THE GIFT OF £.£: //cisyf<^// 3 1924 101 120 883 The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924101120883 THE MECHANICAL THEORY OP HEAT. THE MECHANICAL THEOKY OP HEAT. BY R CLAUSIUS. TRANSLATED BY WALTER E. BROWNE, M.A., LATE FELLOW OF IRINIIT COLLEOS, OAMBKIDGB. Hontjon : MACMILLAN AND CO. 1879 Camirtiige : PEINTED BY C. J. CLAY. M.i AT THE DNIVEESITY PiffiSF. TRANSLATOE'S PREFACE. The following translation was undertaken at the instance of Dr T. Archer Hirst, F.R.S., the translator of the first collected edition (mentioned in the Author's Preface below) of Professor Clausius' papers on the Mechanical Theory of Heat. The former work has however been so completely re- written by Professor Clausius, that Dr Hirst's translation has been found scarcely anywhere available ; and I must there- fore accept the full responsibility of the present publication. I trust it may be found to supply a want which I have reason to believe has been felt, namely, that of a systematic and connected treatise on Thermodynamics, for use in Universities and Colleges, and among advanced students generally. With the view of rendering it more complete for this purpose, I have added, with the consent of Professor Clausius, three short appendices on points which he had left unnoticed, but which still seemed of interest, at any rate to English readers. These are, (1) The Thermo-elastic pro- vi translatoe's preface. perties of Solids ; (2) The application of Thermo-dynamical principles to Capillarity ; (3) The Continuity of the Liquid and Gaseous states of Matter. My best thanks are due to Dr John Hopkinson, F.E..S., both for the suggestion of these three points, and also for the original and very elegant investigation from first principles, contained in the first Appendix, and in the commencement of the second. My thanks are also due to Lord Rayleigh, F.R.S., E. J. Routh, Esq., and Professor James Stuart, for kindly looking through the first proof of the translation, and for various valuable suggestions made in connection with it. WALTER E. BROWNE. 10, ViCTOKiA Chambees, Wesimiksteb, November, 1879. AUTHOR'S PREFACE. Many representations having been made to the author from different quarters that the numerous papers "On the Me- chanical Theory of Heat," which he had published at different times during a series of years, were inaccessible to many who, from the widespread interest now felt in this theory, were anxious to study them, he undertook some years back to publish a complete collection of his papers relating to the subject. As a fresh edition of this book has now become necessary, he has determined to give it an entirely new form. The Mechanical Theory of Heat, in its present development, forms already an extensive and independent branch of science. But it is not easy 'to study such a subject from a series of separate papers, which, having been published at different times, are unconnected in their form, although they agree in their .contents. Notes and additions, however freely used to explain and supplement the papers, do not wholly over- come the difficulty. The author, therefore, thought it best so to re-model the papers that they might form a connected whole, and enable the work to become a text-book of the science. He felt himself the more bound to do this because Vlll AUTHOES PREFACE. his long experience as a lecturer on the. Mechanical Theory of Heat at a Polytechnic School and at several Universities had taught him how the subject-matter should be arranged and represented, so as to render the new view and the new method of calculation adopted . in this somewhat difficult theory the more readily intelligible. This plan also enabled him to make use of the investigations of other writers, and by that means to give the subject greater completeness and finish. These authorities of course have been in every case duly recognized by name. During the ten years which have elapsed since the first volume of papers appeared, many fresh investigations into the Mechanical Theory of Heat have been published, and as these have also been discussed, the contents of the volume have been considerably increased. Therefore in submitting to the public this, the first part of his new investigation of the Mechanical Theory of Heat, the author feels that, although it owes its origin to the second edition of his former volume, still, as it contains so much that is fresh, he may in many respects venture to call it a new work. R. OLAUSIUS., Bonn, December, ]87o. TABLE OF CONTENTS. MATHEMATICAL INTRODUCTION. ON MECHANICAL WOKK, ON ENERGY, AND ON THE TREATMENT OP NON-INTEGEABLE DIFFERENTIAL EQUATIONS. §3. Defiuition and Measurement of Mechanical Work Mathematical Determination of the Work done by variable com- ponents of Force Integration of the Differential Equations for Work done Geometrical interpretation of the foregoing results, and obaer- Tations on Partial Differential Coefficients Extension of the foregoing to three Dimensions On the Brgal General Extension of the foregoing Eelation between Work and Vis Viva . On Energy PAGE 1 9 11 13 17 19 CHAPTER I. FIRST MAIN PRINCIPLE OP THE MECHANICAL THEORY OP HEAT. EQUIVALENCE OF HEAT AND WORK. § 1. Starting Point of the theory 21 § 2. Positive and Negative Values of Mechanical Work ... 22 § 3. Expression for the First Main Principle 23 § 4. Numerical Eelation between Heat and Work .... 24 § 5. The Mechanical Unit of Heat 25 § 6. Development of the First Main Principle 27 § 7. Different Conditions of the quantities J, W, and K . . .27 § 8. Energy of the Body 30 § 9. Equations for Finite Changes of Condition — Cyclical Processes . 32 § 10. Total Heat — ^Latent and Specific Heat 33 § 11. Expression for the External Work in a particular case . . 35 X CONTENTS. CHAPTER II. ON PERFECT GASES. PAOE § 1. The Gaseous Condition of Bodies 38 § 2. Approximate Principle as to Heat absorbed by Gases ... 42 § 3. On the Form which the equation expressing the first Main Prin- ciple assumes in the case of Perfect Gases .... 43 § 4. Deduction as to the two Specific Heats and transformation of the foregoing equations 46 § 5. Eelation between the two Specific Heats, and its appUcatiou to calculate the Mechanical Equivalent of Heat .... 48 § 6. Various Formulse relating to the Specific Heats of Gases . . 52 § 7. Numerical Calculation of the Specific Heat at Constant Volume . 56 § 8. Integration of the Differential Equations which express the First Main Principle in the case of Gases 60 § 9. Determination of the External Work done during the change of Volume of a Gas 64 CHAPTER III. SECOND MAIN PEINCIPLE OF THE MECHANICAL THEORY OF H!eAT. § 1. Description of a special form of Cyclical Process .... 69 § 2. Eesult of the CycHoal Process 71 § 3. Cyclical Process in the case of a body composed partly of Liquid and partly of Vapour 73 § 4. Camot's view as to the work performed during the Cyclical Process 76 § 5. New Fundamental Principle concerning Heat . . . ' . 78 § 6. Proof that the Eelation between the Heat carried over, and that converted into work, is independent of the matter which forms the medium of the change ...... 79 § 7. Determination of the Function (Tj Tj) 81 § 8. Cyclical Processes of a more complicated character ... 84 § 9. Cyclical Processes in which taking in of Heat and change of Temperature take place simultaneously 87 CHAPTER IV. THE SECOND MAIN PEINCIPLE UNDER ANOTHER FORM; OR PRIN- CIPLE OF THE EQUIVALENCE OF TEANSFORJIATIONS. § 1. On the different kinds of Transformations 91 § 2. On a Cyclical Process of Special Form 92 § 3. On Equivalent Transformations 86 CONTENTS. XI PAOB § 4. Equivalence Values of the Transformations 97 § 5. Combined value of all the Transformations wliieh take place in a single CycHeal Process . . . . , . ' . 101 § 6. Proof that in a reversible Cyclical Process the total value of all the Transformations must be equal to nothing . . . 102 § 7. On the Temperatures of the various quantities of Heat ; and the Entropy of the body 106 § 8. On the Function of Temperature, t . . . . . .107 . CHAPTER V. FORMATION QF THE TWO FUNDAMENTAL EQUATIOlfS. § 1. Discussion of the Variables which determine the Condition of the body . , 110 § 2. Elimination of the quantities U and S from the two fundamental equations . ; . 112 § 3. Case in which the Temperature is taken as one of the Independent Variables 115 § 4. Particular Assumptions as to the External Forces . . . 116 § 5. Frequently occurring forms of the differential equations . . 118 § 6. Equations in the case of a body which undergoes a partial change in its condition of aggregation 119 § 7. Clapeyron's Equation and Carnot's Function .... 121 CHAPTER VI. APPLICATIOjST of the mechanical theory of heat to SATURATED VAPOUR. § 1. Fundamental equations for Saturated Vapour .... 126 § 2. Specific Heat of Saturated Steam 129 § 3. Numerical Value of h for Steam 133 § i. Numerical Value of h for other vapours 135 § 5. Specific Heat of Saturated Steam as proved by experiment . . 139 g 6. Specific Volume of Saturated Vapour 142 I 7. Departure &om the law of Mariotte and Gay-Lussac in the case of Saturated Steam ........ 143 5 8. Differential-coefficients of — 148 §9. A Formula to determine the Specific volume of Saturated Steam, and its comparison with Experiment 150 1 10. Determination of the Mechanical Equivalent of Heat from the behaviour of Saturated Steam 155 Xll CONTENTS. PAGE § 11. Complete Differential Ec^aation for Q in the case of a body composed both of Liquid and Vapour 156 § 12. Change of the Gaseous portion of the mass .... 157 § 13. Belation between Volume and Temperature .... 159 § 14. Determination of the Work as a function of Temperature . . 160 CHAPTER VII. FUSION AND VAPORIZATION OF SOLID BODIES. i 1. Fundamental Equations for the process of Fusion . . . 163* i 2. Eelation between Pressure and Temperature of Fusion . . 167 i 3. Experimental Verification of the Foregoing Eesult . . . 168 i 4. Experiments on Substances which expand during Fusion . . 169 i 5. Eelation between the Heat contained in Fusion and the Tempe- rature of Fusion 171 I 6. Passage from the solid to the gaseous condition .... 172 CHAPTEE VIII. ON HOMOGENEOUS BODIES. § 1. Changes of Condition without Change in the Condition of Aggre- gation , 175 § 2. Improved Denotation for the differential coefficients . . . 176 § 8. Eelations between the Differential Coefficients of Pressxire, Volume, and Temperature 177 § 4. Complete differential equations for Q 178 § 5. Specifie Heat at constant volume and constant pressure . . 180 § 6. Specific Heats under other oiroumstanees 184 § 7. laentropic Variations of a body 187 § 8. Special form of the fundamental equations for an Expanded Eod 188 § 9. Alteration of temperature during the extension of the rod . . 190 § 10. Further deductions from the equations 192 CHAPTEE IX. DETEKMINATION OF ENERGY AND ENTROPY. i 1. General equations 195 I 2. Differential equations for the case in which only reversible changes take place, and in which the condition of the Body is determined by two Independent Variables .... 197 CONTENTS. Xlll PAGE § 3. Introdnction of the Temperature as one of the Independent YatiaWes 200 § 4. Special case of the differential equations on the assumption that the only External Force is a uniform surface pressure . . 203 § 5. Application of the foregoing equations to Homogeneous Bodies and in particular to perfect Gases 20S § 6. Application of the equations to a body composed of matter in two States of Aggregation 207 § 7. Kelation of the expressions Dxy and Axy 209 CHAPTER X. ON NON-BBVEESIBLE PKOCESSES. § 1. Completion of the Mathematical Expression for the Second Main Principle 212 § 2. Magnitude of the Uncompensated Transformation . . . 214 § 3. Expansion of a Gas unaccompanied by Work .... 215 § 4. Expansion of a Gas doing partial work 218 § 5. Methods of experiment used by Thomson and Joule . . . 220 § 6. Deyelopment of the equations relating to the above method . 221 § 7. Besults of the experiments, and the equations of Elasticity for the gases, as deduced therefrom 224 § 8. On the Behaviour of Vapour during expansion and under various circumstances 228 CHAPTER XL APPLICATION OF THE THEOKT OF HEAT TO THE STEAM-ENGINE. i 1. Necessity of a new investigation into the theory of the Steam- Bngiue 234 j2. On the Action of the Steam-Engine 236 I 3. Assumptions for the purpose of SimpMoation .... 238 I 4. Determination of the Work done during a single stroke . . 239 i 5. Special forms of the expression fovmd in the last section . . 241 \ 6. Imperfections in the construction of the Steam-Engine . . 241 I 7. Pambour's Eormulse for the relation between Volume and Pres- sure 242 J 8. Pambour's Determination of the Work done during a single stroke 244 ) 9. Pambour's Value for the Work done per unit- weight of steam . 247 i 10. Changes in the Steam during its passage from the Boiler into the Cylinder 248 XIV CONTENTS, PAOE § 11. Divergence of the Eesnlts obtained from Pambour's Assumption 251 § 12. Determination of the work done during one stroke, taking iato consideration the imperfections abeady noticed . . » 252 § 13. Pressure of Steam in the Cylinder during the different Stages , of the Process and corresponding Simplifications of the Equations 255 § 14. Substitution of the Vohune for the corresponding Tempera- ture in certain cases . . 257 § 15. Work per Unit-weight of Steam 259 § 16. Treatment of the Equations 259 § 17. Determination of -^ and of the product Tg .... 260 § 18. Introduction of other measures of Pressure and Heat . . 261 § 19. Determination of the temperatures T^ and Ta . . . . 262 § 20. Determination of c and r . . . ' 264 § 21. Special Form of Equation (32) for an Engine working without expansion 265 § 22. Numerical values of the constants 267 § 23. The least possible value of V and the corresponding amount of Work 268 § 24. Calculation of the work for other values of F . . . . 269 § 25. Work done for a given value of V by an engine with expansion . 271 § 26. Summary of various cases relating to the working of the engine 273 § 27. Work done per unit of heat delivered from the Source of Heat . 274 § 28. Friction . 275 § 29. General investigation of the action of Thermo-Dynamio Engines and of its relation to a Cyclical Process 276 § 30. Equations for the work done during any cychoal process . . 279 § 31. AppHoation of the above equations to the limiting case in which the Cyclical Process in a Steam-Engine is reversible . . 281 § 32. Another form of the last expression 282 § 33. Influence of the temperature of the Source of Heat . . . 284 § 34. Example of the application of the Method of Subtraction . . 286 CHAPTEE XII. ON THE CONCENTRATION OP EATS OF LIGHT AND HEAT, AND ON THE LIMITS OF ITS ACTION. § 1. Object of the investigation 293 I. Beasons why the ordinary method of determining the mutual radiation of two surfaces does not extend to the present case 296 g 2. Limitation of the treatment to perfectly black bodies and to homogeneous and unpolarized rays of heat .... 296 § 3. Kirchhoff's Formula for the mutual radiation of two Elements of surface 297 § 4. Indetermiuateness of the Formula in the case of the Concen- tration of rays 300 CONTENTS. XV PAGE II. Determination of corresponding points and corresponding Elements of Surface in three planes cut by the rays . 301 § 5. Equations between the co-ordinates of the points in which a ray cuts three given planes 301 § 6. The relation of corresponding elements of surface . . . 304 § 7. Various fractions formed out of six quantities to express the Eolations between Corresponding Elements .... .SC9 III. Determination of the mutual radiation when there is no concentration of rays ....... 310 § 8. Magnitude of the element of surface corresponding to ds^ on a plane in a particular position ....... 310 § 9. Expressions for the quantities of heat which da, and ds, radiate to each other 812 § 10. Eadiation as dependent on the surrounding medium . . . 314 IV. Determination of the mutual radiation of two Elements of Surface In the case when one is the Optical Image of the other 316 § 11. Relations between B,D,F,a.ndiE 316 § 12. Application of A and C to determine the relation between the elements of surface 318 § 18. Eelation between the quantities of Heat which ds, and ds^ radiate to each other 319 V. Eelation between the Increment of Area and the Eatio of the two solid angles of an Elementaiy Pencil of Bays . 321 § 14. Statement of the proportions for this case 321 VI. General determination of the mutual radiation of two sur- faces in the case where any concentration whatever may take place 324 § 15. General view of the concentration of rays 324 § 16. Mutual radiation of an element of surface and of a finite surface through an element of an intermediate plane . . . . 3r 5 § 17. Mutual radiation of entire surfaces 3'. 8 § 18. Consideration of various collateral circumstances . . . 329 § 19. Summary of results 330 CHAPTER XIII. DISCUSSIONS ON THE MECHANICAL THEORY OF HEAT AS HEKE DEVELOPED AND ON ITS FOUNDATIONS. j 1. Different views of the relation between Heat and Work . . 332 i 2. Papers on the subject by Thomson and the author . . . 3S3 i 3. On Eankine's Paper and Thomson's Second Paper . . . 3i?4 i 4. Holtzmaan's Objections 337 i 5. Decher's Objections 339 i 6. Fundamental principle on which the author's proof of the second main principle rests . 340 i 7, Zeuner's first treatment of the subject 341 i 8. Zeuner's second treatment of the subject 342 XVI CONTENTS. PAGE § 9. Eankine's treatment of the subject 345 § 10. Him's Objections ... 348 § 11. Wand's Objections . . ... 353 § 12. Tait'a Objeetions 860 Appenhix I. Thermo-elastie Properties of Solids .... 363 II. CapiUarity 369 „ III. Continuity of the Liquid and Gaseous states . . . 373 ON THE MECHANICAL THEORY OP HEAT. MATHEMATICAL INTRODUCTION. ON MECHANICAL WORK, ON ENEHGT, AND ON THE TREATMENT OF NON-INTEGRABLE DIFFERENTIAL EQUATIONS. § 1. JOefinition and Measurement of Mechanical Work. Every force tends to give fiiotion to the body on which it acts; but it may be prevented from doing so by other opposing forces, so that equilibrium results, and the body reniains at rest. In this case "the force performs no work. But as soon as the body moves under the influence of the force, Work is performed. In order to investigate the subject of Work under the simplest possible conditions, we may assume that instead of an extended body the force acts upon a single material point. If this point, which we may call p, travels in the same straight line in which the force tends to move it, then the product of the force and the distance moved through is the mechanical work which the force performs during the motion." If on the other hand the motion of the:point is in any other direction than the line of action of the force, then ibhe work performed is represented by the product of the distance moved through, and the com- ponent of the force resolved in the direction of motion, This component of force in the line of motion may be piopitive or negative in sign, according as it tends in th§ c. 1 ' 2 ON THE MECHANICAL THEORY OF HEAT. same direction in which the motion actually takes place, or in the opposite. The work likewise will be positive in the first case, negative in the second. To express the difference in words, which is for many reasons convenient, recourse may be had to a terminology proposed by the writer in a former treatise, and the force may be said to do or perform work in the former case, and to destroy work in the latter. From the foregoing it is obvious that, to express quan- tities of work numerically, we should take as unit that quantity of work which is performed by an unit of force acting through an unit of distance. In order to obtain a scale of measurement easy of application, we must choose, as our normal or standard force, some force' which is thoroughly known and easy of measurement. The force usually chosen for this purpose is that of gravity. Gravity acts on a given body as a force always tending downwards, and which for places not too far apart may be taken as absolutely constant. If now we wish to lift a weight upwards by means of any force at our disposal, we must in doing so overcojjae the force of-gravity; and gravity thus gives a measure of the force which we must exert for any slow lifting action. Hence we take as our unit of work that which must be performed in order to lift a unit of weight through a unit of length. The units of weight and length to be. chosen are of course matter of indif- ference ; in applied mechanics they are generally the kilo- gram and the metre respectively, and then the unit of work is called a kilogram metre. Thus to raise a weight of a kilograms through a height of b metres ab kilogrammetres of work are required ; and other quantities of work, in cases where gi;avity does not come directly into play, can also be expressed in kilogrammetres, by comparing the forces em- ployed with the standard force of gravity. § 2. Mathematical determination of the Work done hy variable components of Force. In the foregoing explanation it has been tacitly assumed that the active component of force has a constant value; throughout the whole of the distance traversed. In reality' this is not usually true for a distance of finite length. On MATHEMATICAL INTRODUCTION. 3 the one hand the force need not itself he the same at diffe- rent points of space ; and on the other, although the force may remain constant throughout, yet, if the path be not straight but curved, the component of force in the direction of motion will still vary. For this reason it is allowable to ex- press work done by a simple product, only when the distance traversed is indefinitely small, i.e. for an element of space. Let ds be an element of space, and S the component in the direction of ds of the force acting on the point p. We have then the following equation to obtain dW, the work done during the movement through the indefinitely small space ds : dW=Sds (1). If P be the total resultant force acting on the point p, and ^ the angle which the direction of this resultant makes with the direction of motion at the point under consideration, then S = P cos (!^)dx + y{r(a^\dy (3a), a distinction has to be drawn, which is of great importance, not only for this particular case, but also for the equations which occur later on in the Mechanical Theory of Heat j and which will therefore be examined here at .some length, so that in future it will be sufficient simply to refer back to the present passage. ■;■ According to the nature of the functions ^ (my) and ■\fr (o^), differential equations of the form (3) fall-iSto two classes, which differ widely both as to the treatment which they require, and the results to which they lead. To the ilATHEMATICAI/ INTRODUCllOU. 5 first class belong the cases, in which, the fiinctions X and Y fulfil the following condition : dy dx '"" ""^ '' The second class comprises all cases, in which this condition is not fulfilled. If the condition (4) is fulfilled, the expression on the right-hand side of equation (3) or (3a) becomes immediately integrable ; for it is the complete differential of some func- tion of X and y, in which these may be treated as indepen- dent variables, and which is formed from the equations dx , ' dy Thus we obtain at once an equation of the form w = F {xy) + const (5). If condition (4) is not fulfilled, the right-hand side of the equation is not integrable ; and it follows that W cannot be expressed as a function of x and y, considered as independent variables. For, if we could put W= Fiacy), we should have ^^dW^ dF{xy) dib dx ' Y^dW^ dFjxy) dy dy ' whence it follows that dX ^ d'Fjxy) dy dxdy ' dY^ d^Fjxy) , dx dydx But since with a function of two independent variables the oxdisr of differentiation is immaterial, we may put dxdy dydx ' 6 ON THE MECHANICAL THEORY OF HEAT. whence it follows that-y- =-5— j «'e- condition (4) is fulfilled for the functions X and Y; which is contrary to the assump- tion. In this case then the integration is impossible, so long as x and y are considered as independent variables. If however we assume any fixed relation to hold between these two quantities, so that one may be expressed as a function of the other, the integration again becomes possible. For if we put /(^2/) = (6), in which f expresses any function whatever, then by means of this equation we can eliminate one of the variables and its differential from the differential equation. (The general form in which equation (6) is given of course comprises the special case in which one of the variables is taken as constant ; its differential then becomes zero, and the variable itSBlf only appears as part of the constant coefficient). Sup- posing y to be the variable eliminated, the equation (.3) takes the form dW=<^{x) dx, which is a simple differential equation, and gives on integration an equation of the form w = F {x) ■\- comi (7). The two equations (6) and (7) may thus be treated as form- ing together a solution of the differential equation. As the form of the function fixy) may be anything whatever, it is clear that the number of solutions thus to be obtained is infinite. The form of equation (7) may of course be modified. Thus if we had expressed x in terms of y by means of equa- tion (6); and then eliminated x and dx from the differential equation, this latter would then have taken the form dW = <^,{y)dy, and on integrating we should have had an equation ''^=■^1(2')+ const (7a3. This same equation can be obtained from equation (7) by substituting y for x in that equation by means of equation (6). Or, instead of completely eliminating x from (7), we may MATHEMATICAL ' INTRODUCTION. 7 prefer a partial elimination. For if the function F (cc) con- tains X several times over in different teriiis, (and if this does not hold in the original form of the equation, it can be easily introduced into it by writing instead of x an expression such as (l — a)x + ax, —^, &c.) then it is possible to substitute y for X in some of these expressions, and to let x remain in others. The equation then takes the form W=^F^Ilx,y) + const (76), which is a more general form, embracing the other two as special cases. It is of course understood that the three equa^ tions (7), (7a), (76), each of which has no meaning except when combined with equation (6), are not different solutions, but different expressions for one and the same solution of the differential equation. Instead of equation (6), we may also employ, to integrate the differential equation (3), another equation of less simple form, which in addition to the two variables x and y also, contains W, and which may itself be a differential equation ; the simpler form however suffices for our present purpose, and with this restriction we may sum up the results of this section as follows. When the condition of immediate integrability, expressed by equation (4), is fulfilled, then we can obtain directly an integral equation of the form : W=F{x,y) + const (A). When this condition is not fulfilled, we must first assume some relation between the variables, in order to make inte- ^ation possible; and we shall thereby obtain a system of two equations of the following form : /(*,2/) = 0, 1 ,gv W=F{a;.,y) + const. \ ^^^' in which the form of the function F depends not only on that of the original differential equation, but also on that of the function/, which may be assumed at pleasure. 8 ON THE MECHANICAL THEORY OF HEAT. § i. Geometrical interpretation of the foregoing results, and observations on partial differential coefficients. The important difference between the results in the two cases mentioned above is rendered more clear by treating them geometrically. In so doing we shall for the sake of simplicity assume that the function F {x, y) in equation (A) is such that it has only a single value for any one point in the plane of co-ordinates. We shall also assume that in the movement of the point p its original and final positions are known, and given by the co-ordinates x^, y^, and a;,, y^ respec- tively. Then in the first case we can find an expression for the work done by the effective force during the motion,: without needing to know the actual path traversed. For it is clear, that this work will be expressed, according to con- dition {A)i by the difference F(x^p^ — F(x„y„). Thus, while the moving point may pass from one position to the other by very different paths, the amount of work done by the. force is wholly independent of these, and is completely known aS soon as the original and final positions are given. In the second case it is otherwise. In the system of equations (B), which ' belongs to this case, the first equation must be treated as the equation to a curve ; and (since the form of the second depends upon it) the relation between them may be geometrically expressed by saying that the work done by the effective force during the motion of the point p can only be determinedj when the whole of the curve, on which the point moves, is known. If the original and final positions are given, the first equation must indeed • be so chosen, that the curve which corresponds to it may pass through those two points ; but the number of such possible curves is infinite, and accordingly, in spite of their coinci- dence at theii: extremities, they will give an infinite number of possible quantities of "work done during the motion. If we assume that the point p describes a closed curve, so that the final and initial positions coincide, and thus the co- ordinates x^, y^ have the same value as x„, y^, then in the first case the total work done is equal to zero: in the second case, on the Other hand, it need not equal zero, but may have any value positive or negative. The case here examined also illustrates the fact that a .(8). MATHEMATICAL INTBOBUCTION. 9^. quantity, which cannot be expressed as a function of x and y (so long as these are taken as independent variables),, may yet have partial differential coefficients according to x and y, which are expressed by determinate- functions of those vari- ables. For it is manifest that, in the strict sense of the words, the components X and Y must be termed the partial differential coefficients of the work W according to x and y : smce, when x increases by dx, y remaining constant, the, work increases by Xdx ; and when y increases by dy, x re- maining constant, the work increases by Ydy. Now whether TF be a quantity generally expressible as a function of x and . y, or one which can only be determined on knowing the path described by the inoving point, we may always employ the ordinary notation for the partial differential coefficients of W, and write Using this notation we may also write the condition (4), the fulfilment or non-fulfilment of which causes the distinction between the two modes of treating the differential equation, in the following form: d fdW\ _ d rdW\ ,g. dy\dx) dx\dy ) Thus we may say that the distinction which has to be' drawn in reference to the quantity W depends on whether the difference t-(-t- I — -r- (—7— I is dqual to zpro> or has dy\dx J dx\dy J a finite value. , - § 5. Extension of the above to three dimensions. If the point p be not restricted in its movement to one plane, but left free in space, we then obtain for the element of work an expression very similar to that given in equation (3). Let a, b, c be the cosines of the angles which the direc- tion of the force P> acting on ..the, point, makes with three 10 ON THE MECHANICAL THEORY OF HEAT. rectangular axes of co-ordinates ; then the three components X, Y, Z of this force will be given by the equations X=aF, Y=hP, Z=.cP. Again, let a, A 7 be the cosines of the angles, which the element of space ds makes with the axes ; then the three, projections dx, dy, dz of this element on those axes are given by the equations dx = ads, dy = ^ds, ds = 'yds. Hence we have Xdx + Ydy + Zdz = {pa. + &/3 + cy) Fds. But if ^ be the angle between the direction of P and ds^ then aai + &;8 + cy = cos ^: hence Xdx + Ydry + Zdz = cos(j>x Pds. Comparing this with equation (2), we obtain dW = Xdx+Ydy + Zdz (10). This is the differential equation for determining the work done. The quantities X, Y, Z may be any functions what- ever of the co-ordinates oc,y,z; since whatever may be the values of these three components at different points in space, a resultant force P may always be derived from them. In treating this equation, we must first consider the fol- lowing three conditions : dX^dY dY^dZ dZ^dX dy dx ^ dz dy ' dx dz ^ '^' and must enquire whether or not the functions X, Y, Z satisfy them. If these three conditions are satisfied, then the expression on the right-hand side of (11) is the complete differential of a function of x, y, z, in which these may all be treated as independent variables. The integration may therefore be at once effected, and we obtain an equation of the form TF = P(a!^s) + const (12). MATHEMATICAL INTEOBUCTION. 11 If we now conceive the point j> to move from a given initial position (x^, y^, z^ to a given final position (ajj, y^, z^ the work done by the force during the motion will be repre- sented by If then we suppose F{x, y, z) to be such that it has only a single value for any one point in space, the work will be completely determined by the original and final positions; and it follows that the work done by the force is always the same, whatever path may have been followed by the point in passing from one position to the other. If the three conditions (1) are not satisfied, the integra- tion cannot be effected in the same general manner. If, however, the path be known in which the motion takes place, the integration becomes thereby possible. If in this case two points are given as the original and final positions, and various curves are conceived as drawn between these points, along any of which the point p may move, then for each of these paths we may obtain a determinate value for the work dpne; but the values corresponding to these different paths need not be equal, as in the first case, but on the contrary are in general, different. § 6. OniheErgal. In those cases in which equation (12) holds, or the work done can be simply expressed as a function of the co-ordirtates, this function plays a very important part in our calculations. Hamilton gave to it the special name of "force function"; a name applicable also to the more general case where, instead of a single moving point, any number of such points are considered, and where the condition is fulfilled that the work done depends only on the position of the points. In the later and more extended investigations with regard to the quantities which are expressed by this function, it has become needful to introduce a special name for the negative value of the function, or in other words for that quantity, the svh- traction of which gives the work performed; and Rankine proposed for this the term 'potential energy.' This name sets forth very clearly the character of the quantity ; but it 1'2 ON THE MECHANICAL THEOEY OF HEAT. is somewliat long, and the author has ventured to propose in its place the term " Ergal." * Among the cases in which the force acting on a point has an Ergal, the most prominent is that in which the force originates in attractions or repulsions^ exerted on the moving point from fixed points, and the value of which depends only on the distance ; in other words the case ia which the force may be classed as a central force. Let us take as centre of force a fixed, point it, with co-ordinates ^, vj, §", and let p be its distance from the moving point p, so that p=J{^-'cr+{v-yr+{^-^r- as)- Let us express the force which tt exerts on p by <^' {p), in which a positive value of the function expresses attraction, and a negative value repulsion ; we then have for the com- ponents of the force the expressions . ^=fW^; F=.^'(p)^. ^=^'(p)i:rf. r r r But by (13) ^ = - ^-^^ : hence X ' (p)%, and simi- larly for the other two axes. If ^ (p) be a function such that (p)=j<}>'{p)dp.... .(14). we may write the last equation thus : j-_ dj>{p) dp _ (i4>{p) . dp dx dx ^ '' and similarly F = -^, ^=-^ (15a). [ Hence we have .Xdx+Ydy + Zd, = -]^^dx + '^dy + ^ds']. But, since in the expression for p given in equation (13) the quantities x, y, z are the only variables, and ^ (p) may there- fore be treated as a function of those three quantities, the ItitHEMATlOAt INTRODUCTION,' 1'3 expression in brackets forms a perJFect differential, and we may write : Xdx+Ydy + Zdz = -d{p) (16). The element of work is thus given by the negative differen- tial of ^ (p) ; whence it follows that 6 (p) is in this case the £rgal. Again, instead of a single fixed point, we may have any number of fixed points tTj, tt^, w„ &c., the distances of which from p^ are p^, p^, p^. Sec, and which exert on it forces ^'(P»)> ^'W. 0'(Pj). &c. Then if, as in equation (14), we as- sume^j(jo), <^^{p), 4)^(p), &c, to be the integrals of the above Junctions, we obtain, exactly as in equation (15), x^. <^'^i(pi) ^M #.,(p») dx doa dsB ^— ^S^W (17). Similarly r=-|2^(p), Z=-^^t^{p) (I7a), whence Xdx + Ydy + Zdz = - d% <^ {p) (18). Thus the sum S ^{p) is here the Ergal. § 7. General Extension of the foregoing. Hitherto we have only considered a single moving point ; we will now extend the method to comprise a system com- posed of any number of moving points, which are in part acted on by external forces, and in part act mutually on each other. If this whole system makes an indefinitely small move- ment, the forces acting on any One point, which forces we may conceive as combined into a single resultant, will per- form a quantity of work which may be represented by the expression {Xdx + Ydy + Zdz), Hence the sum of all the 14 ON THE MECHANICAL THEOET OF HEAT. work done by all the forces acting in the system may be represented by an expression of the form t{Xdx-^ Ydy + Zdz), in which the summation extends to all the moving points. This complex expression, like the simpler one treated above, may have under certain circumstances the important peculiarity that it is the complete differential of some func- tion of the co-ordinates of all the moving points ; in which case we call this function, taken negatively, the Ergal of the whole system. It follows from this that in a finite move- ment of the system the total work done is simply equal to the difference between the initial and final values of the Ergal; and therefore (assuming that the function which, represents the Ergal is such as to have only one value for one position of the points) the work done is completely deter- mined by the initial and final positions of the points, without its being needful to know the paths, by which these have moved from one position to the other. This state of things, which, it is obvious, simplifies greatly the determination of the work done, occurs when all the forces acting in the system are central forces, which either act upon the moving points from fixed points, or are actions between the moving points themselves. First, as regards central forces acting from fixed points, we have already discussed their effect for a single moving point ; and this discussion will extend also to the motion of the whole system of points, since the quantity of work done in the motion of a number of points is simply equal to the sum of the quantities of work done in the motion of each several point. We can therefore express the part of the Ergal relating to the action of the fixed points, as before, by 2 (^ (p), if we only give such an extension to the summation, that it shall comprise not only as many terms as there are fixed points, but as many terms as there are combinations of one fixed and one moving point. Next as regards the forces acting between the moving points themselves, we will for the present consider only two points p and p\ ■w^Hh co-ordinates co, y, z, and x, y', z\ MATHEMATICAL INTKODXJCTION, 15 respectively. If r be tlie distance between these points, we have r = J{x'-a=y + (y'-yr+[,'-zy (19). We may denote the force which the points exert on each other by/'(r), a positive value being used for attraction, and a negative for repulsion. Then the components of the force which the point p exerts in this mutual action are /W^, /'«^, fir) and the components of the opposite force exerted by p' are /'W^', /'W^'. /'(r)i^'. But by (19), differentiating dr _ x' — X ^ dr _ x — x^ dx r ' dx' r ' so that the components of force in the direction of x may also be written and if /(r) be a function such that f(r)=jf'{r)dr (20), the foregoing may also be written -dfjr) , -df(r) ^ dx ' dd Similarly the components in the direction of y may be written -t?f(r) . :^/(r). dy ' dy' ' and those in the direction of z -df{r)^ -df(r) 16 ON THE MECHANICAL THEOET OF HEAT. That part of the total work done in the indefinitely small motion of the two points, which is due to the two opposite foTcea arising from their mutual ^,ction, may therefore be expressed as follows : dz J But as r depends only on the six quantities x^ ,y, z, oc', y, z, and f{r) can therefore be a function of these six quantities only, the expression in brackets is a perfect differential, and the work done, as far as concerns the mutual action between the two points, may be simply expressed by tl)e function In the same way may be expressed the work due to the" mutual action of every other pair of points ; and the total work done by all the forces which the points exert among themselves is expressed by the algebraical suni -rf/(r)-J/(/)-d/(0-...; or as it may be otherwise written, -<^[/W+/(^')+/(0 + -] or -dS/W; in which the summation must comprise as many terms as there are combinations of moving points, two and two. This sum S/(r) is then the part of the Ergal relating to the mutual and opposite actions of all the moving points. If we now finally add the two kinds of forces together, we obtain, for the .to.tal work done in the indefinitely small motion of the system of points, the equation 2 {Xdx + Ydy + Zdz) =-dt(]>(p)^ dtfir) = ^d[tcl>{p)+-ZAr)] (21), whence it foUows that the quantity %{p) + '^f{r) is the Ergal of the whole of the forces. acting together in ^he systeni,. The assumption lying at the root of the foregoing analy- sis, viz. that central forces are the only ones acting, is of course only one among all .the assumptions mathematically MATHEMATICAL INTEODUCTION. 17 possible as to the forces; but it forms a case of peculiar importance, inasmuch as all the forces which occur in nature may apparently be classed as central forces. § 8. Relation between Work and Vis Viva. Hitherto we have only considered the forces which act on the points, and the change in position of the points them- selves ; their masses and their velocities have been left out of account. We wiU now take these also into consideration. The equations of motion for a freely moving point of mass m are well known to be as follows : If we multiply these equations respectively by dx 7, dy ,^ dz , • dt^'' dt^'' It^'' and then add, we obtain The left-hand side of this equatioii may be transformed into m d 2dt m <%'<%' dt. or, if V be the velocity of the point, ^i(p.dt=%ldt=d(^v^ 2 dt dt V2 y and the equation becomes ■ <'(f«')=(^S+4!+4)* '^')- If, instead of a single freely moving point, a whole system of freely moving points is considered, we shall have for every n 2 18 ON THE MECHANICAL THEORY OF HEAT. point a similar equation to, the above; and by summation we shall obtain the following:- (^S+4!+^S)* (^«'' the meaning of which may be expressed as follows: The Wo^h done during any time hy the forces acting upon a system is equal to the increase of the Yis Viva of the system during the same time. In this expression a diminution of Vis Viva is of course treated as a negative increase. It was assumed at the commencement that all the points were moving freely. It may, however, happen that the poihts are subjected to certain constraints in reference to their motion. They may be so connected with each other that the motion of one point shall in part determine the .motion of others ; or there may be external constraints, as for in- stance, if one of the points is compelled to move in a given fixed plane, or on a given fixed curve, whence it will natur- * Translator's Note. The vis viva of a particle is here defined as half the mass multiplied by the square of the velocity, and not the whole mass, as was formerly the custom. MATHEMATICAL INTEODUCTION. 19 ally follow that all those points, which are in any connection with it, will also be to some extent constrained in their motion. If these conditions of constraint can be expressed by equations which contain only the co-ordinates of the points, it may be proved, by methods which we will not here con- sider more closely, that the reactions, which are implicitly comprised in these conditions, perform no work whatever during the motion of the points ; and therefore the principle given above, as expressing the relation between Vis Viva and Work done, is true for constrained, as well as for free motion. It is called the Principle of the Equivalence of Work and Vis Viva. I'M- § 9. On Energy. In equation (28), the work done in the time from t^ to t is expressed by in which t is considered as the only independent variable, and the co-ordinates of the points and the components of the forces are taken as functions of time only. If these functions are known (for which it is-requisite that we should know the whole course of the motion of all the points), then the inte- gration is always possible, and the work done can also be determined as a function of the time. Cases however occur, as we have seen above, in which it is not necessary to express all the quantities as functions of one variable, but where the integration may still be effected, by writing the differential in the form S (Xdx + l^y -t- Zdz), and considering the co-ordinates therein as independent vari- ables. For this it is necessary that this expression should be a perfect differential of some function of the co-ordinates, or in other words the forces acting on the system must have an Ergal. This Efgal, which is the negative value of the above, function, we will denote by a single letter. The letter U is generally chosen for this purpose in works on Me- chanics: but in the Mechanical Theory of Heat that letter is needed to express another quantity, which will enter as 2—2 20 ON THE MECHANICAL THEOET OF iHEAT. largely into the discussion; we will therefore denote the Ergal by J. Hence we put: %(X.Ax^Ydy-^Ziz) = -dJ...... (29), whence if J^ be the value of the Ergal at time f„i we have: '0 /, ^{Xdx^Ydy^Zdz)=J,-J. (30), which expresses that the work done in any time is equal to the decrease in the Ergal, If we substitute J^ — J for the integral in equation (28), we have: T-T, = J,-Jov T+J=T, + J, (31); whence we have the following principle: The sum of the Vis Viva arid of the Ergal remains constant during the motion. This sum, which we will denote by the letter TJ, so that U=T + J. (32), is called tbe Energy of the system ; so that the above prin- ciple may be. more shortly expressed by saying: The Energy remains constant during the motion. This principle, which in recent times has received a much more extended application than formerly, and now forms one of the chief foundations of the whole structure of physical philosophy, is known by the name of The Principle of the Conservation of Energy. (21) CHAPTER I. FIEST MAIX PRINCIPLE OF THE MECHANICAL THEORY OF HEAT, OR PRINCIPLE OF THE EQUIVALENCE OF HEAT AND WORK. ■ § 1, Nature of Heat. Until recently it M'as the generally accepted view that Heat was a special substance, which was present in bodies in greater or less quantity, and which produced thereby their higher or lower temperature; which was also sent forth from bodies, and in that case passed with immense speed through empty space and through such cavities as ponder- able bodies contain, in the form of what is called radiant heat. In later days has arisen the other view that Heat is in reality a mode of motion. According to this view, the heat found in bodies and determining their temperature is treated as being a motion of their ponderable atoms, in which motion the ether existing within the bodies may also participate ; and radiant heat is looked upon as an undulatory motion propagated in that ether. It is not proposed here to set forth the facts, experiments, and inferences, through which men have been brought to this altered view on the subject ; this would entail a refer- ence here to much which may be better described in its own place during the course of the book. The conformity with experience of the results deduced from this new theory will probably serve better than anything else to establish the foundations of the theory itself We will therefore start with the assumption that Heat consists in a motion of the ultimate particles of bodies and of ether, and that the quantity of heat is a measure of the Vis Viva of this motion. The nature of this motion we 22 ox t:he mechanical theoey of heat. shall not attempt to determine, but shall merely apply to Heat the principle of the equivalence of Vis Viva and Work, ■which applies to motion of every kind ; and thus establish a principle which may be called the first main Principle of the Mechanical Theory of Heat. § 2. Positive and negative values of Mechanical Work. In § 1 of the Introduction the meOhanical work done in the movement of a point under the action of. a force was defined to be The product of the distance moved through and of the component of the force resolved in the direction of motion. The work is thus positive if the component of force in the line of motion lies on the same side of the initial point as the element of motion, and negative if it falls on the opposite side. From this definition of the positive sign of mechanical work follows the principle of the equiva- lence of Vis Viva and Work, viz. The increase in the Vis Viya is equal to the work done, or equal to the increase in total work. The question may also be looked at from another point of view. If a material point has once been set in motion, it can continue this movement, on account of its momentum, even if the force acting on it tends in a direction opposite to that of the motion; though its velocity, and therewith its Vis Viva, will of course be diminishing all- the time. A material point acted on by gravity for example, if it has received an upward impulse, can continue to move against the force of gravity, although the latter is continually diminishing the velocity given by the impulse. In such a case the work, if considered as work done by the force, is negative. Conversely however , we may reckon work as positive in cases where a force is overcome by the momen* tum of a previously acquired motion, as negative in cases where the point follows the direction of the force. Applying the form of expression introduced in § 1 of the Introduction, in which the distinction between the two opposite directions of the component of force is indicated by different words, we may express the foregoing more simply as follows : we may determine that not the work done, but the work destroyed, by a force shall be reckoned as positive. On this method of denoting work done, the principle of EQUIVALENCE OF HEAT AND WORK. 23 the equivalence of Vis Viva and Work takes the following form : The decrease in ike Vis Viva is equal to the increase in the Work done, or The sum of the Vis Viva and Work done is constant. This latter form will be found. very convenient in what follows. In the case of such forces as have an Ergal, the meaning of that quantity was defined (in § 6 of the Introdi^oiiqg) in such a manner that we must say, ' The Work d^e^is 'equal to the decrease in the Ergal.' If we use the method of denot- ing work JMt ^described, we must say on the contrary, 'The work dop?^is equal to the increase in the Ergal;'- and if the constant occurring as one term of the Ergal be determined in a particular way, we may then regard the Ergal as simply an expression for. the work done. § 3. Expression for the first Fundamental Principle. Having fixed as above what is to be the positive sign for work done, we may now state as follows the first main Principle of the Mechanical Theory of Heat. In all cases where work is produced by heat, a quantity of heat is consumed proportional to the work done; and inversely, by the expenditure of the same amount of work tfte same quantity of heat m,ay be produced. This follows, on the mechanical conception of heat, from the equivalence of Vis Viva and Work, and is named The Principle of the Equivalence of Heat and Work. If heat is consumed, and work thereby produced, we may say that heat has transformed itself into work ; and con- versely, if work is expended and heat thereby produced, we may say that work has transformed itself into heat. Using this mode of expression, the foregoing principle takes the following form : Work may transform itself into heat, and heat conversely into work, the quantity of the one bearing always a fixed proportion to that of the other. This principle is established by means of many pheno-' mena which have been long recognized, and of late years has been confirmed by so many experiments of different kinds, that we may accept it, apart from the circumstance of its forming a special case of the general mechanical principle of the Conservation of Energy, as being a principle directly derived from experience and, observation. 