean volume of 1 lb. of liquid water = 0-017 cubic foot nearly. Let it be required to find, in the first place, V B , the volume to which the water must be allowed to expand by partial evaporation under pressure, in order that its temperature may fall to 40° Centigrade; and, secondly, how much power will be developed in all, after the water has been totally reliquefied by refrigeration at constant pressure, at the temperature of 40°. First, by the equation (G3), A $ = ~ (V B - v) = 1398 x hyp. log. |i|| = 402-G24 ; divide by |- = 8-2075 ; then Y B -v = 49-055 cubic feet. addv= 0-017 Aggregate volume of water and steam at 40°, V B = 49-072 ., As the volume of one pound of steam at 40° Centigrade, according to the fourth column of the table in Article 38, is 313 - 5G cubic feet, it appears from this calculation that somewhat less than one-sixth of the water will evaporate. Secondly, it appears, from equation (G5), that after the water has been restored to the liquid state by refrigeration at 40° Centigrade, the whole power developed — that is to say, the area A B a — will be 1398 foot-pounds X j 412°-5 - 312°-5(l + hyp. log. <— |— J j = 1398 ft. lbs. X 10° Centigrade = 13,980 ft. lbs., or one-tenth of the equivalent of the heat expended. The other nine- tenths constitute the heat abstracted during the reliquefaction at 40° Centigrade. ON THERMODYNAMICS. 395 This calculation further shows, that in order that one pound of water and steam at 40° C. may be raised to 140° C. solely by compressing it into the liquid state, it must occupy at the commencement of the operation the volume V B = 4 9 - 07 2 cubic feet; and that the power expended in the compression will be as follows : — Foot-pounds. Area of the curvilinear triangle ABa, Fig. 21, as already calculated, 1 3,9S0 Area of the rectangle aBY B v — P 2 (V B — v) = . . . 7,522 Total, 21,502 42. Proposition XVIII. — Problem. Having the same data as in the last proposition, it is required to draw a curve of no transmission through any point on the diagram of energy for the aggregate of a liquid and its vapour. (Solution.) In Fig. 22, through the given point B draw the straight isothermal line AB corresponding to the absolute temperature r v and cutting the ordinate corresponding to the volume of total liquefaction in A. Through A, according to the last proposition, draw the curve of no trans- mission, A D M. Let E D C be any other isothermal line, corresponding to the absolute temperature r 2 , and cutting the curve A M in D. Draw isothermal lines a b, e cl c at indefinitely-small distances from A B, EDO v A B a c fa 55 ^ \ \d "^^T£- J \ C — jnt d NM A r E V n -X Fk respectively, corresponding to the same indefinitely small difference of temperature dr. Draw the ordinates V D dD, V B bB; then draw the ordinate V c cC at such a distance from V D rfD, that the indefinitely-small rectangles DCcd, ABba shall be equal. Then, as the difference 8 r is indefinitely diminished, C approximates indefinitely to a point on the required curve of no transmission, B N. This is Proposition III. applied to aggregates, mutatis mutandis. 39G ON THERMODYNAMICS. The symbolical representation of this proposition is as follows : — let F l and P 2 be the pressures of the aggregate of liquid and vapour corresponding respectively to the temperatures r r and r 2 ; then the following expressions for the difference between the thermodynamic functions $ of the curves A M, B N are equal, A$ = ^(V -V D ) = ^(V B -»). • (GG.) 43. Corollary. (Absolute Maximum Efficiency of Vapour-Engines.) If the volume V B be that corresponding to complete evaporation at the temperature t u that is to say, if V B = •, then the curve B C N will represent the mode of expansion under pressure, of vapour of saturation in working an engine, and will be defined by the equation YV {v ~ v) V _ y = — . . . . (G7.) (It If in this equation be substituted the value of v — v, in terms of the latent heat of evaporation at the higher temperature, given by equation (60), it becomes h. -'-up- • ■ (° a > In this case the diagram A B C D, Fig. 22, is evidently that of a vapour- engine working with the absolute maximum of efficiency between the absolute temperatures r x and r 2 . The heat expended at each single stroke, per unit of weight of fluid, is the latent heat of evaporation at the higher temperature, or L x ; the area of the diagram is given by the following equation : — T-, — T., E = (r 1 -r 2 )A$ = ^ 3.1* . . (09.) This is the mechanical power developed at each single stroke by a unit of weight of the substance employed. The efficiency is represented by r = T ^ Ll2 , .... (to.) L l T l - « being the expression for the maximum efficiency of thermodynamic engines in general. ON THERMODYNAMICS. 397 The conditions of obtaining this efficiency are the following : — First, That the elevation of temperature from r 2 to t x , during the operation represented by the curve D A on the diagram, shall be produced entirely by compression. The volume at which this heating by compression must commence is given, according to Proposition XVII., by the following equation : — V. = . + ^ . K, h n , log. *=$ . . (71.) dr Secondly, That the expansive working of the vapour shall be carried on until the temperature falls, by expansion alone, to its lower limit ; that is to say, until the volume reaches the following value, obtained by adding together equations (68) and (71) : — Ve = . + / p; • { K, h )T . log. ^ + -h- } . (72.) dr 44. Numerical Example. To exemplify this numerically, let the same data be employed as in Article 41, the substance working being one pound avoirdupois of water. These data, with some additional data deduced from them, are given in the folio win sj table : — Temperature in Centigrade Degrees : — Above melting ice (T), Above zero of gaseous tension (t), Above absolute cold (r — k), Pressure in pounds per square foot (P), . „ „ per square inch, Initial Volume of saturated steam, V B = v\, = 8*004 cubic ft. per lb. Latent Heat of Evaporation : — In degrees, applied to one pound of liquid water, 509 o, l Centigrade. In foot-pounds (Lj), . . . . . 707,445 - 3G At upper limit At lower limit of Actual Heat. . 140° 40° 414-6 314-6 412-5 312-5 . 7557 153-34 52-5 1-065 From these data are deduced the following results : — Absolute Maximum Efficiency ; = - 2424 398 ON THERMODYNAMICS. Duty of one pound of water; being the area of the diagram ABCD, . . 1 71,484-75 ft. lbs. Volume at which the compression must commence; calculated as in Art. 41, . V D = 49"1 cubic ft. per lb. Volume to which the expansion must be car- ried; calculated by equation (72), . V c = 258*1 cubic ft. per lb. V c 258*1 Ratio of Expansion, . . . . = y- = 8 .qq^ = 32-25. 45. Liquefaction of Vapour by Expansion under Pressure. In Fig. 22, let the abscissa? of the curve BFE indicate the volumes corresponding to complete evaporation at the pressures denoted by its ordinates. For most known fluids, a curve of no transmission, B C N, drawn from any point B of the curve of complete evaporation in the direction of X, falls within that curve ; so that by expansion of saturated vapour under pressure, a portion in most cases will be liquefied. To ascertain whether this will take place in any particular case, and to what extent, equation (60), which gives the volume of unity of weight of saturated vapour at the temperature t,, is to be compared with equation (72), which gives the volume at the same temperature of unity of weight of an aggregate of liquid and vapour, which lias expanded under pressure from a state of complete evaporation at the temperature r v The difference 1 letween the volumes given by these equations is as follows (neglecting, as usual, the expansibility in the liquid state) : — '■'•' - v < = tV • { — — - k* • h yp- lo s- T y^ } ( 73 -) d F 2 (. T 2 — K T 1 — K T 2 — K ) JV That this quantity is almost always positive appears from the following considerations. The latent heat of evaporation L, is in general capable of being represented approximately by an expression of this form : L = a- &(r- k), . . . . (74.) (For water, a = 79G° Centigrade x K w = 1,106,122 ft. lbs. ; b = 0-695 X K w = 965-772 ft. lbs. per Centigrade degree). Hence, the second factor in equation (73) is nearly equal to »(r,-rj K L . hyp. logA- ' . (75.) (r x — k) . (t 2 — k) r 2 — k Now hyp. log. — < -. ON THERMODYNAMICS. 399 Therefore, the expression (75) is positive so long as exceeds K L , the specific heat of the liquid. . (75 A.) For water this condition is fulfilled for all temperatures lower than 523^° Centigrade (at which r x — k = 796° Centigrade); and there is reason to believe that it is fulfilled also for other fluids at those tempera- tures at which their vapours can be used for any practical purpose. To determine the proportion of the fluid which is liquefied by a given expansion under pressure, we have the following formula, deduced from equation (58) : — v — V n = -] - c (76.) v 2 — v As a numerical example, we may take the case of Art. 44, where saturated steam at 140° Centigrade is supposed to be expanded under- pressure until its temperature falls to 40° Centigrade. The volume of one pound of water and steam at the end of the expansion has already been found to be V c = 258-1 cubic feet. While, according to the table in Art. 38, the volume of a pound of steam at that temperature is v'. 2 = 313'56 cubic feet. Consequently, the fraction liquefied by the expansion is 313-56 - 258-1 55-46 313-56 - 0-016 313-544 = 0-177. This conclusion was arrived at contemporaneously and independently, by M. Clausius and myself, about four years since. Its accuracy was subsequently called in question, chiefly on the ground of experiments, which show that steam, after being expanded by being " wire-drawn," that is to say, by being allowed to escape through a narrow orifice, is super- heated, or at a higher temperature than that of liquefaction at the reduced pressure. Soon afterwards, however, Professor William Thomson proved that those experiments are not relevant against the conclusion in question^ by showing the difference between the free, expansion of an elastic fluid, in which all the power due to the expansion is expended in agitating the particles of the fluid, and is reconverted into heat, and the expansion of the same fluid under a pressure equal to its own elasticity, when the power- developed is all communicated to external bodies, such, for example, as the piston of an engine. 400 ON THERMODYNAMICS. The free expansion of a vapour will be considered in the sequel. 46. Efficiency of a Vapour-Engine without heating by compression. The numerical example of Art. 44 sufficiently illustrates the fact, that the strict fulfilment of the condition specified in Art. 43, as necessary to the attainment of the absolute maximum of efficiency of a vapour-engine, is impossible in practice. Let us consider, in the first place, the effect of dispensing with the process D A, during which the fluid is supposed to have its high tempera- ture restored solely by compression. The effect of this modification is evidently to add to the heat expended that which is necessary to elevate the temperature of the liquid from r 2 to t v and to add to the power developed an amount represented by the area AD E (Fig. 22). To express this symbolically, we have — The latent heat of evaporation at t v as before, ...... The additional heat expended (K h being the mean specific heat of the liquid between t x and r 2 ), Total heat expended, . Then, for the power developed, we have L, K,.(r.-T 2 ) I^ + K^-r,) (77.) The area ABCD, as in Art. 43, = -* - 2 . L,, the area AD E, as in Art. 40, equation (G5), = K t { ( Tl - K ) - (r 2 - k) (l + hyp. log. T f^j } the sum of which quantities is the total power developed, . The efficiency may be expressed in the following form : — Power developed _ r l — r Heat expended t x — k \ + K L (r x - r 2 ) (78.) y (79.) an equation which shows at once how far the efficiency falls short of the absolute maximum. ON THERMODYNAMIC'S. 401 For a numerical example, the same data may be taken as in Arts. 41 and 44. Then the heat expended, per pound of steam, is thus made up : — Latent heat of evaporation, as in Art. 44, . Heat required to raise the water 100° C, as in Art. 41, Total heat expended per lb. of water, The power developed consists of, — The area A B C D, as in Art. 4 4, The area ADE, as in Art. 41, . Foot-pounds. 707,445-30 139,800-00 847,245-30 Foot-pounds. 171,484-75 13,980-00 Total power developed per lb. of water, 185,464*75 . 185,484-75 • Efficiency 847,245-36 ' Absolute maximum efficiency, as in Art. 44, 0-2189 0-2424 Loss of efficiency by omitting the heating by compression, 0-0235 or about one-tenth part of the absolute maximum. 47. Efficiency of a Vapour-Engine with incomplete expansion. It is in general impossible in practice to continue the expansion of the vapour down to the temperature of final liquefaction ; and from this cause a further loss of efficiency is incurred. Let it be supposed, for example, that while the pressure of evaporation / r ^ Pi 1 i 1 H P* ^\ Cr 3A E P 3 K ^~~-^C D\ -s >iyt -r i l A "> c Fi<r. 23. Pi corresponds to the line AB, in Fig. 23, and the pressure of liquefaction, P 3 , to the line E D C, the pressure at which the expansion terminates, P,, 2c 402 ON THERMODYNAMICS. corresponds to an intermediate line H L G. Let v A, ?•/ B, as before, be the ordinates corresponding to complete liquefaction and to complete evaporation, at the pressure Pj. Draw, as before, the curves of no transmission A M, B N, cutting H L G in L and G, and EDO in D and C; draw also the ordinate V KG, cutting E D C in K. Then the expansion terminates at the volume V (; , and ABGKE is the indicator-diagram of the engine. To find the power represented by this diagram, the area A L H is to be found as in Art. 40, the area A B G L as in Art. 43, and the rectangle HGKE by multiplying its breadth V G — v (found as in Art. 43) by its height H E, which is the excess of the pressure at the end of the expansion P 2 , above the pressure of final liquefaction, P 3 . Hence, we have the following formula for the indicated power developed, per unit of weight of fluid evaporated : — E = area A B G K E = K L j (r, - ic) - (r, - k) 1 +hvp. log. T| "^l + L./ 1- ^ n 5 r 2 - J J ' r x -K y (80.) + (P 2 -P3)^{^ + K thy p.log^ r,-«J ,/■ The heat expended is of course L, + K, (rj — t 3 ). To illustrate this numerically, let the fluid be water; let the temperature of evaporation be 1 40° Centigrade, and that of liquefaction 40°, as in the previous examples; and let the expansion terminate when the pressure has fallen to 100° Centigrade. The numerical data in this case are the following : — 1. 2. 3. During the At the end of During the evaporation, the expansion, final liquefaction. Temperature in Cent, degrees :— Above melting ice, . . 