24 ON THE MECHANICAL THEORY OF HEAT. § '4. Numerical Relation between Heat and Work. While the mechanicd,! principle asserts that the changes in the Vis Viva and in the corresponding Work done are actually equal to each other, the principle which expresses the relation between Heat and Work is one of Proportion only. The reason is that heat and work are not measured on the same scale. Work is measured by the mechanical imit of the kilogramme tre, whilst the unit of heat, chosen for convenience of measurement, is That amount of heat which is required to raise one kilogram of water from 0° to 1" (Centigrade). Hence the relation , existing between heat and work can be one of proportion only, and the numerical value must be specially determined. If this numerical value is so chosen as to give the work corresponding to an unit of heat, it is called the Mechanical Equivalent of Heat ; if on the contrary it gives the heat corresponding, to an unit of work, it is called the Thermal Equivalent of Work. We shall denote the former by E, and the latter by -^ . The determination of this numerical value is effected in different ways. It has sometimes been deduced from already existing data, as was first done on correct principles by . Mayer (whose method will be further explained hereafter), although, from the imperfection of the then existing data, his result must be admitted Hot to have been very exact. At other times it has been sought to determine the number by experiments specially made with that view. To the dis- tinguished English physicist Joule must be assigned the credit cf having established this value with the greatest cir- cumspection and care. Some of his experiments, as well as •determinations carried out at a later date by others, will more properly find their place after the development of the theory ; and we will here confine ourselves to stating those of Joule's experiments which are the most readily understood, and at the same time the most certain as to their results. Joule measured, under various circumstances, the heat generated by friction, and compared it with the work con- sumed in producing the friction, for which purpose he employed descending weights. As accounts of these experi- EQUIVALENCE OP HEAT AND WORK. 25 ments are given in many books, they need not here be described ; and it will suffice to state the results as given in his paper, published in the Phil. Trans., for 1850. In the first series of experiments, a very extensive one, water was agitated in a vessel by means of a revolving paddle wheel, which was so arranged that the whole quantity of water could not be brought into an equal state of rotation throughout, but the water, after being set in motion, was continually checked by striking against fixed blades, which occasioned numerous eddies, and so produced a large amount of friction. The result, expressed in English measures^ is that in order to produce an amount of heat which will raise 1 pound of water through 1 degree Fahrenheit, an amount of work equal to 772'695 foot-pounds must be consumed. In two other series of experiments quicksilver was agitated in the same way, and gave a result of 774'083 foot-pounds. Lastly, in two series of experiments pieces of cast iron were rubbed against each other under quicksilver, by which the heat given out was absorbed. The result was 774'987 foot-* pounds. Of all his results Joule considered those given by water as the most accurate; and as he thought that even this figure should be slightly reduced, to allow for the sound pro- duced by the motion, he finally gave 772. foot-pounds as the most probable value for the number sought. Transforming this to French measures we obtain the result that. To produce the quantity of heat required to raise 1 kiiogramme of water through 1 degree Centigrade, work must be consumed to the amount o/'4!23'55 kilogrammetres. This appears to be the most trustworthy value among those hitherto determined, and accordingly we shall henceforward use it as the mechanical equivalent of heat, and write J5=423-55 (1). In most of our calculations it will be sufficiently accurate to use the even number 424. § 5. The Mechanical Unit of Heat. Having established the principle of the equivalence of Heat and Work, in consequence of which these two may be 26 ON THE MECHANICAL THEORY OF HEAT. opposed to each other in the same expression, we are often in the position of having to sum up quantities, in which heat and work enter as terms to be added together. As, however, heat and work are measured in different ways, we cannot in such a case say simply that the quantity is the sum of the work and the heat, but either that it is the sum of the heat and of the heat-equivalent of the work, or the sum of the work and of the work^equivalent of the heat. On account of this inconvenience Rankine proposed to employ a different unit for heat, viz. that amount of heat which is equivalent to an unit of work. This unit may be called simply the Mechanical Unit of Heat. There is an obstacle to its general introduction in the circumstance that the unit of heat hitherto used is a quantity which is closely connected with the ordinary calorimetric methods (which mainly depend on the heating of water), so that the reductions required are slight, and rest on measurements of the most reliable character ; while the mechanical unit, besides need- ing the same reductions, also requires the mechanical equivalent of heat to be known, a requirement as yet only approximately fulfilled. At the same time, in the theoretical development of the Mechanical Theory of Heat, in which the relation between heat and work often occurs, the method of expressing heat in mechanical units effects siich important simplifications, that the author has felt himself bound to drop his former objections to this method, on the occasion of the present more connected exposition of that theory. Thus in what follows, unless the contrary is expressly stated, it will be always understood that heat is expressed in mechanical units. On this system of measurement the above mentioned first main Principle of the Mechanical Theory of Heat takes a yet more precise form, since we may say that heat and its corresponding work 'are not merely proportional, but equal to each other. If later on it is desired to convert a quantity of heat expressed in mechanical units back again to ordinary heat units, all that wUl be necessary is to divide the number given in mechanical units by H, the mechanical equivalent of heat. EQUIVALENCE OF HEAT AND WOEK. 27 § 6. Develc^ment of the Jtrst main Principle. Let any body whatever be given, and let its, condition as to temperature, volume, &c. be assumed to be known. If an indefinitely small quantity of heat dQ is imparted to this body, the question arises what becomes of it, and what effect it produces. It may in part serve to increase the amount of heat actually existing in the body ; in part also, if in conse- quence of the imparting of this heat the body changes its condition, and that change includes the overcoming of some force, it may be absorbed in the work done thereby. If we denote the total heat existing in the body, or more briefly the Quantity of Heat of the body, by H, and the indefinitely small increment of this quantity by dR, and if we put dL for the indefinitely small quantity of work done, then we can write : dQ = dH-\-dL (I). The forces against which the work is done may be divided into two classes: (1) those which the molecules of the body exert among themselves, and which are therefore dependent on the nature of the body itself, and (2) those which arise from external influences, to which the body is subjected. According to these two classes of forces, which have to be overcome, the work done is divided into internal and external work. If we denote these two quantities by dJ and d W, we may put dL = dJ+dW (2), and then the foregoing equation becomes dQ = dH+dJ+dW (II). § 7. Different conditions of the Quantities J, W, and H. The internal and external work obey widely different laws. As regards the internal work it is easy to see that if a body, starting from any initial condition whatever, goes through a cycle of changes, and finally returns to its original condition again, then the internal work done in the whole process must cancel itself exactly. For if any definite amount, positive or negative, of internal work remained over at the end, there must have been produced thereby either an ■ 28 ON THE MECHANICAIi THEORY OF HEAT. equivalent quantity of external work or a change in the body's quantity of heat ; and as the same process might be repeated any number of times it would in the positive case be possible to create work or heat out of nothing, and in the negative case to get rid of work or heat without obtaining any equivalent for it ; both of which results will be at once admitted to be impossible. If then at every return of the body to its original condition the internal work done becomes zero, it follows further that in any alteration whatever of the body's condition the internal work done can be determined from its initial and final conditions, without needing to know the way in which it has passed from one to the other. For if we suppose the body to be brought successively from the first condition to the second in several difierent ways, but always to be brought back to its first condition in exactly the same way, then the various quantities of internal- work done in difierent ways in the first set of changes must all be equivalent to one and the same quantity of internal work done in the second or return set of changes, which cannot be true unless they are all equal to each other. We must therefore assume that the internal forces have an Ergal, which is a quantity fully determined by the existing condition of the body at any time, without its being requisite for us to know how it arrived at that condition. Thus the internal work done is ascertained by the increment of the Ergal, which we will call J; and for an indefinitely small change of the body the differential dJ of the Ergal forms the expression for the internal work, which agrees with the nota- tion employed in equations 2 and II. If we now turn to the external work, we find the state of things wholly difi^erent. Even when the initial and final conditions of the body are given, the external work can take very different forms. To show this by an example, let us choose for our body a Gas, whose condition is determined by its temperature t and volume v, and let us denote the initial' values of these by t^,v^, and the final values by %,v^; let us also assume that t^>t^, and v^> v^. Now if the change is carried out in the following way, viz. that the gas is first ex- panded, at the temperature \, from v^ to v^, and then is heated, at the volume v^, from under ordinary conditions the exactness of the law is already, so great, that for most purposes of research it may be taken as perfect, we may imagine for every gas an ultimate condi- tion, in which the exactness is really perfect; and in what follows we will assume this, ideal condition to be actually 40 ON THE MECHANICAL THEORY OF HEAT. reached, calling for brevity's sake all gases, in which this is assumed to hold, Perfect Gases. As however the quantity a, according to Eegnault's deter- minations, is not absolutely the same for all the gases which have been examined, and has also somewhat different values for one and the same gas under different conditions, the question arises, what value we are to assign to a in the case of perfect gases, in which such differences can no longer appear. Here we must refer to the values of a which have been found to be correct for various permanent gases. By experiments made on the system of increasing the pressure while keeping the volume constant, Regnault found the fol- lowing numbers to be correct for various permanent gases : Atmospheric Air 0003665. Hydrogen 0003667. Nitrogen 0003668. Carbonic Oxide ' 0-003667. The differences here are so small, that it is of little im- portance what choice we make ; but as it was with atmo- spheric air that Regnault made the greatest number of experiments, and as Magnus was led in • his researches to a precisely similar result, it appears most fitting to select the number 0003665. Regnault, however, by experiments made on the other system of keeping the pressure constant and increasing the volume, has obtaitied a somewhat different value for a in the case of atmospheric, air, viz. 0'003670. He has further observed that rarefied air gives a somewhat smaller, and com- pressed air a somewhat larger, coefficient of expansion than air of ordinary density. This latter circumstance has led some physicists to the conclusion that, as rarefied air is nearer to the perfect gaseous condition than air of ordinary density, we ought to assume for perfect gases a smaller value than 0003665. Against this it may be urged, that Regnault observed no such dependence of the coefficient of expansion on the density in the case of hydrogen, but after increasing the density threefold obtained exactly the same value as before ; and that he also fou©d that hydrogen, in its devia- tion from the laws of Mariotte and Gay Lussac, acts altogether differently, and ior the, most 'part in exactly the opposite ON PERFECT GASES. 41 way, from atmospheric air. In. these ' circumstances the author considers that additional weight is given to the re- sult taken above from the figure for atmospheric air; since it will hardly be questioned that hydrogen is at least as near as atmospheric air to the condition of a perfect gas, and therefore in drawing conclusions relative to that condi- tion the behaviour of the one is as much to be noted as that of the other. It appears therefore to be the best course (so long as fresh observations have not furnished>a more satisfactory start- ing point for wider conclusions) to adhere to the figure which, under the pressure of one atmosphere, has been found to agree almost exactly for atmospheric air and for hydrogen ; and thus to write : a = 0003665 =3j^ (2). he : equation thus ; If we denote the reciprocal - by a we may also write the pv = Pf{a^-t) (3). And if for brevity we put : i?=^5!!o (4) T=a+t .........(5), we then obtain the equation in the form pv = RT (6). .B is here a constant which depends on the nature of the gas and is inversely proportional to its specific gravity*. T represents the temperature, provided this is measured not from the freezing point, but from a zero point lying a degrees lower.. The temperature thus measured from —a we shall term the Absolute Temperature, a name which will be more * For S is proportional to the volume of a unit of weight of the gas at standard pressure and temperature ; and is therefore inyersely proportional to the wgight of a unit of -volume, i.e, to the specific gravity. [Translator^) 42 ox THE MECHANICAL THEORY OF HEAT, fully explained further on. I'aking the value of a given in equation (2) we obtain <^ = l = M (7). T=273 + t] § 2. Approadmate Principle as to Heat absorbed ly Gases. In an experiment of Gay Lussac's, a vessel filled with air was put in communication with an exhausted receiver of equal size, so that half the air from the one passed over into the other. On measuring the temperature of each half, and comparing it with the original temperature, he found that the air which had passed over had become heated, and the air which remained behind had become cooled, to exactly the same degree; so that the mean temperature was the same after the expansion as before. He thus proved that in this kind of expansion, in which no external work was done, no loss of heat took place. Joule, and after him Regnault, carried out similar experiments with greater care, and both were led to the same result. The principle here involved may also be deduced, without reference to special experiments, from certain properties of gases otherwise ascertained, and its accuracy may thus be checked. Gases shew so marked a regularity in their beha- viour (especially in the relation between volume, pressure, and temperature, expressed by the law of Mariotte and Gay Lussac), that we are thereby led to the supposition that the mutual action between the molecules, which goes on in, the interior of solid and liquid bodies, is absent in the case of gases ; so that heat, which in the former cases has to over- come the internal resistances, as well as the external pressure, in order to produce expansion, in the case of gases has to do with external pressure alone. If this be so, then, if a gas expands at constant temperature, only so much heat can thereby be absorbed as is required for doing the external work. Again, we cannot suppose that the total amount of heat actually existing in the body is greater after it has expanded at constant temperature than before. On these assumptions we obtain the following principle ; a permanent gas, if it expands at a constant temperature, absorbs only ON PEEFECT GASES. 43 SO much heat as is required for the eootemal work which it performs in so doing. We cannot of course give to this principle any greater validity than that of the principles from which it springs, but must rather suppose that for any given gas it is true to the same extent only in which the law of Mariotte and Gay Lussac is true. It is only for perfect gases that its absolute accuracy may be assumed. It is on this understanding that the author brought this principle into application, combined it as an approximate assumption with the two main princi- ples of the Mechanical Theory of Heat, and used it for establishing more extended conclusions. More recently Mr, now Sir William Thomson, who at first did not agree with one of the conclusions so deduced, under- took in conjunction with Joule to test experimentally the accuracy of the principle*; and for this purpose instituted with great care a series of skilfully conceived experiments, whichj on account of their importance, will be more fully and exactly discussed further on. These have completely con- firmed the truth not only of the general principle, but also of the remark added by the author as to its degree of exactness. In the permanent gases on which they experimented, viz. aianospheric air and hydrogen, the principle was found so nearly exact that the deviations might for the purpose of most calculations be neglected; while in the non-permanent gas selected for experiment (Carbonic Acid) somewhat greater deviations were observed, exactly as might have been ex- pected from the behaviour of that gas in other respects. After this we may with the less scruple apply the princi- ple, as being exact for actually existing gases in the same degree as the law .of Mariotte and Gay Lussac, and absolutely exact in the case of perf6ct gases. § 3. On the Form which the Equation expressing the first main Principle assumes, in the case of perfect gases. We now return to equation (IV), viz. : dQ = dU+pdv, in order to apply it to the case of a perfect gas, of which we assume as before one unit of weight to be given. » Phil. Trans. 1863, 1854, 1862. 44 ON THE MECHANICAL THEORY OF HEAT. The condition of the gas is completely determined, when its temperature and volume are known ; or it may be deter- mined by its temperature and pressure, or by its volume and pressure. We will at present choose the first-named quan- tities, temperature and volume, to determine the condition, and accordingly treat T and v as the independent variables, on which all other quantities relating to the condition of the gas depend. If then we regard the energy U of the gas as being also a function of these two variables, we may write dr dv ■v^hence equation (IV) becomes dU ,^. fdU ^^-^'^-^if^^y^ («)■ This equation, which in the above form holds not only for a gas, but for any body whose condition is determined' by its temperature and volume, may be considerably simpli- fied for gaseous bodies, on account of their peculiar proper- ties. The quantity of heat, which a gas must absorb in ex- panding at constant temperature through a volume dv, is generally denoted by -r^ dv. As by the approximate assump- tion of the last Section this heat is equal to the work done in the expansion, which is expressed by pdv, we have the equation : dQ, . dQ But from equation (8) dQ^dV^ dv dv ^ ' hence from the last two equations we obtain f-» (»)• ON PERFECT GASES. 45 Hence we conclude that in a perfect gas the energy U is independent of the volume, and can only be a function of the temperature. 77r If in equation (8) we put -5- = 0, and substitute for j-~, the symbol C,, it becomes dQ= C,dT+pdv (10). From the form of this equation we see that C„ denotes the Specific Heat of the Oas at constant volwme, since GjiT expresses the quantity of heat which must be imparted to the gas in order to heat it from 2' to T+dT, when du is equal to zero. As this Specific Heat = -tj,, i.e. is the differ- ential coefl&cient with respect to temperature of a function of the temperature only, it can itself also be only a function of temperature. In equation (10) all the three quantities T, v, and p are found ; but since by equation (6) pv — Rt, it is easy to eliminate one of them ; and by eliminating each in succession we obtain three different forms of the equation. Eliminating p we obtain, dQ'=C,dT + -^d'B (11). 7? 7' Again, to eliminate v we put v =^ — ; whence we have dv = — dT^—T- ip. If we substitute this value of dv in equation (10), and then combine the two terms of the equation which contain dT, we obtain dQ = {C, + B)dT-?^dp (12). Lastly, to eliminate T, we obtain from equation (6), by differentiation, -,_vdp+pdii 4G ON THE MECHANICAL THEOTJT OF HEAT. Substituting -in equation (10) dQ=^vdp + ^^pdv (13). § 4. Deductions as to the two Specific Heats, and transformation of the foregdi ig equations. ^^"VtrfiAKfju^ In the same way as We see from eqiwiion (10) that the quantity C,, which appears as factoi/of dT, denotes the specific heat at constant '^fee mpgratuj e, we may see from equation (12) that the factorof d!Z in that equation, viz. C, + R, expresses the Specific Heat at constant pressiire. If therefore we denote this Specific Heat by G^ we may put (7^=a, + i2.... '....; (14), which eqiiation ^ves the relation between the two Specific Heats. Since 5 is a constant, and C„, as shewn above, is a func- tion of temperature only, it follows from equation (14) that Cp also can only be a function of temperature. When the author first drew in this manner from the Mechanical Theory of Heat the conclusion that the two Specific Heats of a permanent gas must be independent of its density, or in other words of the pressute to which it is subjected, and could depend only on its temperature ; and when he added the further remark that they were thus in all probability constant ; be found himself in opposition to the then prevailing views on the subject. At that time it was considered to be established from the experiments of Suer- mann, and from those of de la Roche and B^rard, that the specific heat of a gas depended on the pressure ; and the circumstance that the new theory led to an opposite conclu- sion produced mistrust of the theory itself, and was used by A. von Holtzmann as a weapon of attack against it. Some years later, however, followed the first publication of the splendid experiments of Eegnault on the specific heat of gases*, in which the influence of pressure and tem- perature on the specific heat was made a subject of special * Comptes Bendm, Vol. xxxn., 1853; also Relation des experiences ■Vol. II. • ' ON PERFECT GASES. 47 investigation. Regnault tested atmospheric air at pressures from 1 to 12 atmospheres, and hydrogen at from 1 to 9 at- mospheres, but could detect no difference in their specific ' heats. He tested them also at different temperatures, viz. between - 30° and + 10°, between 0° and 100°, and between 0° and 200° ; and here also he found the specific heatt always the same *. The result of his experiments may thus be ex- pressed by saying that, within the limits of pressure and temperature to which his observations extended, the specific heat of permanent gases was found to be constant. It is true that these direct explanatory researches were confined to the specific heat at constant pressure; but there will be little scruple raised as to assuming the same to be correct for the other specific heat, which by equation (14) differs from the former only by the constant B. Accordingly in what follows we shall treat the two, specific heats, at least for perfect gases, as being constant quantities. By help of equation (14) we may transform the three equations (11), (12) and (13), which express the first main principle of the Mechanical Theory of Heat as applied to gases, in such a way that they rnay contain, instead of the Specific Heat at constant volume, the Specific Heat at con- stant density ; which may perhaps appear more suitable, since the latter, as being determined by direct observation, ought to be used more frequently than the former. The resulting equations are : .(15). V dQ = C,dT-^^dp dQ = ^^^vdp + ^pdv Lastly, we may introduce both Specific Heats into the equations, and eUminate B, by which means the resulting * The numbers obtained for atmospheric air (Be/, des Exp. Vol. il. , p. 108) are as follows in ordiuary heat units : between - 30» and + 10» 0'23771, „ 00 „ I'OO" 0-23741, „ O" „ , 208» 0-23751, which may be taken as practically the same. 48 ON THE MECHANICAL THEORY OF HEAT, equations become symmetrical as to^ and v, as follows : .(16). G In the above equations the speciiic heats are expressed in mechanical units. If we wish to express them in ordinary- heat units, we have only to divide these values by the Mechanical Equivalent of Heat. Thus if we denote the specific heats, as expressed in ordinary heat units, by c, and Cj,, we may put «.= §. ^ = ^ (17)- Applying these equations to equation (14), and dividing by E, we have c. = c, + |, (18). § 5. Relation between the two Specific Heats, and its application to calculate the Mechanical Equivalent of Heat. If a system of Sound-waves spreads itself through any gas, e.g. atmospheric air, the gas becomes in turn condensed and rarefied; and the velocity with which the sound spreads depends, as was seen by Newton, on the nature of the changes of pressure produced by these changes of density. For very small changes of density and pressure the relation between the two is expressed by the differential coefficient of the pressure with respect to the density, or (if the density, i.e. the weight of a unit of volume, is denoted by p) by the differential coefficient -y-. Applying this principle we obtain for the velocity of sound, which we will call u, the following equa- tion « = V; 4 <'^'' in which g represents the. accelerating force of gravity. ON PERFECT GASES. 49' Now in order to determine the value of the diffe'rential coefficient -J- Newton used the law of Mariotte* according to which pressure and density are proportional to each other. pdp-pd p He therefore put - = constant, whence by differentiation : and therefore whence (19) becomes .- =»■ t-l m: «-\/«:- .(21). The velocity calculated by this formula did not however agree with experiment, and the reason of this divergence, after it had been long sought for in vain, was at last discovered by Laplace. The law of Mariotte in fact holds only if the change of density takes place at constant temperature. But in sound vibrations this is not the case, since in every condensation a heating of the air takes place, and in every rarefaction g, cooling. Accordingly at each condensation the pressure is increased, and at each rarefaction diminished, to a greater extent than accords with Mariotte's law. The question now arises how, under these circumstances, can the value jof -y- dp be determined. Since the condensations and rarefactions follow each other with great rapidity, the exchange of heat that can take place during each short period between the condensed and rarefied parts of the gas must be very small. Neglecting this, we have to do with a change of density, in which the quantity of gas under consideration receives no heat and gives forth none; and we may thus, in applying to this case the differential equa- * Tkis law is commonly known in England as ' Boyle's law,' as being originally due to Boyle. (Translator.) c. 4 So ON THE MECHANICAL THEORY OF HEAT. tions of the last section, put dQ = 0, Hence, e.g. from the last Qf equations (16),, we olstain: or G^vdp + Oj,pdv = 0. Now, since the volume v of one unit of weight 18 the re- ciprocal of the density, we may put v = -, and , therefore r dv =. — j^ ; whence the equation becomes p p or # = £p2 (22). This value of the Differential Coefficient -^ differs from that dp deduced from Mariotte's law, and given in (20), by containing as factor the ratio of the two Specific Heats. If for simplicity we put ^ = 1 (23), the last equation becomes ^ = hP (24), dp p Substituting this value of -^ in equation (19), we get instead of (21) « = \/%^ (25)v From this equation the' velocity of sound u can be calculated if A is known ; or, on the other hand, if the velocity of sound is known by experiment, we can apply the equation to calcu- late k, changing it firs': into the form h^"^^ (26). gp ^ J ON PEBFECT GASES. 51 The velocity of sound iu air has been several times deter- mined with great care by various physicists, -whose results agree with each other very closely. According to the experi- ments of Bravais and Martens* the velocity at freezing tem- perature is 332"4 m. per second (10906 feet). We will in- Siert this value in equation (26). We may also give g its recognized value 9'809 m. (32-2 feet). To determine the quotient - we may give the pressure p any value we please, but we must then assign to the density p the value corre- sponding to that pressure. We will assume p to be the pressure of 1 atmosphere. This must be expressed in the fonnula by the amount of weight supported per unit of surface. As' this weight is equal to that of a column of quicksilver, whose base is 1 sq. m. and height 760 mm., and which therefore has a volume of 760 cubic decimetres, and as, according to Regnault, the Specific Weight of quicksilver at 0°, as compared with water at 4°, is 13"696, we obtain p = l atmosphere = 760 x 13"596 = 10333 kg. per sq. metre. Lastly, p is the weight of a cubic metre of air under the assumed pressure of 1 atmosphere and at temperature 0°, which, according to Regnault, is 1'2932 kg. Substituting these values in equation (26) we obtain (332-4)- X 1-2 932, *~ 9-809x10333 ~*^"- Ha-ving thus determined the quantity h for atmospheric air, we can now use equation (18) to calculate the quantity E, i.e. the Mechanical Equivalent of Heat, as was first done by Mayer. For we have from (18) and, if we again denote by k the quotient -s, which is the » Ann. de Chim. iii., 13, 5; and Fogg. Am. Vol. lxvi., p. 351. 4—2 52 ON THE MECHANICAL THEORY OF HEAT. C ■ C same as -^ , and accordingly substitute -^ for c„ we have ^-of%-: '■'■^- Here we may substitute for k its value 1"410 just found, and for c^ its value as given by Regnault, 02375. It then remains to determine B, or ^-^° . To do this, let us agam take Pn as the pressure of 1 atmosphere, which, as seen above, is equal to 10333, and we then have for v„ the volume in cubic metres of 1 kg. of air under the, above pressure of 1 atmosphere and at temperature 0", which according to Regnault is 0'7733. Lastly we have already assumed the value of a to be 273. The value of B for atmospheric air will therefore be given by the equation P _ 10333 X 07733 __„„ 273 Substituting these values for k, c^, and B in equation (27) we obtain 1-410 X 29-27 _ ^ ~ 0-410 X 0-2375 ~*'^'^^- This figure agrees very closely with that determined by Joule from the friction of water, viz. 423-55. In fact it must be admitted that the agreement is more close than, considering the degree of uncertainty as to the data used in the calculation, we could have had any right to expect; so that chance must have assisted in some degree to produce it. In any case, however, the agreement forms a striking con- firmation of the equations deduced for permanent gases. § 6. Various Formulce relating to the Specific Heats of If in equation (18), p. 48, we consider the quantity ^as known, we may apply that equation to calculate the specific heat at constant volume from that at constant pressure, which is known from experiment. This application is of special importance, because the method of deducing the ratio of the ON PERFECT GASES. 53 two specific heats from the velocity of sound is only prac- ticable in the case of the very few gases for which that velocity has been experimentally determined. For all others, equation (18) offers the only means as yet discovered of calculating the specific heat at constant volume from that at constant pressure. It must here be observed that equation (18) is exactly true only for perfect gases, although it gives at least approxi- mate results for other gases. The circumstance has also to be considered, that the determination of the specific heat of a gas at constant pressure is the more difficult, and therefore the value determined the less reliable, in proportion as the gas is less permanent in its character, and thus diverges more widely in its behaviour from the laws of a perfect gas ; therefore, as there is no need to seek in our calculations a greater accuracy than the experimental values themselves can possibly possess, we may treat the mode of calculation employed as sufficiently complete for our purpose. Accordingly we begin by putting equation (18) in the form o. = c,-§ (28). Here for E we shall use the value 423'55. S is determined by equation (4) It = a where p„v„ are the pressure and volume at the temperature of freezing. Should it be difficult to make observations on the gas at this temperature (as is the case with many vapours) we may also, by equation (6), give B the value i?=5 (29), where p, v, and T are any three corresponding values of pressure, volume, and absolute temperature. This quantity B, as already observed, is only so far depen- dent on the nature of the gas, that it is inversely proportional to its specific gravity. For if we denote by v' the volume of. a unit of weight of air at temperature T and pressure p. 54 O'S THE MECHANICAL THEORY OF HEAT. and by R' the corresponding value of R, we have ■"' ji • Combining this with equation (29), B = R'^, V But -, is the ratio of the volumes of equal weights of the two gases, and is therefore the reciprocal of the ratio of the weights of equal volumes, which ratio is called the Specific Gravity of the gas, as compared with common air. If we call this specific gravity d, the last equation becomes ^ = f (^«)- Substituting this value of B in (28) we obtain '^-'^'-M (^1)- The quantity here denoted by R', i.e. the Value of R for atmospheric air, has been already determined in § 5 to be equal to 29-27. Hence further, whence the equation, which serves to determine the Specific Heat at constant volume, takes this simple form : 0-0691 ''« = ". ^— (32). If in the next place we apply this equation to the case of air, for which d=l, and for the sake of distinction denote by accented letters the two specific heats for air, we get the following equation : c' =c;- 00691 (33), ON PERFECT GASES. 55 and substituting for c'^ its value according to Kegnault, which is 02375, we obtain the result c'. = 0-2375 -0-0691 = 0-1684 (34.)* !For the other gases the equation may be given in the following form : ^^^^-0:0691 ^3.^^ which, as will be seen later, is specially convenient for the application of the values given by Eegnault for specific heats at constant pressure. The specific heats denoted by c^ and c„ relate to a unit of weight of the gas, and have for unit the ordinary unit of heat, i.e. the quantity of heat required to raise a unit of weight of water from the temperature 0° to 1". We may thus say that the gas, in relation to the heat which it requires to raise its temperature either at constant pressure or constant volume, is referred as regards weight to the standard of water. , With gases however it is desirable to refer to the standard of air as regards volume ; i.e. so to determine the specific heat, as to compare the quantity of heat, which the gas Requires to raise its temperature through 1°, with the quantity of heat which an equal volume of air; taken at the same temperature and pressure, requires to raise its tempe- rature to the same extent. We may use this kind of comparison in the case of both the specific heats, inasmuch as we assume in the one case that both the gas under con- sideration and the atmospheric air are heated at constant pressure, and in the other that they are both heated at constant volume. The specific heats thus determined may be denoted by 7^ and 7^. As we denote by v the volume which a unit weight of gas assumes at a given pressure and temperature, the quantity of heat, which a unit-volume of the gas absorbs at constant pressure in being heated through 1°, will be expressed by -^ , • It will be Been that cj and e,' fulfil the condition fonnd , above for perfect gases; — = 1-410. {Translator.) 56 ON THE MECHANICAL THEORY OF HEAT. c or iu the case of atmospheric air by -f ■ The specific heat 7^ is found by dividing the former quantity by the latter, or, ^^ = ^x-^;=%x^ = ^cZ (36). Similarly % = ^d (37). c. In the first of these two equations we may give to c'^ its value as found by Regnault, 0'2375 ; the equation then becomes ^ = 0^ (^«)- In the second we may put for c'„, according to (34)^ the value 0"1684<, and for c, the expression given in (35); whence we have c„d- 00691 ,„Q> § 7. Numerical Calculation of the Specific Heat at con- stant Volume. The formulae developed in the last section have been applied by the author to calculate from the values which Eegnault has determined by his researches for the Specific Heat at constant Pressure of a large number of gases and vapours, the corresponding values of the Specific Heat at constant Volume. In so doing he has in some sort recal- culated one of the two series of numbers given by Regnault himself; who has expressed the Specific Heat at constant Pressure in two different ways, and has brought together the resulting numbers in two series, one of which is superscribed ' en poids,' and the other ' en volume.' The first series con- tains the values which result, if the gases in question are compared weight by weight with water, in relation to the quantity .of heat. required to warm them through 1°; in other words, the values of the quantities denoted above by Cp. The numbers in the second series are simply obtained from those ON PERFECT GASES. 57 in the first by multiplying them by the corresponding specific gravity, i. e. they are the values of the product Cpd. These latter numbers were no doubt those most easily • calculated from the observed values of c^,; but their signi- fication is somewhat complicated. With them the quan- tity of heat has for its unit the ordinary unit of heat, whilst the volume to which they refer is that which a unit-weight of atmospheric air assumes, when under the same tempera- ture and pressure as the gas under consideration. The tedious- ness of the verbal description thus required makes the numbers troublesome to understand and to apply ; moreover this mode of expressing the Specific Heat of gases has been used, so far as the author knows, by no previous writer. In considering gases with reference to volume, it has in all other cases been custoinary to compare the quantity of heat, which a given gas requires to raise its temperature through 1°, with the quantity of heat which an equal volume of atmospheric air requires under the same conditions for the same purpose, or, as briefly expressed above, by comparing the gas, volume for volv/me, with air. The numbers thus obtained are re- markable for their simplicity, and allow the laws which hold as to the specific heats of the gas to be treated with special clearness. It will therefore, the author believes, be found an ad- vantage that he has calculated, from the values given by Eegnault under the heading 'en volume' for the product Gj,d, the values of the quantity 7^,, defined above. All that was required for this, by (38), was to divide the values of c^d by 0-2375. He has further calculated the values of c„ and 7,; calcula- tions which by equations (35) and (39) could be very simply performed, by taking fi:om the values of the product c^d the number 0'0691, and dividing the remainder by d, or by 01684, respectively. The numerical values thus calculated are brought together in the annexed table, in which the different columns have the following signification : Column I. gives the name of the gas. Column II. gives the Chemical composition, and this expressed in such a way that the diminution of volume pro- S8 ON THE MECHANICAL THEORY OF HEAT. duced by the combination can be immediately observed. For in each case those volumes of the simple gas are given, which must combine in order to give Two Volumes of the compound gas. Thus we assume for Carbon, as a gas, such an hypothe- tical volume as we must assume, in order to say that one volume of Carbon unites with one volume of Oxygen to make Carbonic Oxide, or, with two volumes to make Carbonic acid. Again, when, e.g. Alcohol is denoted in the Table by CjjHgO, this means that two volumes of the hypothetical car- bon gas, six volumes of Hydrogen, and one volume of Oxygen, make up together two volumes of Alcoholic vapour. For sulphur-gas the specific gravity used to determine its volume is that found by Sainte-Claire Deville and Troost for very high temperatures, viz. 2"23. In the five last combinations in the Table, which contain Silicon, Phosphorus, Arsenic, Titanium, and Tin, these simple elements are denoted by their ordinary chemical signs, without reference to their volumes in the gaseous condition, because the gaseous, volumes of these elements are partly still unknown, partly hampered with certain irregularities not yet thoroughly cleared up. Column III. gives the Density of the gas, using the values given by Kegnault. Column IV. gives the Specific Heat at constant Pressure as compared, weight for weight, with water, or in other words referred to a unit-weight of the gas and expressed in ordinary units of heat. These are the numbers given by Regnault under the heading ' en poids.' Column V. gives the Specific Heat at constant Pressure compared, volume for volume, with air, calculated by divid- ing by 0'2375 the numbers given by Regnault under the heading ' en volume.' Column VI. gives the Specific Heat at constant Volume compared, weight for weight, with water, calculated by equa- tion (35). Column VII. gives the Specific Heat at constant Volume compared, volume for volume, with air, calculated by equation (39). ON PERFECT GASES. 59 Name of the Gaa. Atmosi)]ierio Air Oxygen Nitrogen Hydrogen Chlorine Bromine Nitric Oxide CarbonicOxide Hydrochloric Acid.... Carbonic Acid Nitric Acid Steam Sulphuric Acid " Hydro-sulphuric Acid. Carbonic di-sulphide . Carburetted Hydrogen Chloroform Olefiant Gas Ammonia Benzine Oil of Turpentine. Wood Spirit Alcohol Ether Ethyl Sulphide.... Ethyl Chloride.... Ethyl Bromide.... Dutch Liquid Aceton Butyric Acid Tri-ohloride of Silicon Tri-chloride of Phos phorus Tri-chlorideof Arsenic Xetra-ohloride of Ti- tanium , Tetra-chloride of Tin. n. Chemical Composi- tion. 0» &. Br, NO CO HCl CO, SOa H^S . OSj •OH4 CHCI3 CjH^ NH3 6 fl CH^O C^HeO C^HioO C^Hju S 0,H,C1 CaHsBr CaH^CI, CjHjO C.HsOa S1CI3 PCI3 AsCIs TiCI^ SnCl^ III. Density. 1 1-1056 0-9713 0-0692 2-4S02 5-4772 1-0384 0-9673 1-2596 1-5201 1-5241 0-6219 2-2113 1-1747, 2-6258 0-5527 4-1244 0-9672 0-5894 2'6942 4-6978 1-1055 1-5890 2-5573 3-1101 2-2269 3-7058 3-4174 2-0036 3-0400 5-8833 4-7464 6-2667 6-6402 8-9654 IV. Specific heat at con- stant Pressure compared weight for weight with Water. 0-2375 0-21751 0-24380 3-40900 0-12099 0-05552 0-2317 0-2450 0-1852 0-2169 0-2262 0-4805 0-1544 0-2432 0-1569 0-5939 0-1567 0-4040 0-5084 0-3754 0-5061 0-4380 0-4534 0-4797 0-4008 0-2738 0-1896 0-2293 0-4125 0-4008 0-1322 0-1347 0-1122 0-1290 0-0939 compared volume for volume with Air. 1 1-013 0-997 0-993 1-248 1-280 1-013 0-998 0-982 1-39 1-45 1-26 1-44 1-20 1-74 1'38 2-72 1-75 1-26 4-26 10-01 2-18 3-03 5-16 5-25 2-57 2-96 3-30 3-48 5-13 3-27 2-69 3-61 3-54 VI. VII. Specific heat at con- stant Volume compared weight for weight with "Water. 0-1684 0-1551 0-1727 2-411 0-0928 0-0429 0-1652 0-1736 0-1304 0-172 0-181 0-370 0-123 0-184 0-131 0-468 0-140 0-359 0-391 0-350 0-491 0-395 0-410 0-453 0-379 0-243 0-171 0-209 0-378 0-378 0-120 0-120 0-101 0-119 0-086 compared volume for volume with Air, 1 1-018 0-996 0-990 1-350 1-395 1-018 0-997 0-975 1-55 1-64 1-36 1-62 1-29 2-04 1-54 3-43 2-06 1-37 5-60 • 13-71 2-60 3-87 6-87 6-99 3-21 3-76 4-24 4-50 6-82 4-21 3-39 3-77 4-67 4-59' 60 ON THE MECHANICA.L THEORY OF HEAT. § 8. Integration of the Differential Equations which ex- press the first main Principle in the case of Oases. The differential equations deduced in sections 3 and 4, which in various forms express the first main principle of the Mechani- cal Theory of Heat in the case of gases, are not immediately integrable, as can be seen by inspection; and must therefore be treated after the method developed in § 3 of the Introduc- tion. In other words, the integration becomes possible as soon as we subject the variables occurring in the equation to some one condition, thus determining the path of the change of condition of the body. We shall here give only two very simple examples of the process, the results of which are important for our further investigations. Example 1. The gas changes its volume at Constant Pressure, and the quantity of heat required for such change is known. In this case we select from the above equations one which contains p and v as independent variables, e.g. the last of Equations (15), which is dQ = ^^vdp + ^piv. As the pressure p is to be constant, we put p=p^, and dp = 0; the equation then becomes dQ = -^p^dv, which gives on integration (if we call v^ the original value oi v) <3 = §Fi(«-«i) (40). Example 2. The gas changes its volume at Constant Temperature, and the quantity of heat required for such change is known. In this case we select an equation which contains y and v as independent variables, e.g. Equation (11), which is dQ = C,dT + — dv. V ON PERFECT GASES. 61 As Tis to be constant, we put T = T^, and dT = ; whence we have 1 V Integrating, V .Q = RT>g^ (41). Hence is_ derived the Principle that if a Gas changes its volume without change of temperature, the quantities of heat absorbed or given off form an arithmetical series, while the volumes form a geometrical series. Again, if we put for R its value ^^ , we have Q=p,v,log^ (42). _ If we suppose this equation.to refer, not directly to a unit weight of the gas, but to a quantity of it such that at pressure p it assumes a volume v^, and then suppose that this volume changes under constant temperature to v, then the equation contains nothing which depends on the special nature of the gas. Therefore the quantity of heat absorbed is independent of the nature of the gas. Further, it does not depend on the temperature, but only on the pressure, being proportional to the original pressure. Another application of the differential equations deduced in sections 3 and 4 consists in making some assumption as to the heat to be imparted to the gas during its change of condition, and then enquiring what course the change of condition will take under such circumstances. The simplest and at the same time most important assumption of this kind is that no heat whatever is imparted to or taken from the gas during its change of condition. For this purpose we may imagine the gas confined in a vessel impermeable to heat, or that the change is so rapid that no appreciable heat can pass to or ffom the gas in the time. 62 ON THE MECHANICAL THEORY OF HEAT. On this assumption we must put dQ = 0. Let us do this for the three Equations (16). Then the first of these becomes C,dT+{G^-G:)^dv = 0. G Dividing by T x (7„ and as before denoting jf by h, we have f+(*-i)?=o. Integrating, log T+(k-l)logv = Const. or Tv^^ = Const. If Tj, Vj are the original values of T, v, we may eliminate the Constant, and obtain .(43). If this equation be applied for example to atmospheric air, then, writing k= 1"410, we can easily calculate the change of temperature which corresponds to any given change of volume. If e.g. we assume a certain quantity of air to be taken at freezing temperature and at any pressure whatever, and to be compressed, either in a vessel impermeable to heat, or with great rapidity, to half its volume, then Tj = 273 (absolute temperature) and — = 2 ; hence the equation be- comes ^'=2»^'» = 1-329, whence T = 273 x 1-329 = 363, or if t be the temperature measured in degrees above freezing point, «= T- 273 = 90°. If a similar calculation is made for the compression of the gas to J and -^ of its original volume, results are obtained, ON PEEFECT GASES. ^3 ■which, combined with the former, are presented in the follow- ing Table. Value of V 1 2 1 4 1 10 T 273 1-329 1-765 2-570 T 863 482 702 t 90» 209» 429» • Again, if in the second of equations (16) we put dQ = 0, we get : F This equation is of the same form as the last, except that p is in the place of v, and that (7^, and C, have their places interchanged. Hence in exactly the same way we shall obtain. whence ©■=(f)" . (44). Finally the last of Equations (16), if d^ be. put = 0, passes into the form already treated in § 5 ; C which may be written and gives on integration P V i>t \vJ .(45). 64 ON THE MECHANICAL THEORY ^ OF HEAT. § 9. Determination of the External Work done during the change ofvolwme of a gas. There is one quantity connected witli the expansion of a gas which still requires to be specially considered, viz. the External Work done in the process. The element of this work, as determined in Equation (6), Ch. I, is d W =pdv. This work may be very clearly set forth by a graphic representation. We will adopt a rectangular system of co-or- dinates, in which the abscissa represents the volume v, and the ordiiiate the pressure p. If we now suppose p to be expressed as a function of v, say p =f (v), then this equation is the equation to a curve, whose ordinates express the values of pr corresponding to the different values of v, and which for brevity we will call the Pressure-curve. In Fig. 3 let rs be this curve, so that, if oe repre- sent the volume v existing at a certain instant, the ordinate ef drawn at e will represent the pressure at the same in- stant. If further eg represent an indefinitely small element of volume dv, and the ordinate gh is drawn at g, then we shall have an indefinitely small para- lellogram efhg, whose area re- presents the external work done in an indefinitely small expansion of the body ; and which differs from the product pdv only by an indefinitely small quantity of the second order, which may be neglected. The same holds for any other indefinitely smaU expansion; and hence in the case of a finite expansion (say from the volume Vj, represented by the abscissa oa, to that otv^, repre- sented by the abscissa oc) the external work, for which we have the equation Fig. 3. W = I pdv , .(46), ON PERFECT GASES. Go IS represented by the quadrangular figure abdc, which is bounded by the difference of abscissae ac, the ordinates ab and cd, and the portion of the pressure-curve bd. In order actually to perform the integration in equation (46) we must know the function of v which expresses the pressure p. On this point we will select as examples the cases already treated in § 8. First, let us assume that the Pressure p is constant. Then the curve of pressure is a straight line parallel to- the axis of X, and abdc is a rectangle (see Fig. 4) whose area is Fig. 4. equal to the product of ac and ab. In this case then we obtain from (46), denoting the constant pressure hy p^, W=p,{v^-v,) •(47), Secondly, let us assume that the Temperature remains constant during the expansion of the gas. Then the. law of Mariotte holds for the relation between pressure and volume, and is expressed by the equation pv= const. From the form of this equation we see that the curve of pressure is an equilateral hyperbola (Fig. 5) having the axes of co-ordinates as asymptotes. A pressure- curve of this kind, which involves the special condition that the temperature is constant, is usually called an Isothermal Curve. To effect the integration in this case we may write for p c. 5 66 ON THE MECHANICAL THEOET OF HEAT. the value ■^' , where 'p^^ is any value obtained for the conr stant in the above equation ; we then get from (46) : ^=PA£''7=i'i^i^og^^ (*®)- We observe that this value of TT coincides with that given in equation (42) for Q ; the reason for this being that the gas, while expanding at constant temperature, absorbs only so much heat as is required for the external work. Joule has employed the equation (48) in one of his deter- minations of the Mechanical Equivalent of Heat. For this purpose he forced atmospheric air into a strong receiver, up to ten or twenty times its normal density. The receiver and pump were meantime kept under water, so that all the heat which was developed in pumping could be measured in the water. The apparatus is represented in Fig. 6, in which H is the receiver, and G the pump. The vessel G, as will be easily understood, was used for the drying of the air, and the vessel with the spiral tube served to give to the air, before its entrance into the pump, an exactly known temperature. From the total quantity of heat given in the calorimeter Joule subtracted the part due to the friction of the pump, the amount of which he determined by working the pump for exactly the same length of time, and under the same mean pressure, but without allowing the entrance of air, and then observing the heat produced. The remainder, after this was subtracted, he took as being the quantity of heat de- veloped by the compression of the air ; and this he compared with the work required for the compression as given by equa- tion (48). By this means he obtained as the mean of two series of experiments the value of 444 kilogrammetres as the Mechanical Equivalent of Heat. This value, it must be admitted, does not agree very well with the value 424 obtained by the friction of water ; the reason of which is probably to be found in the far larger sources of erroi: attending experiments on air. Nevertheless at that time, when the fact that the work required for developing a given quantity of heat was equal under all cir- cumstances was not yet placed on a firm basis, the agreement ON PERFECT GASES. 67 of- the values found by such wholly different methods was close enough to aid considerably in the establishment of the principle. As a third case of determination of work done, we may assume that the gas changes its volume within an envelope c c cy;^ impermeable to heat ; or, which comes to the same thing, that the change of volume takes place too rapidly to allow of the passing of any appreciable quantity of heat to or from the body during the time. 5—2 ON THE MECHANICAL THEORY OP HEAT. In this case the relation between pressure and volume is given by equation (45), viz.: Pi \vj ' "The curve of pressure corresponding to this equation (Fig. 7) falls more steeply than that delineated in Fig. 5. ilankine has given to this special class of pressure- curves, which correspond to the case of expansion within an envelope impermeable to heat, the name of Adiabatic curves (from Sia/3alveiv, to pass through). On the other hand Gibbs {Trans. Connecticut Academy, vol. II. p. 309) has proposed to name them Isentropiccurves, because in this kind of ex- pansion the Entropy, a quan- Fig. 7. tity which will be discussed further on, remains constant. This latter form of nomenclature is the one which the author pro- poses to adopt, since it is both usual and advantageous to designate curves of this kind according to that quantity which remains constant during the action that takes place. To effect the integration in this case, we may put, accord- ing to the above equation, I>=FiK whence (46) becomes (49). (69 ) CHAPTEE III. SECOND MAIN PRINCIPLE OF THE MECHANICAL THEORY OF HEAT. § 1, Description of a special form of Cyclical Process. In order to prove and to make intelligible the second Principle of the Mechanical Theory of Heat, we shall com- mence by following out in all its parts, and graphically repre- senting in the manner already described, one special form of cyclical process. For the latter purpose we will assume that the condition of the variable body is determined by its volume V and its pressure p, and will employ, as before, a rectangular system of co-ordinates, in which the abscissse represent'volumes, and the ordinates pressures. Any point on the plane of co-ordinates will then correspond to a certain condition of the body, in which its volume and pressure have the same volumes as the abscissa and ordinate of the point. Further, every variation of the body's condition will be represented by a line, whose extreme points determine the initial and final condition of the body, and whose form shews the way in which the pressure and volume have simul- taneously varied. In Fig. 8 let the initial condition of the body, at which the cyclical process commences, be given by the point a, so that the abscissa oe = Vi and the ordinate ea=^p^ represent the initial volume and pressure respectively. By means of these two quantities the initial temperature, which we wUl call Tj, is also fixed. 70 ON THE MECHANICAL THEORY OF HEAT. Now let the body in the first place expand, while retain- ing the same temperature T,. If no heat were imparted to it e i f Fig. 8. during expansion, it would necessarily become cooler: we will therefore assume that it is put in communication with a body K, acting as a reservoir of heat, which body has the same temperature T,, and does not appreciably vary from this during the cyclical process. From this body the variable body is supposed to draw during the expansion just sufficient heat to keep itself also at the temperature T^. The curve, which duritig this expansion expresses the change of pressure, is part of an isothermal curve. In order tha,t we may give definite forms to the graphic repre- sentations of this curve, and of others yet to be described, we will, without limiting the investigation itself to any particular bodies, draw the figure as it would appear in the case of a perfect gas. Then the isothermal curve, as ex- plained above, will be an equilateral hyperbola; and, if the expansion take place from the volume oe = v^ to the volume o/= Fj, we shall obtain the part ah of such an equilateral hyperbola. When the volume V^ has been reached, let us suppose the body K^ to be withdrawn, and let the variable body be left to continue its expansion by itself, without any heat hieing imparted to it. The temperature must then fall, and we obtain as curve of pressure an iseniropio curve, which descends more steeply than the isothermal curve. Let this expansion continue till the volume V^ is reached, giving us the portion of an isentropic curve be. The lower temperature thus attained we may call T.^. SECOND MAIN PRINCIPLE. 7l From henceforward let the body be compressed, so as to bring it back, to its original volume. Let the compression first take place at the constant temperature T^, for which purpose we may suppose that the body is connected with a body K^ at temperature T^, acting as a reservoir of heat, and that it gives up to K^ just so much heat as suffices to keep itself also at temperature T^. The pressure-curve correspond- ing to this compression is again an isothermal curve, and in the special case of a perfect gas is another equilateral hyper- bola, of which we obtain the portion cd during the reduction of volume to oh=v^. Finally, let the last compression, which brings the varia- ble body back to its initial volume, take place without the presence of the body K^, so that the temperature rises, and the pressure follows the line of an isentropic curve. We wUl assume that the volume oh = v^, up to which the compression went on according to the first mode, is so chosen, that the compression which begins from this volume and continues to volume oe = v^ is just sufficient to raise the temperature again from T^ to T^. If then the initial tem- perature is thus regained at the same time as the initial volume, the pressure must also return to its initial value, and the last curve of pressure must therefore exactly hit the point a. When the body is thus brought back again to the original condition, expressed by the point a, the cyclical pro- cess is complete. § 2. Result of tJie Cyclical Process. During the two expansions which take place in the course \ \, d c h f S Fig. 10. vaporisation is continually replaced from the body K^, so that the temperature, and with it the pressure of the vapour, remains unaltered. The isothermal curve corresponding to this expansion is therefore a straight line parallel to the abscissa. When the combined volume has increased in this way from ■oe to of, a quantity of external work has been thereby performed, which is represented by the rectangle eabf. Now withdraw the body iT,, and let the envelope enlarge still further, withotit any passage of heat inwards or outwards. Then there will be partly an expansion of the vapour already existing, partly a generation of new vapour ; in consequence the temperature will fall, and the pressure with it. Let this go on until the temperature has changed from Tj to T^, at which time the volume og has been attained. The fall of pressure, which has taken place during this expansion, will be represented by the isentropic curve be, and the external work performed by fbrg. Now let the envelope be compressed, so as to bring the liquid and vapour back again to their original combined SECOND MAIN PRINCIPLE. 7o volume oe ; and let this compression take place-, partly in con- nection with the body K^, of constant temperature T^, to which all the heat produced by condensation of vapour passes over, so that the temperature T^ remains unaltered : partly apart from this body, so that the temperature rises. Let it also be arranged, that the first compression shall extend only so far (to oh) as that the decrease of volume he then remaining may be just sufficient to raise the temperature again from T^ to T^. During this first compression the pressure remains unaltered, at the value gc ; the external work thus absorbed is therefore represented by the rectangle gcdh. During the last compres- sion the pressure increases, and is represented by the isentropic curve da, which must end exactly at the point a, since with the original temperature T^ we must also have the original pressure ea. The external work absorbed in this last operation is represented by hdae. At the end of the operation the liquid and vapour are again in their original condition, and the cyclical process is complete. The surplus of the positive ^bove the negative external work, or the external work W which has been gained on the whole in the course of the process, is represented as before by the quadrangle abed. To this work must correspond the absorption of an equivalent quantity of heat Q ; and if we denote by Q^ the heat imparted during the expansion, and by Qj the heat given out during the contraction, we may put Qj = Q+^j, and the final result of the cyclical process is again expressed by saying, that the quantity of heat Q is converted into work, and the quantity Q.^ has passed over fi;om the hotter body K^ to the colder ^j. ■ This cyclical process may also be carried out in the reverse direction, and then the quantity of heat Q will be generated out of work, and given off to the body K^, while the quantity Q^ will pass over from the colder body /C, to the hotter K^. In a similar manner cyclical processes of this kind may be carried out with other variable bodies, and graphically represented by two isothermal and two isentropic lines ; in which cases, while the form of the curves depends on the nature of the body, the result of the process is always of the same kind, viz. that one quantity of heat is converted into work, or generated out of work, and that another quantity passes over from a hotter to a colder body, or vice versa. 76 ON THE MECHANICAL THEORY OF HEAT. The question now arises. Whether the quantity of heat converted into work, or generated out of work, stands in a generally constant proportion to the quantity which passes over from the hotter to the colder body, or vice versd; or whether the proportion existing between them varies according to the nature of the variable body, which is the medium of the transfer. § 4. Carnot's view as to the work performed during a Cyclical Process. Camot, who was the first to remark that in the produc- tion of mechanical work heat passes from' a hotter into a colder body, and that conversely in the consumption of mechanical work heat can be brought from a colder into a hotter body, and who also conceived the simple cyclical process above described (which was first represented graphically by Clapeyron), took a special view of his own as to the funda- mental connection of these processes *- In his time the doctrine was still generally prevalent that heat was a special kind of matter, which might exist within a body in greater or lesser quantity, and thereby occasion differences of temperature. In accordance with this doctrine it was supposed that heat might change the character of its distribution, in passing from one body into another, and further that it could exist in different conditions, which were denominated respectively 'free' and 'latent'; but that the whole quantity of heat existing in the universe could neither be increased nor diminished, inasmuch as matter can neither be created nor destroyed. Camot shared these views, and accordingly treated it as self-evident that the quantities of heat, which the variable body in the course of the cyclical process receives from and gives out to the surrounding space, are equal to each other, and consequently cancel each other. He lays this down very distinctly in § 27 of his work, where he says : " we shall assume that the quantities of heat absorbed and emitted in these different transformations compensate each other exactly. This fact has never been held in doubt ; admitted at first with- out reflection, it has since been verified in many instances by • Bejlexions lur la puissance motriee du feu. Paris, 1824. " "' SECOND MAIN PfllNCIPLK 77 experiments with the calorimeter. To deny it would be to sub- Vert the whole theory of heat, which rests on it as its basis." Now since on this assumption the quantity of heat exist- ing in the body was the same after the cyclical process as before it, and yet a certain amount of work had been achieved, Carnot sought to explain this latter fact from the circum- stance of the heat falling from a higher to a lower tempera- ture. He drew a comparison between this descending passage of heat (which is especially striking in the steam-engine, where the fire gives off heat to the boiler, and conversely the cold water of the condenser absorbs heat) and the falling of water from a higher to a lower level, by means of which a machine can be set in motion, and work done. Accordingly in § 28, after making use of the expression ' fall of water,' he applies the corresponding expression ' fall of caloric' to the sinking of heat from a higher to a lower temperature. Starting from these premises, he laid down the principle that the quantity of work done must bear a certain constant relation to the 'passage of heat,' i.e. the quantity of heat passing over at the time, and to the temperature of the bodies between which it passes ; and that this relation is indepen- dent of the nature of the substance which serves as a medium for the performance of work and passage of heat. His proof of the necessary existence of this constant relation rests on the principle " That it is impossible to create moving force out of nothing," or in other words, " That perpetual motion is an impossibility." This mode of dealing with the question does not accord with our present views, inasmuch as we rather assume that in the production of work a corresponding quantity of heat is consumed, and that in consequence the quantity of heat given out to the surrounding space during the cyclical process is less than that received from it. Now if for the production of work heat is consumed, then, whether at the same time with this consumption of heat there takes place the passage of another quantity of heat from a hotter to a colder body, or not, at least there is no ground whatever for saying that the work is created out of nothing. Accordingly not only must the principle enunciated by Carnot receive some modifi- cation, but a different basis of proof from that used by him must be discovered. 78 ON THK 'MECHANICAL THEORY OF HEAT. § 5. New Fundamental Principle concerning Heat. Various considerations as to the conditions and nature of heat had led the author to the conviction that the tendency of heat to pass from a warmer to a colder body, and thereby equalize existing differences of temperature (as prominently shewn in the phenomena of conduction and ordinary radia- tion), was so intimately Tsound up with its whole constitution that it must have a predominant influence under all conceiv- able circumstances. He thereupon propounded the following as a fundamental principle : " Heat cannot, of itself, pass from a colder to a hotter body." The words ' of itself,' here used for the sake of brevity, require, in order to be completely understood, a further ex- planation, as given in various parts of the author's papers. In the first place they express the fact that heat can never, through conduction or radiation, accumulate itself in the warmer body at the cost of the colder. This, which was already known as respects direct radiation, must thus be further extended to cases in which by refraction or reflection the course of the ray is diverted and a concentration of rays thereby produced. In the second place the principle must be applicable to processes which are a combination of several different steps, such as e.g. cyclical processes of the kind described above. It is true that by such a process (as we have seen by going through the original cycle in the reverse direction) heat may be carried over from a colder into a hotter body : our principle however declares that simul- taneously with this passage of heat from a colder to a hotter body there must either take place an opposite passage of heat from a hotter to a colder body, or else some change or other which has the special property that it is not reversible, except under the condition that it occasions, whether directly or indirectly, such an opposite passage of heat. This simul- taneous passage of heat ia the opposite direction, or this special change entaUing an. opposite passage of heat, is then to be treated as a compensation for the passage of heat from the colder to the warmer body ; and if we apply this concep- tion we may replace' ^ihe words " of itself" by " without com- pensation," and then enunciate the principle as follows : " A passage of heat from a colder to a hotter body cannot take place without compensation." SECOND MAIN peincipm:. 79 This proposition, laid down as a Fundamental Principle by the author, has met with much opposition ; but, having repeatedly had occasion to defend it, he has always been able to shew that the objections raised were due to the fact that the phenomena, in which it was believed that an uncompen- sated passage of heat from a colder to a hotter body was to be found, had not been correctly understood. To state these objections and their answers at this place would interrupt too seriously the course of the present treatise. In the discus- sions which follow, the principle, which, as the author believes, is acknowledged at present by most physicists as being correct, will be simply used as a fundamental principle ; but the author proposes to retixm to it further on, and then to consider more closely the points of discussion which have been raised upon it. § 6. Proof that the relation between the quantity of heat carried over, and that converted into work, is independent of the nature of the matter which forms the medium of the change. Assuming the foregoing principle to be correct, it may be proved that between the quantity of heat Q, which in a cyclical process of the kind described above is transformed into work (or, where the process is in the reverse order, generated by work), and the quantity of heat Q^, which is transferred at the same time from a hotter to a colder body (or vice versS,), there exists a relation independent of the nature of the variable body which acts as the medium of the transformation and transfer ; and thus that, if several cyclical processes are per- formed, with the same reservoirs of heat K^ and K^^, but with different variable bodies, the ratio j- will be the same for all. If we suppose the processes so arranged, according to their magnitude, that the quantity of heat Q, which is trans- formed into work, has in all of them a constant value, then we have only to consider the nmgnitude of the quantity of heat Qj which is transferred, and the principle which is to be proved takes the following form : " If ^here two different variable bodies are used, the quantity of heat Q transformed into work is the same, then the quantity of heat Q^, which is transferred, will also be the same." 80 ON THE MECHANICAL THEORY OF HEAT. Let there, if possible, be two bodies G and C (e.g. the perfect gas and the combined mass of liquid and vapour, described above) for which the values of Q are equal, but those of the transferred quantities of heat are different, and let these different values he called Q^ and Q\ respectively : (y^ being the greater of the two. Now let us in the first place subject the body (7 to a cyclical process, such that the quantity of heat Q is transformed into work, and the quantity Qjj is transferred from K^ to K^. Next let us subject C to a cyclical process of the reverse description, so that the quantity of heat Q is generated out of work, and the quantity Q'^ is transferred from K^ to K^. Then the above two changes, from heat into work, and work into heat, will cancel each other ; since we may suppose that when in the first process the heat Q has been taken from the body K^ and transformed into work, this same work is expended in the second process in producing the heat Q, which is then returned to the same body K^. In all other respects also the bodies will have returned, at the end of the two operations, to their original condition, with one exception only. The quantity of heat Q\, trans- ferred from K^ to K^, has been assumed to be greater than the quantity Q^ transferred from K^ to K^. Hence these two do not cancel each other, but there remains at the end a quantity of heat, represented by the difference Q\ — Q^, which has passed over from K^ to K^. Hence a passage of heat will have taken place from a colder to a warmer body without any other compensating change. But this contradicts the funda- mental principle. Hence the assumption that Q\ is greater than Qj must be false. Again, if we make the opposite assumption, that Q^ is less than Q^, we may suppose the body C to undergo the cyclical process in the first, and G in the reverse direction. We then arrive similarly at the result that a quantity of heat Q^ — Q\ has passed from the colder body K, to the hotter K^, which is again contrary to the principle. Since then Q^ can be neither greater nor less than Q^ it must be equal to Q^; which was to be proved. We will now give to the result thus obtained the mathe- matical form most convenient for bur subsequent reasoning. Since the quotient ^ is independent of the nature of the SECOND MAIN PRINCIPLE. 81 variable body, it can only depend on the temperature of the two bodies JT^ and ^j, which act as heat reservoirs. The same will of course be true of the sum This last ratio, which is that between the whole heat received and the heat transferred, we shall select for further considera- tion ; and shall express the result obtained in this section as follows : " the ratio -^ can only depend on the temperatures Tj and T^" This leads to the equation : I^'^c^'.t;) (2), in which <^ {T^T^ is some function of the two temperatures, which is independent of the nature of the variable body. § 7. Determination of the Function (T^T^. The circumstance that the function given in equation (2) is independent of the nature of the variable body, offers a ready means of determining this function, since as soon as we have found its form foi; any single body it is known for all bodies whatsoever. Of all classes of bodies the perfect gases are best adapted for such a deterniination, since their laws are the most accu- rately known. We will therefore consider the case of a per- fect gas subjected to a cyclical process, similar to that graphi- cally expressed in Fig. 8, § 1 ; which figure may be here repro- duced (Fig. 11), inasmuch as a perfect gas was there taken as an example of the variable body. In this process the gas takes up a quantity of heat Q during its expansion ah, and gives out a quantity of heat Qj during its com- pression cd. These quanti- ties we shall calculate, and l — f g ■' then compare with each other. For this purpose we must first turn our attention to the volumes represented by the c. 6 Fig. 11. 82 ON THE MECHANICAL THEORY OF HEAT. aBscissse oe, oh, of, off, and denoted by v^, v^, F,, F,, in order that we may ascesrtain the relation between them. Now the volumes v^v^ (represented by oe, oh) form the limits of that change of volume to which the isentropic curve ad refers, and which may be considered at pleasure as an expansion or a compression. Such a change of volume, during which the gas neither takes in nor gives out any heat, has been treated of in § 8 of the last chapter, in which we arrived at the fol- lowing equation (43), p. 62 : 2; \v) ' c„ where T and v are the temperature and volume at any point in the curve. Substituting for these in the present case the final values T„ and v,, we have : "2' S.ft\ !=© (=^ In exactly the same way we obtain for the change of volume represented by the isentropic curve be (of which the initial and final temperatures are also T^T^ ; T /F\*^ fW (* Combining these two equations we obtain : T^ = -, or — '=— ? (o). We must now turn to the change of volume represented by the isothermal curve ab, which takes place at the constant temperature T^, and between the limits of volume v^ and Fj. The quantity of heat received or given off during such a change of volume has been determined in § 8 of the last chapter, and by the equation (41) there given, p. 61, we may put in the present case : Q, = ET,log'^K (6). SECOND MAIN PRINCIPLE. 83 Similarly for the change of volume represented by the iso- thermal curve do, which takes place at temperature T^ between the limits of volume v^ and V^, we have : Q, = i?T,logJ' (7). From these two equations we obtain by division : ^ = ^ (8) V V smcebyfS) —' = —*. The function occurring in equation (2) is now determined, since to bring this equation into unison with the last equation (8) we must have : <^(2;rj = -JJ (9). ■'a We can now use in place of equation (2) the more deter- minate equation (8), which may also be written as follows : &-^^ = (10). , The form of this equation may be yet further changed, by affixing positive and negative signs to Q^, Q^. Hitherto these have been treated as absolute quantities, and the distinction that the one represents heat taken in, the other heat given out, has been always expressed in words. Let us now for convenience agre6 to speak of heat taken in only, and to treat heat given out as a negative quantity of heat taken in. If accordingly we say that the variable body has taken in during the cyclical process the quantities of heat Qj and Q,, we must here conceive Q^ as a negative quantity, i.e. the same quantity which has hitherto been expressed by —Q^. On this supposition equation (10) becomes : % + %' = (11>, 84 ON THE MECHANICAL THEORY OF HEAT. § 8, Cyclical processes of a more complicated character. Hitherto we have confined ourselves to cyclical processes in which the taking in of quantities of heat, positive or negative, takes place at two temperatures only. Such pro- cesses we shall in future call for brevity's sake Simple Cyclical Processes. But it is now time to treat of cyclical processes, in which the taking in of positive and negative quantities of heat takes place at more than two tempera- tures. We may first consider a cyclical process with heat taken in at three temperatures. This is represented graphically by the figure abcdefa (Fig. 12), which, as in the former cases, consists of isentropic and isothermal curves only. These curves are again drawn, by way of example, in the form which they would take in the case of a perfect gas, but this. g ^-^e Fig. 12. is not essential. The curve ab represents an expansion at. constant temperature T^; be an expansion without taking in heat, during which the temperature falls from T, to T^; cd an expansion at constant temperature T^ ; de an expansion without taking in heat, during which the temperature falls from jTj to Tg ; ef a, compression at constant temperature T^ ; and lastly fa a compression withoiit taking in heat, during which the temperature rises from T^ to T^, and which brings back the variable body to its exact original volume. In thp SECOND MAIN PRINCIPLE. 85 expansions ab and cd the variable body takes in positive quantities of beat Q, and Q,, and in the compression ef the negative quantity of heat Q^. It now remains to find a rela- tion between these three quantities. For this purpose let us suppose the isentropic curve 6c produced in the dotted line eg. The whole process is thereby divided into two Simple Processes abgfa and cdegc. In the first the body starts from the condition a and returns to the same again. In the second we may suppose a body of the same nature to start from the condition e, and to r6tum to the same again. The negative quantity of heat Q^, which is taken in during the compression ef, we may suppose divided into two parts q^ and g,', of which the first is taken in during the compression gf, and the second during the compression eg. We can now form the two equations, corresponding to equation (11), which will hold for the two simple processes. These equations are, for the process abgfa, ^+53 = 'p ~ fp ' and for the process cdego -tj ■'■3 Adding these equations we obtain -'i ■'■i ■'■a or, since 93+9a=Qa' .(12). In exactly the same way we may treat a process in which heat is taken in at four temperatures, as represented by the annexed figure abcdefgha. Fig. 13, which again consists solely of isentropic and isothermal lines. The expansions ab and cd, and the compressions ef and gh, take place at temperatures Tj, Tjj, Tj, T^, and during these times the quantities of heat Qi>Qi> Qi> Qi ^^® taken inrespectively ; the two former being 86 ON THE MECHANICAL THEOEY OF HEAT. positive, and the two latter negative. Ptoduce the isentropic curves 6c and fg in the dotted lines ci and gk respectively. Then the whole process is subdivided into three Simple Pro- cesses akghja, hUfk, and cdeic, which may be supposed to be 1^ — ^e Fig. 13. carried out with three exactly similar bodies. We may sup- pose the quantity of heat Q^, taken in during the expansion ah, to be divided into two parts q^ and q^, corresponding to ex- pansions ak and kh ; and the negative quantity Q^, taken ia during the compression ef, to be likewise divided into q^ and q^, corresponding to compressions if and ci. Then we can form the followiag equations for the three simple processes : First, for akgha Secondly, for Icbifk Thirdly, for cdeic 2>- + «5 = ■'a ■'■a SECOND MAIN PRINCIPLE. 87 Adding, we obtaan or rp ~ rp ^ rr "^ m> " ■^1 -^H -^S -^4 .(13). In exactly the same way any other cyclical process, which can be represented by a figure consisting solely of isentropic and isothermal lines, and which has any given number of temperatures at which heat is taken in, may be made to yield an equation of the same form, viz. or generally •^l -^2 -^3 -^4 4-0. .(11). § 9. Cyclical Processes, in which taking m of Heat and change of Temperatv/re take place simultameously. We have lastly to consider such cyclical processes as are represented by figures not consisting solely of isentropic and isothermal lines, but altogether general in form. The mode of treatment is as foUows. Let point a in Fig. 14. Fig. 14 represent any given condition of the variable body ; let pq be an arc of the isothermal curve which passes through 88 ON THE MECHANICAL THEORY OF HEAT. a, rs an arc of the isentropic curve wliicli passes througt thfe same point. Now let the body un4ergo a variation which is expressed by a pressure-curve not coinciding with either of the above, but taking some other course such as be or de. Then we may consider. such a variation as made up of a very grSat number dfvery small variations, in which we have alternately change of temperature without taking in of heat, and taking in of heat without chahge of temperature. This series of successive variations wiU be represented by a dis- continuous line, made up of alternate elements of isothermal and isentropic curves, as drawn in Fig. 15, along the course of Fig. 15. be and de. The smaller the elements of which the dis- continuous curve is made up, the more closely will it coincide with the continuous line, and if these are indefinitely small the coincidence wiU be indefinitely close. In this case it can only make an indefinitely small difference, in relation to the quantities of heat taken in and their temperatures, if we substitute for the variation reprpsented by the continuous line the indefinitely large number of alternating variations, which are represented by the discontinuous line. We are now in a position to consider a complete cyclical process, in which the taking in of heat is simultaneous with changes of temperature, and which may be represented graphically by curves of any form whatever, or merely by a 'single continuoTls and closed curve, such as is drawn in Fig. 16. The area of this closed curve represents the ex- SECOND MAIN PRINCIPLE. 89 ternal work consumed. Let it be divided into indefinitely thin strips by means of adjacent isentropical curves, as shewn by the dotted lines in Fig, 16. Let us suppose these curves joined at the top and bottom by indefinitely small elements Fig. 16. of isothermal lines, which cut the given curve, so that throughout its length we have a broken line, which is every- where in indefinitely close coincidence with it. By the above reasoning we may substitute for the process represented by the continuous line the other process represented by the broken line, without producing any perceptible alteration in the quantities of heat taken in, or in their temperatures. Fur- ther, we may again substitute for the process represented by the broken fine an indefinitely great number of Simple Pro- cesses, which will be represented by the indefinitely small quadrangular strips, made up each of two adjacent isentropic curves, and two indefinitely smaU elements of isothermal curves. If then for each one of these last processes we form an equation similar to (11), in which the two quantities of heat are indefinitely small, and can therefore be denoted by differentials of Q ; and if all these equations be finally added together ; we shall then obtain an equation of the same form as (14), but in which the sign of summation is replaced by the sign of Integration, thus : -f ^ ^^^^- I'- 90 ON THE MECHANICAL THEORY OF HEAT. This equation, which was first published by the author in 1854 (Pogg. Ann. vol. 93, p. 500), forms a very convenient expression for the second main Principle of the Mechanical Theory of Heat, as far as it relates to reversible processes. This Principle may be expressed in words as follows : If in a reversible Cyclical Process every element of heat taken in {positive or negative) he divided by the absolute temperatwre at which it is taken in, and the differential so formed be inte- grated for the whole cov/rse of the process, the integral so ob- tained is equal to zero. If the integral l-m-j corresponding to any given succession of variations of a body, be always equal to zero provided the body returns finally to its original condition, whatever the intervening conditions may be, then it follows that the ex- pression under the integral sign, viz. -^ , must be the perfect differential of a quantity, which depends only on the present condition of the body, and is altogether independent of the way in which it has been brought into that condition. If we denote this quantity by 8, we may put or dQ = Td8..., (VI), . an equation which forms another expression, very convenient in the case of certain investigations, for the second main principle of the Mechanical Theory of Heat. (91) CHAPTER IV. THE SECOND MAIN PRINCIPLE UNDER ANOTHER FORM, OR PRINCIPLE OF THE EQUIVALENCE OF TRANSFORMATIONS. § 1. On the two different hinds of Transformations. In the last chapter it was shewn that in a Simple Cyclical Process two variations in respect to heat take place, viz. that a certain quantity of heat is converted into work (or generated out of work), and another quantity of heat passes from a hotter into a colder body (or vice versS,). It was found fur- ther that between the quantity of heat transformed into work (or generated out of work) and the quantity of heat transferred, there must be a definite relation, which is independent of the nature of the variable body, and therefore can only depend on the temperatures of the two bodies which serve as reservoirs of heat. For the former of these two variations we have already employed the word "transformation," inasmuch as we said, when work was expended and heat thereby produced, or conversely when heat was expended and work thereby pro- duced, that the one had been " transformed " into the other. We may use the word " transformation " to express the second variation also (which consists in the passage of heat from one body into another, which may be colder or hotter than the first), inasmuch as we may say that heat of one tem- perature " transforms " itself into heat of another tempera- ture. On this principle we may describe the result of a simple cyclical process in the following terms : Two transformations are produced, a transformation from heat into work (or vice versS,) and a transformation from heat of a higher tempera- 92 ON THE MECHANICAL THEORY OF HEAT. ture to heat of a lower (or vice versS.). The relation between these two transformations is therefore that which is to be ex- pressed by the second Main Principle. Now, in the first place, as concerns the transformation of heat at one temperature to heat at another, it is evident at once that the two temperatures, between which the trans- formation takes place, must come under consideration. But the further question now arises, whether in the trans- formation from work into heat, or from heat into work, the temperature of the particular quantity of heat concerned plays an essential part, or whether in this transformation the particular temperature is matter of indifference. If we seek to deduce the answer to this question from the consideration of a Simple Cyclical Process, as described above, we find that it is too limited for our purpose. For since in this process there are only two bodies which act as heat, reservoirs, it is tacitly assumed that the heat which is trans- formed into work is derived from (or conversely the heat generated out of work is taken in by) one or other of these same two bodies, between which the transference of heat also takes place. Hence a definite assumption is made from the beginning as to the temperature of the heat transformed into work (or conversely generated out of work), viz. that it coincides with one of the two temperatures at which the transference of heat takes place ; and this limitation prevents us from learning what influence it would have on the relation between the two transformations if the first-mentioned tem- perature were to alter, while the two latter remained un- altered. To ascertain this influence, we may revert to those more complicated cyclical processes, which have also been described in the last chapter, § 8, and to the equations derived from them. But in order to give a clearer and simpler view of the question it is better to consider a single process specially chosen for this investigation, and by its help to bring out the second Main Principle anew in an altered form. § 2. On a Cyclical Process of special form. Let us again take a variable body, whose condition is completely . determined by its volume and pressure, so that THE EQUIVALENCE OF TRANSFORMATIONS. 93 we can represent its variations graphically in the manner already described. We will once more by way of example construct the figure in the form it assumes for a perfect gas, but without making in the investigation itself any limiting assumption whatever as to the nature of the body. Let the body be first taken in the condition defined by the point a in Fig. 17, its volume being given by the abscissa Fig. 17. oh, and its pressure by the 'ordinate ha. Let T be the tem- perature corresponding to these two quantities, and deter- mined by them. We will now subject the body to the follow- ing successive variations : (1) The temperature T of the gas is change.d to T^ , which we will suppose less than T. This may be done by enclosing the gas within a non-conducting envelope, so that it can neither take in nor give out heat, and then allowing it to expand. The decrease of pressure caused by the simul- taneous increase of volume and faU of temperature will be represented by the isentropic curve ah; so that, when the temperature of the gas has reached T,, its volume and pressure have become ovand ih respectively. (2) The variable body is placed in communication with a body iTj of temperature T,, and then allowed to expand still further, but so that all the heat lost in expansion is restored by K^. With respect to the latter it is assumed that, on account of its magnitude or from some other pause, its temperature is hot 94 ON THE MECHANICAL THEORY OF HEAT. perceptibly altered by this giving out of beat, and may tbere- fore be taken as constant. Hence the variable body will also preserve during its expansion the same constant temperature r,, and its diminution of pressure will be represented by an isothermal curve be. Let the quantity of heat thus given off by Z, be called Q^. (3) The variable body is disconnected from K^ and allowed to expand still further, without being able either to take in or give out heat, until its temperature has faUen from T^ to fj. Let this diminution of pressure be represented by the isentropic curve cd. (4) The variable body is placed in communication with a body K^, of constant temperature T^, and is then com- pressed, parting with all the heat generated by the com- pression to jKj. This compression goes on until K^ has received the same quantity of heat Q,, as was formerly abstracted from K . In this case the pressure increases ac- cording to the isothermal curve de. (5) The variable body is disconnected from K^, and compressed, without being able to take in or give out heat, until its temperature has risen from T^ to its original value T, the pressure increasing according to the isentropic curve- ef. The volume on, to which the body is brought by this process, is less than the original volume oh, since the pressure to be overcome, and consequently the external work to be transformed into heat, is less during the compression de than during the expansion be; so that, in order to restore the same quantity of heat Q,, the compression must be con- tinued further than would have been necessary merely to annul the expansion. (6) The variable body is placed in communication with a body-^ of constant temperature T, and allowed to expand to its original volume oh, the heat lost in expansion being restored from K. Let Q be the quantity of heat thus required.- If the body attains the original volume oh at the original tem- perature T, then the pressure must also revert to its original value, and the isothermal curve, which represents this last expansion, will therefore terminate exactly in the point a. The above six variations make up together a Cyclical Pro- cess, since the variable body is finally restored exactly to its THE EQUIVALENCE OF TRANSFOEMATIONS. 95 original condition. Of the three bodies, K, K^, K^, which in the whole process only come under consideration in so far as they serve as sources or reservoirs of heat, the two first have at the end lost the quantities of heat Q, Q^ respectively, whilst the last has gained the quantity of heat Q^ ; this may be expressed by saying that Q^ has passed from K^ to K^, while Q has disappeared altogether. This last quantity of heat must, by the first fundamental principle, have been transformed into external work. This gain of external work is due to the fact that in this cyclical process the pressure during expansion is greater than during compression, and therefore the positive work greater than the negative ; its amount is represented, as is easily seen, by the area of the closed curve abode/a. If we call this work W, we have by equation (5a) of Chapter I. Q=W. It is easily seen that the above Cyclical Process embraces as a special case the process treated of at the commencement of Chapter III., and represented in Fig. 8. For if we make the special assumption that the temperature T of the body K is equal to the temperature 2", of the body K^, we may then dp away witL K altogether, and use K^ instead. The result of the process will then be that one part of the heat given out by the body iT^ has been transformed into work, and the other part has been transferred to the body R^, just as was the case in the process above mentioned. The whole of this cyclical process may also be carried out in the reverse order. The first step will then be to connect the variable body with K, and to produce, instead of the final expansion fa of the former case, an initial compression af: and similarly the expansions fe and ed, and the compressions dc, cb, and ba will be produced one after another, under exactly the same circumstances as the converse variations in the former case. It is obvious that the quantities of heat Q and Qj^ will now be taken in by the bodies K and K^ respectively, and the quantity of heat Qi will be given ovi by the body K^. At the same time the negative work is now greater than the positive, so that the area of the closed figure now represents a loss of work. The result of the reversed process is therefore that 96 ON THE MECHANICAL THEORY OF HEAT. the quantity of heat Q^ has been transferred from K^ to K^^, And that the quantity of heat Q has been generated out of work and given to the body K. § 3. On Uquipalent Transformations. In order to learn the mutual dependence of the two simultaneous transformations above described, viz. the trans- ference of Qi, and the conversion into work of Q, we shall first assume that the temperatures of the three reservoirs of heat remain the same, but that the cyclical processes, through which the transformations are effected, are different. This may be either because different variable bodies are subjected to similar variations, or because the same body is subjected to' any other cyclical process whatever, subject only to the con- dition that the three bodies K, K^ and K^ are the only bodies which receive or give out heat, and also that of the two latter the one receives just as much as the other gives out. These different processes may either be reversible, as in the case considered, or non-reversible ; and the law which governs the transformations will vary accordingly. However the modification which the law undergoes for non-reversible pro- cesses can be easily applied at a later period, and hence for the present we will confine ourselves to the consideration of reversible processes. For all such it follows from the Principle laid d.own in the last chapter (p. 78) that the quantity of heat Q,, transferred from if J to K^, must stand in a constant relation to the quantity Q transformed into work. For let us suppose that there were two such processes, in which, while Q was the same in both, Q was different : then we might successively execute that in which Q^ was the smaller in the direct order, and the other in the reverse. In this case the quantity of heat Q, which in the first process would have been trans- formed into work, would in the second process be transformed again into heat and given back to the body K; and in other respects also everything would at the conclusion be restored to its original condition, with this single exception that the quantity of heat transferred from K^ to K^ in the second pro- cess, would be greater than the quantity transferred from K^ to K^ in the first process. Thus on the whole we have a transfer of heat from the colder body K^ to the hotter K -, THE EQUIVALENCE OF TEANSFORMATIOKS. 97 with nothing to compensate for it. As this contradicts the fundamental principle, it follows that the above supposition cannot be true ; in other words Q must always stand in the same ratio to Q^. Of the two transformations in a reversible process such as the above, either can replace the other, provided this latter be taken in the reverse direction : in other words, if a trans- formation of the one kind has taken place, this can be again reversed, and a transformation of the other kind substituted in its place, without the occurrence of any other permanent change. For example, let a quantity of heat Q be in any way generated out of work, and taken in by the body K; then by the cyclical process above described it can be again withdrawn from the body K, and transformed back into work, but in so doing a quantity of heat Q^ will be transferred from the body JST, to the body K^. Again, if the quantity of heat Q^ has previously passed from K^^ to K^, it can by performing the above process in the reverse order be transferred back again to K^, whilst at the same time the quantity of heat Q, at the temperature of the body K, wiU be generated out of work. It is thus seen that these two kinds of transformation may be treated as processes of the same nature j and two such transformations, which may mutually replace each other in the way indicated, wUl be henceforth called " Equivalent Transformations." § 4. Equivalence-Values of the Transformations. We have now to find the law according to which the above transformations must be expressed mathematically, so that the equivalence of the two may appear from the equality of their values. The mathematical value of a transformation may be termed, thus determined, its " Equivalence- Value." We must £u:st settle the order in which each transforma- tion is to be taken as positive : this may be chosen arbi- trarily for one of the two classes, but it will then be fixed for the other, since clearly we must regard a transformation in the latter class as positive, if it is equivalent to a positive transformation in the former. In all that follows we shall consider the transformation of Work into Heat, and therefore the passage of heat from a higher to a lower temperature, as being positive quantities. It will be seen later why this C. ' 98 ON THE MKCHANICAL THKORT OF HEAT. choice as to the positi-ve and negative sign is preferable to the opposite. With regard to the magnitude of the equivalence-value, it is at once seen that the value of a change from work into heat must he proportional to the quantity of heat generated, and that beyond this it can only depend on its temperature. We may therefore express generally the equivalence-value of the generation out of work of the quantity of heat Q, of temperature T, by the formula Q xf(T), where f{T) is a function of temperature which is the same for all eases. If Q is negative in this formula, what is expressed is that the quantity of heat Q 'has been transformed, not out of work into Tieat, but out of heat into work. Similarly the value of the passage of a quantity of heat Q from ihe temperature Tj to the temperature 2", must be propor- tional to the quantity of heat which passes, and beyond this can only depend on the two temperatures. We may therefore express it generally by Dhe formula Q x F {T^, T^, in which F [T^, T^ is a function of the two temperatures, also constant for all eases, and which we cannot at present determine more closely; but of which it is clear from the commencement that, if the two temperatures are interchanged, it must change its sign, without changing its numerical value. We may therefore write, FiT„T,) = -F{T,,T,) (1). In order to compare these two expressions with each other, we have the condition that in every reversible process of the kind given above the two transformations that take place must be equal in magnitude but of opposite sign, so that their algebraical sum is zero. Thus if we choose for a mo- ment the particular process fully described above (§ 2), the quantity of heat Q, at temperature T, is then transformed into work, giving as its equivalence-value — Qx-fiT); and the quantity of heat Q^ passes from temperature T^ to T^, thus giving as its equivalence-value (?, x F{T^, T^. There- fore the following equation must hold : -Qxf{T) + Q,xF{T„T,)=0 (2). Let us now suppose a similar process performed in the reverse order, and under the conditions that the bodies -K" THE EQUIVALENCE OF TRANSPOBMATIONS. 