140° 100° 40° Above zero of gaseous ten- sion, T — . Above absolute cold, r — k, Pressure, in lb. per square f oot, P = 7557 Pressure, in lb. per square inch, . 14-6 3746 31 4-6 -12-5 372-5 312-5 57 2116-4 153 -34 52-5 14 7 1-065 ON THERMODYNAMICS. 403 1. 2. 3. During the At the end of During the evaporation, the expansion, linal liquefaction. in lb. per square foot per Centigrade degree, . . 21416 75-617 8-2075 Initial Volume of steam in cubic- feet per lb., . . . 8-004 Latent heat of evaporation, L t , in foot-pounds per lb. of steam, 707,445-36 Total heat expended, in foot-pounds per lb. of steam, . . 847,245*36 Mean specific heat of liquid water — Between 40° and 140° Cent., 1398 feet of fall. Between 100° and 140° Cent., 1409 Applying equation (80) to these data, we obtain the following results : — Foot-pounds. AreaALH, 2,818 AreaABGL, 68,601 Area HGKE= (P, - P 3 ) . (V 6 - v) = 1963 lbs. per square foot x 24'58 cubic feet, . . . = 48,250 Total power developed by 1 lb. of water evaporated, 119,669 Efficiency = ^|^| =0-1413 J 847,245 Efficiency computed in the last article, . . . 0"21S9 Difference = loss of efficiency by incomplete expansion, 0-0776 V 24-60 Ratio of expansion — y = —-— = 3-07 nearly. 1 v 8-004 J If the power of the same engine be now computed by the tables and formulae published in the twentieth volume of the Transactions of the Roijol Society of Edinburgh (See p. 278), which were calculated on the supposition that steam is sensibly a perfect gas, the following results are obtained : — Ratio of expansion, " = 2-921 = s in tables. 8'4204 404 ON THERMODYNAMICS. "Action at full pressure " (PiV, in tables), " Coefficient of gross action " (Z in tables) for the ex- pansion 2 "9 2 1 , Gross action (P^ Z), . Deduct for back-pressure of liquefaction, P 3 V G = 1533 4 X 24-6, . ■ Foot-pounds. 63,633 Power developed per lb. of steam, . 122,221 This result is too large by about one forty-seventh part, a difference to be ascribed chiefly to the error of treating steam as a perfect gas. This difference, however, is not of material consecpience in computing theoreti- cally the power of a steam-engine, being less than the amount of error usually to be expected in such calculations. 48. My object in entering thus minutely into the theory of the efficiency of vapour-engines is, not so much to provide new formulae for practical use, as to illustrate the details of the mechanical action of heat under varied and complicated circumstances, and to show with precision the nature and influence of the circumstances which prevent the production, by steam- engines, of the absolute maximum of efficiency corresponding to the tem- peratures between which they work. To illustrate the results of these calculations with respect to the con- sumption of coal, let it De assumed, as in Art. 33, that each pound of coal consumed in the furnace communicates to the water, or air, or other elastic substance which performs the work, an amount of heat equivalent to 6,000,000 foot-pounds, which corresponds to a power of evaporating, in Absolute Theoretical Maximum, being the same for every perfect thermo- dynamic engine working between the same limits of temperature, 140° - 40° L40°+2724° ' Efficiency. Effect per pound of coal in foot-pounds. 2424 1,454,400 Deductions: — For raising the temperature of the feed-water from 40° to 140° Cent., For stopping the expansive working at 3 "07 times the initial volume instead of 32 times, . Reduced Efficiency and Effect, 0-0235 0-0776 o-ioii 141,000 465,600 606,600 01413 847, SOO ON THERMODYNAMICS. 405 round numbers, about seven times its weight of water. Then the following calculation shows the theoretical indicated duty of one pound of such coal, when the limits of working temperature are 140° and 40° Centigrade, at the absolute maximum of theoretical efficiency, and at the reduced efficiency, computed in the preceding article, on the supposition that the expansive working ceases at the atmospheric pressure. The last of these quantities corresponds to a consumption of about 2*34 lbs. of coal per indicated horse-power per hour. The conditions of the preceding investigations are very nearly fulfilled in steam-engines with valves and steam-passages so large, and a velocity of piston so moderate, that the pressure in the cylinder during the admission of the steam is nearly the same with that in the boiler. In many steam-engines, however, the steam is more or less "wire- drawn ; " that is to say, it has to rush through the passages with a velocity, to produce which there is required a considerable excess of pressure in the boiler above that in the cylinder. The power developed during the expansion of the steam from the pressure in the boiler to that in the cylinder is not altogether lost; for, as already stated in Art. 45, it is expended in agitating the particles of the steam, and is ultimately con- verted into heat by friction, so that the steam begins its action on the piston in a superheated state; and both its initial pressure and its expansive action are greater than those of steam of saturation of the same density. The numerical relations of the temperature, pressure, and density of superheated steam are not yet known with sufficient precision to constitute the groundwork of a system of exact formulae representing its action. Some general theorems, however, will be proved in the sequel, respecting superheated vapours, which may be found useful when the necessary experimental data have been obtained. Calculation and experiment concur to prove that in Cornish single- acting engines the initial pressure of the steam in the cylinders is very much less than the maximum pressure in the boilers ; generally, indeed, less than one-half.* It is doubtful, however, whether this arises altogether from wire-drawing in the steam-passages and valves ; for when it is con- sidered that in such engines, even at their greatest speed, the steam-valve remains shut nearly the whole of each stroke, being opened during a small portion of the stroke only, it may be regarded as probable that the sudden opening of this valve causes a temporary reduction of temperature and pressure in the boiler, itself. 49. Composite Vapour-Engines. The steam-and-ether engine of M. du Trembley is an example of what may be called a composite vapour-engine, in which two fluids are era- " See Mr. Pole's work on the Cornish Engine, and Art. 36 of a paper on the Mechanical Action of Heat, Trans. Roy. Soc. o/Edin., Vol. XX. {Vide p. 291.) 400 ON THERMODYNAMICS. ployed, a less and a more volatile; the heat given out during the lique- faction of the less volatile fluid serving to evaporate the more volatile fluid, which works an auxiliary engine, and is liquefied in its turn °y refrigeration. Let the efficiency of the engine worked by the less volatile fluid be expressed in the form »-* so that is the fraction of the whole heat expended which is given out to n the more volatile fluid. Let the efficiency of the engine worked by the mure volatile fluid be i-ii n then the efficiency of the combined engines will be 1-- (81.) nn If both the engines are perfect thermodynamic engines, let Tj be the absolute temperature at which the first fluid is evaporated; t, that at which it is condensed, and the second fluid evaporated ; and r 3 that at which the second fluid is condensed ; then, l-Ul- 1 :-'-*; 1-1 = 1-^^; l-±.= l_Il=^ (81 A.) n t 1 — k n To — k nn t x — k bein" equal to the theoretical maximum efficiency of a simple thermo- dynamic engine working between the limits of temperature r t and r 3 . Composite vapour-engines, therefore, have the same theoretical maximum efficiency with simple vapour-engines, and other engines moved by heat, working between the same temperatures; but they may, nevertheless, enable the same efficiency to be obtained with smaller engines. 50. Curves of free expansion for nascent vapour. By nascent vapour is to be understood that which is in the act of risin^ from a mass of liquid. If this vapour be at once conducted to a condenser, without performing any work, and there liquefied at a tempera- ture lower than that at which it was evaporated, its expansion, from the pressure of evaporation down to the pressure of liquefaction, will take place according to a law defined by a curve analogous, in some respects, but not in all, to the curve of free expansion for a homogeneous substance, referred to in Proposition VI. To determine theoretically the form of this curve, it is necessary to know the properties of the isothermal curves and curves of no transmission for the fluid in question in the gaseous state, OX THERMODYNAMICS. 407 when above the temperature of saturation for its pressure. Having these data, we can solve numerically the following problem : — Proposition XIX. — Problem. To draw the curve of free expansion for vapour nascent under a given pressure. (Solution.) In Fig. 24, let AB, parallel to OX, be the isothermal line of an aggregate of liquid and vapour at the pressure of evaporation P x corresponding to the temperature t x : let A v v B v^ be ordinates parallel to OY; so that v 1 is the volume of unity of weight of the liquid at this temperature, and i\ that of unity of weight of the vapour at saturation. Let D F be a line drawn parallel to X, at a distance representing any lower pressure P 2 corresponding to the temperature r . It is required to find the point where the curve of free expansion drawn from B inter- sects DF. Let v., be the volume of unity of weight of the liquid at the lower pressure and temperature, ?;., D an ordinate parallel to Y, and D A a curve representing the law of expansion of the liquid as the pressure and temperature increase. Draw the curves of no transmission D N, B L indefinitely prolonged towards X; ascertain the indefinitely-prolonged area LBADN; draw a curve of no transmission MC, cutting DF in C, such that the indefinitely-prolonged area MCDN shall be equal to the indefinitely-prolonged area LBADN; then will C be the point required where the curve of free expansion B C intersects the line D F. (Demonstration.) Unity of weight of the fluid being raised in the liquid state from the temperature r 2 and corresponding pressure P 2 , to the temperature t x and corresponding pressure P x ; then evaporated com- pletely at the latter pressure and temperature; then expanded without performing work, until it falls to the original pressure P 2 ; then cooled at this pressure till it returns to the original temperature r 2 , at which it is finally liquefied ; the area A B C D represents the expansive power 408 ON THERMODYNAMICS. developed during this cycle of operations, which, as no work is performed, must be wholly expended in agitating the thud, and reproducing by friction the heat consumed by the free expansion represented by the curve B C, which heat is measured by the indefinitely-prolonged area MCBL, which area is therefore equal to the area A B C D. Subtracting from each of these equal areas the common area B U C, and adding to each of the equal remainders the indefinitely-prolonged area LUDN. we form the areas MCDN, LBADN, which are consequently equal. Q. E. D. 51. Of the total heat of evaporation. The symbolical expression of the preceding proposition is formed in the following manner. The area LBADN represents the total heat of evaporation, at the temperature r v from the temperature r 2 , and is composed of two parts, as follows : — LBADN f 1 K h dr + L 11 . • (82.) of which the first is the heat necessary to raise the liquid, whose specific heat is K L , from t 2 to r v and the second is the latent heat of evaporation at t v Let v£ be the volume of unity of weight of the vapour at the pressure P 2 and temperature of saturation t 2 ; draw the ordinate P 2 'E, meeting DF in E, through which point draw the indefinitely-prolonged curve of no transmission Eli: then is the area MCDN divided into two parts, as follows : — MCDN = MCER + EEDX= K r dr + L 2 , . (83.) in which equation r c denotes the temperature corresponding to the point C on the curve of free expansion, and K p the specific heat of the vapour, at the constant pressure P 2 when above the temperature of saturation; so that the first term represents the heat abstracted in lowering the temperature of the vapour from r c to the temperature of saturation r 2 , at the constant pressure P 2 ; and the second term, the latent heat of evaporation at r 2 abstracted during the liquefaction. By equating the formula? (82) and (83), the following equation is obtained : — f 1 K L dT + L 1 -L. 2 = f TC K v dr, . . (84.) 7"., T.-, ON THERMODYNAMICS. 409 which is the symbolical solution of Proposition XIX., and shows a relation between the total heat of evaporation of a fluid, the free expansion of its vapour, and the specific heat of that vapour at constant pressure. 52. Approximate Jaw for a vapour which is a perfect gas. If the vapour of the fluid in question be a perfect gas, and of very great volume as compared with the fluid in the liquid state, the curve; BC will be nearly a hyperbola, and will nearly coincide with the isothermal curve of the higher temperature r v to which, consequently, r c will be nearly equal; and the following equation will be approximately true : I 1 K l4 dr + L X -L,= j 'ly/r, . . (85.) which, when the difference between the higher and lower temperatures diminishes indefinitely, is reduced to the following: — K, + ^ = K„, .... (86.) that is to say: — Corollary. — Theorem. When a vapour is a perfect gas, and very hilly as compared with its liquid, the rate of increase of the total heat of evaporation with temperature is nearly equal to the specific heat of the vapour at constant pressure. This was demonstrated by a different process, in a paper read to the Royal Society of Edinburgh in 1850. It has not yet been ascertained, however, whether any vapour at saturation approaches sufficiently near to the condition of perfect gas to render the theorem applicable. 53. Concluding Remarks. In conclusion, it may be observed, that the theory of the expansive action of heat embodied in the propositions of this paper contains but one principle of hypothetical origin — viz., Proposition XII., according to which the actual heat present in a substance is simply proportional to its temperature measured from a certain point of absolute cold, and multiplied by a specific constant ; and that although existing experimental data may not be adequate to verify this principle precisely, they are still sufficient to prove that it is near enough to the truth for all purposes connected with thermodynamic engines, and to afford a strong probability that it is an exact physical law. 410 MAXIMUM PRESSURE AND LATENT HEAT OF VAPOURS. XXL— OX FORMULAE FOR THE MAXIMUM PRESSURE AXD LATENT HEAT OF VAPOURS.