99 and K^, amd the quantity of heat Q^ which passes between them, remain the same, but that for the body K of tempera- ture T is substituted another body K' of temperature T': and let us call the heat generated out of work in this case Q'. Then, corresponding to the formier equation, we have the following ; Q' ^/{T') + Q,x F{T„ T,) = (3). Adding (2) and (3) and substituting from (1) we obtain, - Q xf{T) + Q' xf{T') = (4). Now let us consider, as is clearly allowable, that these two successive processes make up together a single process ; then in this latter the two transferences of heat between K^ and K^ will cancel each other and disappear from the result ; we have therefore only left the transformation into work of the quantity of heat Q, given off by K, and the generation out of work of the quantity of heat Q' taken in by K'. These two transformations, which are of the same kind, can however be so broken up and re-arranged as to appear in the light of transformations of different kinds. For if we simply hold fast to the main fact, that the one body K has lost the quantity of heat Q, and the other K' gained the quantity Q", then the heat equivalent to the smaller of these two quanti- ties may be considered as having been transferred directly from K to K', and it is only the difference between the two which remains to be considered as a transformation of work into heat or vice versS,. For example, let the temperature T be higher than T; then the transference of heat on the above view is from a hotter to a colder body, and is therefore .positive. Accordingly the other transformation must be negative, i.e. a transformation from heat into work : whence it follows that the quantity of heat Q given off by K is greater than the quantity Q' taken in by K'. Thus if we divide Q into its two component parts ^ and Q — Q', then the first of these will have passed over from K to K', and the second is the quantity of heat transformed into work. On this view the two processes appear as combined into a single process of the same kindj for the circumstance that the heat transformed into work is not derived from a third body, but from one or other of the same two bodies, between 7—2 100 ON THK MECHANICAL THEORY OP HEAT. whicli the transference of heat takes place, makes no essential difference in the result. The temperature of the heat transformed into work is optional, and can therefore have the same value as the temperature of one of the two bodies ; in which case the third body is no longer r equired . Accord- ingly for the two quantities of heat Q' and Q— Q there must be an equation of the same form as (2), namely : -Q^g xf{T) + Q'xF {T, T) = 0. Eliminating Q by means of equation (4) and then striking out Q', we obtain F{TT')=f{T)-f{T) (5). As the temperatures T and T' are any whatever, the function of two temperatures jP'(T'2"), which holds for the second kind of transformation, is thus shewn to agree, with the function of one temperature/ (2'), which holds for the first kind. For the latter function we will for brevity use a simpler symbol. For this it is convenient, for a reason which will be apparent later on, to express by the new symbol not the function itself, but its reciprocal. We wiU therefore put ^■=7^0^/(20 = ^ (6), so, that T is now ihe unknown function of temperature which enters into the Equivalence-value. If special values of this function have to be written down, corresponding to tempera- tures Tj, T^, etc.', or T', T' , etc., then this can be done by simply using the indices or accents for t itself. Thus equa- tion (5) wiE become i^(r, 2") = -,--. T T Hence the second Main Principle of the Mechanical Theory of Heat, which in this form may perhaps be called the principle of the Equivalence of Transformations, can be expressed in the following terms : "K we call two transformations which may cancel each other without requiring any other permanent change to take place, Equivalent Transformations, then the generation out of work THE EQUIVALENCE OP TRANSFORMATIONS. 101 of the quantity of heat Q of temperature<^nas tne equivalence- value — ; and the transference of the quantity of heat Q from temperature T, to temperature T, has the Equivalence-value <-^> in which t is a function of temperature independent of the kind of process by which the transformation is accomplished." § 5. Combined value of all the transformations which take place in a single Cyclical Process. If we write the last expression of the foregoing section in the form , we see that the passage of the quantity of heat Q from temperature T to T^ has the same equiva- lence-value as a double transformation of the first kind, viz. the transformation of the quantity Q from heat of tem- perature T, into work, and again out of work into heat of temperature T^. The examination of the question how far this external agreement has its actual foundation in the nature of the process would here be out of place ; but in any case we may, in the mathematical determination of the Equivalence-Value, treat every transference of heat, in what- ever way it may have taken place, as a combination of two opposite transformations of the first kind. By this rule it is easy for any Cyclical Process however complicated, in which any number of transformations of both kinds take place, to deduce the mathematical expression which represents the combined value of all these trans- formations. For this purpose, when a quantity of heat is given off by a heat reservoir, we have no need first to enquire what portion of it is transformed into work, and what becomes of the remainder ; but may instead reckon every quantity of heat given off by the hea,t reservoirs which occur in the cyclical process as being wholly transformed into work, and every quantity of heat taken in as being generated out of work. Thus if we assume that the bodies K-^, K^, K^, etc. of temperatures T^, T^, T,, etc. occur as heat reservoirs, and if Q^, Q^, Q^, etc. are the quantities of heat given off during 102 ON THE MECHANICAL THEOET OF HEAT. the Cyclical Process (in wliicli we will now consider quanti- ties of heat taken in as negative quantities of heat given out*), then the combined value of all the transformations, which we may call N, will be represented as follows : j^=_i^_Q._^_etc., Ta ■^2 T3 or using the sign of summation, iV — 2^ (7). T It is here supposed that the temperatures of the bodies K^, K^, K^, etc. are constant,' or at least so nearly so that their variations may be neglected. If however the tempera- ture of any one of the bodies varies so much, either through the taking in of the quantity of heat Q itself,. or through any other cause, that this variation must be taken into account, then we must, for every element of heat dQ which is taken in, use the temperature which the body has at the moment of its being taken in. This naturally leads to an integration. If for thje sake of generality we assume this to hold for all the bodies, th.en the foregoing equation takes the following form : f''^ (8), "—It in which the integral is to be taken for all the quantities of heat given off by the different bodies. § 6. Proof that in a reversible Cyclical Process the total value of all the transformations must he equal to nothing. If th-e Oyclical Process under consideration is reversible, then, however complicated it may be, it can be proved that the transformations which occur in it must cancel each other, so that their algebraical sum is equal to nothing. * This choice of positive and negative signs for the quantities of heat agrees with that which we made in the last chapter, where we considered a quantity of heat taken in by the variable body as positive, and a quantity given out by it as negative ; for a quantity given out by a heat reservoir is taken in by the variable body, and vice versa. THE EQUIVALENCE OF TRANSFORMATIONS. 103 For let us suppose tliat this is not the case, i.e. that this' algebraical sum has some other value ; then let us imagine the following process applied. Let all the transformations which take place be divided into two parts, of which the first has its algebraical sum equal to nothing, and the second is made up of transformations all having the same sign. Let the transformations of the first division be separated out into pairs, each composed of two transformations of equal magnitude but opposite signs. If all the heat reservoirs are of constant temperature, so that in the Cyclical Process there is only a finite number of definite temperatures, then the number of pairs which have to be formed will be also finite; but should the temperatures of the heat reservoirs vary continuously, so that the number of temperatures is indefinitely great, and therefore the quantities of heat given off and taken in must be dis- tributed in indefinitely small elements, then the number of pairs which have to be formed will be indefinitely large. This however, by our principle,, makes no difference. The two transformations of each pair are now capable of being done backwards by one or two Cyclical Processes of the form described in § 2. Thus in the first place let the two given transformations be of different kinds, e.g. let the quantity of heat Q of tem- perature T be transformed into work, and the quantity of heat Q^ be transferred from a body K^ of temperature T, to a body K^ of temperature T^. The symbols Q and Q^ are here supposed to represent the absolute values of the quanti- ties. Let it be also assumed that the magnitudes of the two quantities stand in such relation to each other that the follow- ing equation, corresponding to equation (2), will hold, viz. Then let us suppose the Cyclical Process to be performed in the reverse order, whereby the quantity of heat Q, of tem- perature T, is generated out of work, and another quantity of heat is tra,nsferred from the body K^ to the body K^. Thi§ lattier quantity must then be exactly equal to the quantity Q^, given in the above equation, and the given transformations have thus been performed backwards. Again let there be one transformation from work into 104 lON THE MECHANICAL THEORY OF HEAT. heat and one from lieat into work, e.g. let the quantity of heat Q of temperature T be generated out of work, and the quantity of heat ^ of temperature 2" be transformed into work, and let fbese two stand in such relation to each other that we may put T T Then let us suppose in the first place that the same process as last described has been performed, whereby the quantity of heat Q of teinperature T has been transformed into work, and another quantity Q^ has been transferred from a body K^ to another body K^. Next let us suppose a second process per- formed in the reverse direction, in which the last-named quantity Q^ is transferred back again from K^ to K^, and a quantity of heat of temperature T' is at the same time gene- rated out of work. This transformation from work into heat must, independently of sign, be equivalent to the former trajisformation from heat into work, since they are both equi- valent to one and the same transference of heat. The quantity of heat of temperature T', generated out of work, must there- fore be exactly as great as the quantity Q' found in the above equation, and the given transformations have thus been made backwards. Finally, let there be two transferences of heat, e.g. the quantity of heat Q transferred from a body K^ of tempera- ture Tj to a body K^^ of temperature T^, and the quantity Q\, from a body K\ of temperature T'j to a body K\ of tem- perature T\, and let these be so related that we may put «.(i-i).e'.(^-ij=o. Then let us suppose two Cyclical Processes performed, in one of which the quantity Q,^ is transferred from K^ to K^, and the quantity Q of temperature T thereby generated out of work, whilst in the second the same quantity Q is again transformed into work, and thereby another quantity of heat transferred from K\ to K\. This second quantity must then be exactly equal to the given quantity Q\, and the two given transferences of heat have thus been done backwards. THE EQUIVALENCE OP TRANSFORMATIONS. 105 When by operations of this kind all the transformations of the first division have been done backwards, there then rernain the transformations, all of the like sign, of the second division, and no others whatever. Now first, if these trans- formations are negative then they can only be transformations from heat into work and transferences from a lower to a bigher temperature; and of these the transformations of the *first kind may be replaced by transformations of the second kind. For if a quantity of heat Q of temperature T is transformed into_ work, then we have only to perform in reverse order the cyclical process described in § 2, in which the quantity of heat Q of temperature T is generated out of work, and at the same time another quantity $j is transferred from a body K^ of temperature T^ to another body K^ of the higher tempera- ture Tj^. Thereby the given transfonnation from heat into work is done backwards, and replaced by the transference of heat from K^ to Ky By the application of this method, we shall at last have nothing left except transferences of heat from a lower to a higher temperature which are not com- pensated in any way. As this contradicts our fundamental principle, the supposition that tlie transformations of the second division are negative must be incorrect. Secondly, if these transformations were positive, then since the cyclical process under consideration is reversible, the whole process might be performed in reverse order; in which case all the transformations which occur in it would take the opposite sign, and every transformation of the second division would become negative. We are thus brought back to the case already considered, which has been found to contradict the fundamental principle. As then the transformations of the second division can neither be positive nor negative they cannot exist at all ; and the first division, whose algebraical sum is zero, must em- brace all the transformations . which occur in the cyclical process. We may therefore write iV= in equation (8), and thereby we obtain as the analytical expression of the Second Main Principle of the Mechanical Theory of Heat for reversi- blg processes the equation '"'^^ = 0...,, , ,(VII). /' T 106 ON THE MECHANICAL THEORY OF HEAT, § 7. On the Temperatures of the various quantities of Heat, and the Entropy of the Body. In the development of Equation VII. the temperatures of the quantities of heat treated of were determined by those of the heat reservoirs from which they came, or into which they passed. But let us now consider a cyclical process, which is such that a body passes through a series of changes of condition and at last returns to its original state. This variable body, if placed in connection with the heat reservoir to receive or give off heat, must have the same temperature as the reservoir ; for it is only in this case that the heat can pass as readily from the reservoir to the body as in the reverse direction, and if the process is reversible it is requisite that this should be the case. This condition cannot indeed be exactly fulfilled, since between equal temperatures there can in general be no passage of heat whatever ; but we may at least assume it to be so nearly fulfilled that the small remaining dififerences of temperature may be neglected. In this case it is obviously the same thing whether we consider the temperature of a quantity of heat which is being transferred as being equal to that of the reservoir or of the variable body, since these are practically the same. If how- ever we choose the latter and suppose that in forming Equa- tion VII. every element of heat Q is taken of that tem- perature which the variable body possesses at the moment it is taken in, then we can now ascribe to the heat reservoirs any other temperatures we please, without thereby making any alteration in the expression | — . With this assumption as to the temperatures we may consider Equation VII. as holding, without troubling ourselves as to whence the heat comes which the variable body takes in, or where that goes which it gives off, provided the process is on the whole a re- versible one. The expression — , if it be understood in the sense lust T given, is the differential of a quantity which depends on the condition of the body, and at the same time is fully deter- mined as soon as the condition of the body at the moment is known, without our needing to. know the path by which THK EQUIVALENCE OF TRAKSFOllMATIONS. 107 the body has arrived at that condition ; for it is only in this case that the integral will always become equal to zero as often as the body after any given variations returns to its original con- dition. In another paper*, after introducing a further de- velopment. of the equivalence of transformations, the author proposed to call this quantity, after the Greek word Tpoirrj, Transformation, the Entropy of the body. The complete explanation of this name and the proof that it correctly expresses the conditions of the quantity under consideration . can indeed only "be given at a later period, after the develop- ment just mentioned has been treated of; but for the sake of convenience we shall use the name henceforward. If we denote the Entropy of the body by ;Si we may put T or otherwise dQ = rdS., (VIII). § 8. On the Temperature Fimction t. To determine the temperature function r we will apply the same method as in Chaptei* III. § 7, p. 81, to determine the function 4> (T^, T^. Eor, as the function t is independent of the nature of the variable body used in the cyclical process, we may, in order to determine its form, choose- any body we please to be subjected to the process. We will therefore again choose a perfect gas, and, as in the above-mentioned section, suppose a simple process performed, in which the gas takes in heat only at one temperature T, and gives it out only at another T^. The two quantities of heat which are taken in and given out in this case, and whose absolute values we may call Q and Q^, stand by equation (8) of the last chapter, p. 83, in the following relation to each other : QrT, ^^^- On the other hand, if we apply Equation VII. to this simple cyclical process, whilst at the same time we treat the * Fogg. Ana^ Vol. cxxv. p. 390. 108 ON THE MECHANICAL THEOEY OF HEAT. giving out of the quantity of heat Q as equivalent to the taking in of the negative quantity — Q, we have the follow- ing equation : ^^»■"^^. ™- From equations (9) and (10) we obtain or ^^^T (U). If we now take T as being any temperature whatever and 2\ as some given temperature, we may write the last equation thus : T=Tx Const (12), and the temperature function t is thus reduced to a constant factor. What value we ascribe to the constant factor is indiffe- rent, since it may be struck out of Equation VII. and thus has no influence on any calculations performed by means of the equation. We will therefore choose the simplest value, viz. unity, and write the foregoing equation r=T ,....(13). The temperature function is now nothing more than the absolute temperature itself. Since the foregoing determination of the function t rests on the equations deduced for the case of gases, one of the foundations on which this determination rests will be the approximate assumption made in the treatment of gases, viz. that a perfect gas, if it expand at constant temperature, absorbs only just so much heat as is required for the external work thereby performed. Should anyone on this account have any hesitation in regarding this determination as perfectly satisfactory, he may in Equations VII. and VIII. regard t as the symbol for the tempierature function as yet THE EQUIVALENCK OF TEANSFOEMATIONS. , 109 undetennined, and use the equations in that form. Any such hesitation would not, in the author's opinion, be justifiable, and in what follows T will always be used in the place of t. Equations VII. and VIII. wUl then be written in the following forms, which have already been given under Equations V. and VI. of the last chapter, viz. dQ = TdS. CHAPTER V. FORMATION OF THE TWO FUNDAMENTAL EQUATION* • § 1. Discussion of the Variables which determine the Condition of the Body. In the general treatment of the subject hitherto adopted we have succeeded in expressing the two main principles of the Mechanical Theory of Heat by two very simple equations numbered III. and VI. (pp. 31 and 90). dQ = dU+dW ..(Ill), dQ = TdS (VI). We will now throw these equations into altered forms which make them more convenient for our further calcula- tions. Both equations relate to an indefinitely small alteration of condition in the body, and in the latter it is further assumed that this alteration is affected in such a way as to be reversible. For the truth of the first equation this assump- tion is not necessary : we will howevpr make it, and in the following calculation will assume, as hitherto, that we have only to do with reversible variations. We suppose the condition of the body under considera- tion to be determined by the values of certain magnitudes, and for the present we will assume that two such magnitudes are sufficient. The cases which occur most frequently are those in which the condition of the body is determined by its temperature and volume, or by its temperature and pressure, or lastly by its volume and pressure. We will not however tie ourselves to any particular magnitudes, but will at first FORMATION OF THE TWO FUNDAMENTAL EQUATIONS. Ill assume that the condition of the body is determined by any two magnitudes which may be called x and y ; and these mag- nitudes we shall treat as the independent variables of our calculations. In special cases we are of course always free to take one or both of these yariables as representing either one or two of the above-named magnitudes. Temperature, Volume and Pressure. If the magnitudes x and y determine the condition of the body, we can in the above equations treat the Energy IT and the Entropy S as being functions of the variables. In the same way the temperature T, whenever it does not itself form one of these variables, may be considered as a function of the two variables. The magnitudes W and Q on the con- trary, as remarked above, cannot be determined so simply, but must be treated in another fashion. The differential coefficients of these magnitudes we shall denote as follows : dW dW f = ^= f = ^ (^)- These differential coefficients are definite functions of x and y. For suppose the variable x is changed into x+dx while y remains constant, and that this alteration of condi- tion in the body is such as to be reversible, then we are dealing with a completely determinate process, and the external work done in that process must therefore be also dW determinate, whence it follows that the quotient -^ must equally have a determinate value. The same will hold if we suppose y to change to 1/ + dy while x remains constant. If then the differential coefficients of the external work W are determinate functions of x and y it follows from Equation III. that the differential coefficients of the quantity of heat Q taken in by the body are also determinate functions of x and y. Let us now write for dW and dQ their expressions as functions of dx and dy, neglecting those terms which are of a 112 ON THE MECHANICAL THEORY OF HEAT. higher order than doa and dy. We then have, dW = mdx + ndi/ (3), dQ = Mdx + Ndi/ (4), " and we thus obtain two complete differential equations, which cannot be integrated so long as the variables a; and y are independent of each other, since the magnitudes m, n and M, N do not fulfil the conditions of integrability, viz. dm _dn , dM _dN dy ~ dx dy dx ' The magnitudes W and Q thus belong to that class which was described in the mathematical introduction, of which the peculiarity is that, although their differential coefficients are determinate functions of the two independent variables, yet they themselves cannot be expressed as such functions, and can only be determined when a further relation between the variables is given, and thereby the way in which the varia- tions took place is known. § 2. Elimination of the quantities U and S from the two Ftmdamental Equations. Let us now return to Equation III., and substitute in it for dW and dQ expressions (3) and (4) ; then, collecting to- gether the terms in dx and dy, the equation becomes, Mdx + Ndy = (^+ m^ dx+(~ + «) dy. As these equations must hold for all values of dx and dy, we must have, iV=-5-H-W. FORMATION OF THE TWO FUNDAMENTAL EQUATIONS. 113 Differentiating the first equation according to y, and the second according to x, we obtain, dM^ffU_ dm dy dxdy dy ' dN^ cP£ dn dx dydx dx We may apply to JJ the principle which holds for every function of two independent variables, viz. that if they are differentiated according to both variables, the order of dif- ferentiation is a matter of indifference. Hence we have d'u ^ dru_ dxdy dydx ' Subtracting one of the two above equations from the other we 'obtain, dM dN_ dm_dn dy dx dy dx ^ We may now treat Equation VII. [in the same manner. Putting for dQ and d8 their complete expressions, it be- comes, Mdx + Ndy = T ij- dx -^ -^ dyj , M^ N. dS, dS, or Y'^^'^T^^'^^'^'^d^'^'''- This equation divides itself, like the last, into two, viz. M^dS T dx' F_d8 T dy' Differentiating the first of these according to y, and the second according to x, we obtain, 114' ON THE MECHANICAL THEORY 01? HEAT. T^ _dxdy ' dx dx _ d'S : T^ dydx' d'S d'S But as before, dxdy dydx ' hence subtracting one of tbe above equations from the other we obtain : dy dy dx dx _^ dM dN Ifji^dT ,rdT\ ,. l^-d^ = T[^dy-^dx) ^^^- The two equations (5) and (6) may be, also written in a somewhat different form. To avoid the use of .so many- letters ill the formula, we will replace M and N, which were introduced as abbreviations for -^ and -i^, by those differen- tial coefficients themselves. Similarly for m and n we will dW ' dW write -i— and t— , Then the right hand side of equation, dx dy ° (5) may be wiitten d^ /dW\ _ d^(dW\ dy \dx J dx\dy J ' Thus the magnitude represented by this expression is a func- tion of X and y, which may generally.be considered as known, inasmuch as the external forces acting on the body are open to direct observation, and from these the external work can be determined. The above difference, which will occur very- frequently from henceforward, we shall call "The Work FORMATION OF THE TWO FUNDAMENTAL EQUATIONS. 115 Difference referred to x and y" and shall use for it a special symbol, putting J. _± ( dW \ _ d_ (dW\ . .. ^~dy\dx) dxKdyj ^^^•■ Through these changes equations (5) and (6) are trans- formed into the following : A. f^\ _ i. f^\ - n rfi\ dy\dx) dx\dyj~ '" ^>-' ^(dQ\_±fdQ\^lfdT^dQ_dT^dQ\ dy\dx) dx\dyj T\dy dx dx dyj" '' These two equations form the analytical expressions of the two fundamental principles for reversible variations, in the case in which the condition of the body is determined by two given variables. From these equations follows a third, which is so far simpler as it contains only differential coeffi- cients of Q of the first degree, viz. ^Jx|f-Sx|=-..- (-)• § 3. Case in which the Temperature is taken as one of the Independent Variables. The above three equations are very much simplified if the temperature of the body is selected as one of the independent variables. Let us for this purpose put y = T, so that the two independent variables are the temperature T, and the still undetermined quantity x. Then we have first, ^=1 dy dT Again, referring to the differential coefficient ;,- , it is assumed in its formation that, while x is changed into x + dx, the other variable, hitherto called y, remains constant. But as T is now itself the other variable, it follows that T must remain constant in the above differential coefficient, or in other words dx 8—2 116 ON THE MECHANICAL THEORY OP HEAT. Again, if we form the Work-difference referred to x and T, this wUl run as follows : ^e^m\_d_idw\ ^"^ dT\dx) dxKdTj ^ ' Hence equations (8), (9), (10) take the following form; ^©-s(S)-^« ('^)- ±(dQ\_d_(d^\^ldQ „, dT\dx) dxKdT) Tdx ^ '' S=^^- (1^)' If the product TD^^,, given in equation (14), be substi- tuted for -j^ in equation (12), and then differentiated accord- ing to T, the following very simple equation will be the result : d fdQ\ dD,^ § 4. Particular Assumptions as to the External Forces. We have hitherto made no particular assumptions as to the external forces to which the body is subjected, and to which the external work done during its alterations of condi- tion is related. We will now consider more closely a special case, which occurs frequently in practice, namely' that in which the only external force which exists, or at least the only one which is of sufficient importance to be taken into consideration, is a pressure acting on the surface of the body, everywhere normal to that surface, and of uniform intensity. In this case external work can only be done during changes in the volume of the body. Let p be the pressure per unit of surface, v the volume ; then the external work done, where this volume is increased to ?; + dv, is given by the equation dW=pdv (16). FOEMATION OF THE TWO FUNDAMENTAI, EQUATIONS. 117 Let US now suppose tlie condition of the body to be given by the values of two given variables x and y. Then the pressure p and volume v must be considered as functions of X and y. We may thus write the last equation as follows : ■"^'^(S-^+l*)- whence it follows that dW dv^ dx " dx dW dv dy ~^ dy\ ,(17). dW dW Putting these values for —=— and -5— in equation (7), and performing the differentiations indicated, taking note also of the fact that -^ — =- = , , , we obtain dxdy dydx -. dp dv dp , dv -,_. ^^ = dy''d^-£+d-y (1^)- This value of D^ we have now to apply to equations (9) and (10). Let x and T be the two independent variables ; then equation (18) becomes _dp dv ^ dv . „, ^"'-dT dx dx^ dT ^ '' which value we have to apply to equations (12), (14) and (15). The expression given in (18) takes its simplest form, if we choose for one of the independent variables either the volume or the pressure, or if we choose both for the two independent variables. For these cases equation (18) takes, as is easily seen, the following forms, 118 ON THE MECHANICAL THEOEY OF HEAT. I) =-^, "" dj/ D^ = l ■(21), (22). Lastly in the cases in which either the volume or pressure is taken as one of the independent variables, we may choose the temperature for* the other. We must then put T for y in equations (20) and (21), which then become D. dp 'It- (23), (24). § 5. Frequently occurring Forms of the Differential Equations. In the circumstances described above, where the only external force is a uniform pressure normal to the surface, it is usual to choose as the independent variables, which are to determine the condition of the body, the quantities last mentioned in the foregoing section, viz. volume and temperature, pressure, and , temperature, or lastly, volume and pressure. The systems of differential equations which hold for these three, cases may be easily deduced from the more general systems given above ; but on account of their frequent use it may be well to collect them together in this place. The first system is the one which the author has usually employed in the treatment of special cases. If V and T are taken as the independent variables, d^(dQ\ _d_(^\_dp dT d_ /^\ _ d^ fdQ dT\dv) dv[dT, dQ_ dp dv ~ dT' IdQ T dv d^(dQ dv Q- dr ..(2.5). FORMATlbN- OF THE TWO FUNDAMENTAL EQUATIONS. 119 If p and T are taken : d (dg\ d_ /dQ\ ^_dv^ dT [dp] dp \dTj dT d dQ _± (dQ\_l dQ dT dp dp [dTJ ~ T dp ' dp dT' ' dp [dTj dT' .(26). ■(27). If V and p are taken : dp \dvj dv \dp J ' d_fdQ\_d/dQ\l^rdT dQ _dT dQ\ dp \dv J dv \dpj ~ T \dp dv dv dp J ' dp dv dv dp § 6. Equations in the case of a body which undergoes a Partial Change in its Condition of Aggregation. A case whict permits a still further simplification peculiar to itself, and which is of special interest from its frequent occurrence, is that in which the changes of condition in the body are connected with a partial change in its Condition of Aggregation. Suppo.se a body to be given, of which one part is in one particular state of aggregation, and the remainder in another. As an example we may conceive one part of the body to be in the condition of liquid and the remainder in the condition of vapour, and vapour of the particular density which it assumes when in contact with liquid : the equations deduced will however hold equally if one part of the body is in the solid condition and the other in the liquid, or one part in the solid and the other in the vaporous condition. We shall, for the sake of generality, not attempt to define more nearly the two conditions of aggregation which we are treating of, but merely call them the first and the second conditions. 120 ON THE MECHANICAL THEOEY OF HEAT. Let a certain quantity of this, substance be inclosed in a vessel of given volume, and let one part have the first, and the other the second condition of aggregation. If the specific volumes (or volumes per unit of weight), which the substance assumes at a given temperature in the two dif- ferent conditions of aggregation, are different, then in a given space the two parts which are in the different con- ditions of aggregation cannot be any we please, but can only have determinate values. For if the part which exists in the condition of greater specific volume increases in size, then the pressure is thereby increased which the inclosed substance exerts on the inclosing walls, and con- sequently the reaction which those walls exert upon it, and finally a point will be reached, where this pressure is so great that it prevents any further passage of the substance into this condition of aggregation. When this point is reached, then, so long as tHe temperature of the mass and its volume, i.e. the content of the vessel, remain constant, the magnitude of the parts which are in the two conditions of aggregation can undergo no further change. If, however, whilst the temperature remains constant, the content of the vessel increases, then the part which is in the conditionj of aggregation of greater specific volume can again increase at the cost of the other, but only until the same pressure as before is attained and any further passage from one condition to the other thereby prevented. Hence arises the peculiarity, which distinguishes this case from all others. For suppose we choose the temperature and the volume of the mass as the two independent variables which are to determine its condition; then the pressure is not a function of both these variables, but of the tempera- ture alone. The same holds, if instead of the volume we take as the second independent variable some other quantity which can vary independently of the temperature, and which in conjunction with the temperature determines fuUy the condition of the body. The pressure must then be inde- pendent of this latter variable. The two quantities, tem- perature and pressure, cannot in this case be chosen as the two variables which are to serve for the determination of the body's condition. We will now take, in addition to the temperature T, any into the following: dr' FORMATION OF THE TWO FUNDAMENTAL EQUATIONS. 121 other magnitude x, as yet left undetermined, for the second independent variable which is to determine the body's condition. Let us then consider the expression given in equation (19) for the work-diflference referred to xT, viz. T\ _dp dv dp . dv ''''~dT''d^~drx^dT- In this we must, by what has been said, put -^ = 0, and we thus obtain ^.-J4: (^«)- The three equations (12), (13), (14) are thereby changed 'g: d_ fdQ\ _ d^ fdQ\ _dp dv . . iT\dool dx\dT)~ df dx ^ '' ji/^N_^/rf<3\_l dQ dT\dxj dx\dTj T dx ^ '' S = ^*x| (^'>- § 7. Clapeyron's Equation and Carnot's Function. To conclude the developments of the Fundamental Equa- tions which have formed the subject of the present chapter, we may consider the equation which Clapeyron* deduced as a fundamentajl equation from the theory of Carnot, in order to see in what relation it stands to the equations we have' here developed. As however Clapeyron's equation contains an unknown function of temperature, usually designated as Carnot's function, it will be advisable beforehand to throw our equations, so far as they will come under considera- tion, into the form which they take, if the temperature function t, introduced in the last chapter, is treated as still indeterminate, and is not, in accordance with the value * Journal de VEcole Foly technique, Vol. xiy. (1834). 122 ON THE MECHANICAL THEORY OF HEAT. there determined, put equal to the absolute temperature T. We shall thereby obtain the advantage of fixing the relation between our temperature-function t and Carnot's Function. If instead of equation dQ=Td8, we use the less determinate Equation VIII. of the last chapter, dQ = rdS, and eliminate 8 from this eque-tion, in the same manner as in § 2, we obtain instead of equation (9) the following : £(dQ\_d^(dQ\l^dj^^dQ_dT^dQ\ _ dy\dx) dx\dy) T\dy dx dx dyj" '' Combining this with (8) we obtain instead of equa- tion (16), ^^S-S"!"^- (»^> If we now assume that the only external force is a uniform and normal pressure on the surface, we may use for D^y the expression given in equation (18), and the above equation becomes dr dO dr dQ {dip dv dp dv — X -7^ — -, - X-^=T{-f x^ f X i) w- dy dx dx dy \dy dx dx dy. If further we choose as independent variables v and p, putting x = v, and y = p, we obtain — ^ _ !^ '^^ — r^ "I dp dv dv dp ^ '' But as T is only a function of T, we may put dr^dr^ dT .dT_dT dT dv dT^ dv dp~ dT^ dp- Introducing these values of t- and -j- in the above equation, FOEMATION OF THE" TWO FUNDAMENTAL EQUATIONS. 123 and dividing by ^ , we obtain, instead of the last of equa- tions (27), the following: dp dv dv dp dT ^"' '' dT It is here assumed that the heat is measured in mechani- cal units. To introduce the ordinary measure of heat, we must divide the expression on the right-hand side by the mechanical equivalent of heat, which gives : dT^^dQ dT^JQ r dp dv dv dp zp dr ^ '' dl' Clapeyron's equation agrees in form with this, since it is written* fx^_fx^=(7 (38), dp dv dv dp ^ where C is an undetermined function of temperature, and is the same as Carnot's Function already mentioned. If we equate the right-hand expressions of (.37) and (38), we obtain the relation between C and t, as follows : ^==-fd^ = Zd^7[ ■■^^^^' ^dT ^—df~ If, adhering to the determination of t given by the author, we assume it to be nothing more than the absolute tempera- ture T, then C takes the simpler form, c=| m. As equation (33) is formed by the combination of two equa- tions, which express the first and second Main Principles, it * Pogg. Ann. Vol. lix. p. 574. 124 ON THE MECHANICAL THEORY OF HEAT. • follows that Clapeyron's equation must be considered not as an expression of the second Main Principle in the form assumed by the author, but as the expression of a principle, which may be deduced from the combination of the first and second principles. As concerns the manner in which Clapeyron has treated his differential equation, this differs widely from the method adopted by the author. Like Carnot, he started from the assumption that the quantity of heat which must be imparted to a body during its passage from one condition to another, may be fully ascertained from its initial and final conditions, without its being necessary to know in what way and by what path the passage has taken place. Accordingly he considered ^ as a function of p and v, and deduced by integrating his differential equation the following expression for Q'. Q=F{T)-C,t>{p,v) (41), in which F{T) is any function whatever of the temperature j and ^(p,v) is a, function of p and v which satisfies the follow- ing more simple differential equation : ■ f:.#_f+^/=i (42). av dp dp av ^ To integrate this last equation we must be able to express the temperature T for the body in question as a function of p and V. If we assume that the body in question is a perfect gas, we have 2^=5 (43). whence dT_p dT_v_ dv B dp B' Hence equation (42) becomes ^^-^f = ^ (**)• Integrating, we have ^ {p> v) = Rlogp + ^ ipv), FORMATION OF THE TWO FUNDAMENTAL EQUATIONS. 125 where ^(pv) is any function whatever of the product pv. For this we may by equation (43) substitute any function what- ever of the temperature, so that the equation becomes, (^{pv) = Blogp + ^fr{T) (45). If we introduce this expression for

- Substituting in equations (1, 2, 3) the values given in equa- tions (7, 9, 10) we have SATUBATED VAPOUR. 129 %^C-H=u% (U). ^+a-F=| (12), P = Tu^T ^^^)- These are the fundamental equations of the Mechanical Theory of Heat as relates to the generation of vapour. Equa- tion (11) is a deduction from the first fundamental principle, (12) from the second, and (13) from both together. If it is desired to use the ordinary and not the mechanical measures for the quantities of heat, we need only divide all the members of the foregoing equations by the mechanical equivalent of heat. In this case we will denote the two specific heats and the hjeat of vaporization by new symbols, putting c = ^; ^ = 2;! '' = A' (^*)- The equations then become dr , u fdp\ ,, ., dT^'-^^EKdT) ^^'^' _+C-A.= y (16), § 2. Specific Heat of Saturated Steam. As the foregoing equations (15), (16) and (17), of which however only two are independentjhave thus been obtained by means of the Mechanical Theory of Heat, we may make use.of them in order to determine more closely two magnitudes, of which one was previously quite unknown and the otber only known imperfectly; viz. the magnitude h and the magnitude s contained in m. If we first apply ourselves to the magnitude h, or the Specific Heat of Saturated Steam, it may be worth while in c. 9 ISO ON THE MECHANICAL THEORY OF HEAT. the first place to give some account of the views formerly promulgated concerning this magnitude. The magnitude h is of special importance in the theory of the steam engine, and in fact the first who published, any distinct views upon it was the celebrated improver of the steam engine, James Watt. In his treatment of the subject he naturally started from those views which were based on the older theory of heat. To this class belongs especially the idea mentioned in Chapter I., viz. that the so-called total Jheat, i.e. the total quantity of heat taken in by a body during its passage from a given initial condition to its present condition, depends only on the present condition and not on the way in which the body has been brought into it ; and that it accordingly may be expressed as a function of those variables on which the condition of the body depends. According to this view we must in our case, in which the condition of the body com- posed of liquid and vapour is determined by the quantities T and m, consider this quantity of heat ^ as a function of T and m ; accordingly we have the equation A (^ _ A (i^ = n dT \ a) differs little from », the left-band side of these two equations is very nearly the same, and, to 144 ON THE MECHANICAL THEOttT OF HEAT. ascertain how saturated steam is related to the law of Mari- otte and Gay-Lussac, we have only to enquire whether the right-hand side of the last equation is constant, or varies with the temperature. To ascertain whether the successive values of an expression are equal to each other, or in what way they depart from each other, is a very simple matter ; and the form of equation (35) is very well adapted for this pur- pose. The author has calculated the values of this expression for a series of temperatures from 0° to 200°, applying the numbers given by Regnault to r and p. For r, the heat of vaporization, the equation (28) was used, viz. : r = 606-5 - Q-mU - 0-00002*'' - 0-0000003«». The more simple formula (30) might have been used without any great difference in the results. To obtain p, the author first applied the numbers which Regnault has published in his well-known large Tables, in which the pressure of steam for every degree from — 32° to + 230° is given. He found how- ever some peculiar variations from the regular course of the numbers, which in certain ranges of temperature had quite a different character from what they had in others ; and he soon discovered that the source of these variations lay in the fact that Regnault had calculated his numbers by empirical formulae, and that for different ranges of temperature he had empjoyed different formulae. It then appeared desirable to the author to emancipate himself entirely from the in- fluence of empirical formulae, and to confine himself to those numbers Which express simply the results of the observa- tions, because these are specially adapted for comparison with theoretical results. Regnault, in order to obtain from his numerous observations the most probable values, used the aid of graphical methods. He constructed curves of which the abscissae represent the temperature, and the ordinates the pressure p, and which run from — 33° to ■\- 230°. From 100° to 230° he also constructed a curve in which the ordi- nates represented not p itself, but the logarithm oip. From this have been obtained the following values, which may be considered the most direct results of his observations, and from which were also taken the values which served for the calculation of his empirical formulae. SATURATED VAPOUK. 145 t in Centi- t in Centi- p in Millimetres, grade Degrees of the Air- thermometer. p m Millimetres. grade Degrees of the Air- thermometer. according to the Curve of Numbers. according to the Curve of Logarithms \ -20» 0-91 110" 1073-7 1073-3 -10 2-08 120 1489-0. 1490-7 4-60 130 2029-0 2030-5 10 9-16 140 2713-0 2711-5 20 17-39 150 3572-0 3578-5 30 31-55 160 4647-0 4651-6 40 54-91 170 5960-0 5956-7 50 91-98 180 7545-0 7537-0 60 148-79 190 9428-0 9425-4 70 233-09 200 11660-0 11679-0 80 354-64 210 14308-0 14325-0 90 525-45 220 17390-0 17390-0 100 760-00 230 20915-0 20927-0 In order to make the required calculation with these data, the values of — rr were determined from the above table for p (it the temperatures 5", 15°, 25°, etc., in the following manner, As - -^ diminishes but slowly as the temperature increases, the decrease was taken as uniform for every interval of 10", i.e. from 0° to 10°, from 10° to 20°, and so on ; e.g. the value for 25° was taken as the mean of the two values for 20° and 30°. Then since 1 dp _d (logp) pdt dt ' the following formulae could be used : \p dtJis.' or otherwise 10 /IM _ Logf SO^I^SIPm /OfiS \pdtJ2,o~ lOM ^'^'''' ■where Log signifies the common system of logarithms, and M * In this column are given, instead of the Logarithms given directly by the curve and used by Eegnault, the numbers which correspond with them, in order to compare them with the numbers in the previous column. C. 10 146 ON THE MECHANICAL THEOET OF HEAT. the modulus of that system. By the help of these values of - -'^- and of the values of r given by the equation stated p dt above, and lastly of the value 273 for a, the values were calculated which the expression on the right-hand side of equation (35) and therefore likewise the expression 1 - a assumes for the temperatures 5°, 15", 25°, etc. These values are given in the second column of the table below. For tempe- ratures over 100° the two series of numbers found above for p were both made use of, and the two results thus obtained are placed side by side. The meaning of the third and fourth columns will be more fully explained below. 1. y^'-'''^-a a t in Degrees 4. Centigrade of 2. 3. Differences. the Air- According to According to ■ thermometer. Experiment. Equation (38). 5" 30-93 30-46 -0-47 15 30-60 30-38 -0-22 25 30-40 30-30 -0-10 35 30-23 30-20 -0-03 45 30-10 30-10 0-00 55 29-98 30-00 + 0-02 65 29-88 29-88 000 75 29-76 29-76 0-00 85 29-65 29-63 -0-02 95 29-49 29-48 -0-01 1Q5 » 29-47 29-50 29-33 -0-14 -0-17 115 29-16 29-02 29-17 + 0-01 +0-15 125 28-89 28-93 28-99 + 0-10 +0-06 135 28-88 29-01 28-80 -0-08 -0-21 145 28-65 28-40 28-60 -0-05 +0-20 155 28-16 28-25 28-38 + 0-22 +0-13 165 28-02 28-19 28-14 + 012 -0-05 175- 27-84 . 27-90 27-89 + 0-05 -0-01 185 27-76 27-67 27-62 -0-14 -0-05 195 27-45 27-20 27-33 -0-12 +0-13 205 26-89 26-94 27-02 + 0-13 +0-08 215 26-56 26-79 26-68 + 0-12 -0-11 225 26-64 26-50 26-32 -0-32 -0-18 SATUEATED VAPOUK. 147 1 This table at once shews that ^ « (s — viz. that in which the pressure increases during the heating under the same conditions as in the case of steam, when this retains its maximum density; and this case must be con- sidered for carbonic acid likewise, if we are to establish a com- parison. Steam has at about 108° an expansive force equal to 1 metre of mercury, and at 129J° equal to 2 metres. We will examine what takes place with carbonic acid, if this is also heated by 21^°, and thereby the pressure increased from 1 to 2 metres. Regnault* gives as the coefficient of expansion of carbonic acid at constant pressure 0'00371 if the pressure is 760 mm., and 0"003846 if the pressure is 2520 mm. For a pressure of 1500 mm., the mean between 1 and 2 metres, the coefficient of expansion, if assumed to increase in proportion to the pressure, will have the value '003767. If carbonic acid under this mean pressure is heated from 0° to Sl^'", the quantitv — wiU increase from 1 to 1 + 0-003767 X 21-5 = 1-08099. Again other experiments of Regnault'sf have shewn that, i| * Relation d's Experiences, t. i. Mem. 1. t Relation des Eny^rienees, 1. 1. Mem. 6. 150 ON THE MECHANICAL THEOEY OF HEAT. carbonic acid, which had a pressure of about 1 m. at a tem- perature of aiiout 0°, is loaded with a pressure of 1 '98292 m., the quantity pv decreases in the ratio of 1 : 0'99146 ; or for an increase of pressure from 1 to 2 ms., it will decrease in the ratio of 1 : 0'99131. Now if both of these take place together, viz. a rise of temperature from 0° to 21^°, and a rise of pressure from 1 to 2 ms., then the quantity ^ — must increase from 1 to 108099 x 0-99131 = 1-071596 very nearly, whence we obtain as the mean value of the differen- d / pv^ tial coefficient „ , at \pi\. ^•Q^f^ ^ 000333. 21-5 We thus see that for the case here considered we have a value for carbonic acid which is less than 0003665, and there is therefore less, ground for surprise at obtaining the same result for steam at its maximum density: If we seek to determine on the other hand the actual coefficient of expansion for steam^ or the number which shews how far a quantity of steam expands, if it is taken at a given temperature and at its maximum density and then heated, apart from water, at constant pressure, we shall cer- tainly obtain a value which is larger, and probably much larger, than 0-003665. § 9. Formula to determine the Specifia Volume of Satu- rated Stea/m, and its comparison with experiment. From equation (37), and equally from equation (34), the relative values of s — a; and therefore to a close approxima- tion those of s, may be calculated for different temperatures, without needing to know the Mechanical Equivalent E. If however we wish to calculate from the equations the absolute value of s, we must either know E^ or must attempt to eliminate it by the help of some other known quantity. At the time when the author first began these researches, several values of E had been given by Joule, taken from various methods of experiment: these differed widely from each other, and Joule had not announced which he considered the SATURATED VAPOUR. 