* 1. It is natural to regard the pressure which a liquid or solid and its vapour maintain when in contact with each other and in equiUbrio, as the result of an expansive elasticity in the vapour, balanced by an attractive force which tends to condense it on the surface of the liquid or solid, and which is very intense at that surface, but inappreciable at all sensible distances from it. According to this view, every solid or liquid substance is enveloped by an atmosphere of its own vapour, whose density close to the surface is very great, and diminishes at first very rapidly in receding from the surface ; but at appreciable distances from the surface is sensibly uniform, being a function of the temperature and of the attractive force in question. 2. Many rears since I investigated mathematically the consequences of this supposition, and arrived at the conclusion, that although it is impossible to deduce from it, in the existing condition of our knowledge of the laws of molecular forces, the exact nature of the relation between the temperature and the maximum pressure of a vapour, yet that if the hypothesis be true, it is probable that an approximate formula for the logarithm of that pressure for any substance will be found, by subtracting from a constant quantity, a converging series in terms of the powers of the reciprocal of the absolute temperature, the constant and the coefficients of the series being determined for each substance from experimental data. Such a formula is represented by Log. P = A - - - C , - &a, T T- where P denotes the pressure, r the absolute temperature, that is, the temperature as measured from the absolute zero of a perfect gas-ther- mometer, A the constant term, and B, C, &c, the coefficients of the converging series. 3. On applying this formula to M. Regnault's experiments on the pressure of steam, it was found that the first three terms were sufficient to * Read before the British Association at Liverpool, September, 1854, and published in the Philosophical Magazine, December, 185-4. MAXIMUM PRESSURE AND LATENT HEAT OF VAPOURS. 411 represent the results of these experiments with minute accuracy through- out their whole extent; that is to say, between the temperatures of - 30° and + 230° Centigrade = - 22° and 44G° Fahrenheit, and between the pressures of .,.-} 0() of an atmosphere, and 82 atmospheres. Formula? of three terms were also found to represent the results of Dr. Ure's experiments on the vapours of alcohol and ether, and formulae of two terms those of his experiments on the vapours of turpentine and petroleum, as closely as could be expected from the degree of precision of the experiments. A formula of two terms was found to represent accu- rately the results of M. Eegnault's experiments on the vapour of mercury. 4. These formula?, with a comparison between their results and those of the experiments referred to, were published in the Edinburgh New Philo- sophical Journal for July, 1849, in a paper the substance of which is summed up at its conclusion in the following proposition (See p. 1) : — If the maximum elasticity of any vapour in contact with its liquid be ascer- tained for three points on the scale of the air-thermometer, then the constants of an equation of the form Lo g.P=A- B -^ T T •may be determined, which equation will give, for that vapour, with an accuracy limited only by the errors of observation, the relation behceen the temperature (t), measured from the absolute zero, and the maximum elasticity (P), at all temperatures between those three points, and for a considerable range beyond them. 5. In the case of water and mercury, the precision of the experimental data left nothing to be desired. I have, however, in the table of constants at the end of this paper, so far modified the coefficients for Avater and mercury as to adapt them to a position of the absolute zero (274° Centi- grade, or 493°-2 Fahrenheit below the temperature of melting ice), which is probably nearer the truth than that employed in the original paper, which was six-tenths of a Centigrade degree lower. This modification, however, produces no practically appreciable alteration in the numerical results of the formula?. 6. It was otherwise with respect to the other fluids mentioned, for which the experimental data were deficient in precision, so that the values of the constants could only be regarded as provisional. 7. A summary published in the Comptes Bendus for the 14th of August, 1854,* of the extensive and accurate experiments of M. Eegnault on the * See Phil. Mag., Series 4, Vol. VIII., p. 2G9. 412 MAXIMUM PRESSURE AND LATENT HEAT OF VAPOURS. elasticities of the vapours of ether, sulphuret of carbon, alcohol, chloroform, and essence of turpentine, has now supplied the means of obtaining formula?, founded on data as precise as it is at present practicable to obtain, for the maximum pressures of these vapours. A synopsis of these formulae, and of the constants contained in them, is annexed to this paper. The constants, as given in the table, are suited for millimetres of mercury as the measure of pressures, and for the scale of the Centigrade thermometer; but logarithms are given, by adding which to them they can be easily adapted to other scales. The limited time which has elapsed since the publication of M. Regnault's experiments prevents my being yet able to bring the details of the investigation of the formula?, and of the comparison of their results with those of experiment, into a shape suited for publication ; but I shall here add some remarks on their degree of accuracy and the extent of their applicability. 8. M. Regnault explains, that his experiments were made by two methods; at low temperatures, by determining the pressure of the vapour in vacuo; at high temperatures, by determining the boiling-point under the pressure of an artificial atmosphere. For each fluid the pressures determined by both those methyl- were compared throughout a certain .series of intermediate temperatures. For all Huids in a state of absolute purity, the results of those two methods agreed exactly (as M. Regnault had previously shown to be the case for water). The presence, however, of a very minute quantity of a foreign substance in the liquid under experiment was sufficient to make the pressure of the vapour in vacuo considerably greater than the pressure of ebullition at a given temperature ; and it would appear, also, that a slight degree of impurity affects the accuracy even of the latter method of observation, although by far the more accurate of the two when they disagree. 9. The degree of precision with which it has been found possible to represent the results of the experiments by means of the formula?, cor- responds in a remarkable manner Avith the degree of purity in which, according to M. Regnault, the liquid can be obtained. Sulphuret of Carbon, M. Regnault states, can easily be obtained perfectly pure. For this fluid, the agreement of the pressures computed by the formula with those determined by experiment throughout the whole range of temperature from — 16° Centigrade to + 136°, is almost as close as in the case of steam. Ether and Alcohol are less easy to be obtained perfectly pure. The discrepancies between calculation and experiment in these cases, though still small, are greater than for sulphuret of carbon. For ether the formula may be considered as practically correct through- MAXIMUM PRESSURE AND LATENT HEAT OF VAPOURS. 413 out the whole range of the experiments, from — 20° Centigrade to + 116°; but for alcohol below 0° Centigrade, the discrepancies, though absolutely small quantities, are large relatively to the entire pressures; and the formula can be considered applicable above this temperature only. Essence of Turpentine has been discovered by M. Eegnault to undergo a molecular change by continued boiling. For this fluid the agreement between the formula and the experiments is satisfactory above 40° Centigrade, and up to the limit of the experiments, 222°, but not below 40°. It is impossible to obtain Chloroform free from an admixture of foreign substances. Accordingly, M. Eegnault has found that the two methods of determining the pressure of the vapour of this fluid give widely different results, neither of which can be represented accurately by the formula now proposed below the temperature of 70° Cent. From this temperature, however, up to 130° Cent., the limit of the experiments, the agreement is close. 10. In the cases of alcohol and turpentine, the discrepancies between the formulae and the experiments at very low temperatures are such as to indicate that they might be removed by introducing a fourth term into the formulae, inversely proportional to the cube of the absolute temperature; but the trifling and uncertain advantage to be thus obtained would be outweighed by the inconvenience in calculation, and especially by the necessity for solving a cubic equation in computing the temperature from the pressure; whereas, with formula? of three terms, it is only necessary to extract a square root, as the formula No. 2 shows. 1 1 . Although, for the mere determination of the maximum pressure of a vapour at a given temperature, or its temperature at a given pressure, a table, or a curve drawn on a diagram may be sufficient, still there are many questions of thermodynamics respecting vapours for the solution of which a formula is essential. Amongst these is the computation of the latent heat of evaporation, which is equivalent to the potential energy or work exerted by the vapour in overcoming external pressure, added to that exerted in over- coming molecular attraction. For unity of iceight of a given substance, this is a function of the pressure, temperature, and density; but for a quantity of the substance such that its volume when evaporated exceeds its volume in the liquid or solid state by unity of cubic space, the latent heat of evaporation is simply the differential coefficient of the pressure with respect to the hyperbolic logarithm of the absolute temperature, as shown in the formula No. 3 ; so that, although the densities of the vapours of the seven fluids referred to in this paper are yet known by conjecture only, and not by direct experiment, we can, from the relation between the pressure and the temperature, determine accurately how much 414 MAXIMUM PRESSURE AND LATENT HEAT OF VAPOURS. heat must be expended in the evaporation of so much of each of them as is necessary in order to propel a piston through a given space under a given constant pressure, and thus to solve many problems connected with engines driven by vapours of different kinds. 12. It is somewhat remarkable, that the coefficients of the reciprocal of the temperature (B) in the formulae for ether, sulphuret of carbon, and alcohol, are nearly equal; as also those of the square of the reciprocal of the temperature (C) for ether and sulphuret of carbon. In consequence of this, the pressure of the vapour of ether, and its latent heat for unity of space, as above denned, at a given temperature, exceed the corresponding quantities for sulphuret of carl ion at the same temperature, in a ratio which is nearly, though not exactly, constant, and whose average value is somewhat less than 1*5. Synopsis of the Formula, &c. Notation. t = absolute temperature = temp. Cent. + 274 c C. = temp. Fahr. + 4G1°2 F. P = maximum pressure of vapour at the absolute tempera- ture T. r — volume of unity of weight of the liquid. V = volume of unity of weight of saturated vapour. L = latent heat of evaporation of unity of weight of the fluid expressed in units of work. A, B, C, constants. Formula. 1. To find the maximum pressure from the temperatu:c. Com. \o r ' P = A — — ... ° T T 1 2. To find the temperature from the maximum pressi n 1 / f A - com, log. P B^ ) _ B r ~ V t ~ C + 4 C- i 2 G MAXIMUM PRESSURE AND LATENT HEAT OF VAPOURS. 415 3. To find the latent heat of evaporation (expressed in units of -work) of so much of the fluid that its hulk when evaporated exceeds its bulk in the liquid state by an unit of space, that is to say, of the weight ^ — of fluid. In this formula the pressure must be expressed in units of weight per square unit. = r— =P(- + ■:, X hyp. log. 10. V - o d; (Hyp. log. 10 = 2-30258509, the common logarithm of which is 0-3622157.) Units of work are reduced to units of heat (degrees in unity of weight of liquid water) by dividing by Joule's equivalent of the specific heat of liquid water, which has the following values, according to the units of temperature and length employed : — Logarithms. Centigrade scale, and metres, . . 423 - 54 2-6268969 Centigrade scale, and feet, . . 1389-6 3'1428898 Fahrenheit's scale, and feet, . . 772'0 2-8876173 Constants in the Formulae for Pressures in Millimetres of Mercury, and Temperatuf.es in Centigrade Degrees. Fluids. A. Log. B. Log. C. B 2 0" B2 4C-" Ether, 7 1284 3-0596504 4-7065130 0011275 0-00012712 Sulphuret of carhon, . G-S990 3-0520049 4-7078426 0-011044 000012197 Alcohol above 0° C, . 7 3259 3-0570610 5-2426805 0032610 0-0000 10634 Water, 7-8143 3-1S11430 5-0881S57 0-0061934 0-000038358 Essence of turpentine I above 40° C.,. . \ 6 2522 2 9025209 5-3712157 0-0019511 0000038067 Chloroform above 70° C, 5-S075 2-4007279 5-3919420 00051022 0-00000026032 Mercury up to 35S° C, 7-5243 3-4675637 To adapt the formula? to other scales of pressure, add the following logarithms to the constants A : — 41 6 MAXIMUM PRESSURE AND LATENT HEAT OF VAPOURS. For inches of mercury, . . . . . • 2 5951/ For kilogrammes on the square metre, . . . 1*1 3341 For pounds avoirdupois on the square foot, . . 0*44477 To adapt the formula' to the scale of Fahrenheit's thermometer multiply B by 1*8, and C by (1*8) 2 = 3-24 ; that is to say, Add to log. B, 0-2552725 Add to log. C • 0-5105450. THE DENSITY OF STEAM. 417 XXIL— ON THE DENSITY OF STEAM.* 1 . The object of the present paper is to draw a comparison between the results of the mechanical theory of heat, and those of the recent experi- ments of Messrs. Fairbairn and Tate on the density of steam, published in the Philosophical Transactions, for 1SG0. General Equation of Thermodynamics. 2. The equation which expresses the general law of the relations between heat and mechanical energy in elastic substances was arrived at indepen- dently and contemporaneously by Professor Clausius and myself, having been published by him in Poggendorff's Annalen for February, 1850, and communicated by me to the Royal Society of Edinburgh in a paper which was received in December, 1849, and read on the 4th of February, 1850 (See p. 23 Jf). The processes followed in the two investigations were very different in detail, though identical in principle and in results ; Professor Clausius having deduced the law in question from the equivalence of heat and mechanical energy as proved experimentally by Mayer and Joule, combined with a principle which had been previously applied to the theory of substantial caloric by Sady Carnot, while by me the same law was deduced from the " hypothesis of molecular vortices," otherwise called the " centrifugal theory of elasticity." 