151 most probable. In this uncertainty the author determined to attempt the determination of the absolute value of s from another starting point, and he believes that his method still possesses interest enough to merit description. The specific weight of gases and vapours is generally ex- pressed by comparing the weight of a unit of volume of the gas or vapour with the weight of a unit of volume of at- mospheric air at the same pressure and .temperature. Simi- larly the specific volume may be expressed by comparing the volume of a unit of weight of the gas or vapour with the volume of a unit of weight of atmospheric air at the same pressure and temperature. If we apply this latter method to saturated steam, for which we have denoted the volume of a unit of weight by s, and if we designate by v' the volume of a unit of weight of atmospheric air at the same pressure and temperature, then the quantity under consideration is given by the fraction — . For s we have the following equation, obtained from (37) by neglecting ) = p7-(TO-«e*'°). 152 ON THE MECHANICAL THEORY OF HEAT. If by the help of this equation we eliminate the constant ■p factor T=rr- from (42), we obtain Ka t^(s_\ m-n^ ^^g. ■ The question is now whether, for any given temperature t„, the quantity f-j or its reciprocal (- ) , which expresses the specific weight of the steam at temperatute t„, can be determined with sufficient certainty. The ordinary values given for the specific weight of steam refer not to saturated but to highly superheated steam. They agree very well, as is known, with the theoretical values which may be deduced from the well-known law as to the relation between the volume of a compound gas, and those of the gases which compose it. Thus e.g. Gay-Lussac found for the specific weight of steam the experimental value 0"6235; whilst the theoretical value obtained by assuming two units of hydrogen and one unit of oxygen to form, by combining, 2 units of steam, is 2 X 006926 X 1-10563 „^„„ ^ = 0-622. This value of the specific weight we cannot in general apply to saturated steam, since the table in the last section, which gives the values of -ri ] < indicates too large a divergence from the law of Mariotte and Gay-Lussac. On the other hand the table shews that the divergences are smaller as the temperature is lower; hence, the error will be insignificant if we assume that at freezing temperature saturated steam follows exactly the law of Mariotte and Gay- Lussac, and accordingly take 0-622 as the specific heat at that temperature. In strict accuracy we must go yet further and put the temperature, at which the specific weight of saturated steam has its theoretical value, still lower than freezing point. But, as it would be somewhat questionable to use equation (37), which is only empirical, at such low temperatures, we shaU content ourselves with the above SATURATED VAPOUH. 153 assumption. Thus giving to t^ the value 0, and at the same time putting (-) = 0-622 and therefore (-)\ = jr^ , we ob- \« /o \v Jo 0622 tain from equation (43) m — we M .(44). v 0-622 (to- w) From this equation, using the values for m, n, and h given in (37a), the quantity -7, and therefore the quantity s, may be calculated for each temperature. The foregoing equa,tion may be thrown into a more convenient form by putting. V .(45), and by giving to the constants M, iV, and a the following values, calculated from those of to, n, and k : -3f= 1-663; iV=0-05527,- a = 1-007164.. .(45a). To give some idea of the working of this formula, we give in the following table certain values of -, and of its re- V . v * ciprocal - , which for the sake of brevity we'stall denote by the letter d, already used to designate specific weight. t 0» 50» I00» 150» 20()« i v' 1-608 1-585 1-550 1-502 . 1-433 d 0-622 0-631 0-645 0-666 0-698 The result that saturated steam diverges, so widely as the above formulae and tables indicate, from the law of Mariotte and Gay-Lussac, which had been previously applied to it without reserve, met at first, as mentioned in another place, with the strongest opposition, even from very, competent 154 ON THE MECHANICAL THEOEY OF HEAT. judges. The author believes however that it is now generally accepted as correct. It has also received an experimental verification by the experiments of Fairbairn and Tate*, published in 1860. The results of their experiments are compared in the follow- ing table, on the^ one hand with the results previously ob- tained by assuming the specific weight to be 0-622 at all temperatures, and on the other hand with the values cal- culated by equation (4-5)'. Temperature in Degrees Centigrade. Volume of a Kilogramme of Saturated Steam in Cubic metres. Values previously obtained. By Equation (45). By Experi- ment. 58-21" 6.S-52 70-76 77-18 77-49 79-40 83-50 86-83 92-66 117-17 118-23 118-46 124-17 128-41 130-67 131-78 134-87 137-46 139-21 141-81 142-36 144-74 8-38 5-41 4-94 3-84 3-79 3-52 3-02 '2-68 2-18 0-991 0-961 0-954 0-809 • 0-718 0-674 0-654 0-602 ,0-562 0-537 0-502 0-495 0-466 8-23 5-29 4-83 3-7f 3-69 3-43 2-94 2-60 2-11 0-947 0-917 0-911 0-769 0-681 0-639 0-619 0-569 0-530 0-505 .0-472 0-465 0-437 8-27 5-33 4-91 3-72 3-71 3-43 3-05 2-62 2-15 0-941 0-906 0-891 0-758 0-648 0-634 0-604 0-583 0-514 0-496 0-457 0-448 0-432 This table shews that the values given by experiment agree much better with those calculated by the author's equation than jwith the values previously obtained ; and that the ex-^ * Proc. Royal Soc. 18S0, and PMV. Mag., Series iv. Vol. xxi. SATURATED VAPOUR. 155 perimental values are in general yet further removed from those previously obtained than are the values derived from the author's formula. § 10. Determination of the Mechanical Equivalent of Heat from the behaviour of Saturated Steam. Since we have determined the absolute values of s, •with- out assuming the mechanical equivalent of heat to be known, we may now applj' these values, by means of equation (17), to determine the mechanical equivalent itself. For this purpose we may give that equation the following form : £= — T^Cs-.') («). The coefficient of s — o- in this equation may be calculated for different temperatures by means of Regnault's tables. For example, to calculate its values for 100", we have given for -ij the value 27"20, the pressure being reckoned in milli- metres of mercury. To reduce this to the measure here em- ployed, viz. kilogrammes per square metre, we must multiply by the weight of a column of mercury at temperature 0", 1 square metre in area and 1 millimetre in height, that is by the weight of 1 cubic decimetre of mercury at 0". As Regnault gives this weight at 13"o96 kilogrammes, the multiplication gives us the number 369 "8. The values of {a + t) and of r at 100° are 373 and 536'5 respectively. Hence we have / a. rt ^£ ^"' "^ ' ~dt _ 373 X 369-8 _ _.^ ;: 536-5 "^''^' and equation (46) becomes ^=257(s-o-) (47). We have now to determine the quantity (s — a), or, since o- is known, the quantity s for steam at 100°. The method formerly pursued, i.e. to use for saturated steam the same specific weight, which for superheated steam had been found by experiment or deduced theoretically from the condensation of 156 ON THE MECHANICAL THEOET OF HEAT. water, led to the result, that a kilogram of steam at 100° should have a volume of 1'696 cuhic metres. From the fore- going however it appears that this value must be consider- ably too large, and must therefore give too large a value for the mechanical equivalent of teat. Taking the specific heat as calculated by equation (45), which for 100° is 0"645, we obtain for s the value 1'638. Applying this value of s we get from equation (47) ^ = 421 ....(48). This method therefore gives for the mechanical equivalent of heat a value which agrees in a very satisfactory manner with the value found by Joule from the friction of water, and with that deduced in Chapter II. from the behaviour of gases ; both of which are about equal to 424. This agree- ment may serve as a verification of the author's theory as to the density of saturated steam. § 11. (complete Differential Equation for Q m the case of a body composed both of liquid and vapour. In § 1 of this chapter we expressed the two first differ- ential coefficients of Q, for a body consisting both of liquid and vapour, by equations (7) and (8), as follows: dm ^' ^ = m{E-G) + Ma Hence we may forai the complete differential equation of the first order for Q, as follows : dQ = pdm + o]. m{n-G) + MG\dT., (49). By equation (12) we may put TT p_dp P whence equation (49) becomes dQ=pdm+ m(^^-^ + Ma dT (50). satueated vapour. 157 Since p is a function of T only, and therefore we have dQ = d{tnp) + {-'^ + MC\dT (51), or dQ = Td{^^^+MCdT (52). These equations are not integrable if the two quantities, whose dififerentials are on the right-haad side, are independent of each other, and the mode of the variations thus left undeter- mined. They become integrable as soon as this mode is deter- mined in any way. We can then perform with them calcu- lations exactly similar to those given for gases in Chapter II. We will for the sake of example take a case which' on the one hand has an importance of its own, and on the other derives an interest from the fact that it plays a prominent part in the theory of the steam-engine. The assumption is that the mass consisting both of liquid and vapour changes its volume, without any heat being imparted to it or taken from it. In this case the temperature and magnitude of the gaseous portion also suffers a change, and some external work, positive or negative, must at the same time be per- formed. The magnitude m of the gaseous portion, its volume V, and the external work W, must now be determined as functions of the temperature. § 12. Change of the Gaseous Portion of the Mass. As the mass within the vessel can neither receive nor give off any heat, we may put dQ = 0. Equation (52) then becomes: Td (^-P^+MCdT=0 (53). If we divide this equation by E, the quantities p and C, which relate to the mechanical measure of heat, change into r and c, which relate to the ordinary measure of heat. If we also divide the equation by T, it becomes: dl-jrj+Mc-^=0 (53a). ., 158 ON THE MECHANICAL THEORY OF HEAT. The first member of this equation is a simple differential, and may at once be integrated : the integration of the second is also always possible, since c varies only with the tempera- ture T. If we merely indicate this integration, and denote the initial values of the various magnitudes by annexing the figure 1, we obtain the following equation: "^^^-mQ^ (54). mr m,r, ,, f^ dT or Actually to perform the integration thus indicated, we may employ the empirical formula for c given by Regnault, For water this formula, already given in (27), is as follows : c = 1 + 000004 + 00000009*'. Since c is thus seen to vary v6ry slightly with the tempera- ture, we will in our calculations for water assume c to be constant, which will not seriously affect the accuracy of the results. Hence (54) becomes: ~^=-^-Mclog-^ (oo), whence m = f(^-^clogJj (56). If we here substitute for r the expression given in (28), or in a simpler form in (30), then m will be determined as a func- tion of tenaperature. To give a general idea of the values of this function, , some values have been calculated for a special case, and col- lected in the following table. The assumption is that the vessel contains at first no water in a liquid condition, but is filled with steam at its maximum density, so that in equation (56) we may put m^ = M. Let there now be an expansion of the vessel. A compression would not be ad- missible, because on the assumption of the absence of water at the commencement, the steam would not remain at its SATURATED VAPOTJE. 159 maximum density, but would be superheated by the heat developed in the compression. In expansion on the other hand the steam not only remains at its maximum density, but a part of it is precipitated as water ; and the diminution of m thus produced is exhibited in the table. The initial temperature is taken at 150° C, and the values of -^ are given for the moments when the temperature has sunk through expansion to 125", 100°, etc. As before, the tem- perature is reckoned from freezing point, and is denoted by t, to distinguish it from the absolute temperature T. t 150» 125" 100" 75» 50" 25" m M 1 0-956 0-911 0-866 0-821 0-776 § 13. Relation between Volume and Temperature. To express the relation which exists between the volume V and the temperature, we may first apply equation (5) : V = mu + Ma: The quantity o-, which expresses the volume of a unit, of weight of the liquid, is a known function of temperature, and the same is therefore true of Ma: It remains to determine mu. For this purpose we need only substitute in equation (55) the expression for r given in equation (17). Thus we obtain mU' X JE d'l dp m.r, ,, , T T, .(-57). whence mu = |(T-*'°^-?.) <'^)- dp dT dp The differential coefficient -pp may be considered as known, since ^ is a known function of the temperature ; and there- fore this equation determines the product mu, and thence, by the addition of Ma, the required quantity v. 160 ON THE MECHANICAL THEORY OF HEAT." The following table gives a series of values of the quotient — , calculated by this equation for the same case as was treated in the last table. Under these are placed for the sake of comparison the values of — , which would hold if the two ordinary assumptions in the theory of the steam-engine were correct : viz. (1) that steam in expansion remains at its maxi- mum density without any part of it condensing; (2) that steam follows the law of Mariotte and Gay-Lussac. On these assumptions we shall have .t 150« 125» 100" V5« 50" 25" V 'P T, 1 1 1-88 1-93 , 3-90 4-16 9-23 10-21 25-7 29-7 88-7 ■ 107-1 § 14. Determination of ike Work as a function of Tem- perature. It remains to determine the work done during the change of volume. For this we have the equation W = pdv ,(59). But by equation (5), taking the magnitude a (which is generally small and very, slightly variable) as constant, we have : dv = d [mu), whence pdv=pd{mu), which may be also written thus pdv = d (mup) — mu -JL dT (60). SATUEATED VAPOUK. 161 In this equation we may substitute for mu -— the expres- sion given in equation (57), and may then perform the inte- gration. The result however is obtained in a more convenient form as follows. By (13) we have: and from (53) we obtain hence '^dT = d(mp) + MCdT; mu^dT=d (mp) + MCdT. Equation (60) now becomes pdv = d {mup) — d (mp) — MCdT = -d[m(p-up)]-MGdT (61). Integrating this equation we obtain W = m;(p^-u^;)-m(p-wp) + MC{T^-T)...(Q2). If in this equation we substitute for p and (7, according to equation (14), the values JEr and JEc, and collect together the terms which contain ^ as a factor, we have : W = mup - m^^p^ + E [m^r^ -mr+ Mc{T^- T)]... (63). From this equation we may calculate W, since mr and mu are already known from the equations (56) and (58). This calculation has been made for the same case as before, and W the values of -jr, i.e. of the work done by a unit of weight during expansion, are given in the following table ; the unit of weight is here a kilogram and the unit of work a kilo- grammetre. The value used for H is 42355, as found by Joule. As a comparison with the numbers of this table it may be c. 11 162 ON THE MECHANICAL THEORY OF HEAT. again mentioned that the value obtained for the work done during the actual formation of steam, as this overcomes the external pressure, is 18700 kilogrammetres per kilogram of water, evaporated at temperature 150° and at the correspond- ing pressure. t 1500 125» 100» 750 50" 250 w M 11300 23200 35900 49300 63700 , CHAPTER VII. FUSION AND VAPORIZATION OF SOLID BODIES. § 1. Fundamental Ilqiiations for the process of Fusion. Whilst in the case of vaporization the influence of the ex- ternal pressure was eaxly observed, and was everywhere taken into account, it had hitherto been left out of account in the case of fusion, where it is much less easily noticed. A little consideration however shews that, if the volume of a body changes during fusion, the external pressure must have an influence on the process. For, if the volume increases, an increase of pressure will make the fusion more difficult, whence it may be concluded that a higher temperature is necessary for fusion at a high than at a low pressure. If on the other hand the volume decreases, an increase of pressure will facilitate the fusion, and the temperature required will be less, as the pressure is greater. To examine more exactly the connection between pressure and fusion-point, and the peculiar changes which are some- times connected with a change of pressure, we must form the equations which are supplied for the process of fusion by the two fundamental principles of the Mechanical Theory of Heat. For this purpose we pursue the same course as for vaporization. We conceive an expansible vessel containing a certain quantity iHf of a substance, which is partly in the solid, and partly in the liquid condition. Let the liquid part have the magnitude m, and therefore the solid part the mag- nitude M—m. The two together are su^osed to fill the vessel completely, so that the content of tffi; vessel is equal to V, the volume of the body. If this volume v and the temperature T are given, thie magnitude m is thereby determined. To prove this, let us 11—2 164 ON THE MECHANICAL THEORY OF HEAT. first suppose that the body expands during fusion. Let it be also in such a condition that the temp'ei-ature T is exactly the melting temperature at that particular pressure. Now if in this condition the magnitude of the liquid part were to increase at the expense of the solid, the expansion which must then result would produce an increase of pressure against the walls of the vessel, and therefore an increased reaction of the walls against the body. This increased pres- sure would produce a rise in the fusion-point, and since the existing temperature would then be too low for fusion, a solidification of the liquid part must begin. If on the con- trary the solid part were to increase at the expense of the liquid, the point of fusion would thereupon sink, and since the existing temperature would then be higher than the fusion -point, a fusion of the solid part must begin. Next, if we make the opposite assumption, viz. that the volume decreases during fusion, then if the solid part increase there must be a rise of pressure and in consequence a partial melting, and if the liquid part increase there must be a fall in pressure and in consequence a pai-tial solidification. Thus on either assumption we have the same result, viz. that only the original proportions of the liquid and solid parts (which proportions correspond to the pressure which gives a tem- perature of fusion equal to the given temperature) can be permanently maintained. Since then the magnitude m is determined by the temperature and volume, this volume will conversely be determined by the temperature and the mag- nitude m; and we may choose T and m as the variables which serve to determine the condition of the body. It now remains to express ^ as a function of T. Here we may again apply -equations (1), (2), (3) of the last chapter, viz. : dT [dm] dm \dTJ '^'dT^d^' .d_ fdQ\ d_ fdQ\ _ jL^ dQ dT\dm) dm\dT) T^dm' iQ^rpdp ^ dv_ dm dT dm ' If we denote by o-, as before, the specific volume (or volume of a unit of weight) for the liquid condition of the body, FUSION AND VAPORIZATION OP SOLID BODIES. 165 and the specific volume for the solid condition by t, we have for the total volume v of the body, V = ma- + {M — m) t, or v=m{a--T) + MT (1), ^^^e^°e ^='^-'^ (2). If further we denote the heat effusion for a unit of weight by p', we may put d^-p (^^- To express -^, the other differential coefficient of Q, we must employ symbols for the specific heat of the body in the liquid and in the solid condition. Here, however, we must make the same remark as in the case of vaporization, viz. that it is not the specific heat at constant pressure which is treated of, but the specific heat for the particular case in which the pressure alters with the temperature in such a way that the temperature shall always be the temperature of fusion for that particular pressure. In the case of vaporiza- tion, where the changes of pressure are generally small, it was possible to neglect the influence of the change of pressure on the specific heat of the liquid body, and to consider the specific heat of a liquid body, as found in the formula, to be equivalent to the specific heat at constant pressure. In the present case small changes of temperature produce such great changes of pressure, that the influence of these oq the specific heat must not be neglected. We will, therefore, under the present circumstances, denote by 0" the specific heat of the liquid, which in the formula for vaporization we denoted by C. The specific heat of the solid body may be de- noted in this case by K'. Applying these symbols we may write '^ = mC' + {M-m)K', ^=m{0'-K')+MK' (4). 166 ON THE MECHANICAL THEOET OF HEAT. From equations (3) and (4) we have dT\dm)~dT ^ '• U^)-^-""' (^)- ,dm \< Inserting these sralues, and the ralue for -^ given in (3), in the above differential equations, we obtain %+,K'-0' = i.-r)^ (7), %-^^'-^'4 ^'\ / = T(■ g,+ A'-c' = J (12), T(a--T) fdp\ '■=-^® ('«)■ These are the equations required, of which the first corre- sponds to the first Fundamental Principle, and the second to the second, whilst the third is a combination of the other two. FUSION AND VAPOEIZATIOK OF SOLID BODIES. 167 § 2. Relation between Pressure and Temperature of Ficsion. The foregoing equations, only two of -which are inde- pendent, maybe applied to determine two quantities hitherto unknown. We will first use the last equation to determine the way in which the temperature of fusion depends on the pressure. The equation may be written ^T Tj.-r) dp Er' ^^*>- This equation in the first place justifies the remark already made, that if a body expand during fusion the point of fusion rises as the pressure increases ; whereas if the body contracts the point of fusion falls. For according as cr is greater or less than t so is the difference a- — t, and therefore also the dT differential coefficient -^ .. positive or negative. Again, by this equation we may calculate the numerical value of —i- . We will calculate this value for the case of water. The vo- lume in cubic metres, or the value of dp'o^'^dp ^ ^'• If here we give to 0, its value from (7a), we obtain d„v _d,v T /d^v\' , . -d^—d^^-oXdr) ^^^>- If we take the reciprocal of (12), we obtain the equation ^^ = g.x^ (14). dv C, dv This equation, if transformed in the same way as (12), gives dv dv C,\dTj ^ '' These differential coeflScients between volume and pres- sure, for the case of the entropy being constant, have been applied to calculate the velocity of propagation of sound in gases and liquids, as has been already described in Chap- ter II. for the case of perfect gases. * § 8. Special Forms of the Fimdamental Equations for an Expanded Bod. Hitherto we have always considered the external force to be a uniform surface pressure. We will now give an ex- ample of a different kind of force, and will take the case of an elastic rod or bar, which is extended lengthwise by a tensional strain, e.g. a hanging weight, whilst no forces act Upon it in a ti'ansVerse direction. Instead of a tensional we may take a compressive strain, so long as the rod is not thereby bent. This we should simply treat in the formulae as a negative tension. The condition that no transverse force acts on the rod would be exactly fulfilled only if the rod were ON HOMOGENEOUS BODIES. 189 placed in vacuo and thus freed from the atmospheric pres- sure. But, since the longitudinal strain, which acts on the cross section, is very large in comparison with the atmos- pheric pressure upon an equal area, the latter may be neglected. Let P be the force, and I the length of the rod, when acted on by the force and at temperature T. The length, and in general the whole condition, of the rod is under these conditions determined by the quantities P and T; and we may therefore choose these as independent variables. Let us now suppose that by an indefinitely small change in the force or temperature or both, the length I is increased by dl. The work Fdl will then have been done by the force P. Since however in our formulae we have taken as positive not the work done but the work destroyed by a force, the equation for determining the external work must be written dW=-Pdl (16). Taking Z as a function of P and T, we may write this equa- tion as follows : dW=-F{gdF+§dT); dW ^dl dW „ dl whence _=-P_; _=-P_. Differentiating the first of these equations according to T and the second according to P, and observing that, since dP P and T are independent variables, -jfjj = 0, we have A(W] p. d'l dT\dP) dPdT' dP\dT)~ dT dTdP- If we subtract the second of these from the first, and substitute for the difference on the left-hand side of the resulting equation the symbol already employed for the same purpose, viz. Df^, we have I>rT = §,-- (17). 190 ON THE MECHANICAL THEORY OP HEAT. This value of 2)pj, we will apply to equations (12), (13), (14), (15) of Chapter V., substituting P for x in this parti- cular case throughout. We then obtain the fundamental equations, in the following form : dT\dP) dP\dT) dT ^ ^' ±(dQ\_±(dQ\_l_dQ dT\dP) dP\dT)~ T dT ^ '' g=^§ (^«)' dP [dTj~-^ dT' ^ '■ § 9. Alteration of Temperature during the extension of the Mod. The form of equation. (20) indicates a special relation between two processes, viz. the alteration in temperature produced by an alteration in length, and the alteration in length produced by an alteration in temperature. Thus if, as is usually the case, the rod lengthens when heated under a constant strain, and -^ is therefore positive, the equation shews that t= is also positive ; whence it follows that, if the rod is lengthened by an increase in the external force, it must take in heat from without if it is to keep its temperature constant, or in other words, if no heat is im- parted to it, it will cool during extension. On the other hand if, as may happen in exceptional cases, the rod shortens when heated at constant pressure, and therefore -^^ is nega- tive, then the equation shews that -j^ is also negative. In this case the rod must give out heat, when lengthened by an increase of strain, if it is to preserve a constant tempera- ture ; and if no giving out of heat takes place it must grow warmer in lengthening. ON HOMOGENEOUS BODIES. 191 The magnitude of the alteration of temperature which takes place if the force is varied, without any heat being imparted to or taken from the rod, is easily determined if we form the complete differential equation of the first order for Q, in the same way as we have already done in the case of bodies under a uniform surface pressure. The differen- f? (1 tial coefficient -jn is determined by equation (20), in which we will write for -ttt, the fuller form -^ . In order to ex- dl dl press the other differential coefficient -^ in a convenient ^ dT form, we may denote the specific heat of the rod under constant strain by C>, and the weight of the rod by M. Then we have dT -^^^^' and the complete differential equation is as follows : dQ = MG^dT+T^ (22). If we now assume that no heat is imparted to or taken from the rod, we must put dQ = 0, which gives = MC^dT+T^dP. dT If we divide this equation by dP, the quotient -^p ex- presses that differential coefficient of T according to P, in the formation of which the entropy is taken as constant; it should therefore be written more fully -f^- We thus obtain the following equation : d,T_ T dj 'dF~~Mcr,''dT ^^^^■ This equation was first developed, though in a slightly different form, by Sir William Thomson, and its correctness was experimentally verified by Joule*. The agreement of • Phil. Traiw., 1869. 192 ON THE. MECHANICAL THEORY OF HEAT. observation and tteory was specially brought out by a pbe- nomenon occurring with India rubber, which had already been noticed by Gough, but was then observed also by Joule and verified by accurate measurements. So long as India rubber is not extended at all, or only by a very small force, it behaves, with regard to alterations in length produced by alterations in temperature, in the same way as other bodies ; i. e. it lengthens when heated and shortens when cooled. When however it is extended by a greater force its behaviour is the opposite; i.e. it shortens when heated and lengthens when cooled. The differential coefficient -^ is thu^ positive in the first case and negative in the second. In accordance with this it exhibits the peculiarity that it is cooled by an increase of the strain, so long as the strain remains small, but is heated by an increase of the strain when the strain is large. This agrees with equation (23), according to which j^- must always have the opposite sign to -~^ . § 10. Further Deductions from the Equations, The complete differential equation (22) may also be so formed as to present T and I, or I and P, as the independent variables. For this purpose we must first state the relation which, holds between the differential coefficients of the quantities T, I, and P. This relation will be expressed by an equation of the same form as (2), viz, : ' U/^Jr (tpt a^ J. _ , . W''df''~dP^~^ ^^*''- First, to -form the complete differential equation which contains T and I as independent variables, we must consider P as a function of T and I, and accordingly write (22) in, the form dQ = MG,dT+ T ^ (g dT + ^ dl) -[MO.^Tf/^)dT + T%'-§dl ON aoMOGENEOTD'S! BODIESi 193^ Transforming the last term by means of equation (24), we have dQ=^(MG, + T^x^^)dT-T^,dl (25). If we denote by 0, the specific heat at constant length, the coefficient of dT in this equation must be equal to MCi ; whence • ^'=^'+J4^f » Transforming this by means of (24), we have T UtI ^'-^'■-M^'-dj; (27)- dp Equation (25) assumes then the following simplified form: dQ = MO,dT-T^dl (28). Secondly, to form the complete differential equation which contains I and P as independent variables, we must consider T as a function of I and P. Equation (22) then becomes dQ = MC^{^+%dP) + T'^dP = Mc/-^dl + (Mc/^^T'f^dP. Transforming the coefficient of dP, we have dQ = M-G/-^dl,-(MO^+T'^.^^)l^dP. . By equation (26) MOi can be substituted^for the expression in brackets. The equation then becomes dQ = MG,^dl + MC,^dP (29). We will again apply equations (28) and (29) 'fo the case c. 13 aad the secoead 194 ON THE MKCHANICAL THEORY OF HEAT. of the rod when it neither receives nor gives out any heat, and therefore dQ = 0. The first equation then becomes '-i-m^i? ^''^•' dl But by equation (24) we may write the latter thus : ds^ _ C^, dji ,„ w V dp-o,''dP ^^^^' Giving to (7,' its value according to (27), we have dl_dl T_ /dpZy , „. dP~dP MOMt) " ^ '' The relation between length and stretching force which is expressed by the diflferential coefficient ^4,, as here de- termined, is that which has to be applied to calculate the velocity of sound in an elastic rod, in place of the relation Expressed by the differential coefficient -5^, which is commonly used, and which is determined by the coefficient of elasticity. In the same way, to calculate the velocity of sound ii;i gaseous and liquid bodies, we must use the relation between volume and pressure expressed by -~ in place of that ex- pressed by -J- . We may however remark that in treating of the propagation of sound, in cases where the force P is not large, we may in equation (82), which serves to deter- mine -J-, substitute for the specific heat at constant tension, denoted by Cp, the specific heat at constant pressure, as measured in the ordinary way under the pressure of the atmosphere. CHAPTER IX. DETERMINATION OF ENEBQT AND ENTBOPY. § 1. General Equations. In former chapters we have repeatedly spoken of the Energy and Entropy of a body as being two magnitudes of great importance in the Science of Heat, which are determined by the condition of the body at the moment, without its being necessary to know the way in which the body has come into this condition. Knowing these magnitudes, we can easily make by their aid various calculations relating to the body's changes in condition, and the quantity of heat thereby brought into action. One of these, the Energy, has already been made the subject of many valuable researches, especially by Kirchhoff*, and the method of determining it is therefore more accurately known. We will here treat of Energy and Entropy simultaneously, and set forth side by side the equations which serve to determine them. In Chapters I. and III. the two following fundamental equations, denoted by (III.) and (VI.) were developed : dQ = dU+dW. (Ill), dQ^TdS. (YI). Here U and S denote the Energy and Entropy of the body, and dU and dS' the changes produced in them by an indefinitely small change in the body's condition : dQ is the quantity of heat taken in by the body during its change; * Pogg. Am,., Vol. cm. p. 177. 13—2 196 ON THE MECHANICAL THEORY OP HEAT. dW the external work performed; and T tlie absolute tem- perature at which the change takes place. The first equa- tion is applicable to any indefinitely small change of con- dition, in whatever way it takes place, but the latter can be applied only to such changes as are in their nature reversible. These two equations we will now write in the form : dU=dQ-dW (1), dS = ^ (2). Their integration will then determine U and 8. Here we must first notice a point which has already been mentioned with regard to energy in Chapter I., § 8. It is not possible to determine the whole energy of a body, but only the increase which the energy has received, whilst the body was passing into its present condition from some other which we choose as its initial condition ; and the same is also true of the Entropy. Now to apply equation (1). Let us suppose that the body- has been brought into its present condition from the given initial condition, the energy of which we will denote by U„, by any convenient path, and in any way reversible or not reversible ; and let us suppose dU to he integrated through the range of this change in condition. The value of this integral will be simply IT— t/,. The integrals oi dQ and dW represent the whole quantity of heat which the body has taken in, and the whole external work which it has per- formed, during the change in condition. These we will de- note by Q and W. Then we have the equation U=U,+ Q-W... (3). Hence it follows that if for any mode of passing from a given initial condition to the present condition of the body, we can determine the heat taken in and the work performed, , we thereby know also the energy of the body, except as regards one constant depending on the initial condition. Next to apply equation (2). Let us suppose that the body has been brought into its present condition from the given initial condition, the entropy of which we denote by S„, by any path whatever, but by a process which is reversible; DETERMINATION OP ENERGY AND ENTROPY. 197 and let us suppose the equation integrated for this change in condition. The integral of dS will have the value 8 — 8„: whence we have S-S, + j^ (4). Hence it follows that if for any passage -of the body, by a reversible method but by any path whatever, from a given initial condition to its present condition, we can determine -fp- 1 we shall thereby know the value of the entropy, ex- cept as regards one constant depending on the initial con- dition. / § 2. Differential Equations for ike Case in, which only Reversible Changes take place, a/nd. in which the condition of the Body is determined by two Independmi Variables. If we apply both the equations (III.) and (VI.) to one and the same indefinitely small and reversible change in the body's condition, the element dQ will be the same in both equations, and may therefore be eliminated. Hence we have ^dS=dU+dW (5). We will now assume that the condition of the body is determined by two variables, wMch, as in Chapter VI., we will generally denote by x and y, signifying by these certain magnitudes to be fixed later on, such as temperature, volume, pressure. If the condition is determined by x and y, then all magnitudes, the values of which are fixed by the con- dition of the body at the moment, without its being neces- sary to know the way in which the body has come into that condition, are capable of being represented by functions of these variables; in which functions the variables must be' considered as independent of each other. Accordingly the entropy 8 and the energy U must be looked upon as functions of x and y. On the other hand, the external work W, as we have repeatedly observed, holds a completely dif- ferent position in this relation. It is true that the differ-!, ential coefficients of W, so far as concerns reversible changes, may be considered as known functions of x and y : W itself however caimot be represented by such a function, $,nd can 198 ON THE MECHANICAL THEORY OF HEAT. only be determined, if we have given not only the initial and final conditions of the body, but also the path by which it has passed from one to the other. If in equation (5) we put dU^-^da, + ~^dy, ,^^ dW. , dW. that equation becomes ^dS , , rpdS , [dU ,dW\. ^{dU dW\, As this equation must hold for any values whatever of dx and dy, it must hold for the cases alnongst others in which one or other of these differentials is equal to zero. Hence it divides into the two following equations : .(6). rpdS^md^ doe dx dx ^dS^dJl dW. dy dy dy From these equations either S or U may be eliminated by a second differentiation. We will first take U, as this gives rise to the simplest equation. For this purpose we must differentiate the first of equations (6) according to y, and the second according to x. We shall write the second differential coefficients of 8 and U in the ordinary manner : but the differ- dW dW ential coefficients of -y- and -^ — we will write as follows : dx dy -r- (^- land-^- f-i— ). This is with the same obiect as in ay \dx/ dx \dy J. '' Chapter V., viz. to shew that they are not second differential coefficients of a function of x and y. Finally we may observe that Tf the absolute temperature of the body, which in this DETERMINATION OF ENERGY AND ENTROPY. 199 investigation we assume to be uniform throughout the body, may also be considered as a function of a; and y. We thus obtain dT^dS ^ d'S ^ • 202 ON THE MECHANICAL THEOEY OF HEAT. which is equation (15) of Chapter V. For the second equa- tion the condition is dx\dT)~dx\dT)~ dT ^''' This equation can be easily shewn to depend on the last. For by (12) Differentiating this equation according to T, we have dK,_ dB^^ d (dW\ 'df~-^ dT '^^^'~dT\dx)- Now, remembering that '^~dT\dx) dx\dTJ' we may write this equation as follows : dA^^ dD,,_d^(dW\ dT dT dx\dTj' On substituting this value of -~ in equation (17), we are brought back to the form of equation (16). We have now to determine 8 and U themselves, by integrating equations (15). Let us suppose that the body has been brought into its present condition, by any path we please, from an initial condition for which the quantities T, 8, X, TJ have the values T^, 8^, x„, U„ respectively : and let this particular change of condition give the range of the integration. As an example, let us suppose that the body is first heated from the temperature T„ to the temperature T, while the other variable keeps its initial value x^, and then that this other variable changes its value from «„ to x, while the temperature remains constant. Then we have (18). (19). DETERMINATION OF ENERGY ANB ENTROPY. 203 In botli these equations the first integral on the right-hand side is a simple function of T, whilst the second is a function of T and x. Let us now make the opposite assumption, viz. that the change of x first takes place at the initial value of T, and then the change of T at the final value of x. Then we obtain In both these equations the first integral on the right-hand side is a simple function of x, and the second of T and x. By what has been said above, we may choose any other path whatever, instead of that which we have taken as our example, in which path the changes of T and x may be transposed in any way, or may both take place at once according to any law. We should naturally in each special case choose that path, for which the data requisite to perform the calculation are most accurately known. § 4. Bpedal case of the Differential equations on the assumption that the only external force is a Uniform Surface Pressure. If we assume as the only External Force a Uniform Pressure normal to the surface, we must put dW=pdv. „ dW dv , dW dv Hence t^ — Pt- ^^^d —r- =Pj-. dx ^ dx dy "■ dy The expressions for D^ and A^ then assume peculiar forms. Those for D^ have been already considered in Chapter V. We have first _ _ d ( ^\ _^f ^^ ■^^~d^V'&) di[Pd^)' "* ~ Uy \T dx) . dx\T^ dy) 204 ON THE MECHANICAL THEORY OF HEAT. In the last of these equations we will put for th^ sake of brevity : ^ = f, (20), whereby it becomes A — T^\ ^ { ^'''\ ^ ^ ( ^'^\ " \jly\ dx) dx \ dyl _ Performing the differentiation in these equations, and remembering that , , ■ = , , , we have ° dxay dydx ^■.'%^i-t-% w. If the temperature T be selected as one independent variable, whilst the other remains x as before, the expressions become _dp du d^ d^ ^'■'~ dT dx - dx^ dT ^ '^'' ^.'=^-®4-£-S <^*)^ or, restoring to ir its value ■'^ , ^^'-^Uy d^ dx dT) ^dx ^'^^'''■ The equations (15) then assume the following forms : db t"" dT'^^^\dT''dx dx'^dTr"' ^^''^' .jy fdQ dv\ ^ rmfdir dv dir dv\ , ,„_. DETEBMINATION OF ENERGY AND ENTEOPT. or written in another form, 205 .(26a). If we further choose for the second variable, as yet unde- termined, the volume v, and thus put x = v, we have , dv ^ dv Hence the preceding equations become (4^i')^ .(27). If the pressure p be chosen as the second independent variable, so that x=p, we have and the equations become '^-■VS'^^-pp' ^H'dT-^S)'H'TT-4)'' . (28). § 5. Application of the foregoing Equations to Homo- geneous Bodies, and in particular to Perfect Gases. For Homogeneous Bodies, where the only external force is a uniform pressure normal to the surface, it is usual, as at the end of the last section, to choose for independent variables two of the quantities T,v, p; and -j^ then takes the simple significations which we have several times alluded to. Thus if T and -v are the independent, variables, and if 206 ON THE MECHANICAL THEORY OF HEAT, the weight of the body be a unit of weight, -^ signifies the specific heat at constant volume : or, if T and p. are the independent variables, the specific heat at constant pressure. Equations (27) and (28) become in these cases dU=C,dT+(T^-p)dv .(29), d8 = ^dT- dv (30). ^rpdp, dU={c^-p§)dT-{T%.p'^)dp\ If we wish to apply these equations to a perfect gas, we may use the following well known equation : pv=IiT. Hence, if T and v be selected as independent variables, dT~ v' and equations (29) then become d8=G,f^E'^' dU=C,dT As in this case G, must be regarded as a constant, these equations can at once be integrated, and give .(31). .(32). 8 = S,+ CJog^+Rlogl^ If we choose T and p as independent variables, we may put dv R , dv ET dT = — and ^- = P dp P DKTERMINATION OF ENERGY AND ENTROPY. 207 accordingly equations (30) become '^'''.f-^fl (S3,. Whence "we obtain by integration 8 = S. + C^lo4-Blogl\ ^3^^_ The integration of the general equations (29) and (30) can of course only be accomplished if, in (29), p and G, are known functions of T and v, or if, in (30), v and. C^ are known functions of T and p. § 6. Application of the Equations to a Body composed of matter in two Different States of Aggregation. As another special case we may select the state of things treated of in Chapters VI. and VII., viz, the case in which the body under consideration is partly in one state of aggre- gation and partly in another, and when the change, which the body may undergo at constant temperature, is such that the magnitudes of the parts in the two different states of aggregation are altered, with a corresponding change in the volume, but no change in the pressure. In this case the pressure jp depends only on the temperature ; and we may therefore put -^ = 0, by which equations (25) and (26) are transformed as follows : (35). As in Chapters, VI. and VII., let us denote by ilf the weight of the whole mass, and by m the weight of the part in the second state of aggregation ; and let us take m in place of x 208 ON I'HE MECHANICAL THEORY OF HEAT. for the second independent variable; then equation (6), Chapter VI., becomes dv _ dm ' for which, by equation (12), Chapter "VI., we may substitute dv dm J, dp ' dT Then the ahove equations become <^S = -^^dT+jjdm, ''H§-''S)''*f-^y- > (36). To integrate these equations we may take as a starting point the condition that the whole mass M is in the first state of aggregation, that its temperature is T^, and that its pressure is the pressure corresponding to that tempera:ture. The passage from this to its present condition (in which the temperature is T, and in which the part m of the whole mass is in the second, and the part Jf — m in the first state of aggregation) may be supposed to take place in the following way : — First let the mass, still remaining entirely in the first state of aggregation, be heated from T„ to T, and let the pressure change at the same time, in such a way that it is always the pressure corresponding to the temperature at the moment : then let the part m pass at temperature T from the first to the second state of aggregation. The integration has to be performed according to these' two successive stages. During the first stage dm = 0, and thus it is only the first term on the right-hand side which has to be integrated. Here -^ has the value MG, where C signifies the specific heat of the body in its first state of aggregation, and for the case in which the pressure changes during the heating in the way described above. This kind of specific heat has been already discussed several times, and the conclusions drawn in Chap. VIII., § 6, shew that where the first state of aggre- DETERMINATION OP ENERGY AND ENTROPY. 209 gation is the solid or liquid, and the second the gaseous, it may safely be taken, for purposes of numerical calculation, as equal to the specific heat at constant pressure. It is only at very high temperatures, for which the vapour tension increases very rapidly with the temperature, that the difference between the specific heat and the specific heat at constant pressures is important enough to be taken into account. Further, during the first change the volume v has the value Ma, where a is the specific volume of the substance in the first state of aggregation. During the second stage dT= 0, and therefore it is only the second term on the right-hand side of equation (3a) which has to be integrated. This inte- gration can be at once performed for both equations, since the coefficient of dm is a constant with regard to m. The resulting equations therefore are S = S, + Mf^^dT+^, U=U„ + M /:(--!?) dT + mp .(37). If in these equations we put m = or m = il/, we obtain the entropy and energy for the two cases in which the mass is either entirely in the first or entirely in the second state of aggregation, under the temperature T, and under the pressure corresponding to that temperature. For example, if the first state is the liquid and the second the gaseous, then if we put m = 0, the expressions Prelate to .the case of liquid under tem- perature T, and under a pressure equal to the maximum vapour tension at that temperature ; or if we put m = M, they relate to saturated vapour at temperature T. § 7. delations of the Eocpressions T>^^ and A^^. In concluding this chapter it is worth while to refer again to the expressions D, following meanings : _ d fdW\ d (dW\ '^~dy\dxl dx\dy)' ^ and A„, which by (7) and (9) have~the A„, = r d^ (l d'W\ _ d_ /I dM\ dy\Tdx) dx\T^dy}_ 14 210 ON THE MECHANICAL THEORY OF HEAT. These are both functions of x and y : but if to determine the condition of the body we choose instead of x and y any two other variables which we may call ^ and ?;, we may form corresponding expressions D^^ and Aj, as follows : I>i. 'dvKd^J d^\dr,)' Hn-- mi d (l dW\ _d^(l dM\ _d^n\T d^J d^\T dr, ,(38). These are of course functions of f and 17, as the former were of x and y. But if we compare one of them, e.g. that for D^^, with the corresponding expresssion for D^^, we find that these are not simply two expressions for one and the same magnitude referred to different variables, but are actually two different magnitudes. For this reason D^ has not been called simply the work difference, but the work difference referred to xy, so that it may be distinguished from Di,,, the work difference referred to ^t]. The same holds true of Aj,j, and A|,. The relation which exists between D^^,. and Z)j, may be found as follows. The differential coefficients which occur in the expression for i)^, in (38), may be derived by firfjt forming the differential coefficients according to x and y, and then treating each of these as a function of ^ and rj. Thus we have om^^dW dx dW d^ d^ dx d^ dy d^' dW^dW dx dW dy dr] ' dx drj dy drj' Differentiating the first of these equations according to 77 and the second according to ^, and again applying the same artifice, we have d (dW\_ dvKd^J- d^ fdW\ dxdx d fdWX dx dy dx\dx) d^ dr] dy\dx J d^ dr] dW d^ d /dW\ dx dy dx d^d/r] dx\dy)dr]d^ d_ fdW\dydy dW d'y dy\dy) d^ dr] dy d^dr] '" DETERMINATION OF ENERGY AND ENTROPY. 211 d_ /dW\ ^ d_ (dW\ dxdx d^ /dF\ dm dy dx \ djo J d^ drj dy\dx J d/q d^ dW d^x d fdrW\dxdy dx d^dt) dx\dy )■ d^ drf d /dW\dydy dW d^y dy\dy) d^ d/rj dy d^d/ij ' If we subtract the second of these equations from the first, all the terms on the right-hand side disappear except four, which may be expressed as the product of two binomial terms in the following equation : dv\d^J d^[dvJ~\d^^dv' dx dy\ 'dv^'d^J ±fdM\ dy\dx) dx \dy ) \' Here the expression on the left side is jDj,, and the expres- sion in the square bracket is D^^ Hence we have finally i'.-C dx dy dx dy d^ dr) drj d^, )l>. .(39). Similarly we may obtain fdx Hi' d^ dr) dx dy\ . 'd^'^t^)^"'- .(39a). If we substitute one new variable only, e.g. if we keep the variable x, but replace y by 17, we must put a; = |^ in the two dcs dss last equations, whence -tz. = ^ and 3- = 0- The equations then become --^D and A -^A A, .(40). If we retain the original variables, but change their order of sequence, the expressions simply take the opposite sign, as is seen, at once on inspection of (7) 'and (9). Hence (41). D,^ = - J?^, and A,^ = - A^^,. 14—2 CHAPTER X. ON NON-REVEKSIBLE PE0CESSE3. § 1. Completion of the Mathematical Expression for the second main Principle. In the proof of the second main principle, and in the investigations connected therewith, it was throughout assumed that all the variations are such as to be reversible. We must now consider how far the results are altered, when the investigations embrace non-reversible processes. Such processes occur in very different forms, although in their substance they are nearly related to each other. One case of this kind has already been mentioned in Chapter I., viz., that in which the force under which a body changes its condition, e.g. the force of expansion of a gas, does not meet with a resistance equal to itself, and therefore does not perform . the whole amount of work which it might perform during the change in condition. Other cases of the kind are the generation of heat by friction and by the resistance of the air, and also the generation of heat by a galvanic current in overcoming the resistance of the wire. Lastly the direct passage of heat from a hot to a cold body, by conduction or radiation, falls into this class. We will now return to the investigation by which it was proved in Chapter IV. that in a reversible process the sum of all the transformations must be equal to zero. For one kind of transformation, viz. the passage of heat between bodies of different temperatures, it was taken as a fundamental principle depending on the nature of heat, that the passage fropi a lower to a higher temperature, which represents negative transformation, cannot take place without compensation. On ON NON-REVERSIBLE PROCESSES. 213 this rested the proof that the sum of all the transformations in a cyclical process could not be negative, because, if any negative transformation remained over at the end, it could always be reduced to the case of a passage from a lower to a higher temperature. It was finally shewn that the sum of the transformations could not be positive, because it would then only be necessary to perform the process in a reverse order, in order to make the sum a negative quantity. Of this proof the first part, that which shews that the sum of the transformations cannot be negative, still holds without alteration in cases where non-reversible transformations occur in the process under consideration. But the argument which shews that the sum cannot be positive is obviously inappli- cable if the process is a non-reversible one. In fact a direct consideration of the question shews that there may very- well be a balance left over of positive transformations ; since in many processes, e.g. the generation of heat by friction, and the passage of heat by conduction from a hot to a cold body, a positive transformation alone takes place, unaccompanied by any other change. Thus, instead of the former principle, that the sum of all the transformations must be zero, we must lay down our prin- ciple as follows, in order to include non-reversible variations : — The algebraic svmi of all the transformations which occur in a cyclical process must always he positive, or in the limit equal to zero. ' ' We may give the name of uncompensated transformations to such as at the end of a cyclical process remain over without anything to balance them ; and we may then express our principle more briefly as follows ; — Uncompensated transformations must always he positive. In order to obtain the mathematical expression for this extended principle we need only remember that the sum of all the transformations in a cyclical process is given ' 7^ . Thus to express the general principle, we must write in place of equation V. in Chapter III., .,-/< /f-io... ,.(ix). 214 ON THE MECHANICAL THEORY OF HEAT. Equation (VI.), Chapter III., then becomes dQ As the first and simplest case we may take that in which a gas expands without doing any external work. We may sup- pose a quantity of the gas to be contained in a vessel and that this vessel is put in connection with another in which is a va- cuum, so that part of the gas can pass from one to the other without meeting any external resistance. The quantity of heat which the gas must in this case take in, in order to keep its temperature unaltered, is determined by putting W=0 in the last equation ; thus we have «=£;(^S-^)'^" (^>- If we make the special assumption that the gas is a perfect one, and therefore that^w —RT, we have dp _R dT~ v' whence dT~^ v~ R v~P' whence (6) becomes <3 = (7). As alreiady mentioned, Gay-Lussac, Joule, and Eegnault have experimented on expansion apart from external work. Joule annexed to his experiment^, described in Chapter II., by which he determined the heat. generated in the compression ON NON-REVERSIBLE PROCESSES. 217 Pig. 18. of air, other experiments upon the expansion of air. The re- ceiver R, shewn . in Fig. 6, was filled with air condensed to 22 atmospheres, and was then con- nected, in the manner shewn in Fig. 18, with an empty receiver B', so that the communication between the two was only closed by the cock. The two receivers were placed together in a water calorimeter, and the cock was then opened, whereupon the air passing over to the receiver B' expanded to about twice its former volume. The calorimeter 'shewed no loss of heat, and thus, so far as could be measured by this apparatus, no heat seemed to be required for the expansion of the air. The above result however holds only for the process as a whole, and not for its individual parts. In the first receiver, in which the expansion takes place and the motion originates, heat is required; in the second, on the contrary, in which the motion ceases, and the air which rushes in first is compressed by that which follows, heat is generated ; and so also in the places where friction has to be overcome during the passage.. Since however the heat generated and the heat required are equal, they cancel each other; and we may say, so far as the general result of the whole process is concerned, that no expenditure of heat takes place. Fig. 19. To observe specially the different parts of the process. Joule varied his experiment by placing the two receivers and 218 ON THE MECHANICAL THEORY OF HEAT. the pipe carrying the cock in three different calorimeters, as shewn in Fig. 19. Then the calorimeter in which was the receiver containing the air shewed a loss of heat, and the two other calorimeters a gain. The whole gain and the whole loss were so nearly equal that Joule considers the difference to be within the limits of error of the observation. § 4. Eaypansiori of a Gas doing Partial Work. If a gas in expanding has a resistance to overcome, but one which is less than its expansive force, then an amount of work will be performed less than the amount which the gas could perform during the expansion. An example of this is the case of a gas rushing into the atmosphere out of a vessel in which it has a pressure higher than atmospheric pressure. In this case also the process is a complicated one. We have not only to deal with the work necessary for the expansion and the corresponding consumption of heat, but in addition heat is consumed in producing the velocity with which the gas escapes ; and heat is again generated when this velocity is subsequently checked. Similarly, heat is con- sumed in overcoming the resistance of friction, and is generated by the friction itself. To investigate accurately all these individual parts of the process would involve us in great difficulties. If however we only wish to determine the quantity of heat, which on the whole must be taken in from without in order to keep the temperature of the gas constant, the case is simple. We can then leave out of account those parts of the process which balance each other, and need only consider the initial and final volume of the gas, and so much of the work done as is not transformed back again into heat. Then the internal work is the same as in any other case of the gas expanding at the same temperature and between the same initial and final volumes ; while the external work is simply represented by the product of the increase of volume and the atmospheric pressure. To determine the required quantity of heat, we start again from equation (5), and there substitute for W the expression for the external work performed in the present case, viz. p^ (v^ — v^, where p^ is the atmospheric pressure. ON NON-SEVERSIBLE PROCESSES. The equation thus becomes 219 <^-f!^{''%-py^-^p^(^^-^:)- .(8). If the gas is a perfect one, the integral on the right-hand side, as shewn in the last section, will = 0, and the equation takes the simpler form Q=pA%-'"d- •(9), which expresses that in this case the heat taken in is only that corresponding to the work required for overcoming the external pressure of the air. If the heat is to be measured according to the ordinary, not the mechanical unit, we must divide the right-hand side of (8) and (9) by the mechanical equivalent of heat, whence we have Q=^(v,-v,). .(9a). This kind of expansion has also been experimented on by Joule. Having as before compressed air to a high pressure in a receiver, he allowed it to escape under atmospheric pressure. In. order to bring the escaping air back to the original temperature, he caused it, after leaving the receiver, to pass through a long coil of pipe, as shewn in Eig. 20, which was placed together with the receiver in a water calorimeter. There then remained in the air only a small reduction of temperature, which it shared in common with the whole mass of the calo- rimeter. The cooling of the calorimeter gave the quantity of heat given off to the air during Fig. 20. 220 OS THE MECHANICAL THEORY OF HEAT. its- expansion. Applying equation (9a) to this quantity of heat, Joule was able to use this experiment as a means of calculating the mechanical equivalent of heat. The numbers obtained from three series of experiments gave a mean value of 438 (in English measures 798) ; a value which agrees closely with the value 444 found by the compression of air, and does not differ from the value 424, found by the friction of water, more widely than can be explained by the causes of error inherent in these experiments. § 5. Method of Experiment used hy Thomson and Joule. The above-mentioned experiments of Joule, in which air contained in a receiver was expanded either by espaping into another receiver or into the atmosphere, shewed that the conclusions drawn under the assumption that air is a perfect gas are in close accordance with experience. If however we wished to know to what degree of approximation air or any other gas obeys the laws of perfect gases, and what are the laws of kny variations that may occur from, the conditions of a per- fect gas, then the above mode of experiment is not sufficiently accurate ; since the mass . of the gas is too small compared with that of the vessels and other bodies which take part in the variation of heat, and therefore the sources of error derived from these have too great an influence on the result. A very ingenious method of making more accurate experi- ments was devised by W. Thomson, and the experiments were carried out by him and Joule with great care and skill. Let us imagine a pipe, through which is forced a continuous current of gas. At one place in this let a porous plug be inserted, which so impedes the passage of the gas, that even when there is a considerable difference between the pressure before and behind the plug, it is only a moderate amount of gas, suitable for the experiment, which can pass through in a unit of time. Thomson and Joule used as plug a quantity of cotton wool or waste silk, which, as shewn in Fig. 21, was compressed between two . pierced plates, AB and CD. Let us now take two sections, EF and GH, one before and one behind the plug, but at such a distance that the unequal motions which may occur in the neighbourhood of the plug ON NON-EEVERSIBLE PROCESSES. 221 H Pig. 21. are not discernible, and there is only a uniform current of gas to deal with. Then the whole process of expansion, corresponding to the differ- ence of pressure before and behind the plug, takes place in the small space between these two sec- tions. If then the current of gas is kept uniform for a considerable time, a state of steady motion is produced, in which all the fixed parts of the apparatus keep their temperature unaltered, and neither take in nor give off heat. Then if, as was done by Thomson and Joule, we surround this space with a non-conducting substance, so that no heat can either pass into it from without or vice versa, the gas can only give out or take in the quantity of heat expended or generated in the process; and thus, even where this quantity is very small, a difference of temperature may exist sufficient to be easily noticed and accurately measured. § 6. Development of the Equations relating to the above method. In order to determine theoretically the difference of temperature in the above case, we will first form the general equations determining the quantity of heat which the gas must have taken in, if the temperature at the second section is to have any required value. From this we can readily find the temperature at which the heat imparted will be nothing. The separate parts of the process in the present case are connected partly with consumption, partly with generation of heat. Heat will be consumed in overcoming the frictional resistance due to the passage through the porous plug; whilst by the friction itself the same amount of heat will be generated. At certain points in the passage heat is consumed in increasing the velocity; whilst at other points heat is generated as the velocity decreases. To determine the quantity of heat which on the whole must be imparted to the gas, we may leave out of account the parts of the process which balance each other ; since it is sufficient for our purpose to know what is the work which remains over as external work done or con- sumed, and at the same time the actual perngianent change in 222 ON THE MECHANICAL THEORY OF HEAT. the vis viva of the current. For this we need only consider the work done at the entrance of the gas into the space between the sections, i.e. at section EF, and also at the exit from that space, i.e. at section GH; and similarly the velocities of the current at those two sections. With regard to the velocities, the difference between their vis viva can readily be calculated. If however they are at each section so small as they were in Thomson and Joule's experiments, their vis viva may be altogether neglected. It then remains only to determine the work done at the two sections. The absolute values of these quantities of work may be obtained as follows. Let us denote the pressure at section ISF by Pj, and suppose the density of the gas at this section to be such that a unit-weight at this density has the volume v^. Then the work done during the passage through the section of a unit- weight of the gas equals p^v^. .Similarly the work done at section GJS will be p.jv^, where p^ is the pressure and v^ the specific volume at that section. These two quantities must however be affected with opposite signs. At section OH, where the gas is escaping from the given space, the external pressure has to be overcome, in which case the work done must be taken as positive ; while in section SF, where the gas is entering the space and thus moving in the same direction as the external pressure, the work must be con- sidered as negative. Thus the net external work per- formed on the whole will be represented by the difference We have now further to determine the quantity of heat, which a unit-weight of the gas must take in while it passes through the distance between the two sections; suppos- ing the gas to have at the first section, where the pressure is p^, the temperature T^, and at the second section, where the pressure is p^, the temperature T^. For this purpose we must use the equation which applies to the case in which a unit-weight of the gas passes from a condition determined by the magnitudes ^^ and T^ into that determined by the magnitudes p^ and T^, and performs in so doing the work Pi%~Pi"v ^® therefore recur to equation (2), in which the symbol w, denoting the external work, must be replaced by P-2^2 ~Pi''i > l^snce we have Q=U,-U^+p,v^-p,v^ (10). ON NON-REVERSIBLE PROCESSES. 223 Here we need only to determine U^— U^, for which purpose we can again use one of the differential equations for U set forth in the last chapter. In this case it is convenient to choose the differential equation in which T and p are the independent variables, i.e. equation (30) of Chapter IX. : In this equation we may put J3 ^£^2'+ p^ cii? =pdv = d{pv)-vdp. It thus takes the following form : dU=C,dT-(T^-v)dp-d{pv) (11). This equation must be integrated from the initial values T^, p^, to the final values T^, p^. The integration of the last term can be performed at once, and we may write : U,-U,==j^C^dT- {t^-v) dp']^ -p,v,+p,v, ....(12). Substituting this value of U^- U^ in equation (10), we obtain Q^j^O,dT-{T^^-v)dp'j (13). Here the expression under the integral sign is the differ- ential of a function of T and p, since C^ satisfies equation (6) of Chapter YIII. : c (?<, _ y d'v d^~ dr- And thus the quantity of heat Q is completely determined by the initial and final values of T and p. If we now introduce the condition corresponding to Thomson and Joule's experiments, viz. that Q = 0, then the difference between the initial and final temperatures is no longer independent of the difference between the initial and final pressures, but on the contrary the one can be found 224 ON THE MECHANICAL THEORY OF HEAT. from the other. If we suppose both these differences indefi- nitely small, we may use instead of (13) the following differential equation : If we here put dQ = 0, we obtain the equation which expresses the relation between dT and dp, and which may be thus written : %-ki'TT-') (-)■ If the gas were a perfect gas, and therefore pv = It T, we should have dv M _ V hence the above equation would become dT dp Thus in this case an indefinitely small difference of pressure produces no difference of temperature; and the same must of course hold if the difference of pressure is finite. Hence one and the same temperature must exist before and behind the porous plug. If on the contrary some difference of temperature is observed, it follows that the gas does not satisfy the law of Mariotte and Gay-Lussac, and by observing the values of these differences of temperature imder various circumstances, definite conclusions may be formed as to the mode in which the gas departs from that law. § 7. Results of the Experiments, and Equations of Elasticity for the gases, as deduced therefrom. The experiments made by Thomson and Joule in 1854* shewed that the temperatures before and behind the plug, were never exactly equal, but exhibited a small difference, which was proportional to the difference of pressure in each * Phil. Trans., 1854, p. 321. . : ON KON-EEVERSIBLE PROCESSES. ^ 225 case. With air at an initial temperature of about 15V losses of temperature were observed, which, if the pressure were measured in atmospheres, could be expressed by. the equation With carbonic acid the losses of heat were somewhat greater ; with an initial temperature of about 19° they satis- fied the equatioii ... The differential equations corresponding to these two equa- tions are as follows : ^=0-26 and ^ = 1-15 (15.) dp dp ^ ' In a later series of experiments, published in 1862*, Thomson and Joule took special pains to ascertain how the cooling effect varies when different initial temperatures are chosen. For this purpose they caused the gas, before reach- ing the porous plug, to pass through a long pipe surrounded by water, the temperature of which could be kept at will to anything up to boiling point. The result showed that the cooling was less at high than at low temperatures, and in the inverse ratio of the squares of the absolute temperatures. For atmospheric air and carbonic acid they arrived at the following complete formulae, in which a is the absolute temperature of freezing point, and the unit of pressure is the weight of a column of quicksilver 100 English inches high: ^=0-92^ and ^ = 4-64 (yj. If one atmosphere is taken as unit of pressure, these formulae become ©■andf.l.a(-;)- C). dp \TJ dp With hydrogen Thomson and Joule observed in their later researches that a slight heating effect took place instead of cooling. They have however deduced no exact formula * Phil: Tram., 1862, .p. 579, C. 13 226 ON THE MECHANICAL THEOBY OP HEAT, for this gas, because tlie observations were not sufficiently accurate. dT If in the two formulae for -j- -. given in (16), we substi- tute for the numerical factor a general symbol A, they combine into one general formula, viz, f=^©' ■ (")• Substituting in equation (14), we obtain dv r^,-t; = ^a(j) (18). According to Thomson and Joule, this equation should be employed for gases as actually existing, in place of the equation referring to perfect gases, T—-v = Q ^ dT "^ "' if we wish to express the relation which exists between the change of volume and. temperature when the pressure is kept constant. If G^ is taken as constant, equation (18) can be integrated immediately. Now it is only for perfect gases that it has really been proved that the specific heat C^ is independent of the pressure; and similarly it is only for perfect gases that the conclusion derived from Regnault's experiments is strictly true, viz. that G^ is al'feo independent of the temperature. If however a gas differs very slightly from the condition of a perfect gas, 0^, will have values differing very slightly from a constant, and these differences may be taken as quantities of the same order. Since in addition the whole term containing G^ is only another- small quantity of the same order, the differences produced in the equa- tion by the differences of G^ will be small quantities of a higher order, and such may in what follows be neglected ; thus we may take C?, as constant. Then multiplying the dT equation by -^ , and integrating, we have ON NON-REVERSIBLE PROCESSES, 227 «=pr-i^a,(jy (19), where P is the constant of integration, which in the present case may be considered as a function of the pressure p. According to the law of Mariotte and Gay-Lussac we should have «=f2^- (20); and it is therefore advantageous to give the function P the form. P=:^ + 7r, P where ir represents another function of p which however can only be very small. Equation (19) then becomes v=^R^+^T-iAC^{^^J (21). This equation Thomson and Joule further simplified as fbllows. The mode in which the pressure and volume of a gas depend on each other varies less from the law of Mariotte according as the temperature is higher. Those terms of the foregoing equation which express this varia- tion must thus become smaller as the temperature rises. The last term is the only one which actually fulfils this con- dition ; the last but one, irT, does not fulfil it. Accordingly this term should not appear in the equation, and putting TT = 0, we obtain . = i2|-iJC7,(jy .(22). This is the equation which according to Thomson and Joule must be used for gases actually existing, in place of equation (20) which holds for perfect gases. An exactly similar equation was previously deduced by Rankine*, in order to represent the variations from the law of Mariotte and Gay-Lussac, found by Eegnault *■ Phil. Trans., 18.54, p, 336. 15—2 228 ON THE MECHANICAL THEORY OF HEAT. in the case of carbonic acid. This, equation in its simplest form may be written „™ a ,an\ ^ pv = BT--^ (23), in which a like M is constant. If we divide this equation by p, and in the last term, which is yery small, replace the product pv by the yery nearly equal product BT, and finally write jS for the constant „ , we obtain which is an equation of the same form as (22). § 8. On the Behaviour of Vapour during Eacpansion under Various Circumstances. As a further example of the different results- which may be produced by expansion, we will consider the behaviour of saturated vapour. We will assume two conditions : (1) that the vapour expanding has to overcome a resistance equal to its" whole force of expansion ; (2) that it escapes into the atmosphere, and thus has only to overcome the atmospheric pressure. Under the last condition we may make a distinc- tion according as the vapour is separate from liquid in the vessel from which it escapes, or is in contact with liquid, which continually replaces by fresh evaporation the vapour which is lost. In all three cases we will determine the quantity of heat, which must be given to or taken from the vapour during expansion, in order that it may -continue throughout at maximum density. First then let us suppose a vessel to contain a unit- weight of saturated vapour, and let this vapour expand, e. g. by pushing a piston before it. In so doing let it exert upon the piston the whole expansive force which it possesses at each stage of its expansion. For this it is requisite only that the piston should move so slowly that the vapour which follows should always be able to equalize its expansive force to that of the vapour which remains behind in the vessel. The quantity of heat Q, which must be imparted to this vapour, if it expands so far as that its temperature falls from a given initial value T^ to a value T^, is simply found by the equation .. g^r^j^dT,.,, (24). ON NON-REVERSIBLE PROCESSES, 229 Here h, is the iiiagnitude introduced ia Chapter VI., and named the Specific Heat of Saturated Vapour. If, as is the case with jpaost vapours, h has a negative value, the foregoing integral, in which the upper limit is less than the lower, represents a positive quantity. In tlie case of "water, h is given by formula (31) of Chapter VI., viz., 8003 n = I'OU rp— • Applying this formula it is easy to calculate the value of Q for any two temperatures 7', and T^. For example let us assume that the steam has an initial pressure of 5 or of 10 atmospheres, and that it expands until its pressure has fallen to one atmo- sphere; then by Regnault's tables we must put T^= a+152"2, or = a -1- 180".3 respectively, and T^ = a + 100; we thus obtain the values Q = 52'1 or = 74"9 units of heat respectively. In the second case we suppose that a vessel contains a unit-weight of saturated vapour apart from liquid, and at a temperature T^ which is above the boiling point of the liquid; and that an opening is made in the vessel, so that the vapour escapes into the atmosphere. Let us proceed to a distance beyond the opening such that the pressure of the vapour is there only equal to the atmospheric pressure. To insure that the current, of vapour shall expand in the proper manner, let the vessel be fitted at the opening with a trumpet-shaped mouth KJPQM (Fig. 22.) This mouth is not actually needed in order that the equations which follow may hold, but merely serves to facilitate the conception. Let KLM be a surface within this mouth, such that the pressure of the vapour is there only equal to atmospheric pressure, and its . velocity . so small that its vis viva may be neglected. We will further assume that the heat gene- rated by the friction of the vapour against the edge of the .opening and the surface of the mouth is not dissipated, but again imparted to the vapour. Now to determine the quantity " 230 ON THE MECHANICAL THEORY OF HEAT. of heat wliich must be imparted to the vapour during ex- pansion, if it is to remain throughout in the saturated con- dition, we will again apply the general equation (2) ; which gives, if in this case we denote the heat by ^', Q'^U,-U,+ W. (25); here U^ is the energy of the vapour in its initial condition within the vessel, If^ the energy of the vapour in its final condition at the surface KLM, and W the external work done in overcoming the pressure of the atmosphere. The energy of a unit-weight of saturated vapour at temperature T is given by the value of U in equation (37) of Chapter IX., if we there put m = M=l, It is , P" [dTj; and yjj First give to T the initial value T^, and let be the values of ^, -^ , and p corresponding to this tempera- ture. Again let T have the final value T^, and let^^, (—] , and Pj be the corresponding values. Then subtracting these two equations from each other we have ^^-^-/I;(^-^S)^^-^^^ 1--4- ^^© -Pi 1- A ^.(S. .(26). The external work which results from the overcoming of the atmospheric pressure p^, during an expansion from volume s, to volume s^, is given by the equation ON NON-EEVERSIBLE PEOCESSE'S. 231 We will give another form to this expression. If, as in -Chapter VI., we put s = u + a; where cr is the specific. volume of the liquid, the equation becomes Substituting for tt the expression given in equation (13), Chapter VI., we have F=i>, ^■m. <^\ +pA'^.-'^^) (27). Now substituting in (25) the value of U^~ U^ from (26), iand of W from (27), we arrive at the equation +P^iF.-<^d (28). Here the heat is expressed in mechanical units. To express it in ordinary heat units the right-hand side must be divided hy E. As before we wiU put ^ = c;^ = r. At the same time, since o- is a small quantity and varies very slightly, we will neglect the quantities -7^ and (ff* — o-^). Thus we obtain g=\^cdT + r,-r,^--^^[p,-p,) (29). AdT), This equation is well adapted for the numerical calcula- tion of Q, since the quantities which it contains have all been determined experimentally for a considerable number of liquids. For water we have according to Regnault. 0+1^=0-305; whence '"''" cdT+ r^- r. = - 0-305 (2; - T^j. J x 232 ON tn^ MECHA]- This is moSt accurate between 2 and 5 atmospheres, and may be applied anywhere between 1 J and 10 atmospheres. § 8. Pambour's Determination of the Work done during a single stroke. The quantities needed for determining the work done, and depending on the dimensions of the engine, will be here denoted in a manner somewhat differing from that of Pambour. The whole space within the cylinder, including the waste space, which is left open for steam during a single stroke, we shall call v'. The waste space we shall call ev' and the space swept through by the piston (1 — e) v. That part of the whole space, which is left open for the steam up to the moment when the cylinder is shut off from the boiler, again inclusive of the waste space, we shall call ev'. Then the space swept through by the piston during the entrance of steam will be denoted by (e — e) v, and that swept through during expansion by (1 — e) v. First to determine the work done during the entrance of the steam. For this purpose we must know the actual pressure in the cylinder at this time, which must be less than that in the boiler, otherwise there would be no flow from one into the other. The ampunt of this difference cannot how- APPLICATION TO THE STEAM-ENGINE. 243 ever be fixed in general terms, because it depends not only on the construction of the engine, but also on how widely the valve in the steam pipe has been opened by the engineer, and on the speed with which the engine is moving. By a change in these circumstances the difference may be made to vary within wide limits. Again the pressure in the cylinder may not remain constant during the whole time the steam is entering, because the speed of the piston, and also the opening left by the valve, may be made to vary during this time. With reference to this latter point Pambour assumes that the mean pressure, to be used for determining the work done, may with sufficient accuracy be taken to be the same as the final pressure which exists in the cylinder at the moment when it is shut off from the boiler. The author does not think it desirable to introduce into the general formulae an assumption of this kind, although in the absence of more exact data it may fairly be resorted to for the purpose of actual calculations ; but he is bound to follow Pambour's method, in order to complete the exposition of his theory. The actual pressure at the moment when the steam is shut off, Pambour determines by means of his equation, as given above, between volume and pressure ; assuming that special observations have been made to determine the quantity of steam which passes from the boiler to the cylinder during an unit of time, and therefore during each stroke. We wUl, as before, denote by M the whole quantity which passes into the cylinder during one stroke, and by m the portion which is in the condition of steam. As this quantity M, of which Pambour only recognizes the part which is in the condition of steam, fills at the moment when the cylinder is closed the space ev', we have by equation (11), e^==^ (12), where p^ is the pressure in the cylinder at that moment. Hence P.-'^-i (12^). If we multiply this equation by (e — e) ?;', which is the 246 ON THE MECHANICAL THEOET OF HEAT. space swept through by the piston up to this moment, we have the following equation for the first part of the wort done: W.^mBx^-^-i/ie-e)!) ..,,(13). The law of Tariation of the pressure during the exp£|,nsion is also given by equation (11). If v is the volume and p the pressure at any moment, then mB , This expression we must substitute in fpdv, and then integrate this from v^ev to v = v'. Thence we obtain for the second part of the work done W^ = mB\og^-^v' 0.-e)l (14). Next, to determine the negative work done by the resis- tance during the return stroke, we must know the value of that resistance. Without entering at present into the ques- tion how this resistance is related to the pressure in the condenser, we will denote the mean pressure by p,; then the work done will be given by. Tf, = -«'(l-6)^„ (15). Finally there remains the work which must be expended in forcing back into the boiler the quantity of liquid M. Pambour has taken no special account of this work, but included it with the friction of the engine. Since, however, for the sake of completing the cycle of operations, it has been included in the author's formulae, it will be investigated here in order to facilitate the comparison. If p^ be the pressure in the boiler, and p„ in the condenser, then equations (4) and (5) show, as- in the example already considered, that this work is on the whole given by W, = -M„". It may be greater or less than p^, according as the steam is shut off from the condenser a little before or after "the end of the return stroke ; for in the former case the steam ■will be compressed still further, while in the latter it will have time to escape partially into the condenser and so to expand again. Finally the mass M must be returned from the condenser into the boiler, during which as before the pressure p„ assists the operation, and the presstire p^ has to be overcome. The quantities of work done during these processes will be represented by expressions very similar to those in the simpler case already considered : a few obvious changes must be made in the suiBxes to the letters, and the quantities added which refer to the waste space. We thus obtain the following equations : During the admission of steam we have, as in § 10, only writing «," instead of w„, W, = {m,u^ +Mcr- m,u;') pi (22). During the expansion from pressure p^ to p^, we have as in equation {^^) of Chapter VI., writing M-^yi, instead of M, TF', = m^ujp^ - m,uj)^ + 'm^Pi- m^P^ +{M+fi)C (T, - T^ (2S). 254 ON THE MECHANICAL THEOfiY OF HiiAT. During the return stroke of the piston, where the whole, space swept through by the piston equals the whole space occupied by ilf + /* at pressure p,, less the waste space which is represented by /AoMj" + /mt, we have F3 = - Km, +M, - m,p, + MG {T, - T,) + fi^, - f,C {T,-T,) ] m^. = «».?. + MO (T, - T,) + ^„p„ -,iG{T,- T,) + /^oWo {jPi -P,) + Ma- {p^ -p^ § 14. Substitution of the Volwme for the porresponding Temp&rature in certain cases. In the above equations it is assumed that, besides the masses M,m^, fju, and /*,,, of which the two first must be found by direct observation, and the two latter can be approxi- mately determined from the size of the waste space, we have also given the four pressures p^, p^, p„ and p„, or, which is the same thing, the four temperatures T^, T^,- 1\, and T^, In practice however this condition is only partially fulfilled, and we must therefore bring in other data to assist us. ' Of the four pressures here mentioned, two only, p^ and p^, may be taken as known : of these the first is given directly by the gauge on the boiler, and the latter can be at least approximately fixed by the gauge on the condenser. The two others, p^ and p^, are not given directly ; but we know the dimensions of the cylinder and the point of cut-ofi", and can thence deduce the volume of the steam at the moment of cut-off and at the end of the stroke. We may then take these volumes as our data in place of the pressures p^ and p^. For this purpose we must throw the equations into a form, which will enable us to use the above data for calculation. Let us now, as before in the explanation of Pambour's theory, denote by v' the whole space within the cylinder which is open to the steam during one stroke, including the waste space; by ev' the space opened to the steam up to the moment of cut-off; and by e/ the waste space. Then in the same way as before we have the following equations : mijj ttj -f- (i¥-t- /t) o- = ev', c. 17 258 ON THE MECHANICAL THEOST OF HEAT. The quantities fjb and a are both so small that their pro- duct may be at once neglected ; whence we have m^ u^ = ev' — Ma m, u, = v' — Mcr /*o = ev .(29). Further, we have by equation (13) of Chapter VI. P = Tug, writing the single letter g for -^, which Vyill occur very frequently in what follows. We can therefore replace p^ and Pj in equations (28) by u^ and u^. Then the masses m, and TWj occur only in the products vn^u^ and m^u^, for which we may substitute the values given in the two first of equa- tions (29). Again, by the last of these equations we may eliminate the mass fi^; and as regards the qther mass n, though this may be somewhat larger than fi^, yet, since the terms which contain (i are generally insignificant, we can without serious error use the value found for /t„: in other words we may drop for the purposes of calculation the as- sumption made for the sake of generality, that the original mass in the waste space is partly liquid and partly gaseous, and consider the whole of it to be in the gaseous form. The above substitutions may be effected in the more general equations (27), as well as in the simplified equations (28). This substitution presents no difficulty, and we will here confine ourselves to the latter set, in order to have the equations in a form adapted for calculation. With these changes they read as follows : W' = mJ>, + MCC^,-T,)-{v'-M, = 3800 ^^0=152 (44). To be able to calculate the work done, we only now need id know the value of V. To assist us in choosing this, we must first know the least possible value of V. This is found exactly as in the case of the engine without expansion, by putting p^^ for p^ in the second of equations (32), and 272 ON THE MECHANICAL THEORY OF HEAT. altering the otter magnitudes, whicli' are ' in combination with Pj, in the same way. We thus obtain in the present case the value 1-010. Starting from this, we will first suppose that the actual speed of the engine exceeds the least possible in the ratio 3 : 2. We may then put in round numbers V= 15, and we will calculate the work done at this speed. . We first determine the two temperatures t^ and t^, by substituting this value of V in the two last of equations (32). Of these t^ has already been approximately found for the non-condensing engine; and since the present case only differs from the former inasmuch as the quantity e, which was there taken equal to 1, has here a different value, we need not go through the process again, but will merely give the final result, which is t, = 137-43», To determine tg we have equation (36), which takes in this case the form ^3 = 26-604 + 51-515 Log ^ (45). From this we obtain the following successive approxima- tions : t' =99-24° T t" =101-93 t'" =101-74 t"" = 101-76 J The last of these, from which further approximations would only diverge in higher places of decimals, we will consider to be the correct value of „ f,,p„, MG{T,- T,) and -/*C7(2;- T,). The three first of these are imparted at the constant temperatures T^, T^ and T^ respectively : and the correspond- ing parts of the integral are ^\ -"^ and ^. The two last are imparted at temperatures which vary contmu- ously between T^ and T^, and between T^ and T^ respectively; and the corresponding parts of the integral are T T MC log-™' , and - fiC log -^^ . If we substitute the sum of these quantities for the integral, the last equation becomes- jVr = _^. + ?M»_if01og|-^» + /^Clog|...(55). Multiplying this expression for JV by Tj, and subtracting the product from the expression given above for the maxi- mum work, we obtain finally for W' the equation W = mj>,-^ m^, + MC (T, - T,) - (M+fj) GT„ log | + ^L,p (56). To compare this expression for W with that given in the first of equations (28), we have only to substitute in the latter the value of m^pg given in the last of those equations, and then put T, = r„. The expression thus obtained agrees exactly with that in equation (56). 288 ON THE MECHANICAL THEORY OF HEAT. In the same way we may also deduct the loss of work due to incomplete expansion. For this purpose we must cal- culate the uncompensated transformation which arises during the passagejof steam from the cylinder to the condenser, and then include this in the expression for W. By this calcula- tion, which we shaiU not here perform at length, we arrive at the same expression for the work as is given in equation (28). NOTE ON THE VALUES OF ^. at Since the differential coefficient ■— occurs frequently in researches upon steam, it is important to know how far the convenient method of determination employed by the author is allowable ; and for this purpose a few values are here placed side by side for comparison. The, formula employed by Eegnault to calculate the pressiures of steam in his tables, for temperatures above 100°, is the following : Log^ =a-ba' - c^, where Log denotes the common system of logarithms, m denotes the temperature from -20" to the value under consideration, so that a;=t + 20, and the five constants have the following values : a =6-2640348, Log 6=0-1397743, Log 0=0-6924351, Log 0=9-994049292 - 10, Log yS= 9-998343862 - 10. From this formula we obtain the following equation for -^ : where .a and /3 have the same values as before, and A and B have the following values : Log 4=8-5197602 -10, Log 5= 8-6028403 - 10. Suppose tbat from this equation we calculate the value of ^ for the dt temperature 147°, then we obtain m = \dtjw 90-115. APPLICATION TO THE STEAM-ENGINE. .289 Next for the approximate mettod of determination, we obtain .from Begnault's tables the pressures ^iffi=3392'98, ^i6=3214'74, whence £yiZ£L« = l!2^ =90.13. 2 2 This approximate valiie agrees so closely with the exact value deduced above, that it may be safely used for it in all investigations upon the steam-engine. With regard to temperatures between 0" and ,100°, the formula used by Begnault to calculate the pressures of steam between these' limits is the following : log^=a + 6a'-c|3'. Here the constants, by Morjtz's improved tables, have the following values : a=4-7393707, Log 6=8-13199071I2 -;10, Logc=0-6X174Q7675, Log 0=^0-006864937152, Log i3= 9-996726536856 - 10. From this formula we can again obtain for -^ an equation of the form where the values of a and ^ are the same as given above, and those of A and B are as follows : Log.4 = 6-6930586-10, Log 5= 8-8513123 -10. If we calculate from this equation the value of ~ corresponding to a temperature of 70", we obtain =10-1112. By the approximate method of determination we obtain p„ -io^_ 243-380- 223-154 _^^. 2 2 This result again agrees sufficiently closely with that obtained from the more exact equation. c. 19 ON THE MECHANICAL THEORY OP HEAT. Table giving the values for steam of the pressure p, its differential coefficient -^ = g, and the product Tg, for different temperatures: all expressed in millimetres of mercury. t in degrees Oiinti- grade. P Difi. 3 Diff. Tg Difi. 0» 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 4-600 4-940 5-302 5-687 6-097 6-534 6-998 7-492 8-017 8-574 9-165 9-792 10-457 11-162 11-908 12-699 13-536 14-421 15-357 16-346 17-391 18-495 19-659 20-888 22-184 23-550 24-988 26-505 28-101 29-782 31-548 33-406 35-359 37-411 0-340 0-362 0-385 0-410 0-437 0-464 0-494 0-525 0-557 0-591 0-627 0-665 0-705 0-746 0-791 0-837 0-885 0-936 0-989 1-045 1-104 1-164 1-229 1-296 1-366 1-438 1-517 1-596 1-681 1-766 1-858 1-953 2-052 0-329 0-351 0-373 0-397 0-423 0-450 0-479 i 0-5«9 , 0-541 0-574 0-609 0-646 0-685 0-725 0-768 0-814 0-861 0-910 0-962 1-017 1-074 1-134 1-196 1-262 1-331 1-402 1-477 1-556 1-638 1-723 1-812 . l-'9d5 2-002 2103 0-022 0-022 0-024 0-026 0-027 0-029 0-030 0-032 0-033 0-035 0-037 039 0-040 0-043 0-046 0-047 0-049 0-052 0-055 0-057 0-060 0062 0-066' 0O69 0O71 0-075 0O79 0-082 0-085 0-089 0-093 0-097 0-101 90 96 103 ' 110 117 125 134 143 152 ■ 162 172 183 195 207 220 234 249 264 280 297 315 333 353 374 395 418 442 467 493 520 549 579 611 644 6 7 7 7 8 9 9 9 10 10 11 12 12 13 14 15 15 16 17 18 18 20 21 21 23 24 25 26 27 29 30 32 33 APPLICATION TO THE STEAM-ENGINE. 291 t in degrees centi- P grade. 33 37-411 34 39-565 35 41-827 36 44-201 37 46-691 38 49-302 39 52-039 40 54-906 41 57-909 42 61-054 43 64-345 44 67-789 45 71-390 46 75-156 47 79091 48 83-203 49 87-497 50 91-980 51 96-659 52 101-541 53 106-633 54 111-942 55 117-475 56 123-241 57 129-247 58 ■ 135-501 59 142-011 60 148-786 61 155-834 62 163-164 63 170-785 64 178-707 65 186-938 66 195-488 67 204-368 68 213-586 69 223-154 70 233-082 71 243-380 72 254-060 73 265-132 74 276-608 75 288-500 DifE. 2-154 2-262 2-374 2-490 2-611 2-737 2-867 3-003 3-145 3-291 3-444 3-601 3-766 3-935 4-112 4-294 4-483 4-679 4-882 5-092 5-309 5-533 5-766 6-006 6-254 6-510 6-775 7-048 7-330 7-621 7-922 8-231 8-550 8-880 9-218 9-568 9-928 10-298 10-680 11-072 11-476 11-892 2-103 2-208 2-318 2-432 2-550 2-674 2-802 2-935 3-074 3-218 3-367 3-522 3-683 3-850 4-023 4-203 4-388 4-581 4-780 4-987 5-200 5-421 - 5-649 5-886 6-130 6-382 6-642 6-911 7-189 7-475 7-771 8-076 8'390 8-715 9-049 9-393 9-748 10-113 10-489 10-876 11-274 11-684 12-106 DifE. Tg 0-105 0-110 0-114 0-118 0-124 0-128 0-133 0-139 0-144 0-149 0-155 0161 0-167 0-173 0-180 0-185 0-193 0-199 0-207 0-213 0-221 0-228 0-237 0-244 0-252 0-260 0-269 0-278 0-286 0-296 0-305 0-314 0-325 0-334 0-344 0-355 0-365 0-376 0-387 0-398 0-410 0-422 644 678 714 751 791 832 874 919 965 1014 1064 1116 1171 1228 1287 1349 1413 1480 1549 1621 1695 1773 1853 1936 2023 2112 2205 2301 2401 2504 2611 2722 2836 2954 3077 3203 3334 3469 3608 3752 3901 4054 4213 19—2 292 ON THE MECHANICAL THEORY OF HEAT. t iw degrees centi- P grade. 75 . 288-500 76 300-820 77 313-579 78 326-789 79 340-464 80 354-616 81 369-258 82 384-404 83 400-068 84 416-262 85 433-002 86 450-301 87 468-175 88 486-638 89 • 505-705 90 525-392 91 545-715 92 566-690 93 588-333 94 610-661 95 633-692 96 657-443 97 681-931 98 707-174 99 733-191 100 760-00 101 787-59 102 816-01 103 845-28 104 875-41 105 906-41 106 938-31 107 971-14 108 1004-91 109 1039-65 110 1075-37 111 1112-09 112 1149-83 113 1188-61 114 1228-47 ,115 1269-41 116 1311-47 117 1354-66 Difi. Diff. Tg 12-320 12-759 13-210 13-675 14-152 14-642 15-146 15-664 16-194 16-740 17-299 17-874 18-463 19-067 19-687 20-323 20-975 21-643 22-328 23-031 23-751 24-488 25-213 26-017 26-809 27-59 28-42 29-27 30-13 31-00 31-90 32-83 33-77 34-74 35-72 36-72 37-74 38-78 39-86 40-94 42-06 43-19 12-106 12-539 12-984 13-442 13-913 14-397 14-894 15-405 15-929 16-467 17-019 17-586 18-168 18-765 19-377 20-005 20-649 21-309 21-985 22-679 23-391 24-119 .24-865 25-630 26-413 27-200 28-005 28-845 29-700 30-565 31-450 32-365 33-300 34-255 35:230 36-220 37-230 38-260 39-320 40-400 41-500 42-625 43-775 0-433 0-445 0-458 0-471 0-484 0-497 0-511 0-524 0-538 0-552 0-577 0-582 0-597 0-612 0-628 0-644 0'660 0-676 0'694 0'712 0-728 0-747 0-765 0-783 0-787 0-805 0-840 0-855 0-865 0-885 0-915 0-935 0-955 0-975 0-990 1-010 1-030 1-060 1-080 1-100 1-125 1-150 4213 4376 4544 4718 4897 5082 5272 5469 5671 5879 6093 6313 6540 6774 7014 7262 7516 7778- 8047 8323 8608 8900 9200 9509 9826 10146 10474 10817 11167 11523 11888 12266 12654 13051 13458 13872; 14296 14730 15178 15635 16102 16581 17072 APPLICATION TO THE STEAM-ENGINE. 293 tin degrees centi- V grade. 117 1354-66 118 1399-02 119 1444-55 120 1491-28 121 1539-25 122 1588-47 123 1638-96 124 1690-76 125 1743-88 126 1798-35 127 1854-20 128 1911-47 129 1970-15 130 2030-28 131 2091-90 132 2155-03 133 2219-69 134 2285-92 135 2353-73 136 2423-16 137 2494-23 138 2567-00 139 2641-44 140 2717-63 141 2795-57 142 2875-30 143 2956-86 144 3040-26 145 3125-55 146 3212-74 147 3301-87 148 3392-98 149 3486-09 150 3581-23 151 3678-43 152 3777-74' 153 3879-18 154 3982-77 155 4088-56 156 4196-59 157 4306-88 158 4419-45 159 4534-36 Difl. 44-36 45-53 46-73 47-97 49-22 50-49 51-80 53-12 54-47 55-85 57-27 58-68 60-13 61-62 63-13 64-66 66-23 67-81 69-43 71-07 72-77 74-44 76-19 77-94 79-73 81-56 83-40 85-29 87-19 89-13 91-11 93-11 95-14 97-20 99-31 101-44 103-59 105-79 108-03 110-29 112-57 114-91 43-775 44-945 46-130 47-350 48-595 49-855 51-145 52-460 53-795 55-160 56-560 57-975 59-405 60-875 62-375 63-895 65-445 67-020 68-620 70-250 71-920 73-605 75-315 77-065 78-835 80-645 82-480 84-345 86-240 88-160 90-120 92-110 94-125 96-170 98-255 100-375 102-515 104-690 106-910 109-160 111-430 113-740 116-085 Difi. 1-170 1-185 1-220 1-245 1-260 1-290 1-315 1-335 1-365 1-400 1-415 1-430 1-470 1-500 1-520 1-550 1-575 1-600 1-630 1-670 1-685 1-710 1-750 1-770 1-810 1-836 1-865 1-895 1-920 1-960 1-990 2-015 2-045 2-085 2-120 2-140 2-175 2-220 2-250 2-270 2-310 2-345 la 17072 17574 18083 18609 19146 19693 20253 20827 21410 22009 22624 23248 23881 24533 25199 25877 26571 27277 27997 28732 29487 30252 31030 31828 32638 33468 34312 35172 36048 36939 37850 38778 39721 40680 41660 42659 43671 44703 45757 46830 47915 49022 50149 Difl. 1012 1032 1054 1073 1085 1107 1127 294' ON THE MECHANICAL THEORY OF HEAT. t lU centi- grade. 159 4534-36 160 4651-62 161 4771-28 162 4893-36 163 5017-91 164 5144-97 165 5274-54 166 5406-69 167 5541-43 168 5678-82 169 5818-90 170 5961-66 171 6107-19 172 6255-48 173 6406-60 174 6560-55 175 6717-43 176 6877-22 177 7039-97 178 7205-72 179 7374-52 180 7546-39 181 7721-37 182 7899-52 183 8080-84 184 8265-40 185 8453-23 .186 8644-35 187 8838-82 188 9036-68 189 9237-95 190 9442-70 191 9650-93 192 9862-71 193 10078-04 194 10297-01 195 10519-63 196 10745-95 197 10976-00 198 11209-82 199 11447-46 200 11688-96 Diff. 117-26 119-66 122-08 124-55 127-06 129-57 132-15 134-74 137-39 140-08 142-76 145-53 148-29 151-12 153-95 156-88 159-79 162-75 165-75 168-80 171-87 174-98 178-15 181-32 184-56 187-83 191-12 194-47 197-86 201-27 204-75 208-23 211-78 215-33 218-97 222-62 226-32 230-05 233-82 237-64 241-50 116-085 118-460 120-870 123-315 125-805 128-315 130-860 133-445 136-065 138-735 141-420 144-145 146-910 149-705 152-535 155-415 158-335 161-270 164-250 167-275 170-335 173-425 176-565 179-735 182-940 186-195 189-425 192-795 196-165 199-565 203-010 206-490 210-005 213-555 217-150 220-795 224-470 228-185 231-935 235-730 239-57b 243-455 Diff. 2-375 2-410 2-445 2-490 2-510 2-545 2-585 2-620 2-670 2-685 2-725 2-765 2-795 2-830 2-880 2-920 2-935 2-980 3-025 3-060 3-090 3-140 3-170 3-205 3-255 3-280 3-320 3-370 3-400 3-445 3-480 3-515 •3-550 3-595 3-645 3-675 3-715 3-750 3-795 3-840 3-885 Tg 50149 51293 52458 53642 54851 56073 57317 58582 61182 62508 63856 65228 66618 68030 69470 70934 72410 73912 75441 76991 78561 80160 81779 83421 85091 86779 88493 90236 91999 93791 95605 97442 99303 101192 103111 105052 107018 109009 111029 113077 115154 Diff, 1144 1165 1184 1209 1222 1244 1265 1286 1314 1326 1348 1372 1390 1412 1440 1464 1476 1502 1529 1550 1570 1599 1619 1642 1670 1688 1714 1743 1763 1792 1814 18S7 1861 1889 1919 1941 1966 1991 2020 2048 2077 CHAPTER XII. ox THE CONCENTRATION OF KAYS OF LIGHT AND HEAT AND ON THE LIMITS OF ITS ACTION. § 1. Object .of the Investigation. The principle assumed by the author as the ground of the second main principle, viz. that heat cannot of itself, or without compensation, pass from a colder to a hotter body, corresponds to everyday experience in certain very simple cases of the exchange of heat. To this class belongs the conduction of heat, which always takes place in such a way that heat passes from hotter bodies or parts of bodies to" colder bodies or parts of bodies. Again as regards the ordi- nary radiation of heat, it is of course well known that not only do hot bodies radiate to cold, but also cold bodies conversely to hot ; nevertheless the general result of this simultaneous double exchange of heat always consists, as is established by experience, in an increase of the heat in the colder body at the expense of the hotter. Special circumstances may however occur during radia- tion, which cause the rays, instead of continuing in the same straight line, to change their direction ; and this change of di- rection may be such, that all the rays from a complete pencil of finite section meet together in one point, and there combine their action. This can he accomplished, as is well known, by the use of a burning-mirror or burning-glass ; and several mirrors or glasses may even be so an-anged, that several pencils of rays from different sources of heat meet together iii one point. 