3. Although, since the appearance of the paper to which I have referred, the notation of the general equation of thermodynamics has been improved and simplified in my own researches, as well as in those of others, I shall here present it, in the first place, precisely in the form in which I first communicated it to this Society, in order to show the connection between that equation in its original form, and the law of the density of steam, which has since been verified by the experiments of Messrs. Fairbairn and Tate. The equation, then, as it originally appeared in the twentieth volume of the Transactions of the Royal Society of Edinburgh (See p. 24-9) is as follows : — * Read before the Eoyal Society of Edinburgh on April 2S, 1S62, and published in Vol. XXIII. of the Transactions of that Society. 2 D 418 THE DENSITY OF STEAM. in which the symbols have the following meanings : — r, the absolute temperature of an elastic substance as measured from the zero of gaseous tension, a point which was then estimated to be at 274 C- G Centigrade below that of melting ice, but which is now considered to be more nearly at 274° Centigrade, or 493 0, 2 Fahr. below that temperature ; k, A constant, expressing the height on the thermometric scale of the temperature of total privation of heat above the zero of gaseous tension. This constant was then only known to be very small ; according to later experiments, it is either null or insensible ; n M, The ideal or theoretical weight, in the perfectly gaseous state, of an unit of volume of the substance, under unity of pressure, at the temperature of melting ice ; C, The absolute temperature of melting ice, measured from zero of gaseous tension (that is to say, according to the best existing data C = 274° Centigrade, or 493°-2 Fahr.); V, The actual volume of unity of weight of the substance ; 8 V, An indefinitely small increment of that volume; <5 r, An indefinitely small increment of temperature ; U, A certain function of the molecular forces acting in the substance ; + B Q', The quantity of heat which appears, or — B Q', the quantity of heat which disappears, during the changes denoted by o V and Bt, through the actions of molecular forces, independently of heat employed in producing changes of temperature ; such quantity of heat being expressed in equivalent units of mechanical energy. The equation having been given in the above form, it is next shown (See p. 252), that the differential coefficients of the function U have the following values : — - = ---C«M/cZV.^. . (3.) 4. The physical law of which the general equation just cited is the symbolical expression, may be thus stated in words : — The mutual trans- THE DENSITY OF STEAM. 419 formation of heat and mechanical energy during any indefinitely small change in the density and temperature of an elastic substance, is equal to the temperature, reckoned from the zero of absolute cold, multiplied by the complete differential of a certain function of the pressure, density, and temperature; which function is either nearly or exactly equal to the rate of variation with temperature of the work performed by indefinite expansion at a constant temperature. 5. It may be remarked that the quantity, — cp = il hyp. log. r + ^-^ (hyp. log. V - U) j r ■ (±0 = ft hyp. log. r + / — d V J a t J (ft being the real specific heat of the substance in units of mechanical energy), is what, in later investigations, I have called the " thermo- dynamic function;" and that by its use, and by making k = 0,, equation (1) is reduced to the simplified form, -8Q' = rS0-feSr; . . . (5.) but the following notation is more convenient : Let $ h denote the whole heat absorbed by the substance, not in units of mechanical energy, but in ordinary thermal units, and J the value of an ordinary thermal unit in units of mechanical energy, commonly called "Joule's equivalent," so that JSh = k$T- SQ'; then the general equation of thermodynamics takes the form J3/i = rS<£ (6.) G. For the purposes of the present paper, the most convenient form of the thermodynamic function is that given in the second line of equation (i) ; but it may nevertheless be stated, that in a paper read to this Society in 1855, and which now lies unpublished in their archives, it was shown that another form of that function, via. : — P„V„\, , fdV fc+^)hyp.log.r-/^P, . . (7.) P V was useful in solving certain questions ; — - ° denoting the same thing u with - — — in equation (1). C n JM 420 THE DENSITY OF STEAM. Application of the General Equation of Thermodynamics to the Latent Heat and Density of Steam. 7. At the time when the general equation (1) was first published, sufficient experimental data did not exist to warrant its application to the computation of the density of a vapour from its latent heat. But very soon afterwards, various points, which had previously been doubtful, were settled by the experiments of Mr. Joule and Professor William Thomson ; and in particular Mr. Joule, by his experiments, published in the Philoso- phical Transactions for 1850, finally determined the exact value of the mechanical equivalent of a British unit of heat, to which he had been gradually approximating since 18-43, viz.: — J = 772 foot-pounds; and Messrs. Joule and Thomson in 1851, 1852, and 1853, made experiments on the free expansion of gases, especially dry air and carbonic acid, which established the very near, if not exact, coincidence of the true scale of absolute temperature with that of the perfect gas-thermometer ; that is to say, those experiments proved that k in the equation (1) is sensibly = 0. When, with a knowledge of these farts, equation (1) is applied to the phenomenon of the evaporation of a liquid under a constant pressure, and at a constant temperature, it takes the following form : — Jh = r^(Y-v), . . . (8.) where J denotes Joule's equivalent, or 772 foot-pounds per British unit of heat (a degree of Fahrenheit in a pound of liquid water) ; h, The heat which disappears during the evaporation of 1 lb. of the liquid; that is, its latent heat of evaporation in British units; r, The absolute temperature (= temperature on Fahrenheit's scale + 461°'2 Fahr.); P, The pressure under which the evaporation takes place ; V, The volume of 1 lb. of the vapour ; v, The volume of 1 lb. of the liquid. As the latent heat of evaporation of various fluids is much more accurately known by direct experiment than the volume or density of their vapours, the most useful form in which the equation (8) can be put, is THE DENSITY OF STEAM. 421 that of a formula for computing the volume of a vapour from its latent heat, viz.: — v-. + H .... (0.) r dr 8. Results of this formula were calculated by Messrs. Joule and Thomson, . and by Professor Clausius for steam, and showed, as had been expected, a greater density and less volume than the law of the perfectly gaseous condition would give. Some results of the same kind, and leading to the same conclusion, were also computed by me, and published in the Philoso- phical Transactions for 1853-54. But for some years no attempt was made by any one to make a table for practical use of the volumes of steam from equation (9); because the scientific world were in daily expectation of the publication of direct experimental researches on that subject by M. Eegnault. 9. At length, in the spring of 1855, having occasion to deliver to the class of my predecessor, Professor Gordon, a course of lectures on the mechanical action of heat, and finding it necessary to provide the students with a practical table of densities of steam founded on a more trustworthy basis than the assumption of the laws of the perfectly gaseous condition, I ventured upon the step of preparing a table of the densities of steam for every eighteenth degree of Fahrenheit's scale, from 86° to 410° inclusive, with the logarithms of those densities and their differences, arranged so as to enable the densities for intermediate temperatures to be calculated by interpolation. Those tables were published in a lithographed abstract of the course of lectures before mentioned, which is now out of print. The same tables, however, have since been revised, and extended to every ninth degree of Fahrenheit, from 32° to 428°, and have been printed at the end of a work On Prime Movers. An account of the original tables was read to the British Association in 1855.* 10. In the unpublished paper before mentioned as having been read to this Society in 1854, the densities of the vapours of ether and bisulphuret of carbon, under the pressure of one atmosphere, as computed by equation (9), are shown to agree exactly with those calculated from the chemical composition of those vapours. 11. The method of using equation (9) to calculate the volume of one pound of steam, is as follows : — I. Calculate the total heat of evaporation of steam from 32°, at a given temperature T° on Fahrenheit's scale, by Eegnault's well-known formula, * The reason for using 9° Fahr. as the interval of temperature is, that it is equal to 5° Centigrade and to 4° Reaumur, so that the tables can be applied with ease to any- one of those three scales. 422 THE DENSITY OF STEAM. 1091-7 + 0-305 (T°- 32°) . . . (10.) II. From that total heat subtract the heat required to raise 1 lb. of water from 32° to T° Fahr., viz., f cc/T; J 32' c being the specific heat of water, the remainder will be the latent heat of evaporation of 1 lb. of steam at T°; that is to say, h = 1091-7 + 0-305 (T - 32°) - P cdT. . (11.) J 32° In computing the value of the integral in this formula, use has been made of an empirical formula founded on M. Kegnault's experiments on the specific heat of water, as to which, see the Transactions of this Society for 1851, viz. : — P c d T = T - T' + 0-000000103 {(T - 39°-l) 3 - (T - 39°-l) 3 } (11 A.) III. The absolute temperature is given by the formula, r = T + 461°-2 Fahr. . . . (12.) IV The value of r - is deduced from the following formula, first a t published in the Edinburgh Philosophical Journal for July, 1849 (See p. 1) :— Com. log. P = A -»; . . ■ (13.) r t- from which it follows that /' P = 2-302GP( B + ^); . . (14.) dr t V t- / the values of the constants for steam being, — A, for pounds of pressure on the square foot, . 8'2591 log. B for Fahrenheit's scale . . . =3-43642 log. C „ „ . . . =5-59873. V. The volume of a pound of liquid water at the temperature T may be computed with sufficient accuracy for the present purpose by the following formula : — v nearly = 0-00801 Q^ + ^). . . (15.) VI. These preliminary calculations having been made, the formula 9 can now be applied to the calculation of the volume of one pound of THE DENSITY OF STEAM. 423 steam (making J = 772); and by this process the tables already mentioned were computed. Comparison of the Eesults of Theory with those of Messrs. Fairbairn and Tate's Experiments. 12. The experiments of Messrs. Fairbairn and Tate on the density of steam are described in a paper which was read to the Eoyal Society of London, as the Bakerian Lecture, on the 10th of May, 1860, and published in the Philosophical Transactions for ^hat year. The results of those experiments give what is called the " relative volume " of steam : that is, the ratio which its volume bears to that of an equal weight of water at the temperature of greatest density, 39° - l Fahr.; but in the following table of comparison, each of those relative volumes is divided by 62"425, the weight of a cubic foot of water at 39 0, 1 in lbs., so as to give the volume of 1 lb. of steam in cubic feet. The numbers of the experiments are the same as in the original paper; those made at temperatures below 212° being numbered from 1 to 9, and those made at temperatures above 212° from 1' to 14'. Comparison of the Theory with the Experiments of Messrs. Fairbairn and Tate. Volume of One Pound of Steam in Cubic Feet Difference Number of Experiment. Temperature Fahrenheit. Difference. Exper. Vol. By Theory. By Exper. 1 136-77 182-20 132-60 -0 40 i 2 155 33 8510 85-44 -0-34 i 3 159-36 77-64 78-86 - 1'22 _ 1 6 5 4 170-92 6016 59 62 + 0-54 + TT7 5 17148 5943 59-51 -0-08 1 ~TTT 6 174 92 5518 55 07 + 011 + T5T 7 1S2-30 47-28 48-87 -1-59 _ 1 S 188-30 41-81 42 03 -0 22 i T5T 9 198-78 33 94 34 43 -0-49 1 ?T7 1' 242 90 15-61 15 23 + 0-12 + Tf^ 2' 244-82 14-77 14 55 + 0-22 + CTT 3' 245-22 1467 14-30 + 0-37 + & 4' 255-50 12-39 12-17 + 0-22 + eV 5' 263-14 10-96 10-40 + 0-56 + tV 6' 267 21 10-29 10-18 + 0-11 + A 7 269-20 9 977 9-703 + 0-274 + A 8' 274 76 9-158 9-361 -0 203 i 9' 273-30 9 367 8-702 + 0-665 + A 10' 279-42 8-539 8-249 + 0-290 + Tg 11' 2S2-58 8-145 7-964 + 0-181 + A 12' 287*25 7-603 7 340 + 0-263 + T5 13' 292-53 7 041 6-93S + 0-103 + BIT 14' 288-25 7-494 7-201 + 0-293 + 3fV 424 THE DENSITY OF STEAM. Remarks on the Differences between the Theoretical and Experimental Results. 13. The differences between the theory and the experiments as to the volumes of steam at temperatures below 212° are, with few exceptions, of very small relative amount; and they are at the same time so irregular as to show that they must have mainly arisen from causes foreign to the data used in the theoretical computations. 1-1. Above 212° also, the differences show irregularity, especially in the case of experiments 8' and 9', where a fall of temperature is accompanied by a diminution instead of an increase in the volume of one pound of saturated steam, as determined by experiment. But still those differences, presenting as they do, in every case but one, an excess of the theoretical above the experimental volume, show that some permanent cause of discrepancy must have been at work ; although they may not be regular enough to determine its nature and amount, nor large enough to constitute errors of importance in practical calculations relating to steam-engines. 15. So far as it is possible to represent those differences by anything like a formula, they agree, in a rough way, with a constant excess of about 0-21: of a cubic foot in the theoretical volume of one pound of steam above the experimental volume; and this also represents, in a rough way, the difference between the curves whose ordinates express respectively the results of the theoretical formula and those of an empirical formula deduced from the experiments, so far as those curves, as shown in a plate annexed to the paper referred to (See Plate II.), extend through the limits of actual experiment on steam, above 212°. 1G. As the principles of the mechanical theory of heat may now be considered to be established beyond question, it is only in the data of the formula that we can look for causes of error in the theoretical calculation. I shall now consider those data, with reference to the probability of their containing numerical errors. I. Total Heat of Evaporation. — It is very improbable that this quantity, as computed by M. Regnault's formula, involves any material error. II. Sensible Heat of the Liquid Water. — The empirical formula from which this quantity is computed was determined from experiments by M. Regnault which agree extremely well amongst themselves. (For the investigation of the formula, see Trans. Boy. Soc. Edin., Vol. XX., p. 441). The subtraction of the sensible heat from the total heat leaves the latent heat, upon which the increase of volume depends; hence, to account for an error in excess of the formula for the volume by means of an error in THE DENSITY OF STEAM. 425 the computation of the sensible heat, it must be supposed that the specific heat of liquid water above 212° increases much more rapidly than M. Eegnault's experiments show, so as to produce a correspondingly more rapid diminution in the latent heat of evaporation. It is easily computed, for example, that to account for an error in excess of 0*24 of a cubic foot in the volume of one pound of steam at 266°, by an error in defect in the sensible heat, we must suppose that error to amount to about 2 1 British thermal units per pound of water ; but such an error is altogether improbable. III. Absolute Temperature. — The position of the absolute zero may be considered as established with a degree of accuracy which leaves no room for any error sufficient to account for the differences now in question. d P IV. Function r . — The same may unquestionably be said of this (It function ; which represents the mechanical equivalent of the latent heat of evaporation of so much water as fills one cubic foot more in the vaporous than in the liquid state. V. The volume of one pound of the liquid water is itself too small to affect the question. VI. The received value of the mechanical equivalent of a unit of heat cannot err by so much as -^ part of its amount. Conclusions. 1 7. It appears, then, that none of the data from which the theoretical calculations are made are liable to errors of a magnitude sufficient to account for the differences between the results of those calculations and the results of Messrs. Fairbaim and Tate's experiments, small as those differences are in a practical point of view. Neither does there appear to have been any cause of error in the mode of making the experiments. There remains only to account for those differences, the supposition that there was some small difference of molecular condition in the steam whose density was measured in the experiments of Messrs Fairbairn and Tate, above 212°, and the steam whose total heat of evaporation, as measured by M. Eegnault, is the most important of the data of the theoretical formula, — a difference of such a nature as to make a given weight of steam in Messrs. Fairbairn and Tate's experiments occupy somewhat less space, and therefore require somewhat less heat for its production, than 42G THE DENSITY OF STEAM. the same weight of steam in M. Regnault's experiments at the same temperature. That difference in molecular condition, of what nature soever it may have been, was in all probability connected with the fact, that in the experiments of Messrs. Fairbairn and Tate the steam was at rest, whereas in those of M. Eegnault it was in rapid motion from the boiler towards the condenser. It is obvious, however, that in order to arrive at a definite conclusion on this subject, further experimental researches are required. THE SECOND LAW OF THERMODYNAMICS. 427 XXIII— ON THE SECOND LAW OF THERMODYNAMICS.* 1. It has long been established that all the known relations between heat and mechanical energy are summed up in two laws, called respectively the first law and the second law of thermodynamics : viz. — First Law. — Quantities of heat and of mechanical energy are con- vertible at the rate very nearly of 772 foot-pounds to the British (or Fahrenheit- avoirdupois) unit, or 424 kilogrammetres to the French (or Centigrade-metrical) unit of heat. Second Law. — The quantity of energy which is converted from one of those forms to the other during a given change of dimensions and condition in a given body, is the product of the absolute temperature into a function of that change, and of the kind and condition of the matter of the body. By absolute temperature is here to be understood temperature measured according to a scale so graduated that the temperature of a homogeneous body shall vary in the simple proportion of the quantity of energy it possesses in the form of sensible or thermometric heat. 2. The laws of thermodynamics, as here stated, are simply the condensed expression of the facts of experiment. But they are also capable of being viewed as the consequence of the supposition, that the condition of bodies which accompanies the phenomena of sensible heat consists in some kind of motion amongst their particles. 3. The first law would obviously follow from the supposition of any kind of molecular motion whatsoever, and it therefore affords of itself no reason for preferring one supposition as to the kind of molecular motion which constitutes sensible heat to another. 4. But if there be molecular motions in bodies, it is certain that, although all such motions are capable of conversion into that which constitutes sensible heat, some of them are not accompanied by sensible heat. For example, the motion (supposed to be vibratory and wave-like) which constitutes radiance, whether visible or invisible, is not accompanied by sensible heat, and only produces sensible heat by its absorption; that * Read before the British Association at Birmingham, September, 1S65, and published in the Philosophical Magazine, October, 1805. 428 THE SECOND LAW OF THERMODYNAMICS. is, in the language of hypothesis, by its conversion into some other hind of motion; while, on the other hand, in the production of radiance sensible heat disappears. 5. The object of the present paper is to give an elementary proof of the proposition, that the second law of thermodynamics folloivs from the supposition that sensible hint consists in any land of steady molecular motion villi in limited spaces. G. The term " steady motion " is here used in the same sense as in hydrodynamics, to denote motion, whether of a continuous fluid or of a system of detached molecules, in which the velocity and direction of motion of a particle depend on its position only; so that each particle of the series of particles which successively pass through a given position assumes in its turn the velocity and direction proper to that position. In other words, steady motion may be denned as motion in a set of streams of invariable figure. "When steady motion takes place in matter that is confined within a limited space, the streams in which the particles move must necessarily return into themselves, and be circulating streams, being in that respect of the nature of whirls, eddies, or vortices. 7. Steady motion keeps unaltered the distribution of the density of the moving matter; and it therefore keeps unaltered the forces depending on such distribution, whether of the nature of pressure or of attraction. In that respect it differs from unsteady motion, such as vibratory and wave-like motion. 8. Conceive a limited space of any figure whatsoever to be filled with matter in a state of steady motion. The actual energy of any particle of that matter is the product of its mass into the half-square of its velocity; and the actual energy of the whole mass of matter is the sum of all those products; and because of the steadiness of the motion, the actual energy of the particle which at any instant whatsoever occupies a given position is some definite fraction of the whole actual energy, depending upon that position, and upon the distribution of matter within the space; but the scale of absolute temperature is defined as being so graduated that the whole actual energy of the matter within the space is the product of the absolute temperature, the mass of matter, and some function of the sort and distribution of the matter ; therefore, the half-square of the velocity of the particle which at any instant occupies a given position in the space con- sidered, is equal to the absolute temperature multiplied by some function of that position, and of the sort and distribution of the matter. 9. Suppose now that the dimensions of the limited space in which the moving matter is enclosed, and the distribution of that matter, undergo an indefinitely small change by the application of suitable forces, and that after that process the motion becomes steady as it was before. THE SECOND LAW OF THERMODYNAMICS. 429 Then the dimensions and position of each circulating stream will have been altered; and the work done in effecting that alteration will consist of energy converted between the forms of potential energy of the applied forces, and actual energy of the molecular motions — that is, between the forms of mechanical energy and of heat. Consider now a point in one of the circulating streams before the^ change, and let fall from it a perpendicular upon the same stream after the change. The work done in shifting the path of the particle which at any instant occupies that point, is the product of the perpendicular displacement of the stream into the force exerted along that perpendicular. But the perpendicular displacement of the stream is a function of the position of the point shifted, the distribution of matter in the space, and the change of dimensions and distribution; and the force is equal and opposite either to the centrifugal force of the particle or to one of its components, and is therefore proportional to the square of the velocity of the particle, and to some function of its position, and of the sort and distribution of matter in the body. Therefore, the energy transformed in shifting the path of any particle is proportional to the square of its velocity, and to some function of its position, of the sort and distribution of matter in the space considered, and of the change in dimensions of that space and in the distribution of the matter. 10. But the square of the velocity of the particle which at any instant occupies a given position has already been shown to be proportional to the absolute temperature, and to some function of that position and of the sort and distribution of the matter ; therefore, if sensible heat consists in any kind of steady molecular motion within limited spaces, the conversion of energy during any change in the dimensions of such spaces, and in the distri- bution of matter in them, is the product of the absolute temperature into some function of that change and of the sort and distribution of the matter. 11. In a paper "On the Mechanical Action of Heat," published in the Transactions of the Royal Society of Edinburgh for 1850 (See p. %3Jj), the author deduced the second law of thermodynamics, in the form above stated, from the hypothesis of a particular sort of steady molecular motion viz., revolution in circular streams or vortices. In a paper " On the Centrifugal Theory of Elasticity," published in the same Transactions for 1851 (Seep. 4.9), he deduced the same law from the hypothesis of steady molecular motion in circulating streams of any figure whatsoever, being a proposition substantially identical with that set forth in the present paper; but as the demonstration in the paper of 1851 involved tedious and intri- cate symbolical processes, he has written the present paper in order to show that the demonstration can be effected very simply. 12. It is obvious that the steadiness of the supposed molecular motions is the essential condition which makes the second law of thermodynamics deducible from a mechanical hypothesis; and that no kind of unsteady 430 THE SECOND LAW OF THERMODYNAMICS. motion, such as vibratory or wave-like motion, would lead to the same results. If, then, it be admitted as probable, that the phenomena of heat are due to unseen molecular motions, it must also be admitted, that while the motions which constitute radiance may be vibratory and wave- like, the motions which constitute sensible or thermometric heat must be steady and like those of circulating streams. 13. The function by which the absolute temperature is multiplied in calculating the conversion of energy between the mechanical and the thermic forms, is the variation of what the author has called the mcta- morphic function* being one term of the thermodynamic function,} which corresponds to what Professor Clausius calls cntrop'tv.% A P P E X D I X. The following is the symbolical expression uf the demonstration given in the paper. Let m stand for the specific properties of the sort of matter which is in a state of steady motion within a limited space ; f for the figures and dimensions of that space, and of the paths described by the particles contained in it ; and 8/ for any indefinitely small change of such figures and dimensions ; p for the position, relatively to the centre of the matter contained in the space, of a point which is fixed so long as S/ = 0. Because the motion is steady, each particle of matter which successively arrives at the point p assumes the velocity, direction, and curvature of motion proper to that point. Let v be that velocity, and r the radius of that curvature ; then for a particle of mass unity, in the act of traversing p, •> Actual energy of mass 1 = — = kr, . . (1.) where r is a quantity upon whose uniformity throughout the space the steadiness of the motion depends, and k a function of (m, f, p) ; and r 2 2 k T Centrifugal force of mass 1 = = — — ; . . (2.) r r v 2 Tt in which r, and consequently — , are functions of (m, f, p). * See " On the Science of Energetics," page 209. + Philosophical Transactions, 1S54. X Ueber verschiedene fur die Anwendung beqneme Formen der Hauptghkhungen der mechanischen Warmetheorie, April, 1865. THE SECOND LAW OF THERMODYNAMICS. 431 Now, let the change denoted by 8/ take place, and let the steadiness of motion be restored : let 8 n be the length of a line drawn through the original position of the point p, so as to be perpendicular to the path of A the stream of particles which formerly traversed p; and let rn be the angle made by 8 n with r. Then 8 n and r n are both functions of (to, f, p, bf). Also, the work done, or energy converted, for a unit of mass at the point p, while the path of the particles that traverse p is shifted through 8 n, is as follows : — A v 2 ~ A 2 k t. 8 n. cos rn »« ,,» -. c n. cos rn = = t X function of (m,f,p, bf). (3.) The energy converted during the change Sf, throughout the whole space considered, is the sum of the quantities of energy converted for each unit of mass within the space. But r by definition is uniform ; and the sum of a set of functions of p is a function of / and m ; therefore, the whole energy converted is A _ 2 h. Sn. cos r n . ,, , <* fX ,., r.2. = X functions of (?'/?,/, bj); . (4.) r and because bf is indefinitely small, the preceding expression is equivalent to the following : Energy converted = r . function (m,f) .Bf= t . 