296 ON THE MECHANICAL THEORY OF HEAT. For cases of this kind there is no experimental proof that it is impossible for a higher temperature to exist at the point of concentration than is possessed by the bodies from which the rays emanate. Rankine* accordingly, on a special occasion, of which we shall speak in another place, has drawn a parti- cular conclusion, which rests entirely on the assumption that rays of heat can be concentrated by reflection in such a way, that at the focus thus produced a body may be raised to a higher temperature than is possessed by the bodies which emit the rays. If this assumption be correct, the principle enunciated above must be false, and the proof, deduced by means of that principle, of the second fundamental principle of the Mechanical Theory of Heat would thus be overthrown. As the author was anxious to secure the principle against any doubt of this kind, and as the concentration of rays of heat, with which is immediately connected that of rays of light, is a subject which, apart from this special question, is of much interest from many points of view, he has attempted a closer mathematical investigation of the laws which govern the concentration of rays, and of the influence which this con- centration can have on the exchange of heat between bodies. The results are contained in the following sections. I. Reasons why the ordinary method of deter- mining THE mutual radiation OF TWO SURFACES does NOT EXTEND TO THE PRESENT CASE. § 2. Limitation of the treatment to perfectly black bodies, and to homaffeneous and unpolarized rays of heat. When two bodies are placed in a medium permeable to heat rays, they communicate heat to each other by radiation. Of the rays which fall on one of these bodies, part is in gene- ral absorbed, part reflected, part transmitted; and it is known that the power of absorption stands in a simple relation to the power of emission. As it is not here our object to inves- tigate the differences between these relations and the laws to which they conform, we wiU take one simple case, viz. that in which the bodies are such that they completely ab- sorb all the rays which fall upon them, either actually on the • On the Ee-concentration of the Meehanioal Energy of the Universe, P/iiJ. Mag., Series iv., Vol. it. p. 358. CONCENTEATION OF BATS OF LIGHT AND HEAT. 297 surface, or in a layer so thin that its thickness may be neg- lected. Such bodies have been named by Kirchhoff, in his well-known and excellent paper on the relation between emission and absorption, "perfectly black bodies*." Bodies of this kind have also the maximum power of emission. It was formerly assumed to be beyond question that their intensity of emission depended only on their tem- perature; so that all perfectly black bodies, at the same temperature and with the same extent of surface, would radiate exactly the same quantity of heat. But as the rays emitted by the body are not homogeneous, but differ accord- ing to the scale of colours, the question of emission must be studied with special reference to this scale ; and Kirchhoff has extended the foregoing principle, by laying down that per- fectly black bodies at equal temperatures send out not only the same total quantity of heat, but also the same quantity of each class of ray. As the distinctions between the rays according to colour have no place in our investigation, we will assume throughout that we have to do with only one known class of ray, or, to speak more accurately, with rays whose wave-length only vaaries within indefinitely small limits. Whatever is true of this class of rays must similarly be true of any other class; and thus the results obtained from homogeneous heat may easily be extended to heat which contains a mixture of different classes of rays. With the same object of avoiding unnecessary complica- tions, we will abstain from discussing polarization, and assume that we have only to do with unpolarized rays. The mode of taking polarization into account in such cases has already been explained by Helmholtz and Kirchhoff. § 3. Kirchhoff' s formula for the mutwil Radiation of two Elements of Swface. Let Sj and s, be the surfaces of two perfectly black bodies of the same given temperature ; and let us consider two elements of these surfaces ds^ and ds^, in order to determine and to compare the quantities of heat which they mutually send to each other by radiation. If the medium, which surrounds the bodies and fills the intervening space, • Pogg. Ann., Vol. oix. p. 275. 298 ON THE MECHANICAL THEOET OF HEAT. is homogeneous, so that the rays simply passin straight lines from one surface to the other, it is easy to see that the quan- tity of heat which ds^ sends to ds^ must be the same as that which ds^ sends to ds^ ; if on the contrary this medium is not homogeneous, but there are variations in it which cause the rays to be broken up and reflected, the process is less simple, and a closer investigation is needed to prove whether the same perfect reciprocity holds in this case also. This investigation has been performed in a very elegant manner by Kirchhoff; and his results will be briefly stated here, so far as they relate to the ease in which the rays on their way from one •element to the other suffer no diminution of strength ; in other words in which the breakings up and re- flections of the ray take place without loss of power, and there is no absorption during its passage. A few variations will alone be made in the denotation and in the system of co- ordinates, to produce a better accordance with what follows. If two points are given, only one of the infinitely large number of rays sent out by one point can in general attain the other*; or if the rays are so broken up and reflected, that several of them m^et in the other point, yet they form in general only a limited number of separate rays, each of which can be treated by itself. The path of such a ray from one point to the other is determined by the condition that the time expended in traversing this path is a minimum, com- pared with the times which would be expended in traversing all other neighbouring paths between the same points. This minimum time is determined, if where, several separate rays meet we investigate only one at a time, by the position of the two points between which it passes ; and we will denote it, as Kirchhoff has done, by T. * The form of expression tliat a point sends out an infinitely large num- ber of rays is perhaps, in the strict mathematical sense, inaccurate, since heat and light can only be sent forth from a surface, and not from a mathematical point. It would be more accurate to refer the sending out of the heat or light not to the point itself, hut to the element of area at the point. As, however, the conception of a ray is itself only a mathematical abstraction, we may, without fear of misconception, retain the statement tiat an infinitely large number of rays proceed from each point of the surface. K it were our objeet to determine quantitatively the heat or light radiated by a surface, it is evident that the size of the surface must be taken into account, and that its elements must he considered, not as points, but as indefinitely small surfaces ; the area of which must appear as a factor in the formula expressing the quantity of heat or light radiated from ah element of surface. CONCENTRATION OF rAtS OF LIGHT AND HEAT. 299- Returning now to the elements of surface ds^ and dSj, we will suppose a plane tangential to the surface to be drawn' through one point of each element; and we will treat ds^ and ds^ as elements of these planes. In each of these planes let us take any system of rectangular co-ordinates, which we will call aTj, 2/1 in the one case, and os^, y^ in the other*. If we now take a point on each plane, the time T^, expended by the ray in passing from one point to the other, is determined, as stated above, by the position of the two points; and this time may therefore be treated as a function of the four co- ordinates of the two points. On these assumptions Kirchhoffs expression for the quantity of heat, which the element ds-^ sends to the element rfsjj per unit of time, is as follows i"; •7r\i d^'T d'T d'T d'T dx^dx^ dy^dy^ dx^dy^ dy^dxj ' "' where ir is the well-known ra.tio between the circumference, and diameter of a circle, and e, is the intensity of emission of the surface Sj, in the locality of the element ds^, so that e^ dsj^ represents the whole quantity of heat radiated by ds^ per unit, of time. Conversely to express the quantity of heat which ds^ sends to ds^, we need only substitute for e^ in the above, expression the quantity e,, which is the intensity of emission of the surface s,. Everything else remains unaltered, as being symmetrical with regard to the two elements ; for the time T, which a ray expends in traversing the path between two points of the two elements, is the same in whichever direction it is moving. If we now assume that the surfaces, which are supposed to be at the same temperature, radiate eqi^al quantities of heat in the same time, then 61 = e^ ; and there- fore the quantity of heat sent by ds^ to ds^ is exactly the same as that sent by ds^ to ds^. * Kirohhoff has chosen two planes at right angles to the directions of the rays in the neighbourhood of the two elements; he has taken axes of 00-' ordinates in these planes, and has projected the elements of surface upon them. t Pogg. Ann., Vol. oix. p. 286. 300 ON THE MECHANICAL THEORY OF HEAT. § 4. Indeterminateness 0/ the Formula in the case of the Concentration of Rays. We have already observed that in general only one ray, or a limited number of separate rays, can pass between two given points. In special cases, however, it may happen, that an in- definitely large number of rays, forming a pencil in either two or three dimensions (i.e. in the latter case forming a cone), and starting from the first point, may again unite in the second. This of course holds with rays of light as well as of heat, and it is usual in Optics to call such a point, in which all the rays of a certain conical pencil sent out from a given point unite agairi) the image of the given point ; or, since conversely the first point may become the image of the second, the two points are called conjugate foci. When what is here described in the case of two particular points holds of all the points of two surfaces, so that every point of the one surface is the conjugate focus of some poinff on the other surface, then the one surface is called the optical image of the other. We may now ask how the exchange of rays takes place between the elements of two such surfaces; whether the above-mentioned reciprocity holds, so that at equal tempera- tures any element of the one surface sends to any element of the other exactly the same amount of heat as it received from it, and therefore one body cannot heat the other to a higher temperature than its own ; or whether in such cases the concentration of the rays makes it possible for one body to heat another to a higher temperature than it possesses itself. To this case Kirchhoff's expression does not directly apply. For let the surface s^ be an optical image of the surface s-^. Then all the rays, which a point p^, lying upon s^, sends out within a certain cone, unite on some point j)^ of the surface s^, and all the surrounding points of s^ receive no rays whatever from^i. Hence if the co-ordinates x^, y^ of the point ^^ are given, then the co-ordinates x^, 3/, of the point p^ are no longer to be taken at pleasure, but are also fixed ; and similarly, if x^, y^ are given, then x-^, y^ are also determined. A differential coefficient of the form , . , where x^ is con- CONCENTRATION OF RATS OF LIGHT AND HEAT. 301 sidered as variable when difFerentiating according to «!, whilst the second co-ordinate y^ of the same point, and the co-ordinates x^, y^ of the other, are assumed to be constant, — and wh^re similarly in differentiating according to x^, this is taken as the only variable — can thus represent no real quan- tity of finite value. Therefore in this case we must find an expression of somewhat dififerent form from Kirchhoff 's ; and for this purpose we must first consider some questions similar to those considered by Kirchhoff in arriving at his expres- sion. II. Determination of corresponding points and coRr RESPONDING ELEMENTS OF SURFACE IN THREE PLANES CUT BY THE RAYS. § 5. Equations between the co-ordinates of the points in which a ray cuts three given planes. Let there be three given planes a, b, c (Fig. 25), and in each of them let there be a system of rectangular co-ordinates, which we may call respectively oo^ya, x^y^, and x^^. Let us take a point p^ in plane a, and a point p^ in plane b, and consider a ray as passing from one of these to the other ; then to determine its path we have the con- dition that the time, which the ray expends in traversing it, must be less than it would expend in travers- ing any other neighbouring path. Call this minimum time T^y It is a function of the co-ordinates of p^ and Pi, i.e. of the four quantities x^y^, x^y^. Similarly let T^ be the time of the ray's passage between two points p^ and p„, in planes a and c ; and let T^^ be the time of its pas- sage between two points p^ and p^, in planes b and c. T^ is a function of the co-ordinates x^y^, x^^; and T^^ is a function of the co-ordinates x^y^, x^y^. Now as a ray, which passes between two of these planes, will in general cut the third plane also, we have for each ray 302 ON THE MECHANICAL THEORY OF HEAT. three points of section, which are so related to each other that any one of them is in general determined by the other two. The equations which serve for this determination may be easily deduced from the above-mentioned condition. Let us first suppose that, points p^ andp„ are given beforehand, and that the point is still unknown in which the ray cuts the intermediate plane h. This point, to distinguish it from other points in the plane, we will call p\. Take any point whatever p^ in this plane, and consider two rays, which we may call auxiliary rays, passing the one from p^ to p^, and the other from p^ to p„. In Eig. 25 these rays are shewn by dotted lines, and the primary ray, which goes direct from p^ to p^, by a full line*. If, as before, we call the times ex- pended by these two rays T^^ and T^^, the value of the sum of these times T„j + T^^ will depend on the position of jjj, and therefore, since the points p^ and p^ are assumed to be given, it may be considered as a function of the co-ordinates x^y^ of point Pj. Among all the values, which this sum may assume if we give to point p^ various positions in the neigh- bourhood of p 5, the minimum value must be that which is obtained by making p^ coincide with p\, in which case the auxiliary rays merely become parts of the direct ray. We therefore have the following two equations to determine the co-ordinates of this point p\ : i(2kt^ = 0- ^i^kt^U^O (1) As T^ and T^^, in addition to the co-ordinates of the un- known point Pj, contain also the co-ordinates of the known points p^ and p^, we may consider the two equations thus established as being simply two equations between the six co-ordinates of the three points iu question. These equations, therefore, can be applied, not merely to determine the co- ordinates of the point in the intermediate plane from those * These lines are shewn canred in the figure, to indicate that the path taken by a ray between two given points need not be simply the straight line drawn between the two points, but a different Une determined by the re- fractions or reflections which the ray may undergo: it may thus be either a broken bne, made up of several straight lines, or (if the medium through which it passes changes its character continuously instead of suddenly) a continuous curve. CONCENTEATION OF BAYS OF LIGHT AND HEAT. 303 of the two other points, but generally to determine any two of the six co-ordinates from the other four. Next let us assume that the points p^ a,ndp^ (Fig, 26),. in which the ray cuts planes a and h, are given, and that the point is yet un- known in which it cuts plane c. This point let us call p""^, to distinguish it from other points in the same plane. Take any point p^ in plane c, and con- sider two auxiliairy rays, one of which goes from p^ to p^, and the other from Pi to p^. In Fig. 26 these are again shewn dotted, while the primary ray is shewn full. Let T^^ and T^„ be the times of passage of these auxiliary rays. Then the value of the differ- ence T^^ — Tj^ depends on the position of p^ in plane c. Among the various values obtained by giving to p„ various positions in the neigh- bourhood of p\, the maximum must be that obtained by making p^ coincide with p\. For in that case the ray passing from j?^ to p^ cuts the plane b in the given point jp^, and is there- fore made up of the ray which passes from p^ to p^ and of that which passes from p^ to p^. Accordingly we may put Hence the required difference is given in this special case by T —T =T ■'■as -'■Sc -^aS* If on the other hand p^ does not coincide with p'^, then the ray which passes from p^ to p^ does not coincide with the two which pass from p^ to p^ and from p^ to p^ ; and since the direct ray between p„ and p^ travels in the shortest ■ time, we must have and therefore we have in general for the required difference the inequality - T — T - If we lastly assume that the points p^ and p^ in the planes h and c are given beforehand, while the point in which the ray cuts the plane a is still unknown, we obtain by an exactly similar procedure the two following conditions : d \Tm ~ TJ) _ ^ _ a[I^~- l^i) _ » f,„^ dx^ ' dy. In this way we arrive at three pairs of equations, each of which serves to express the corresponding relation bet-^een the three points in which a ray cuts the three planes a,h,c; so that if two of the points are given the third can be found, or, more generally, if of the six co-ordinates of the three points four are given the other two may be deter- mined. § 6. Belation between Corresponding! Elements of Sur- face. We will now take the following case. Given on one of the planes, say a, a point p^, and on another, say b, an element of surface ds'^; then if rays pass from pa to the different points of ds^, and if we suppose these rays produced till they cut the third plane c, they will all cut that plane in general within another indefinitely small element of surface, which we will call dsg (Fig. 27). Let us now deter- mine the relation between ds^ and dso. * In KirchhofE's paper (p. 285) it ia stated of the quantity there con- sidered, which is essentially the same as the difference here treated of, except that it refers to four planes instead of three, that it must be a minimum. This may possibly be a printer's error, and in any case an interchange of max- imum and minimum in this place would have no further influence, because the principle used in the calculations which follow, viz. , that the dierential coefficient =:0i holds equally for a maximum or a minimum. CONCENTRATION OF KAYS OF LIGHT AND HEAT. 305 Fig. 27, In this case, of the six co-ordinates which relate to each ray (viz. those of the three points in which the ray cuts the three planes) two, viz. x^ and y^, are given beforehand. If we now take any values we, please for 00^ and y^, the co-ordinates x^ and y^ are in general thereby deter- mined. Thus in this case we may consider x„ and y^ as two functions of x^ and y^. As the form of the element ds,, may be any whatever, let it be a rect- angle da?5, dy„, and let us find the point in plane c correspondisng to every point in the outline of this rectangle. We shall then have on plane c an indefinitely small parallelogram which forms the corresponding element of surface. The magnitude of this parallelogram is determined as follows. Let \ be the length of the side which corresponds to the side dx„ of the rectangle in plane h, and let (Xa;„) and (\y„) be the angles which this side makes with the axes of co-ordinates. Then \ cos (\a:.) = £^ dx^ ; \ cos l^y^) = ^dx^. Again, let /* be the other side of the parallelogram, and let (>*%„) and (jiy^ be the angles it makes with the axes. Then we have li cos {^uc,) = -£j dy, ; fi cos {fiyc) = ^ '^3'^ • Let (V) be the angle between the sides X and /*. Then we have cos (V) = cos {\x,) cos i/ix,) + cos {\y,) cos (fiy,) 'dx, dx dy, dy\ dx.dy^ \d£t dyt dx^ dyj dyt dx^dyj X/* C. 20 306 - ON THE MECHANICAL THEORY OP HEAT. Now to determine the area cfo, of the parallelogram, we may write ds^ = \fi sin (\/i) = \fijl— cos' (\/t) = ^\y-cos'(X/i)\y. Here we may substitute for cos (X/i) the expression just given, and for X" and fi" the following expressions derived from the above equations : — ,_{fdfoX Then several terms under the root caricel each other, and the remainder form a square as follows : . V \dx^ dy^ dy^ dxj " ' This quadratic equation can therefore be solved at once. But it must be observed that the' difference within the brackets may be either positive or negative, and, as we have only to do with the positive root, we will denote this by putting the letters v.n. (valor numericus) before this difference. We can then write To ascertain how a;„ and y„ depend upon ir^ and y^, we must apply one of the three pairs of equations in § 5. We will first choose equations (1). If we differentiate those equations according to ss^ and ^j, remembering that'eacli of the quantities denoted by T contains two of the three pairs of co-ordinates x^, y^, x^, y^', x^, y„, as denoted. by the indices; and if in differentiating we treat x, and .y.-as functions of CONCENTRATION OF BAYS OF LIGHT AND HEAT. 307 ajj and y„ -whilst we take x^ and y^ as constant; then we obtain the following four equations ; {dx,f X ? _L dx^dx^ tfoj dx^dy, + (fr.. + dx^dy^ d'(T^.+T,:) (.dy>T + dy^dx dx„ dy^ dx^dy, dXi dy. dx. dy^dy, dy^dx^ dy^ dy^dy^ dyi dxt x^ = dy^ .(5). If by help of these equations we determine the four differential coefficients -j^° , z^' , -p , ^ and subsititute toj dy^ dx^ dy,, the values thus found in equation (4), we obtain the required relation between ds^ and ds„. To be able to write the result more briefly, we will use the following symbols : A = v.n.(^ dx^dx^ d^Z, d'Z, dy^dy, dx^dy. E= v.n. . idx,r "" {dy,Y dy.dxj W. dx^dy^ •in Then the required relation may be written as follows : dsg E 'A' ■(8). Again, if we suppose in like manner that a point p^ is given on plane c (Fig. 28), and find on plane a the element da^, which corresponds to the given element ds^ on plane b, then the result can be derived frofla that last given by simply interchanging the indices a and c. If for brevity we write a = vn fi^x^^ ■ Kdx^dXi dyjiy„ dxj,y^ dyM\ -)...(9). 20—2 308 ON THE MECHANICAL THEORY OF HEAT, then we have ds, C .(10). ib Pig. 28. Lastly, suppose a point p^ to be given on plane h (Fig. 29) Fig. 29 and choose any element of surface ds^ on plane a. Let us suppose that rays from different points of this element pass through p^, and that they are produced to the plane c. Then the magnitude of the element of surface dse, in which all these rays meet plane c, is found, using the same symbols as before, to be as follows : q A' .(11). From this we see that the. two corresponding elements in this case bear exactly the same relation to each other, as the CONCENTRATION OF BATS OF LIGHT AND HEAT. 309 two elements which are obtained when we have an element rfSj given in plane b, and then, having assumed as the origin of the rays a point first in plane a, and secondly in plane c, determine in each case the element of surface in the third plane corresponding to the element da^, § V. Fractions formed out of six qucmtities to express the Relations between Corresponding Elements. In the last section we have only employed the first of the three pairs of equations in § 5. We can however employ the two other pairs (2) and (3) in the same manner. Each pair leads us to three quantities of the same kind as those already denoted by A, G, and E. These quantities serve to express the relations between the elements of surface. Of the nine quantities thus obtained, hoTvever, there are four which are equal to each other, whereby the actual number is reduced to six. The expressions for these six are here placed together for the sake of convenience, although three of them have been already given. '{dx^dx^ dyjiy^ dx^dij, dy„dxj '\dx^dx^ dyjiy^ dxjiy^ dyJ^J \dxjx^ dyj,y^ dxjdy^ dyjxj n ^„ {^{T^-TJ) ^^ d^{T^-T^) \ d^{T.-TJ Y\ ^~ l~~W^ (dyj^ L ^ady. J I ^-^•^i [dx:f "" . {dy,r L dx,dy, J J ^ = ^-°i {dx:f (dyj L dx^y, j] } (I-). By help of these six quantities every relation between two elements of surface can be expressed by three different fractions, .as may be shewn in "tabular form as foUows : 310 ON THE MECHANICAL THEORY OF HEAT. E A G A F~B B D ~E-A"C A F B ~G~ B D .(II.). It is easily seen that the three horizontal rows relate to the three cases, in which the given point through which the rays must pass is taken either in plane a, plane c, or plane b. Of the three vertical rows of fractions, which express the relations between the elements of surface, the first is deduced from equation (i) of § 5, the second from (2) and the third from (3). Since the three fractions, which express a given relation between two elements of surface, must be equal to each other, we have the following equations between the six quantities r BG_ GA^ AB A' = EF; B' = FD; G'=DE (13). Our further investigations will be performed by means of these six quantities ; and since every relation between two elements of surface is expressed by three different fractions, we can always choose amongst these the fraction most suit- able for each special case. III. DeTEBMINATION of the mutual EADLiTION, WHEN THERE IS NO CONCENTRATION OF EATS. § 8. Magnitvde of the Element of Surface corresponding to ds„ on a plane in a particular position. We will first consider the case to which Kirchhoff's ex- pression refers, and seek to determine how much heat two elements send out to each other, on the assumption that every point of one element receives from every point of the other one ray and only one ; or at most a limited number of particular rays, which may be considered separately. CONCENTRATION OF RATS OF LIGHT AND HEAT. 311 Given two elements ds^ and ds^ lA planes a and c (Fig. 30), we will first determine the heat which ds^ sends to ds^. For this pur- pose let us suppose the intermediate plane h to lie parallel to plane a at a distance p, which is so small, that the part which lies between these two planes of any ray passing from ds^ to ds^ may be considered as a straight line, and the medium through which it passes as homogeneous. Let us now take any point in element ds^, and consider the pencil of rays which passes from this point to the element ds^. This pencil will cut plane h in an element ds^ whose magnitude is given by one of the three fractions in the uppermost horizontal row of equations (II.). Choosing the last of these we have the equation , Fig. 30. ^ds. (14). The quantity G may in this case be brought into a specially simple form, on account of the special position of plane h. For this purpose let us follow Kirchhoff in choos- ing the system of co-ordinates in h so as to correspond exactly with that in the parallel plane a; i.e. let the origins of both lie in a common perpendicular to the two planes and let each axis of one system be parallel to the correspond- ing axis of the other. Let r be the distance between two points lying on the two planes, and having co-ordinates x^, y^ and x^, y^ respectively. Then r^Jp'+ix.-xS + iy-yJ' .(15). Let us now suppose a single ray to pass from one of these points to the other ; then, since its motion between the two planes is supposed to be rectilinear, the length of its path will be simply represented by r ; and if we denote by v^ its velocity in the neighbourhood of plane a, which by the assumption, will remain nearly constant between a and b, 312 ON THE MECHANICAL THEORY Ot HEAT. then the time which the ray expends in the passage will be given by the equation "' v.- The expression for C may therefore be written 1 / d'r ^^ cPr ^V ^^ CPr \ ~^'^' v^Kdx^dx^ dy^dy^ dx^dy^ dy^dxj' Substituting for r its value as given by (15), we obtain G = l-^^'? W Hence equation (14) becomes ds, = v:x-,Bds^ .(17). r If we denote by the angle which the indefinitely small pencil of rays, which starts from a point on ds^, makes with the normal at that point, then cos ^ = - ; and the last equa- tion also takes the form ' "'""''^.Bds, (18). ""' cos=^^ § 9. Expressions for the quantities of Heat which ds„ and ds„ radiate to each, other. When the magnitude of the element of surface ds^ is determined, the quantity of heat which ds^ sends to ds, can be easily expressed. ^ From every point of ds^ an indefinitely small pencil of rays goes to .d&/, and the solid angle of the cone made by the pencil from each of these points may be taken as the same. ' The magnitude of this angle is determined by the magnitude and position of that element ds^ in which the cone cuts plane b. To express this angle geometrically, let us suppose that a sphere of radius p is drawn round the point from which the rays start, and that within this sphere we may consider the path of the rays as being rectilinear. If da is CONCENTRATION OF RATS OF LIGHT AND HEAT. 313 the element of surface in -which this sphere is cut by the pencil of rays, then — ^ represents the angle of the cone. But since the element ds^ is at the distance r from the vertex of the cone, and since the normal to the surface at ds^, which is parallel to the already mentioned normal to the surface at ds^, forms with the indefinitely small cone of rays the angle 6, we have the equation da- _ c os X ds^ . . — T i \1"J' If we substitute for «?«, its value from (18), we obtain d^^^Bds^ (20). We have now to determine the magnitude of that part of the heat sent from ds^ which corresponds to this indefinitely small cone; or in other words, how much heat ds^ sends through the given element da- upon the spherical area. This quantity of heat is proportional (1) to the magnitude of the radiating element ds^, (2) to the angle of the cone, or to -^, and (3), according to the well-known law of radiation, to r the cosine of the angle 6, which the indefinitely small cone makes with the normal. It may therefore be expressed by e cos ^ -^ ds^, P where e is a factor depending on the temperature of the surface. To determine this factor we have the condition that the whole quantity of heat which tfe„ radiates, or which it sends to the whole surface of any hemisphere above plane a, must equal e^ds^ where e„ is the intensity of emission from plane a at the position of ds^. Hence we have ?!' ^ ! cos 0da- = e^ The integration extends over the whole of the hemisphere, and gives 314 ON THE MECHANICAL THEORY OF HEAT. Substituting this value of e in the above expression, we obtain for the quantity of heat which ds^ sends through da the formula — cos a —2 ds^. TT ft We have only to substitute in this formula the value for —J given in equation (20), and we obtaiin the required expres- sion for the quantity of heat which ds^ sends to ds^, viz.: If conversely we require the quantity of heat which ds^ sends to ds^, and if we denote by e„ the intensity of emission from plane c at the position of ds^, and by t'^the velocity of the rays in the neighbourhood of ds^, we obtain the expression : in § 10. Radiation as dependent on the surrovrndingMediwrn. The expressions obtained in the last section are in general the same as Kirchhotf's expression given in § 3, and differ only inasmuch ' as they contain as factor the square of the velocity, which does not occur in Kirchhoff's expression, because' he considers nothing but the velocity in vacuo, and takes this as unity. Since however the bodies, whose mutual radiation we are considering, may often be in different media, where the velocity of the rays is different, this factor is not without importance; and its introduction leads also to a special conclusion of some theoretical interest. As mentioned in § 2, it has been hitherto assumed that with perfectly black bodies the intensity of emission depends only on the temperature, so that two such bodies of equal temperature would radiate equal quantities of heat from equal areas of surface. As. far as the author knows, the ques- tion whether the surrounding medium has also an influence on the intensity of emission has never been considered. Since however the two expressions given above for the mutual radiation of two elements contain a factor which CONCENTEATIOlJ OF EATS OF LIGHT AND HEAT, 315 depends on the nature of the medium, it becomes necessary to consider this medium, and the method of determining its influence. If from the above two expressions we omit the factor which is common to both, viz. — ds^ds^, we have the result that the quantity of heat, which the element ds^ sends to the element ds^, bears to that which ds^ sends to ds„ the ratio ^a^a '• ^a"- If ■w^e now assume that at equal temperatures the radiation is always equal, even when the media in contact with the two elements are- different, then for equal tempera- tures we must put e^=e^', and the quantities of heat, which the two elements radiate to each other, would then not be equal, but would be in the ratio u/ : v^. It would follow that two bodies which are in different media, e.g. one in water and the other in air, would not tend to equalize their temperatures by mutual radiation, but that one would be able by radiation to raise the other to a higher temperature than that which itself possesses. If on the other hand we maintain the universal correct- ness of the fundamental principle- laid down by the author, viz. that heat cannot of itself pass from a colder to a hotter body, then we must consider the mutual radiations of two perfectly black elements of equal temperature as being themselves equal, and must therefore put eav: = e,v:^ (21). Hence e,, :c, ::^;/^^^/ (22). Since the ratio of the velocities is the reciprocal of that of the coefficients of refraction, which we m&y call n, and m„, this proportion may be written e.:c„ ::«/:< (23). Hence the radiation of perfectly black bodies at equal temperatures is different in different media, and varies in- versely as the squares of the velocities in those media, or directly as the squares of the coefficients of refraction. Thus the radiation in water must bear to that in air the ratio {^y : 1, nearly. 316 ON THE MECHANICAL THEORY OP HEAT. > We have also to temember that in the heat radiated frofia perfectly black bodies there are rays of very different wave- lengths; and if we assume that the equality of mutual radiation holds, not merely for the heat ais a whole, but also for each wave-length. in particular, we must have for each of these a proportion similar to (22) and (23) but in which the quantities in the right-hand ratio have somewhat dif- ferent values. Lastly, instead of perfectly black bodies, let us consider bodies in which the absorption of the rays received is partial only. We must then introduce in the formula, in place of the emission, a fraction having the emission as numerator and the coefficient of absorption as denominator. For this fraction we can obtain relations similar to those obtained for the emission alone. This generalization, in which the influence of the direction of the rays upon the emission and absorption must also be taken into account, need not here be entered upon. IV. Determination of the mutual radiation of TWO ELEMENTS OF SURFACE, IN THE CASE WHEN ONE IS THE OPTICAL IMAGE OP THE OTHER, § 11. Relations between B, T>, F and E, We have hitherto assumed that the planes a and c, so far as we are concerned with them, give out their rays in such a way, that one ray and one only, or at most a limited quantity of individual rays, pa^ss from any point in the one to any point in the other. We will now pass on to the case in which this does not hold. The rays which diverge from points in the one plane may be made to converge by reflections or refractions, and to meet again on the other plane ; so that for any point p^ on plane a there may be one or more points or lines on plane c, in which an indefinitely large number of the rays coming from p^ cut that plane, whilst other parts of the same plane receive no rays whatever from p^. The same of course holds of the rays which start from plane e and arrive at plane a, since the rays passing to and fro be- tween the same points describe the same paths. Among the innumerable cases of this description we will, CONCENTRATION OF RAYS OF LIGHT AND HEAT. 317 Kg. 31. for the sake of clearness, consider fixst the extreme case illustrated in Fig. 31. In this case all the rays sent out by J3„ within a cer- tain definite cone meet again in a single point p^ of plane c. This case occurs for example, when the deflection of the rays is effected by a lens, or by a spherical mirror, or by any system of concentric lenses or mirrors. We are here supposed to neglect the spherical and chromatic aberration, which we have a right to do with regard to the latter, inasmuch as we have confined ourselves to homogeneous rays. Two points thus corresponding to each other, as the points of starting and of re-union of the rays, are called, as already mentioned, conjugate foci. For each ray in such a case the co-ordinates «„, y„ of the point p^ in which it strikes plane c, are determined by the co-ordinates x^ y„ of the starting point p^. The other points on plane c in the neighbourhood of p^ receive no rays from p^; for there is no path to them which has the property that the time in which the ray would traverse it is a minimum, as compared with the time in which it would traverse any other adjacent path. Hence the quantity T^ which expresses this minimum time, can have a real value only for p^, and not for any of the points round it. The differential coeffi- cients of 2'^, in which the co-ordinates x^, y^ are assumed to be constant and one of the co-ordinates x^ y^ to be variable, (or conversely x^, y^ to be constant and one of the co-ordinates x^ 2/o *o be variable) can thus have no real finite values. It follows that of the six quantities A, B, G, D, E, F, which have been determined by equations (I.), the three B,^ D, F are not applicable to the. present case, inasmuch as they contain differential coefficients of r„; whilst the three others A, G, E contain only differential coefficients bf T^ and T^„. Let us now assume that plane h is so chosen that between it and the planes a and c, so far as we are concerned with them, the radiation takes place on the former system, so that one ray and one only, or at most a 318 OK THE MECHANICAL THEORY OF HEAT. limited number of rays, pass from any point on plane b to any point on plane a or c. Then for all the points with which we are concerned, the quantities T^ and Tj„ and their differential coefficients have real values not indefinitely large. The quantities A, C and E are then as applicable in this case as in the former. One of these quantities, E, takes in this case a special value, which may at once be found. For -the three points in which the ray cuts the three planes a, h, «, the two equa^ tions given in (1) must hold, viz. : In the present case the position of the poirCt in which the ray cuts plane h is not determined hj the position of the two points p^ and p^, but plane h may be cut in all points of a certain finite area. Hence the two equations above must hold for all these points, and therefore the equations obtained by differentiating these according to ajj and y^ must also hold, viz. : cP (?:.+ rj _.. d' (T,,+ TJ _ d' (T^+ TJ _ ^ da,: -"' dx,dy, -"' d^^ -U...C^4;. ■ If we apply this to the equation determining E in equa--, tions (I.), we obtain E = Q (25). The two other quantities A and C have in general finite values, which differ in different circumstances, and which we must now use in our further investigation, ' • § 12. Application of A and O to determine the Relation bstween the Elements of Surface. Let us suppose that the element ds^ on plane a has an optical image ds^ on plane c, so that every point of ds^ has a -point on ds^ as its conjugate focus, and vice vers^. We have now to enquire whether the quantities of heat, which these elements send to each other, when taken as elements of the surfaces of two perfectly blaclc bodies of equal tem- peratures, are equal or not,. CONCENTBATION OF RAYS OP LIGHT AND HEAT. 319^ First, to determine tlie position and magnitude of the image ds„ corresponding to the given element ds^. Take any point p^ on the intermediate plane h, and consider all the rays starting from points on ds^, which pass through p^. Each of these rays strikes plane c in the conjugate focus to that from which it starts ; and therefore the element of surface in which this pencil of rays cuts plane c is pre-i cisely the optical image ds„ of the element ds^. Therefore, to express the ratio between the ^reas of ds„ and ds^, we may use one of the three fractions in the lowest row of equations (II.)> which express the relation between the two elements of surface, in which an indefinitely small pencil, passing through a single point ^j of plane b, cuts the two planes a and c. Of these three fractions the first is alone applicable in this case, since the other two are undetermined. We have thus the equation 1=^ ••• -f^^)- . This equation is interesting also from an optical point of view, as being the . most general equation which can be given to determine the ratio between the area of an object and that of its optical image. It should be remarked that the intermediate plane h, to which the quantities A and G are related, may be any whatever, and can therefore in any particular case be chosen as is most convenient for calcula- tion. § 13. Relation between the Quantities of Heat which ds^ and dSj radiate to each other. Having thus determined the element of surface ds^ which forms the image of ds^, let us take on plane b, instead of a single point, an element of surface ds^, and let us con-^ sider the rays which the two elements ds„ and ds^ send through this element dsy All the rays, which start from one point of ds^ and pass through ds^, unite again in on^ point of ds^ ; and thus all the rays which ds„ sends through? dsj exactly strike the element ds,i and vice yers^. The two quantities of heat which ds^ and ds„ send to ds^ are thus the same as the quantities of heat. which ds^ and ds^ sea^ 320 ON THE MECHANICAL THEORY OF HEAT, to each other through the intermediate element ds^. These quantities of heat are therefore given at once by what has gone before. Thus for the quantity of heat which ds^ sends to ds^, the same expression will hold, as held in § 9 for the quantity of heat which ds^ sends to ds^, provided we substitute G for B and ds^ for ds^. The expression thus becomes eji^^ — ds^ ds^. IT Similarly for the quantity of heat which ds^ sends to ds^, the expression will be the same as that for the quantity of heat which ds^ sends to ds^, provided we substitute A for 5 and dsj for ds^, or will be e„«/ — dsjds^. IT Kemembering that by equation (26) Gds^ = Ads^, we see that these two expressions stand to each other in the ratio e„w/ : e,v'. We obtain precisely the same result, if we take any other element ds,, on plane b, and consider the quantities of heat which the two elements ds^ and ds^ send to each other through this new element. These will , always be found to stand to each other in the ratio e^t;/ : e„i;/. Since the quan- tities of heat, which ds^ and ds„ send to each other on the whole, are made up of those which they send to each other through all the different elements of the intermediate plane, the same ratio "must hold for the whole ; and we thus obtain the final result, that the total quantities of heat which ds^ and ds^ radiate to each other, stand in the ratio ej!^ : e^vf. This is the same relation as was found in sections 8 and 9 for the case where there is no concentration of rays ; it thus follows that the concentration of rays, however much it alters the absolute magnitudes of the quantities of heat which two elements radiate to each other, leaves the ratio between them exactly the same. It was shewn in § 10, that if in the case of ordinary un- concentrated radiation the principle holda that heat cannot pass from a colder to a hotter body, then the radiation must CONCENTRATIOK OF RAYS OF LIGHT AND HEAT. 321 differ in different media, and must be^uch that for perfectly black bodies of equal temperatures If this equation is satisfied, then in the present ease also, where the elements ds^ and ds^ are images one of the other, the quantities of heat which they mutually radiate must be equal; and therefore, in spite of the concentration of the rays, one element cannot raise the other to a higher temperature than its own. V. Relation between the increment of area and the RATIO or THE TWO SOLID ANGLES OP AN ELEMENTARY PENCIL OF RAYS. § 14. Statement of the Proportions for this Case, As an immediate result of the foregoing we may here develope a proportion, which appears to have some general interest, inasmuch as it illustrates a peculiar difference in the behaviour of a pencil of rays in the case of an object and of its image. This difference' must always exist and have a de- terminate value, when the object and the image have differ- ent areas. Consider an indefinitely small pencil of rays, which starts from a point on ds^, passes through the element ds^ on the intermediate, plane, and then unites again in a point on cfo„. We may compare the divergence of the rays at their starting point with the convergence of the same at their point of re- union. This divergence and convergence (or, to use the ordinary phrase, the solid angles of the indefinitely small cones, which the pencil forms at its points of starting and re-union) are given directly by the same method which we have used in§ 9, d,g follows:— •' ■ • Suppose that round each of the points there is described a sphere of so small a radius, that we may consider the i ays' as going in straight lines as far as the surface of this sphere: and then consider the element of surface in which the pencil bif rays cuts the sphere. Let do- be this element and let p be the radius of the sphere; then the angle of the indefinitely small cone, which contains the rays so far as they are recti- c. 21 322 ON THE MECHANICAL THEOET OF HEAT. linear, will' be expressed by —^ . This fraction we have de- termined for a similax case in § 9 by equation (20), and in the expression there given we have only to alter the letters in order to transform it into the expression for the present case. In order to express the angle of the cone for that starting; point of the rays which lies 4n pl^ie a, we have only to substitute in that expression ds^ for ds^ and G for B. In addition,, ;the symbol ^,. which expresses the angle between the pencil of rays and the normal to the surface of ds^, may be changed to 6^ , so as to express more clearly that it relates to plane a; and for the same reason the suffix a may be added to — j, ., Thus we obtain ^4]=^Gds, (27), To obtain the other equation, which gives the angle of the cone at the point of re-union on plane c, we have only to change the S'uffia^ a into c throughout, and also to substitute A for G. Thus we have , ^°'^ "" Ads^ (28). !da ./)V„ cos^, From these two equations we obtain the proportion, cos 5„ (dek bodies, we consider such as only partially absorb the rays falling on them; or if instead of homogeneous heat we consider heat. which con- tains systems of waves ©f different lengths ; or lastly, if instead of taking all the rays as unpolarized we include the phenomena of polarization; still in all these cases we have to do only with facts, which hold equally for the heat sent out by any one body,. and for that which it receives from other bodies. It is not necessary to consider these facts more closely, since they also take place with ordinary radiation without concentration, and the object of the present investi- gation was only to consider the special actions which might possibly be produced by concentration of the rays. § 19. Sumrmiry of Restdts, The main results of this investigation may be briefly stated as follows : (1) In order to bring the action of ordinary radiation, without concentration, into accordance with the fundamental principle, that heat cannot of itself pass from a colder to a hotter body, it is necessary to assume that the intensity of emission from a body depends not only on its own composi- tion and temperature, but also on the nature of the sur- rounding medium; the relation being such, that the in- tensities of emission in different media stand in "the inverse ratio of the squares of the velocities of radiation in the CONCENTRATION OF RAYS OF LIGHT AND HEAT. 331 media, or in the direct ratio of the squares of the coefScients of refraction. (2) IF this assumption as to the influence of the sur- rounding media is correct, the above fundamental principle is not only fulfilled in the case of radiation without concen- tration, but must also hold good when the rays are concen- trated in any way whatever by reflections or refractions ; since this concentration may indeed change the absolute magnitudes of the quantities of heat, which two bodies radiate to each other, but not the ratio between these quantities. CHAPTER XIII. DISCUSSIONS ON THE MECHANICAL THEOET OF HEAT AS HERE DEVELOPED, AND ON ITS FOUNDATIONS. § 1. Different Views as to the Relation between Heat and Work. The papers of the author on the Mechanical Theory of Heat, as reproduced in all essential particulars in this volume, have frequently met with opposition; and it may be desirahle to give here some of the discussions which have taken place on the question, since many points are raised in them, on which the reader may even yet be in doubt. The removal of these doubts may be facilitated by his learning what has been already written upon these points. As already mentioned in Chapter III., the first important attempt to reduce to general principles the action of heat in doing work was made by Carnot. He started from the assumption that the total quantity of heat existing was in- variable, and then supposed that the falling of heat from a higher to a lower temperature brought about mechanical work, in the same way as the falling of water from a higher to a lower level. But simultaneously with this conception the view gained ground that heat is a mode of motion, and that in producing work heat is expended. This view was set forth at intervals from the end of the last century by individual writers, such as Rumford, Davy, and Seguin*; but it was not. tin 1840 that the law corresponding to this view, viz. that of * In a paper by Mohr, publiBhed 1837, heat is in some places called a molion, in otfieis a force. , DISCUSSIONS ON THE THEORY, 333 tVie equivalence of neat and work, was definitely laid down by Mayer and Joule, and proved by the latter to be correct by means of numerous and brilliant experiments. Soon afterwards the general principle of the conservation of energy was laid down by Mayer* and Helmholtzf (by the latter in a specially clear and convincing manner), and was applied to various natural forces. A fresh starting point was thus given to researches on the science of heat ; but in carrying these out great difficul- ties presented themselves, as was natural with so widely extended a theory, which was intertwined with all branches of natural science, and influenced the whole range of physi- cal thought. In addition, the wide acceptation which Car- net's treatment of the mechanical action of heat had won for itself, especially after being brought by Clapeyron into an elegant analytical form, was unfavourable for tlie reception of the new theory. It was believed that there was no alter- native but either to hold to the theory of Carnot, and reject the new view according to which heat must be expended to produce work, or conversely to accept the new view and reject Carnot's theory. § 2. Papers on the Subject hy Thomson and the Author. A very definite utterance on the then position of the question was given by the celebrated English physicist, now Sir William Thomson, in an interesting paper which he pub- lished in 1849 (when most of the above-mentioned researches of Joule had already appeared and were known to him), under the title, "An Account of Carnot's Theory of the Motive-Power of Heat, with numerical results deduced from Kegnault's ex;periments on steam J." He still maintains the position of Carnot, that heat may do work without any change in the quantity of heat taking place. He however points out a difficulty in this view, and goes on to say, p. 545 : " It might appear that this difficulty might be wholly removed, if we gave up Carnot's fundamental axiom, a view which has been strongly urged by Mr Joule," He adds,, <' but if we do * Die organiscke Bewegung in ihrem Zusammenhange mit dem Stoff^ weeheel. Heilbronn, 1845. + Ueber die Erhaltung der Kraft, 1837. T Trans. Royal Soc, of Edin., Vol. xvi. p. 541, 834 ON THE MECHANICAL THEORY OP HEAT. this we stumble over innumerable other difficulties, which are insuperable without the aid of further experimental researches, and without a complete reconstruction of the Theory of Heat. It is in fact experiment to which we must look, eitber for a confirmation of Carnot's axiom, and a clearing up of the difiSculty whicb we have noticed, or for a cornpletely new foundation for the Theory of Heat." At the time when this paper appeared the author was writing his first paper " On the Mechanical Theory of Heat," which was brought before the Berlin Academy in 1850, and printed in the March and April numbers of Poggendorff's Annalen. In this paper he attempted to begin the recon- struction of the theory, without waiting for further experi- ments; and he succeeded, he believes, in overcoming the difficulties mentioned by Thomson, so far at least as to leave the way plain for any further researches of this character. He there pointed out the way in which the fundamental conception, and the whole mathematical treatment of heat,, must be altered, if We accepted the principle of the equiva^' lence of heat and work ; and he further shewed that it was not needful wholly to reject the theory of Carnot, but that- we might adopt a principle, based on a different foundation from Carnot's, but differing only slightly in form, which might be combined with the principle of the equivalence of heat and work, to form with it the ground-work of the new theory. This theory he then developed for the special cases of perfect gases and saturated vapour, and thereby obtained; a series of equations, which have been universally employed in the form there given, and which will be found in Chap' ters II. and VI. of this volume. § 3. On Rcunhine's Paper and Thomson's Second Paper. In the same month (February, 1850) in which the author's papej* was read before the Berlin Academy, a valu- able paper by Eankine was read before the Royal Society of Edinburgh, and was afterwards published in their Trans- actions (;VoL 20, p. 147)*. Rankine there proposes the hypothesis, that heat consists in a rotary motion of the molecules ; and thence deduces with * In 1854 it was reprinted -with some alterations, in the Fhil. Mag,, Series 4, Yol. vn. pp. 1, HI, 172. DISCtrSSIONS ON THE THKORT. 