8F (m,f). (5.) Let t be called absolute temperature, and this is the second law of thermo- dynamics. It is to be observed that / may be, and often is, a function of r. 432 THE SECOND LAW OF THERMODYNAMICS. XXIV.— OX THE WANT OF POPULAR ILLUSTRATIONS OF THE SECOND LAW OF THERMODYNAMICS * 1. The science of thermodynamics is based on two laws, the first of which states the fact of the mutual convertibility of heat and mechanical energy, while the second shows to what extent the mutual conversion of those two forms of energy takes place under given circumstances. In the course of the last few years the first law has been completely "popularised;" it has been amply explained in books and lectures,! composed in a clear and captivating style, and illustrated by examples at once familiar and interesting, so as to make it easily understood by those who do not make science a professional pursuit. 2. The second law, on the other hand, although it is not less important than the first, and although it has been recognised as a scientific principle for nearly as long a time, has been much neglected by the authors of I id] mlar (as distinguished from elementary) works;! and the consequence is that most of those who depend altogether on such works for their scientific information remain in ignorance, not only of the . second law, but of the fact that there is a second law ; and knowing the first law only, imagine that they know the whole principles of thermodynamics. The latter is the worst evil of the two : " a little learning " is not " a dangerous thing " in itself, but becomes so when its possessor is ignorant of its littleness. 3. In the present paper I do not pretend to supply that want, but rather to point out its existence to authors who possess the faculty of popularising ; in order that they, by means of lectures, writings, and lecture-room experiments, may convey a general knowledge of the nature and results of the second law of thermodynamics to those who feel an interest in science without making it a regular study. 4. Before considering how the second law can best be stated and explained, it may be well to point out how far it is possible to proceed * From The Engineer of June 28, 1867. f Such, for example, as Dr. Tyndall's Heat Considered as a Mode of Motion. X The explanation of the second law of thermodynamics in Dr. Balfour Stewart's excellent treatise on heat is elementary ; but it is not, nor does it profess to be, popular. THE SECOND LAW OF THERMODYNAMICS. 433 with the solution of questions in thermodynamics by means of the first law alone, without the aid of the second law. The first law informs us that when mechanical work is done by means of heat, a quantity of heat disappears, bearing a constant ratio to the quantity of work done — viz., that of one British unit of heat (or one degree of Fahrenheit in a pound of water) for every 722 foot-pounds of mechanical work done. In order, therefore, to calculate how much heat will disappear during a given change in the dimensions of a substance under the action of given forces, it is necessary to know the quantity of work done during such change ; and the cases in which the expenditure of heat can be calculated by means of the first law alone, are those and those only, in which the work done can be directly measured ; that is to say, in which the work is sensibly altogether external, or done against forces exerted between the body under consideration and other bodies, and in which no part or no sensible part of the work is internal, or done against forces exerted upon each other by the particles of the body, and therefore incapable of direct measurement. 5. The only phenomenon which fulfils that condition is the expansion of a perfect or sensibly perfect gas ; that is, of a substance in a condition such that its pressure at a given temperature is proportional, or sensibly proportional, to its density simply. To illustrate this by an example : let us suppose one pound of atmospheric air (which, though not absolutely a perfect gas, may be treated as such for practical purposes) at the temperature of melting ice (32° Fahr.), to be contained in a cylinder whose sectional area is equal to one square foot, being confined in that cylinder by means of a piston loaded Avith a pressure amounting to 4233 pounds. Then, from the experiments of Regnault on the pressure and density of air, it is known that the length of the cylindrical space occupied by the air is 6*1 93 ft. Next, let the load on the piston be gradually diminished until it is reduced to 2116| lbs., being one-half of its original amount, and let the question to be solved be — what quantity of heat must be communicated to the air, in order that its temperature may remain constant during the expansion which accompanies the diminution of pressure ? The solution is as follows: — As the temperature is constant, and the air is treated as perfectly gaseous, the product of the pressure on the piston into the volume of the air • that is to say, 4233 x 6-193 = 26215, remains constant during the expansion; and the external work done in driving the piston against the gradually diminishing load is found by multiplying that product by the hyperbolic logarithm of 2, the rate of expansion, that is to say — Work done = 26215 x '69315 = 18171 foot-pounds. 2 E 434 THE SECOND LAW OF THERMODYNAMICS. That is the external work, and the internal work is practically inappreci- able ; therefore, that also is the mechanical equivalent of the quantity of heat required in order to keep the temperature of the air constant; and dividing 18171 by 772, that quantity of heat in ordinary British thermal units is found to be 23*54 j that is, as much heat as would raise the temperature of 23'54 lbs. of water one degree of Fahrenheit; and such is the value of the latent heat of expansion of 1 lb. of air in doubling its volume at the constant temperature of 32° Fain-. Here, then, the solution has been obtained by means of the first law of thermodynamics only, without the aid of the second. 6. It is otherwise Avhen the external work is accompanied by the performance of internal work to a practically important extent; in other words, when we deal with substances which cannot be treated as perfectly gaseous, such as fluids in the act of evaporating. For example, let there be, as before, a cylindrical vessel, whose sectional area is one square foot, and let it contain 1 lb. of water in the liquid state, at the temperature of 212° Fahr., which will occupy a length of the cylinder equal to "017 of a foot. Let the water be confined by means of a piston, the load upon which, in order to be just sufficient to confine the water, must be equal to the mean atmospheric pressure on a square foot, that is to say, 211G3 lbs. If additional heat be now communicated to the water, without altering the load on the pistan, it is well known that its temperature does not rise, but that it passes by degrees into the state of steam, driving the piston before it. By the time the water has entirely assumed the state of steam, it occupies 2G"3G ft. in length of the cylinder; so that the piston has been driven through 26*343 ft. against the constant load of 21 16*3 lbs. Let the question proposed be, to calculate from those data the expenditure of heat. The external work has the following value : — 211G-3 x 2G-343 = 55750 foot-pounds, and this is equivalent to 7 2" 2 units of heat nearly. But besides the external work done in driving the piston, there is internal work done in overcoming the cohesion of the particles of water ; and that internal work is incapable of direct measurement. 7. Here it is that the second law of thermodynamics becomes useful ; for it informs us how to deduce the whole amount of work done — internal and external — from the knowledge which Ave have of the external work. That law is capable of being stated in a variety of forms, expressed in different words, although virtually equivalent to each other. The most convenient form for the present purpose appears to be the following: — To find the whole work, internal and. external, multiply the absolute tempera- ture at which the change of dimensions takes place, oy the rate per degree at which the external work is varied ly a small variation of the temperature. THE SECOND LAW OF THERMODYNAMICS. 435 By absolute temperature is meant temperature measured from the absolute zero, or point of total privation of heat, which is known by theoretical deduction from experimental data to he about 461 0, 2 below the ordinary zero of Fahrenheit's scale. 8. To apply the second law to the present problem, suppose the tempera- ture at which the given increase of volume (viz., 26 - 343 cubic feet) takes place, to be lowered by one degree of Fahrenheit. Then, from tables and formulas founded on Regnault's experiments, we know that the pressure of the steam is diminished by 42*05 lbs. on the square foot. Hence, the external work is diminished by reduction of temperature at the following rate : — F = 42"05 x 26'343 = 1 107'7 foot-pounds per degree of Fahrenheit. The absolute temperature is t = 212° x 461°-2 = G73°-2 Fahr.; and, therefore, the whole work, internal and external, done during the evaporation of 1 lb. of water at the temperature of 212° Fahr. is, t F = 673°-2 x 1107-7 = 745,800 foot-pounds very nearly. To reduce this quantity to British thermal units, divide by 772; the result is 745,800 -r- 772 = 966, being the latent heat of evaporation of steam at 212° Fahr. The total work just calculated is made up as follows : — Foot-pounds. External Avork, computed in Article 6, . . 55,750 Internal work, 690,050 Total work, 745,800 from which it appears that the external work done in evaporating water under the mean atmospheric pressure is less than *l\ per cent, of the whole work; the remainder, or 92^ per cent., being internal work. 9. The second law may also be applied to solve the inverse problem, of deducing from the expenditure of heat in a given process, and from the relations between pressure and temperature, the change of dimensions with which that process is accompanied; and such has been the use chiefly made of that law in the actual history of thermodynamics. Previous to the publication, in September, 1859, of the experiments of 43G THE SECOND LAW OF THERMODYNAMICS. Messrs. Fairbairn and Tate, there did not exist any accurate determinations of the density of saturated steam at different temperatures ; and, therefore, some of the writers on thermodynamics found it necessary to calculate that density theoretically, by the help of the second law, from the latent heat of evaporation, which was very accurately known through the experi- ments of Regnault. The following is an example of such calculations : — Latent heat of evaporation of 1 lb. of water at 212° Fahr., as known by experiment, in ordinary British thermal units, 9GG Applying the first law, that is, multiplying by Joule's equivalent, . . . . . . . . . 772 The value of that latent heat in foot-pounds of work is found to be ........ 7 1 -"» 752 Dividing that quantity of work by the absolute temperature, G73° - 2 Fahr., the work per degree of absolute temperature is found to be, in foot-pounds, . . . . .1107*7 But the variation of pressure per degree, in pounds on the square foot, is ....... . 42'0r> Therefore, the increase of volume of 1 lb. of water in the act of evapo- rating, at 212° Fahr., is 1107-7 -=- 42-05 = 26-343 cubic feet ; To which, adding the volume of the water in the liquid state ....... - 017 „ There is found the volume of 1 lb. of atmospheric steam, 26-360 cubic feet. 10. The example by which the second law of thermodynamics has here been illustrated, has been puposely chosen of a very simple kind ■ but that law enables the most complex questions respecting the expenditure of heat required to produce a given mechanical effect to be solved ; and the solution is always effected in the same manner : that is to say, by deducing the total work, internal and external, from the manner in which a small variation of temperature affects the external work. 11. The law of the efficiency of a perfect heat engine, may be stated thus : If the substance — for example, air or water — which does the work in a perfect heat engine receives all the heat expended at one fixed temperature, and gives out all the heat which remains unconverted into work at a THE SECOND LAW OF THERMODYNAMICS. 4-37 lower fixed temperature, the fraction of the whole heat expended which is converted into external work is expressed by dividing the difference between those temperatures by the higher of them, reckoned from the absolute zero. Now this is, in fact, the second law of thermodynamics expressed in other words ; but whether the demonstration of that fact, that is, of the substantial identity of the second law, as stated in Article 7, with the law stated in the preceding sentence, is capable of being put in a popular form is doubtful, seeing that it involves the notion of limiting ratios. The applications, however, of the law of the efficiency of a perfect heat engine are very simple and easy. For example, it informs us that if the steam which drives an engine receives all the heat expended upon it at temperatures not exceeding 248° Fahr. (corresponding to the absolute temperature 248° + 461°-2 = 709°-2), and if all the heat not converted into external work is given out by that steam at temperatures not below 104° Fahr. (being an ordinary temperature of condensation), the efficiency of that engine, being the fraction of the whole heat expended that is converted into external work, cannot possibly exceed the following value — 248 - 104 144 248 + 461-2 709-2 •203. The same law informs us that in order that the whole heat expended in a heat engine may be converted into external work, it is necessary that the temperature of the condenser or refrigerator should be the absolute zero — a temperature unattainable by human means. Thus, a knowledge of the second law of thermodynamics, as applied to the efficiency of heat engines, is a safeguard against the formation of projects for increasing the perfor- mance of such engines beyond the highest possible limits. 1 2. There seems, then, to be no difficulty in explaining and illustrating, in a popular way, the applications of the second law to various scientific and practical questions, and the agreement of its results with those of experiment, which agreement is the real proof of its being true. But it appears by no means so easy to demonstrate popularly the connection between the second law of thermodynamics with the idea of " heat as a mode of motion." That connection consists in the fact that the second law of thermodynamics necessarily follows from the established laws of dynamics when they are applied to the supposition that the sort of motion which constitutes heat is a whirling or circulating motion of the jwticles of bodies— & supposition otherwise called the " hypothesis of molecular rortkes." The original demonstrations of that fact, which appeared in February, 1850, and December, 1851 {Seep. 49), involve algebraical processes that are quite beyond the range of a popular explanation ; and to understand even the elementary proof without algebra, which was read to the British 438 THE SECOND LAW OF THERMODYNAMICS. Association in 18G5 (See p. 4-27), requires the habit of scientific reasoning. It is much to be wished that some means could be devised for making that proposition as widely understood as the first law of thermodynamics now is; for as matters now stand, the popular knowledge of thermodynamics is, as regards the second law, eighteen years behind the actual state of that science. THE SECOND LAW OF THERMODYNAMICS. 43*) XXV.— EXAMPLES OF THE APPLICATION OF THE SECOND LAW OF THERMODYNAMICS TO A PERFECT STEAM- ENGINE AND A PERFECT AIR-ENGINE* 1 . The following examples are intended to illustrate the application of the second law of thermodynamics to a perfect steam-engine and a perfect air- engine, expending the same quantity of heat, and working between the same limits of temperature, viz. : — Quantity of heat expended per stroke, reduced to an equivalent quantity of mechanical work, in foot-pounds, 68420. Degrees of Fahrenheit. Temperatures. Ordinary scale. Absolute scale. Upper limit, . . . .266° 727 0, 2 Lower limit, .... 104° 565°'2 Difference, or range, . . 1G2° 162 O- 2. The phrase " perfect engine " is here used to denote an engine which realises the greatest quantity of mechanical work possible with the given expenditure of heat and between the given limits of temperature, being the limit towards which actual engines may be made to approach through the progress of practical improvements. Such an engine must fulfil the following conditions : — There must be no waste of heat through conduction or radiation, or of work through friction; and the whole heat expended must be communicated to the working substance at the higher limit of temperature. In other words, the elevation of the temperature of the working substance from the lower to the higher limit must be effected without any expenditure of heat ; for whatever heat is expended in pro- ducing elevation of temperature, is either wholly or partially lost as regards the performance of mechanical work. 3. There are two ways of effecting the elevation of temperature without expenditure of heat. One is to raise the temperature by compression of the working substance in a non-conducting cylinder ; the mechanical work necessary for that purpose being obtained by means of the expansion of the working substance in a non-conducting cylinder while its temperature is falling back from the higher to the lower limit. The other way is by conduction, viz. : — to make the working substance, while its temperature is falling, communicate its heat to a set of metal tubes, or to a network, * From The Engineer of August 2, 1S67- 440 THE SECOND LAW OF THERMODYNAMICS. called a " regenerator " or " economiser," from which the heat is given out again at the proper time to the working substance in order to raise its k, temperature. Supposing those two processes to be carried out in a theoretically perfect manner, the re- sults as regards the economy of heat are exactly the same.* In the follow- ing examples the steam-engine will be supposed to act by compression, and the air-engine by the aid of a perfect regenerator. 4. According to the second law of thermodynamics, the efficiency of each of those engines is the same, viz. : — 162 727"2 = 0-2228; and the work realised per also the same, viz. : — stroke is 68420x0-2228 = 15244 foot-pounds. The object of the following calcula- tions is to show in detail what becomes of the difference between the whole energy expended in the form of heat and that obtained in the form of mechanical work, viz. ; — G8420— 15244 = 531 7G foot-pounds. 1 ieing the rejected or necessarily lost energy. 5. The diagrams of both engines are represented in the figure. Abso- lute pressures are supposed to be re- presented in pounds on the square foot by ordinates measured parallel to OP; volumes occupied by the working substance, in cubic feet, by distances measured parallel to V. • The fact that those results are the same is illustrated in the case of air-engines by numerical examples, which may be found in A Manual of the Steam-Eiufine and other Prime Movers, pages 347 to 369. In practice the regenerator answers best for an air- engine, because of the very large space required for the other process. THE SECOND LAW OF THERMODYNAMICS. 441 The diagram of the steam-engine is marked by full lines and capital letters ; that of the air-engine by dotted lines and italic letters. In the case of the steam-engine the whole work done by the steam in driving the piston is represented by the area FCDE; the work expended in com- pressing part of the steam and feeding the boiler by the area FBAE; and the indicated work by the area BCDA. In the case of the air-engine the whole work done by the air in driving the piston is represented by the area, f bed; the work expended in raising the pressure of the air and feeding the engine by the area fade ; and the indicated work by the area b e d a. 6. To begin in detail with the steam-engine. From the limits of temperature given in Article 1, it is found, by means of the proper table or formula, that the limits of pressure are as follows : — Lb. on the Lb. on the Sq. In. Sq. Ft. Absolute pressure of admission, F, 39 - 25 5652 Absolute back pressure, E, . . 106 152 - G 7. The volume of water in the liquid state which is used per stroke is represented by F B, a distance too small to be seen in the diagram ; the volume of the same water when in the state of steam at the higher limit of pressure by F C ; and, consequently, the increase of volume of the water in the act of evaporating by B C. That increase of volume is produced by communicating to the water the whole heat expended. Now 68420 foot-pounds is the mechanical value of the latent heat of evaporation, under the higher of the given pressures, of so much Avater as fills one cubic foot more in the state of steam than it does in the liquid state ; * so that B C represents one cubic foot. The steam then expands in a non-conduct- ing cylinder, without receiving or giving out heat, until its pressure and temperature fall to their lower limits. C D represents the expansion curve, and E D the volume occupied at the end of the expansion, partly by steam and partly by a small quantity of water which spontaneously liquefies during the expansion. Part of the steam represented in volume by D A, is then condensed by conduction of heat, at the lower limit of temperature ; and a volume of steam represented by A E, less the volume of the water in the liquid state, is left uncondensed, in order that by its compression into the liquid state heat enough may be produced to raise * Formula for that latent heat, in units of work : — ■where p is the absolute pressure and t the absolute temperature. 442 THE SECOND LAW OF THERMODYNAMICS. the temperature of the feed-water to the higher limit. The curve of com- pression, showing how the pressure increases as the volume is diminished, is represented by A B, and this completes the double stroke or revolution of the engine. The following are the successive volumes occupied by the steam at different periods, neglecting the volume of the liquid water, as being so small compared with that of the steam that it is unnecessary to take it into account for the present purpose.* Cubic Feet. Volume on admission, F C, taken as sensibly equal to BC, 100 Volume at end of expansion, E D, .... 24*68 Volume at beginning of compression, E A, . . . t"08 The weight of water expended per stroke is 0'095G lb. 8. The areas shown in the diagram, as computed by the proper formula 1 , given in the footnote to the preceding article, are as follows: — * The following are formulae for these volumes, and for the work represented by the areas in the diagram, first demonstrated by the author in the Philosophical Trans- actions for December, 1853, and given also in A Manual of the Steam-Engine and other Prime Mov< rs. Let t and t' denote the higher and lower limits of absolute tempera- ture ; p and \> the corresponding pressures ; L and L' the corresponding values of l> ; J, Joule's equivalent of the specific heat of liquid water (772 foot-pounds per degree of Fahrenheit in a pound of water) ; H, the value in foot-pounds of the latent heat of evaporation of 1 lb. of water at the higher limit (— 745,S00 - - 7 J (T - A) nearly, where A is the boiling point under one atmosphere) ; then — E A J t' L . . t bc= wn' ******'* E D _ E A t' L BC~BC + fL' ; „ ,, _ „ JL \ . ./, . . t\) L(7 - t') F C D E = H -\t - t ( 1 + hyp. log. t ,J { + -^— ^ ; FBAE=^j«-*'(l + hyp.log.^j; B C D A = L£_Z_*1>. All these formula; were independently demonstrated by Clausius in 1S55. (See his Abhandlungen iiber die mechanische Wdrmetheorie.) If the water at the lower limit of pressure in the example Mere all in the state of steam, it would occupy 29 "9 cubic feet. The difference between this and 24 - 6S — viz., 5 - 22 cubic feet, shows what proportion of the steam is spontaneously liquefied during the given expansion in a non-conducting cylinder. THE SECOND LAW OF THERMODYNAMICS. 4i3 Foot-pounds. Total work obtained through the action of the steam in driving the piston, allowing for back pressure, FCDE = 16690 Work expended through compressing part of the steam into the liquid state, F B A E = . . . .1446 Difference, being the indicated work per stroke, BCDA = 15244 as already calculated in Article 4, by the second law of thermodynamics. The lost or rejected heat, 68420 - 15244 = 53176 foot-pounds, is the heat abstracted during the condensation of the volume of steam represented by A D ; and it is impossible to get back any part of that heat, because it is all abstracted at the lower limit of temperature, and heat will not pass from a colder to a hotter body. 9. To proceed now to the case of the air-engine. The limits of pressure do not, as in the case of steam, depend on the limits of temperature, but may be fixed according to convenience. In general, the lower limit of absolute pressure is the atmospheric pressure, which we will estimate in the present example at 2116-3 lbs. on the square foot, or 14*7 lbs. on the square inch; and this is represented by Oe in the diagram. The upper limit of absolute pressure being arbitrary, we will assume it in the present case to be three atmospheres, or 6348-9 lbs. on the square foot, or 44*1 lbs. on the square inch. It will presently be seen that the weight of air to be used per stroke depends on the proportion borne by the upper limit of pressure to the lower. That weight of air — to be afterwards determined — is drawn into the air-pump at the atmospheric pressure and at the lower limit of temperature ; and it occupies a volume represented in the diagram by e d. It is then compressed into a smaller volume, represented by fa, so that it rises to the higher limit of pressure, and transferred from the pump to the working cylinder ; and, in order that this compression and transfer may cause the smallest possible expenditure of work, the air must be kept during the compression at the lower limit of temperature by means of a proper refrigerating apparatus for abstracting all the heat that the compression generates. None of that heat can be got back, for it is all abstracted at the lower limit of temperature. The temperature, then, being uniform, the volume varies inversely as the absolute pressure ; fa, in the present example, is one-third of e d, and the curve d a is a common hyperbola. The work expended in the air-pump, which is the exact 444- THE SECOND LAW OF THERMODYNAMICS. equivalent of the heat generated there and abstracted by the refrigerating apparatus, is represented by the area fade, and is computed in foot-pounds by multiplying together the following factors: — The constant 53 - 15; the hyperbolic logarithm of the ratio compression (hyp. log 3 = 1'0986); the absolute temperature at which the compression takes place in degrees of Fahrenheit (in this case 565 0, 2 Fahr.); and the weight of air used, in pounds. 10. The air on its way from the pump to the working cylinder passes, without change of pressure, through a perfect regenerator, in which all the heat given out by the previous supply of air is stored up; and it thus rises to the higher limit of absolute temperature, and at the same time undergoes dilatation from the volume fa to the volume///, which volumes are to each other in the ratio of the limits of absolute temperature; viz.: — fa : fh : : 5G5-2 : 727*2 : : 1 : 128G7. 11. The next process is that the air continues to expand and drive the piston until its pressure falls to the lower limit, its volume at the same time increasing to that represented by c c; and it is then finally expelled, giving out to the regenerator to be used over again the heat correspond- ing to the difference between the upper and lower limits of temperature. In order that the greatest quantity of work possible may be obtained from the expansion represented by the curve be, and in order also that the air during its expulsion may give out to the regenerator a quantity of heat sufficient to raise the temperature of the next supply of air to the higher limit, the temperature must, by the supply of a sufficient quantity of heat, be maintained uniform during the expansion represented by be; and that quantity of heat constitutes the whole expenditure in a perfect air-engine. Such being the case, the volume varies inversely as the pressure ; e c, in the present example, is three times fb, and the curve be is a common hyper- bola. It is evident, moreover, that the volumes represented by e d and e c are to each other in the ratio of the limits of absolute temperature, already given. The whole work obtained by the action of the air in the cylinder, which is the exact equivalent of the whole heat expended, is represented by the area/Jce, and is computed in foot-pounds by multiplying together the following factors. The constant 53 - 15 ; the hyperbolic logarithm of the ratio of expansion (hyp. log. 3 = 1 - 098G); the absolute temperature at which the expansion takes place (in this case 727° - 2 Fahr.) : and the weight of air used, in pounds. 1 2. The area bed a represents the indicated work per stroke, being the excess of the work obtained in the cylinder above the work expended in the air-pump ; and the proportion which it bears to j 'bee (representing the whole expenditure of heat) is obviously that of the range of temperature THE SECOND LAW OF THERMODYNAMIC'S. 445 to the higher absolute temperature, as already stated in Article 3, viz. : 162 — — - = 0'2228. In other words, the areas shown in the diagram repre- 727-2 ° l sent the following quantities : — ■ Foot-pounds. Total expenditure of heat per stroke, / b c e = . . 68420 Heat produced by compression and abstracted by the refrigerator, fa de = . . . . . .53176 Indicated work per stroke, b c d a = . . . .1524 + The indicated work per stroke may be calculated independently by multi- plying together the following factors: — The constant 53 - 15; the hyperbolic logarithm of the ratio of expansion (hyp. 3 = 1-0986); the range of absolute temperature (162° Fahr.); and the weight of air used, in pounds. 1 3. The weight of air used per stroke is determined by the considera- tion, that it is to be sufficient to absorb when expanding to three times its initial volume at the absolute temperature 7 2 7° '2 Fahr., a quantity of heat equivalent to 68420 foot-pounds. The calculation is as follows : — Constant factor depending on the nature of the gas in foot-pounds per degree of Fahrenheit, . . 53- 15 Multiply by the absolute temperature, 727 3 -2F. Product in foot-pounds, being equal to the product of the absolute pressure in pounds on the square foot, and the volume of 1 lb. of air in cubic feet, 38651 Multiply by the hyperbolic logarithm of the rate of expansion, hyp. log. 3 = . . ■ • 1*0986 Product being the heat absorbed by each pound of air in foot-pounds, . . . . • • 42516 Heat expended 68420 _ 1-6113 lb dividedby - 42510 " of air used per stroke. With a different ratio of expansion the weight of air required per stroke, would be different, varying inversely as the logarithm of that ratio. Had any other gas been employed instead of air the only difference in this quantity would have been that arising from a 410 THE SECOND LAW OF THERMODYNAMICS. different value of the constant factor. The following are the volumes occupied by the air at different periods : — ed J" J h Cubic Feet. ^2-87 7-62 981 29-43 14. The following is a summary of the comparison between the steam- engine and the air-engine in the example given: — Per Stroke. Steam-Engine. POO) in >umls. Air- Engine. Foot-pounds. Difference. Heat expended, ..... Heat rejected Heat transformed into mechanical work, Work obtained by expansion, Work expended in compression. . Indicated work, ( 16420 53176 • 120 53176 l :.•_• 1 1 15244 0-2228 0-2228 16690 1440 GS420 53176 51730 51730 15244 15244 The quantities in the column headed " difference," represent a diminu- tion of the work obtained by expansion, and an exactly equal saving in the work expended in compression, produced by the mutual attraction of the particles of water. 15. It may be useful to point out what effects Avould be produced by departing from that condition of maximum efficiency which requires that the elevation of temperature shall be produced without the expenditure of heat. In the case of the steam-engine there would be a gain of indicated work, represented by the area FBAE = 144G foot-pounds, so that the indicated work per stroke would be 16690 foot-pounds. At the same time there would be an additional expenditure of heat, calculated as follows : — THE SECOND LAW OF THERMODYNAMICS. 4-47 Joule's equivalent of the specific heat of liquid water, 772 Number of degrees of Fahr. by which the temperature of the feed-water has to be raised, . . . 162° Weight of water used per stroke (lb.), . . . 0-0956 Product of those three factors, being the additional expenditure of heat per stroke in foot-pounds. . 11956 Expenditure of heat as formerly calculated, . . 68420 Total, 80376 Efficiency, as diminished by omitting the compression, 16690 _ 80376 ~ 0-208 being about fifteen-sixteenths of the greatest possible efficiency with the given limits of temperature. 16. In the case of the air-engine, suppose the regenerator omitted, so that the whole elevation of temperature of the air has to be produced by heat supplied from the furnace ; then, in the present example, there would be an additional expenditure of heat, calculated as follows : — Dynamical equivalent of the specific heat of air under constant pressure, in foot-pounds per degree of Fahr., 183-45 Elevation of temperature, . . . . 162° F. Weight of air used per stroke, the rate of expansion being 3, 1-6113 lb. Product of those three factors, being the additional expenditure of heat in foot-pounds, . . . 47886 Expenditure of heat as formerly calculated, . . 68420 Total, 116306 Efficiency, as diminished by altogether omitting the 1 5944 regenerator . '" - = . . . . .0-131 nearly, 116300 448 THE SECOND LAW OF THERMODYNAMICS. or about six-tenths of the greatest possible efficiency with the given limits of temperature. 17. It is evident that the regenerator, or some equivalent apparatus, cannot be omitted consistently with economy of heat in an air-engine. An actual regenerator, however, will never succeed in storing and giving out the whole supply of heat required for the elevation of temperature. Suppose, for the sake of illustration, that in the present example the regenerator stores and gives out nine-tenths of the heat required, leaving one-tenth to be supplied from the furnace, then we have the following expenditure of heat per stroke : — Foot-pounds. Expenditure a* originally calculated, . . . G8420 Additional expenditure through imperfect action of the regenerator, ...... 4789 Total 73209 Efficiency, as diminished by imperfect action of the 15244 regenerator = 0-208 nearly, or about fifteen-sixteenths of the greatest possible efficiency with the given limits of temperature. 18. The law which the preceding examples illustrate leads to the con- clusion that if by means of air-engines greater economy of fuel than in steam-engines is to be attained, it must be by the following means : — Working with a greater range of absolute temperature than is practicable or safe in steam-engines, and using the products of combustion directly to drive the piston, so as to save nearly the whole of the heat that is wasted in a steam boiler, or in an air-engine in which the products of combustion are not so used ; and it is probable that the latter is the more easily practicable of the two means of economising heat. 19. The following examples illustrate the relation between the efficiency of a heat engine and the consumption of fuel per indicated horse-power per hour. The fuel is supposed to be nearly pure carbon, the mechanical equivalent of the whole heat produced by the combustion of one pound of it being 11,000,000 foot-pound'. THE SECOND LAW OF THERMODYNAMICS. 449 Efficiency. Indicated Work per Pound of Carbon. • Pound Carbon per Indicated Horse-power per Hour. 1 11,000,000 0-18 0-5 5,500,000 0-36 0-25 2,750,000 0-72 0-20 2,200,000 0-90 0-15 1,650,000 1-20 0-12:. 1,375,000 1-44 o-ioo 1,100,000 1-80 0-075 825,000 2-40 0-050 550,000 3-60 0-025 275,000 7-20 SUPPLEMEX T. 20. I think it desirable to add the following explanations and illustra- tions of the principle, that in an air-engine, under all circumstances whatsoever, the heat produced hj the compression of the air is wholly and unavoidably lost — a principle which is a necessary consequence of the fact that heat never passes directly from a colder to a hotter body. 21. The heat produced by the compression must either be abstracted from the air as fast as it is produced, so as to keep the temperature of the air constant during the compression, or allow r ed to accumulate and raise the temperature of the air. In the former case (which is that described in Article 9) it is at once evident that the heat, being abstracted by means of an external substance, such as water or air, that is colder than the lowest temperature of the working substance, can never be transmitted to the working substance again. 22. In the latter case, when the heat produced by the compression is not abstracted, but allowed to accumulate and raise the temperature of the air, the air is passed through the regenerator or econo miser on its way 2 F 450 THE SECOND LAW OF THERMODYNAMICS. towards the working cylinder at an increased temperature higher than the lower limit. Therefore, the air which has done its work and is escaping through the regenerator in the opposite direction is not reduced below the same increased temperature, for it is only through the entering air being at a lower temperature that any heat can be abstracted from the escaping air. Therefore, the escaping air carries off to waste a quantity of heat equal to the quantity of heat produced by compression and allowed to accumulate in the compressed air, so that the effect of such accumulation is neutralised, and the total expenditure of heat remains unaltered. 23. Moreover, the increase of temperature of the air undergoing com- pression causes an increase in the quantity of work expended in compressing it above the quantity of work which would be required if the temperature were kept constant ; and thus the indicated work is diminished, and the engine ceases to be one of maximum efficiency between the given limits of temperature. 24. To illustrate this by an example, I will suppose that the compres- sion of the air takes place in a non-conducting cylinder, and is carried to an extent such that the temperature of the air is raised from the lower limit to tin- higher limit by means of the compression alone. Then it is known that the pressure of the air, instead of varying inversely as the volume simply, varies inversely as that power of the volume whose index is l - 408, while the absolute temperature varies inversely as that power of the volume whose index is 0'408. In Fig. 2 (as in the former figure) let volumes in cubic feet be represented by distances parallel to OV, and pressures in pounds on the square foot by ordinates parallel to OP, so that the areas of diagrams represent quantities of work in foot-pounds. Let O E represent the original Fit absolute pressure, E D the original volume . of the air, and F the increased pressure at the end of the compression. Then, if the tempera- ture were kept uniform, the compression curve of the diagram woidd be a common hyperbola D A, such that OF xFA= OE x ED. But the THE SECOND LAW OF THERMODYNAMICS. 451 rise of temperature causes the compression curve to assume a steeper figure, D B, of which the law has already been stated. The absolute FB temperature at the end of the compression is increased in the ratio ^rr', and the work done in driving the compressing pump is represented by the area of the diagram EFBD, and is equivalent to the quantity of heat required in order to produce the rise of temperature from the lower to the higher limit under constant pressure. Now, taking the same data as before, viz. : — Degrees of Fahrenheit. Temperatures. Ordinary scale. Absolute scale. Upper limit, . . . .266° 727°*2 Lower limit, .... 104° 565°*2 Difference, . . . .162° 162°*0 Ratio in which the absolute temperature is to be increased, FB 727°*2 FA 565°*2 = 1-2867; Dynamical value of the specific heat of air under constant pressure, 183*45 foot-pounds per degree of Fahrenheit. Original pressure, OE = 2116*3 lbs. on the square foot ; Original volume of one pound of air, 14*19 cubic feet; We obtain the following results : — ED Ratio of volumes, :=-= — 1*8546 ; Jb x> ■p.* * 0F ED ED FB i-^±r Ratio of pressures, ^ = ^ = ^ ' YJ. = X 1*2867 = 2*3862; Pressure at end of compression OF = 5050 pounds on the square foot; Volume of one pound of air at end of compression, 7*65 cubic feet; Work done in compressing pump per pound of air, 162° x 183*45 = 29719 foot-pounds. 25. The air being now at the upper limit of temperature, let it be transferred to the working cylinder, and there expanded without allowiug 4-32 THE SECOND LAW OF THERMODYNAMICS. the temperature to fall below that limit, until the pressure falls to the atmospheric pressure E, and then let it be discharged. The expansion- curve B C will, as in Article 11, be a common hyperbola, so that the ratio of expansion will be ^-^-2-3862 BF - OE- Jd8bJ ' and the volume of one pound of air at the end of the expansion will be 18 '2 6 cubic feet. The area of the diagram FBCE will represent at once the work done in driving the piston, and the whole expenditure of heat. The method of computing that area has already been stated in Article 11, and the calculation in the present case is as follows, for each pound of un- used : — Constant factor, . . . . . . . 53-15 Absolute temperature at which the expansion takes place, 727°-2 F. Hyperbolic logarithm of the ratio of expansion; hyp. log. 2*3862 = 0*8697 Product of those three factors, being the work done and heat expended in the working cylinder, per pound of air used, in foot-pounds, . . . 336 K> 26. The air finally escapes at the higher limit of temperature, and therefore carries to waste the whole of the heat which was employed in raising its temperature, having been produced by compression. The value of that heat in foot-pounds per pound of air has been already found, in Article 24, to be 29719. 27. The indicated work represented by the area B C D is consequently, in foot-pounds per pound of air, 33615 - 29719 = 389G ; and the efficiency of the engine, 3896 33615 = ' 11Gnear1 ^' or about one-half of the greatest possible efficiency with the given limits of temperature. 28. In order that the present example may be the more easily com- pared with the former examples, we will suppose that the heat to be expended per stroke is, as before, equivalent to 68420 foot-pounds. THE SECOND LAW OF THERMODYNAMICS. 453 Then the expenditure of air per stroke required in order to take up that heat during its expansion is found to be ||^= 20354 lb,; whence the following results are obtained : — Foot-lbs. Waste heat per stroke equivalent to the work of driving the compressing pump, 162° x 183'45 X 2-0354 = G0490 Indicated work per stroke, 3896 X 2-0354 = . . 7930 7930 Efficiency,—- — = 0T16, as already found. Volumes successively occupied by the air in cubic feet per stroke : ED = 2889; FB = 15'58; EC = 3717. 29. The case of Article 9 — where the whole of the heat generated in the compressing pump is abstracted as fast as it is produced — and that of the example just described — in which the whole of that heat is at first employed in raising the temperature of the air while an equal quantity of heat goes to waste with the escaping air that has done its Avork — form two extremes, and between those extreme cases there may lie an indefinite number of intermediate cases, in which part of the heat generated by com- pression is abstracted at once, and part employed in raising the tempera- ture of the air. It will be readily understood that in all those intermediate cases the result is the same as in the two extreme cases — the whole of the heat generated by the compression of the air goes to waste either at once or with the escaping air, and none of it is available for conversion into indicated work; nor would it be possible that it should become so available, unless it were possible for heat to be directly transferred from a colder body to a hotter body. 454 THE WORKING OF STEAM IN COMPOUND ENGINES. XXVI.— OX THE WORKING OF STEAM