335 much skill a series of principles on the action of heat, which agree with those deduced by the author from the first main principle of the mechanical theory*. The second main prin- ciple was not touched by Rankine in this paper, but was treated in a second paper, which was brought before the Eoyal Society of Edinburgh a year afterwards (April, 1851). In this he remarka-f- that be had at first felt doubtful of the correctness of the mode erf reasoning by which the author had arrived at this second principle ; but that having com- municated his doubts to Sir W, Thomson, he was induced by him to investigate the subject more closely. Ke then found that it ought not to be treated as an independent principle in the theory of heat, but might be deduced from the equations, which he had given in the first section of his former paper. He proceeds to give this new proof of the principle, which however, as will be shewn further on, is in opposition, , for certain very important, cases, with his own views, as elsewhere expressed. This paper of 1851 Rankine added as a fifth section to his. former paper on account of the connection between them. Thence has arisen with some authors the mistaken idea that this new paper was actually part of the former one, and that Rankine had therefore given a proof of the second main principle at the same time as the author. From the fore- going it will be seen that his proof (waiving the question how far it is satisfactory) appeared a year later than the author's. In March of the same year, 1851, a second paper by Sir W. Thomson on the Theory of Heat was laid before the Royal Society of Edinburgh f. In this paper he abandons his former position with regard to Carnot's theory and assents to the author's exposition of the second main principle. He then extends the treatenent of the subject. For whilst the author had confined himself in the mathematical treatment of the question to the case of gases, of vapours, and of the process of evaporation, and had only added that it was easy to see how to make similar applications of the theory. to other • Edinb. Trans., Yol.,xx. p. 205; Phil. Mag., Series i, Vol. vii. p. 2i9. t Phil. Mag., Vol. vii. p. 250. j Edin. Trans., Vol. xx. p. 261 ; Phil. Mag., Series 4, Vol. iv. pp. 8, 105, 168. 330. ON THE MECHANICAL THEORY OF HEAT. cdses, Thomson developes.a series of general equations, whieh are independent of the body's condition of aggregation, and only then passes on to more special applications. On one point this second paper still falls short of the author's. For here also Thomson holds fast by the law of Mariotte and Gay-Lussac in the case of saturated vapour, and hesitates to accept an hypothesis with respect to perma- nent gases, which the author had made use of in his investi- gation (see Chapter II., § 2, of this work). On this he remarks* : " I cannot see that any hypothesis, such as that adopted by Clausius fundamentally in his investigations on , this subject, and leading, as he shews, to determinations of the densities of saturated steam at different temperatures, which indicate enormous deviations from the gaseous laws of variation with temperature and pressure, is more probable, or is probably nearer the truth, than that the density of sa- turated steam does follow these laws, as it is usually assumed to do. In the present state of science it would perhaps be wrong to say that either hypothesis is more probable than the other." Some years later, after he had proved by his joint experiments with Joule that this hypothesis is correct within the limits assigned by the author, hfe used the same method as the author to determine the density of saturated vapour "I". Rankine and Thomson, so far as the author knows, have always recognized most frankly the position here assigned to the first labours of themselves and the author on the me- chanical theory of heat. Thomson remarks in his paper J: " The whole theory of the moving force of heat rests on the two following principles, which are respectively due to Joule and to Carnot and Clausius." Similarly he introduces the second main principle as follows: "Prop, II. (Carnot and Clausius)." He then proceeds to give a proof discovered by himself, and goes on§: "It is with no wish to claim priority that I make these statements, as the merit of first establish- ing the proposition on correct principles is entirely due to Clausius, who published his demonstration of it in the month • Edin. Trans., Vol. xx. p. 277; Phil. Mag., Yol. iv. p. 111. t Phil. Trans., 1854, p. 321. $ Mdin. Trans., Vol. xx. p. 264 ; Phil. Mag., Vol. iv; p. 11; § Edin. Trans., Vol. xx. p. 266 ; Phil, Mag., Vol. iv. pp. 14, 242. ' ' DISCUSSIONS ON THE THEOET. 337' of May last year, ia the second part of his paper on the Motive Power of Heat." § 4, Holtzmann's objecticms. From other quarters repeated and in some cases violent objections were raised to the author's first paper, to which, in the same and following years, a series of other papers, serving to complete the theory, were added. The earliest of these objections came from Holtzmann, who had published in 1845 a short pamphlet* on the subject. In this it would at first appear as if he wished to treat the question from the point of view, that for the generation of work there was necessaiy not merely a change in the distribution of heat, but also an actual destruction of it, and that conversely by destroying work heat could be again generated. He remarks (p. 7): " The action of the heat which has passed to the gas is thus either a raising of temperature, combined with an increase of the elastic force, or a certain quantity of mechanical work, or a mixture of the two ; and a certain quantity of mechanical work is equivalent to the rise in temperature,. Heat can only be measured by its effects; of the two above-named effects mechanical work is especially adapted for measurement, and it will be chosen accordingly for the purpose. I call a unit of heat that amount of heat, which by its entrance into a gas can perform the mechanical work a, or, using definite mea- sures, which can raise a kilograms to a height of 1 metre." Further on (p. 12) he determines the numerical value of the constant a by the method previously used by Mayer, and explained in Chapter II., § 5; the number thus obtained cor- responds perfectly with the mechanical equivalent of heat, as determined by Joule by various other methods. In extend- ing his theory however, i.e. in developing the equations by which his conclusions are arrived at, he follows the same method as Clapeyron; so that he tacitly retains the assump- tion that the total quantity of heat which exists is invariable; and therefore that the quantity of heat which a body takes in, while it passes from a given initial condition to its present condition, must be expressible as a function of the variables, which determine that condition. * Ueber die Warme und Etasticitdt der Gase und Dampfe; von C. Holtzmann. Mannheim, 1845 ; also Pogg. Am., Vol. lxxii a. Q. 22 338 ON THE MECHANICAL THEOEY OF HEAT. In the author's first paper the inconsequence of this method was pointed out, and the question treated in another way.; on which Holtzmann wrote an article*, in which he endeavoured to shew that this method of treatment, and specially the assumption that heat was expended in pro- ducing work, was inadmissible. The first objection which he raised was of a mathematical character. He carried out an investigation similar to that in the author's paper, in order first to determine the excess of the heat which a body takes in over that which it gives out, during a simple cyclical process consisting of indefinitely small variations, and secondly to compare this excess with the work done. But iii such a process both the work done and the excess of heat must be indefinitely small quantities of the second order ; and there- fore in the whole investigation, care must be taken that all quantities of the second order, which do not cancel each other, shall be taken into account. This Holtzmann neg- lected to do ; and he was thus led to a final equation, which contained a self-contradiction, and in which he therefore imagined that he had found a proof of the inadmissibility of the whole method. This objection was easily disposed of by the author in his reply. He further brought forward as an obstacle to the theory, that, according to the formulae given, the specific heat of a perfect gas must be independent of its pressure, whereas the experiments of Suermann, and also those of De la Roche and B^rard, shewed that the specific heat of gases increased as the pressure diminished. On this conflict between his own theory and the experiments which were then known and supposed to be correct, the author remarked in his reply as follows : "On this point I must first point out that, even if these observations are perfectly correct, they yet say nothing against the fundamental principle of the equivalence of heat and work, but only against the approximate assumption which I have made, viz. that a permanent gas, if it expand at constant temperature, absorbs only so much heat, as is required for the external work which it thus performs. But besides it is sufficiently known how unreliable are in general the determinations of the specific heats of gases ; and all the * Pogg. Ann., Vol. lxxxii. p. 445. DISCUSSIONS ON THE THEORY. 339 more in those few observations -whicli have been hitherto made at varying pressures. I did not therefore conceive myself bound to abandon the above assumption on account of these observations, although they were well known to me at the time when I wrote my former work; because the other grounds, which may be alleged for the correctness of the assumption within the limits which I had there laid down, are not wholly destroyed by the grounds which may be alleged against it." This remark found its full confirmation in Eegnault's Mesearches, published some years a,ftervvards, on the specific heats of gases, which actually led to the result that these earlier, observations were inaccurate, and that the specific heat of permanent gases is not visibly dependent on the pressure. § 5. Decker's Objections. Another most energetic attack on the author's theory was made in 1858, by Professor G. Decher, in a paper " On the Nature of Heat," published in Dingier 's Polytechnischer Journal, Vol. 148, pp. 1, 81, 161, 241. He characterizes the author's mathematics, in the first half of his paper of 1850, and in another paper of 1854, as an abuse of analysis, and bungling nonsense ; he quotes the equations and principles there cited with single or double notes of admiration, and finally, after proving completely to his own satisfaction that the results are untenable, concludes thus: "These then are the data on which the fundamental principles of the new theory of Heat should rest, and by which its agreement with ex- perience should be proved ; they shew in the clearest light that the celebrated work of Herr Clausius, on which he him- self and other physicists have built £(,s on a secure foundation, is nothing more than a rotten nut, which looks well from the outside, but in reality contains nothing whatever." Of the second half of the paper of 1850, which relates to the second main principle of the theory, Herr Decher ob- serves (p. 163), that having mastered the first half, he saw no inducement to consider the second any further. On examining more closely the objections raised by Herr Decher against the author's mathematical investigation, it is seen that they are due to the fact that he has not understood 22—2 340 ON THE MECHANICAL THEORY OF HEAT. the differential equations there formed, which, though not generally int^grable, become so as soon as one further rela- tion is assumed to exist among the variables. In spite of all •which the author has said, he has throughout treated the quantities to which these equations refer, viz. the quantities of heat taken in by a body in passing from a given initial condition to its present condition, as mere functions of the variables which determine the condition of the body. After quoting the author's equation for gases, viz. dt\dv) dvKdt) y '^ '' •where A is the heat-equivalent of the unit of work, i.e. the reciprocal of E, he remarks, page 243: "In equation (1) ( -^1 and (-5^) are fully determined as the differential co- efficients of a known function of v and t, viz- Q, taken ac- cording to V and t respectively as sole variables ; and in whatever •way this function may be formed, aud whatever relation may be supposed to exist between v and t, the right side of the equation must always equal zero." This incorrect conception, thus formed by a professed mathematician, convinced the author that the meaning and treatment of this kind of differential equation, although long before established by Monge, was not so generally known as he had supposed; accordingly in his reply*, after a brief notice of some other points raised by Decher, he treated the subject more fully, giving a mathematical explanation, which seemed to him sufficient to obviate any such misunderstand- ings in future. This was afterwards prefixed to the collection of the author's papers as a mathematical introduction ; and the essential part of it has been imported into the mathe- matical introduction to the present work. § 6. Fundamental Principle on which the Author^s Proof of the Second Main Priticiple rests. The more recent objections to the author's theory, and the departure from his views in more recent treatises, chiefly • Dinglei's PolyUchnucher Journal, Vol. ci. p. 29, DISCUSSIONS ON THE THEOET. 341 refer to "his method of proving tte second main principle of the theory. This proof rests, as shewn in Chapter III., on the following fundamental principle: — Heat cannot of itself (or without compensation) pass from a colder to a hotter body. This fundamental principle has been very variously re- ceived by the scientific public. Some appear to consider it so self-evident that it is needless to state it as a specific principle, whilst others on the contrary doubt its correctness. § 7. Zewner's first Treatment of the Subject. The first of the two modes of viewing the question men- tioned in the last section appears in Zeuner^s valuable paper of 1860 " On the Foundations of the Mechanical Theory of Heat." Here Zeuner gives the author's proof of the second main principle essentially in the same form as it has also been given by Reech*. The two differ only in one point. Reech gives the principle, that heat cannot of itself pass from a colder to a hotter body, expressly as a fundamental principle laid down by the author, and bases his proof upon it. Zeuner on the contrary does not mention this principle at all : he shews that if for any two bodies the second main principle of the theory did not hold, then by means of two cychcal processes performed with these two bodies in opposite directions, heat could be made to pass from a colder to a hotter body without any other special change, and he then goes on " as we may repeat both pro- cesses as often as we please, using the two bodies alternately in the way described, it would follow that we might, with the aid of nothing and without using either work or heat, continually transfer heat from a body of lower to one of higher temperature, which is an absurdity." Few readers would probably assent to the opinion that the impossibility of transferring heat from a colder to a hotter body is so self-evident, as is here indicated by the short remark "which is an absurdity." Taking the facts of conduction, and of radiation under ordinary circumstances, we may undoubtedly say that this impossibility is established by daily experience. But even with radiation the question * Secapitulatian tris-succincte des recherches algebriqties faites sur la theorie des ejfects m^caniques de la chaleur par diffSrents auteurs : Jmirn. de Limville, Ser. n. Vol. i. p. 58. Sii ON THE MECHAN'ICAL THEORY OF HEAT. arises, whether it is not possible to concentrate the rays of heat artificially by means of mirrors or burning-glasses, so as to produce a higher temperature than that of the radiating , bodies, and thus to effect the passage of the heat into a hotter body. The author has, therefore, thought it necessary to treat this question in a special paper, the contents of which are given in Chapter XII. Matters are still more complicated in cases when heat is transformed into work, and vice versi, whether this be by effects such as .those of friction, resistance of the air, and electrical resistances, or whether by the fact that one or more bodies suffer such changes of condition, as are connected partly with positive and partly with negative work, both internal and external. For by such changes heat, to use the common expression, becomes latent or free, as the case maybe; and this heat the variable bodies may draw from or impart to other bodies of different temperatures. If for all such cases, however complicated the processes may be, it is maintained that without some other permanent change, which may be looked upon as a compensation, heat can never pass from a colder to a hotter body, it would seem that this principle ought not to be treated as one altogether Self-evident, but rather as a newly-propounded fundamental principle, on whose acceptance or non-acceptance the validity of the proof depends. § 8. Zeuner's later Treatment of the Subject. The mode of expression employed by Zeuner was criticized by the author on the grounds stated in the last section, in a paper published in 1863. In the second edition of his book, published in 1866, Zeuner has therefore struck out another way of proving the second main principle. Assum- ing the condition of the body to be determined by the pressure p and volume v, he forms, for the quantity of heat d Q, taken in by the body during an indefinitely small varia- tion, the differential equation dQ = A{Xdp+7dv).... (2), where X and Y are functions of p and v, and A is the heat- equivalent of work. This equation, as is well known, cannot DISCUSSIONS ON THE THEOET. 343 be integrated so long as p and v are independent variables. He then proceeds (p. 41) : "But let /S be a new functioti of jo and v, the form of ■which may be taken for the present to be known as little as. that of X and 3^, but to which we will give a signification, which will appear immediately from what follows. Multiply- ing and dividing the right-hand side of the equation by B, we have dQ = ^/S&2> + 1^^1)1 (3). We may now choose S, so that the expression in brackets is a perfect differential ; in other words, so that -„ may be the integrating factor, or B the integrating divisor, of the expres- sion within brackets of equation (2)." From this it follows that in the following equation derived from (3), the whole right-hand side is a perfect differential, and there- fore for a cyclical process we must have ■(4), / f-o (* In this way Zeuner arrives at an equation similar to equation (7) of Chapter IV., viz. {dq_ f = 0. T The resemblance, however, is merely external. The essence of this latter equation consists in this, that t is a function of temperature only, and further a function which is inde- pendent of the nature of the body, and is therefore the same for all bodies. Zeuner's quantity 8, on the contrary, is a function of both the variables, f and v, on which the bodies' condition depends ; and further, since the functions X and Y, in equation (2), are different for different bodies, it must be true of S also that it may be different for different bodies. So long as this holds with regard to B, equation (5) has done nothing for the proof of the second main principle ; 344 ON THE MECHANICAL THEORY OF HEAT. since it is self-evident that there must in general be an integrating factor, which may be denoted by -q, and by which the expression within brackets in equation (2) may be converted into a complete differential. Accordingly, in Zeuner's proof, as he himself concludes, everything depends on the fact that iS* is a function of temperature only, and a function which is the same for all bodies, so that it may be taken as the true measure of the temperature. For this purpose he supposes a body to undergo different variations, which are such that the body takes in heat whilst 8 has one constant value, and gives out heat whilst 8 has another constant value ; and which together make up a cycli- . cal process, shewing a gain or loss of heat. This procedure he compares with the lifting or dropping of a weight from one level to another, and with the corresponding mechanical work ; and he proceeds (p. 68) : "A further comparison leads to the interesting result that we may consider the function 8 as a length or a height, and the expression -^ as a weight ; in what follows therefore I shall call the above value the Weight of the Heat." Since a name has here been introduced for a magnitude containing 8, in which name there is nothing which relates to the body under consideration, it appears that an assumption has here been tacitly made, viz. that 8 is independent of the nature of the body, which is by no means borne out by the earlier definition. Zeuner then carries still further the comparison between the processes relating to gravity and those relating to heat, and .transfers to the case of heat some of the principles which hold for gravity ; in so doing he treats 8 as a, height, and -j^ as a weight, just as before. Then, having finally observed that the principles thus obtained are true if we take 8 to mean the temperature itself, he proceeds (p. 74) : " We are there- fore justified in taking as the basis of our further researches the hypothesis that 8 is the true measure of temperature." It appears from this that the only real foundation of the reasonings, which in bis second edition Zeuner puts forward as the basis of the second main principle, is the analogy DISCUSSIONS ON THE THEOBY. 345 between the performance of work by gravity and by heat ; and moreover that the point "which has to be proved is in part tacitly assumed, in part expressly laid down as a mere hypothesis. § 9. RanMne's Treatment of the Subject. We may now turn to those authors who have considered that the fundamental principle is not sufficiently trustworthy, or even that it is incorrect. Here we must first examine somewhat more closely the mode of treatment which, as already mentioned, Bankine considered must be substituted for that of the author. Rankine, like the author, divides the heat which must be imparted to the body, in order to raise its temperature, into two distinct parts. One of these serves to increase the heat actually existing in the body, and the other is absorbed in , work. For the latter, which comprises the heat absorbed in the internal and in the external work, Rankine uses an expression, which in his first section he derives from the hypothesis that matter consists of vortices. Into this method of reasoning we need not enter further, since the circum- stance that it rests on a particular hypothesis as to the nature of molecules and their mode of motion, makes it sufficiently clear that it must lead to the consideration of com- plicated questions, and thus leaves much room for doubt as to its trustworthiness. In the author's treatises he has based the development of his equations, not on any special views as to the molecular constitution of bodies, but only on fixed and universal principles; and thus, even if the above fact were the only one which could be alleged against Rankine's proof, the author would stiU expect his own mode of treating the subject to be finally established as the most correct. But yet more uncertain is Rankine's mode of determining the second part of the heat to be imparted, viz. that which serves to increase the heat actually existing in the body. Rankine expresses the increase of the heat within the body, when its temperature t changes by dt, simply by the product Kdt, whether the volume of the body changes at the same time or not. This quantity K, which he calls the real specific heat, he treats in his proof as a quantity inde- pendent of the specific voltime^ Any sufficient ground for this 346 ON THE MECflANlCAL THEORY OF HEAT. procedure will be sought in vain in his paper ; on the contrary, data are found which stand in direct opposition to it. In the introduction to his paper he gives, under equation (13), an expression for the real specific heat K, which contains a factor k, and of which he speaks as follows*: "The co- efficient Ic (which enters into the value of the 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 kind of substance, depending in a manner yet unknown on some circumstance in the con- stitution 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." Hence it appears to be Rankine's view that the real specific heat of the same sub- stance may be different in difi:erent states of aggregation ; and even for the assumption that it may be taken as in- variable for the same state of aggregation he produces no other ground than that there is no experimental proof to the contrary. In a later work, A Manual of the Steam Engine and other Prime Movers, 1859, Rankine speaks yet more distinctly on this point as follows (p. 307): "A change of real specific heat, sometinles considerable, often accompanies the change between any two of those conditions" (i.e. the three con- ditions of aggregation). How great a difference Rankine conceives to be possible between the real specific heats of one and the same substance in different states of aggregation, is shewn by his remark at the same place, that in the case of water the specific heat as determined by observation, which he calls the apparent specific heat, is nearly, equal to the real specific heat. Now Rankine well knew that the observed specific heat for water is twice as great as that for ice, and more than twice as great as that for steam. Since then the real specific heat for ice and steam can never be greater than the observed, but only smaller, Rankine must assume that the real specific heat of water exceeds that of ice and steam by 100 per cent, or more. If we now ask the question how on this supposition the * Phil. Mag., Ser. 4, Vol. vn. p. 10. • DISCUSSIONS ON THE THEORY. 347 increase of heat actually present in a body, whicli occurs when its temperature t increases by dt, and its volume v by dv, is to be expressed, the answer will be as follows : When the body, during its change of volume, suffers no change in its state of aggregation, we shall be able to express the increase of heat, as Rankine has done, by a simple product of the form Kdt ; but the factor K must have different values for different states of aggregation. In cases, however, where the body during its change of volume also changes its state of aggregation (e.g. the case we have treated so often, where we have a certain quantity of matter partly in the liquid and partly in the gaseous condition, and where the magnitude of these two parts changes with the change in volume, either by the evaporation of part of the liquid, or by the condensation of part of the vapour), we can then no longer express the increase of heat connected with a simultaneous change in temperature and volume by a simple product Kdt ; but must use an expression of the form Kdt + K^dv. For if the real specific heat of a substance were different in different states of aggregation, it would be necessary to con- clude that the quantity of heat existing in it must also depend on its state of aggregation ; so that equal quantities of the substance in the solid, liquid, and gaseous condition would contain different amounts of heat. Accordingly, if part of the substance change its state of aggregation without any change of temperature, there must also be a change in the quantity of heat contained in the substance as a whole. Hence it follows that Bankine by his own admission can only treat the mode in which he expresses the increase of heat, and the mode in which he uses that expression in his proof, aS being allowable for the cases in which there are no changes in the state of aggregation ; and, therefore, his proof holds for those cases only. For all cases where such changes occur the principle remains unproved ; and yet these cases are of special importance, inasmuch as it is chiefly to these that the principle has hitherto been applied. In fact we must go further, and say thtii the proof thus loses all strength even for cases where there is no change in !the state of aggregation. If Rankine .assumes that the real 348 ON THE MECHANICAL THEOET OF HEAT. specific heat may be different in different states of aggrega- tion, there seems no ground whatever left for supposing that it is invariable in the same state of aggregation. It is known that with solid and liquid bodies changes may occur in the conditions of cohesion, apart from any change in the state of aggregation. With gaseous bodies also, in addition to their great variations in volume, we have the distinction, that the more or less widely they are removed from their condensation-point the more or less closely do they follow the law of Mariotte and Gay-Lussac. How then, if changes in the state of aggregation may have an influence on the real specific heat, can we refuse to ascribe a similar, even if a smaller, influence to changes like the above? Thus the proposition, that the real specific heat is invariable in the same state of aggregation, is not only left unproved by Rankine, but, if we accept his special assumption, becomes in a high degree improbable. To this criticism on his proof, which appeared in a paper of the author's, published in 1863*, Rankine made no reply; but in a later article on the subject-f* he expressly maintained the truth of his view, firequently before stated, that the real specific heat of a body may be different in different states of aggregation; whereby the force of his proof is limited to the cases in which no change in the state of aggregation takes place. § 10. Hvriis Objection. A yet more definite attack upon the author's funda- mental principle, that heat cannot of itself pass from a colder to a hotter body, was made by Him in his work, pub- lished in 1862, Exposition ATudytique et Eocperimentale de la theorie mecanique de la chaleur, and in two subsequent articles in Cosmosl. He has there described a particular operation which gives at first sight an altogether startling result. After a reply irom the author §, he explained || his attack as having for its object only to mark an apparent objection to the principle, whilst in reality he agreed with the author; and he has expressed himself to the same effect in the second and third editions of his valuable work. * Pogg. Ann., Vol. cxx. p. 426. + Phil. Mag., Ser. 4, Vol. xxx. p. 410. J Tol. XXII. pp. 283, 413, § Vol. xxii. p. 560. || VoL xxii. p. 734. DISCUSSIONS ON THE THEORY. 349 , In spite of this it seems worth while to state here the attack and the reply, because the conception of the subject there expressed is one very near the truth, and which might easily hold under other circiimstances. An objection thus raised has a real scientific value of its own ; and when it is put in so clear and precise a light, as Hirn has done in this case by means of his skilfully-conceived operation, it can only be advantageous for science : since the fact that the apparent objection is defined and placed clearly in view will greatly facilitate the clearing up of the point. In this way we shall attain the advantage that a difficulty, which other- wise might probably lead to many misunderstandings, and necessitate repeated and long discussions, will be disposed of once and for ever. In^ thus referring once more to this question, the author is far from wishing to make the objec-. tion a ground of complaint against Hirn, but rather believes that this objection has increased the debt which the Mechani- cal Theory of Heat owes to him on other accounts. The operation alluded to, on which Hirn has based his observations, is as follows : Let there be two cylinders A and B (Fig. 32) of equal area, which are connected at the bottom by a compara- tively narrow pipe. In each of these let there be an air-tight piston; and let the piston-rods be fitted with teeth engaging on each side with the teeth of a spur wheel, so that if one piston descends the other must rise through the same distance. The whole space below the cylinders, including the connecting pipe, must thus remain invariable during the motion, because as the space diminishes in one cylinder it increases in the other by aa equal amount. First, let us suppose the piston in £ to be at the bottom, and therefore that in ^ at the top ; and let cylinder A be filled with a perfect gas of any given density and of temperature t,. Now let the piston descend ia A, and Fig. 32, S50 ON THE MECHANICAL THEORY OF HEAT. rise in B, so that the gas is gradually driven out of A into B. The connecting ' pipe through which it must pass is kept at a constant temperature t^, which is higher than i„, so that the gas in passing is heated to temperature t^, and at that -temperature enters cylinder B. The walls of both cylinders, on the other hand, are non-conducting, so that within them the gas can neither receive nor give off heat, but can only receive heat from without as it passes through the pipe. To fix our ideas let the initial temperature of the gas be that of freezing, or 0°, and that of the connecting pipe 100°, the pipe being surrounded by the steam of boiling water. It is easy to see what will be the result of this operation. The first small quantity of gas which passes through the pipe will be heated from 0° to 100°, and will expand by the cor^ responding amount, i. e. ^^ of its original volume. By this means the gas which remains in A will be somewhat com- pressed, and the pressure in both the cylinders somewhat raised. The next small quantity of gas which passes through the pipe will expand in the same way, and will thereby com^ press the gas in both cylinders. Similarly each successive portion of gas will act to compress still further not only the gas left in A, but also the gas which has previously expanded in B, so that the latter will continually tend to approach its initial density. This compression causes a heating of the gas in both cylinders ; and as all the gas which enters B enters at a temperature of 100°, the subsequent temperature must rise above 100°, and this rise must be the greater, the more the gas within B is subsequently compressed. Let us now consider the state of things at the end of the (pperation, when all the gas has passed from A into B. In the topmost layer, just under the piston, will be the gas which entered first, and which, as it has suffered the greatest subsequent compression, will be the hottest. The layers below will be successively less hot down to the lowest, which will have the same temperature, 100°, which it attained in passing the pipe. For our present purpose there is no need to know the temperature of each separate layer, but only the mean temperature of the whole, which is equal to the temr perature that would exist if the temperatures in the different layers were equalized by a mixing up of the gas. This mean temperjiture will be about 120°. DISCUSSIONS ON THE THKOEY. 351 In a later article published in Cosmos, Him lias com- pleted this operation, by supposing that the gas in B is finally brought into contact with mercury at 0°, and thereby cooled back again to 0°; that it is then driven back from B to A under the same conditions as from A to B, and is therefore heated in the same manner ; that it is then again cooled by mercuTy, again driven from A to B, and so on. Thus we have a periodical operation, in which the gas is continually brought back to its original condition, and all the heat given off by the source of heat passes over to the mercury employed for cooling. Here we will not enter into this extension of the. process, but confine ourselves to the first simple operation, in which the gas is heated from 0" to a mean temperature of 120°; for this operation comprises the essence of Hirn's ob- jection. In this operation it is clear that no heat is gained or lost ; for the pressure in both cylinders is always equal, and there- fore both pistons are always pressed upwards with equal force. These forces are communicated to the wheel which gears with the piston-rods ; and thus, neglecting friction, an indefinitely small force will be sufficient to turn the wheel in one or the other direction, and thereby move one piston up and the other down. The excess of heat in the gas cannot therefore be created by external work. The process, as is easily seen, is as follows. Whilst a quan- tity of gas, which is a very small fraction of the whole, is heated and expanded in passing through the pipe, it must take in from the source sufficient heat to heat it at constant pressure. Of this, one part goes to ' increase the heat actually existing in the gas, and another part to do the work of expansion. But since the expansion of the gas within the pipe is followed by a compression of the gas within the cylinders, the same quantity of heat will be generated in the one place as is absorbed in the other. Thus that part of the heat derived from the source, which is turned into work within the pipe, appears again as heat within the cylinders; and serves to heat the gas left in A above 0°, its initial temperature, and the gas which has passed into B above 100°, the temperature at which it entered ; in other words to produce the rise in temperature already mentioned. Accordingly, without con- sidering the intermediate process, we may say that all the 352 ON THE MECHANICAti THEORY OF HEAT. heat, which the gas contains at the end of the operation more than at the beginning, comes from the source of heat attached to the connecting-pipe. Hence we have the sin- gular result, that by means of a body whose temperature is 100°, i.e. the steam surrounding the pipe, the gas within the cylinders is heated above 100°, or, looking only to the mean temperature, to 120°. Here then, is a contradiction of the fundamental principle that heat cannot of itself pass from a colder to a hotter body, inasmuch as the heat imparted by the steam to the gas has passed from a body at 100° to a body at 120°. One circumstance however has been forgotten. If the gas had had an initial temperature of 100° or more, and had then been raised to a still higher temperature by steam, whose temperature was only 100°, this would no doubt be a contradiction of the fundamental principle. But this is not the real state of things. In order that the gas may be above 100° at the end of the operation, it must necessarily be below 100° at the beginning. In our example, in which the final temperature is 120°, the initial is 0°, The heat, which the steam has imparted to the gas, has therefore served in part to heat it from 0° to 100°, and in part to raise it from 100° to 120°. But the fundamental principle refers only to the temperatures of the bodies between which heat passes, as they are at the exact moment of the passage, and not as they are at any subsequent time. Accordingly we must conceive the passage of heat in this operation to take place as follows. The one part of the heat given off by the steam has passed into the gas, whilst its temperature was still below 100°, and has therefore passed into a colder body ; and only the other part, which has served to heat the gas beyond 100°, has passed into a hotter body. If then we compare this with the funda- mental principle, which says that, when heat passes from a colder to a hotter body, without any transformation of work into heat or any change in molecular arrangement, then of necessity there must be in the same operation a passage of heat from a hotter to a colder body, we easily see that there is complete agreement between them. The peculiarity in Him's operation is only this, that there are not two differ- ent bodies concerned, of which one is colder and the other hotter than the source of heat ; but one and the same body^ DISCUSSIOKS ON THE THEORY. 353 the gas, takes in one part of the operation the place of the colder, and in another part that of the hotter body. This in- volves no departure from the principle, but is only one special case out of the many which may occur. Dupre has raised similar objections against the fundamental principle ; but as there is nothing in them essentially new, they will not here be entered upon. § 11. Wands Objections. Some years later Th. Wand treated of the same principle in a paper entitled "Kritische Darstellung des zweiten Satzes der Mechanische Warmetheorie*." He gives his conclusions in the three propositions following: (1) "The second prin- ciple of the mechanical theory of heat, i.e. the impossibility of a passage of heat to a higher temperature without a conver- sion into work or a corresponding passage of heat to a lower temperature, is false." (2) "The deductions from this prin- ciple are only approximate empirical truths, which hold only so far as they are established by experiment." (3) "For technical calculations the principle may be taken as correct, since experiments on the substances used for the generation of work and of cold shew a very close agreement with it." The placing of such propositions side by side seem.i in itself a doubtful measure. If a principle has been found to agree with fact in so many cases, as to compel us to say that for technical calculations it may be taken as correct, it seems dangerous to conclude nevertheless that it is false, in the face of the probable supposition that the apparent objections which yet remain would be cleared up by closer examination. The following appear to be the chief grounds on which Wand bases his rejection of the principle; excluding those which relate to internal work and electrical phenomena, be- cause these subjects are not here treated of. " If we suppose," he says on p. 314, " that in the bring- ing of a certain quantity of heat from a lower to a higher temperature a certain quantity. of work must of necessity be destroyed, it follows that, if the same quantity of heat falls from a higher to a lower temperature, the same quantity of work must re-appear. Now suppose that this fall takes • Karl's Repertorium der Bxjtenmental-Phynk, Yol. iv. pp. 281, 369. c. 23 354 ON THE MECHANICAL THEOET OF EEAT. place by simple conduction or by a non-reversible cyclical process. Then the above is not true, because the falling of heat by conduction goes on without any other change what- ever. Therefore for equalizations of temperature by simple conduction there is nothing equivalent to the second prin- ciple ; and this from the logical point of view is one of the weakest points in that principle, and leads to much subse- quent inconvenience." The circumstance here mentioned, that compensation is required only in the passage of heat to a higher temperature, and not to a lower, has been frequently stated above ; and in Chapter X. is expressed in the general form, that negative transformations cannot take place with- out positive, but that positive transformations can take place without negative. From this circumstance the second main principle becomes doubtless less simple in form than the first, but it would be hard to shew that it is logically im- perfect. The inconveniences mentioned by Wand iri the above paragraph he arrives at by the following considerations. He supposes a simple cyclical process to be carried out, during which the two bodies between which the heat passes, and which he calls the heating and cooling body, have tempera- tures which are close to 0°, and difier from each other by an indefinitely small quantity, which he calls dt. For this symbol, which will appear with another signification in the analysis which follows, we will substitute h, and will call the temperatures of the two bodies, reckoned from freezing-point, and S respectively. Further, Wand supposes the cyclical process to be so arranged, that one unit of heat passes over from the hotter to the colder body, and therefore that the quantity of heat ^^ is transformed into work. He then pro- ceeds : " The process being ended, I will heat the whole ap- paratus, comprising both the heating and the cooling body, by 100°. The difference of temperature between the two bodies will remain unaltered. If we now wish to destroy the heat thus obtained by means of the reverse cyclical process, we must take from the colder body the heat f^f. The colder body thus loses the heat ^^, and giv.es it up to the hotter body ; and if by the reverse process all is cooled back again to the initial temperature 0", the initial condition of things DISCUSSIONS ON THE THEOKT. 355 is restored. There is no work performed or consumed, and yet a passage of heat has taken place from the second body, which has remained the colder, to the first which has remained the hotter throughout the process. This however is no refu- tation of the second principle. For to obtain this result there must be a continuous succession of alternate heatings and coolings of the apparatus, i.e. heat must pass from a hotter to a colder body; but this passage takes place by conduction, for which there is no equivalent. Hence it follows from the process here described, that with regard to the distribution of the heat it is by no means the same thing, whether we do nothing at all, or carry out a compound cyclical process as here described." We have here the case of two opposite cyclical processes carried out at different temperatures, in which the work done and the work consumed cancel each other, but more heat passes from the hotter to the colder body than vice vers4 ; and "Wand holds that the passage of the surplus heat from the colder to the hotter body has taken place without compensation. He has however neglected certain differences of temperature, which occur in this somewhat complicated operation. For after the first process, in which the hotter body has given off and the colder body taken in heat, he heats the whole apparatus and the two bodies by 100°; and he cools them by 100° after the second process, in which the colder body has given off heat and the hotter taken it in. But in giving off and taking in heat the two bodies alter their temperature somewhat, and the reservoirs of heat, which perform the heating and cooling, do not therefore take back the heat during the cooling at the same tempera- ture as they gave it out during the heating. Hence arise passages of heat of which Wand has taken no account. These differences of temperature are of course very small, since the two bodies must be assumed so large, that the varia- tions of temperature produced in them by the cyclical process may remain small compared to the difference of the original temperatures. But then the quantities of heat, which the bodies take from and give back to the reservoirs during their heating and cooling, are also very large; and since to determine the heat which has passed we must multiply the differences of temperature by the actual 23—3 356 OK THE MECHANICAL THEORY OF HEAT. quantities of heat, we arrive at magnitudes wliicli are quite large enough to compensate for the surplus heat which has passed between the bodies. To prove this point we will make the calculation itself. First, with regard to the actual surplus heat which has passed between the bodies, since the temperatures are and B, and the quantity of heat equals -^i^ , this has the equivalence value 2^g ( 273 + 8 " 273) °^' ^^S^^^^^^S terms of a higher order than the first with regard to S, — ^=^ S. We have now to determine the equivalence value of the passages of heat, which take place during the heating and coohng of the body by 100°. By Chapter IV., § 5, we must divide the element of heat taken in by one of the two bodies from a reservoir of heat (reckoning heat given off as negative heat taken in) by the absolute temperature which the body has at the moment, and we must then form the negative integral for the heating and cooling. Let M be the mass of each body, and C its specific heat, which we suppose constant ; then the quantity of heat, which it takes in during a rise of temperature dt, equals MCdt, and this we shall take as expressing the element of heat. If for convenience we put ^~MG ■(6). the element will be expressed by - dt. We must consider e MG as very large, and therefore e as very small, so much so as to be small even in comparison with the small difiference of temperature h. If we now take the first cyclical process, the colder body has, to begin with, the temperature 0, and the hotter the temperature S. During the process the former takes in the s quantity of heat 1, and the latter loses the quantity 1 + ^=^ . These must be divided by MC, or multiplied by e, to obtain DISCUSSIONS ON THE THEORY. 357 the changes of temperature which they produce in the bodies ; thus at the end of the process 'wie colder body has the temperature e, and the hotter the temperature From these temperatures both bodies are now to be heated by 100°. The negative integral relating to the heating of "the colder body is /■lOO+e 1 ^"~i. 273+T If we put T = t — e, then /■lOO A .100 1 h 273 H K 273 + T + e* Neglecting higher terms in the expansion of e, we have 1 1 6 273 + T + e~273+T (273+t)'" whence ^ = --J^ 273T-r+Jo (273T^^ ^'^- The negative integral relating to the heating of the hotter body is rioo+!-(i+4)' 1 f" dt e .(,,i-). 273 + r Whence we obtain in the same way as before „_ 1 /•""+» dr / B Ui°«+ ° cZt . ^ 7L 273T^'"l^"^273JJs (273 + Tf ^''^- During the second cyclical process the colder body gives 373 off a quantity of heat = gs^ , and the hotTter body receives a 358 ON THE MECHANICAL THEOBY OP HEAT. quantity ^r^^ + ^f^ . The temperatures of the two bodies, after the second cyclical process, become respectively 100 + e-||. = 100-|«-«e, 100+S-(l+4)e+(g|+4)e = 100 + S + 100 273^' From these temperatures both bodies are cooled down by 100°. The negative integral relating to this cooling is for the colder body f 1"" 1 Ann lOI 1 — ~\ 100 ^ = I 100 ^ ■'"o-are" 273+^ •'"are' 273+i 273+ « whence as before we have ^~eJo 273 + T + 273J0 (273 + t)^ ^''^• For the cooling of the hotter body we have similarly 1 |-ioo+8 dr _ 100 /•"°+ » dr , ^~ejs 273 +T 273 is (273 + t7 ••••^^"^• By adding together A, B, G, D, we obtain the equivalence value of all the transferences of heat during the heating and cooling. In this addition the integrals which have the factor - cancel each other, and two of the others may be combined together. Whence we have A+JB + G + I)-- 373 /•I"" dr ''27BJo (273 + T)'' V273^273/'J8 (273 + t)' ^^^> DISCUSSIONS ON THE THEORY. 359 Performing the iategration, the right-hand side becomes 1 373 /_ _1_ 1_\ _ /373 _S_\ / 273 V 373 "*" 27ii) V273 "^ 273/ [ 373 + S "^ 273 + S. If we expand the second product in terms of S to the first order, most of the resulting- terms cancel each other, and the expression for the equivalence value of the transferences of heat during the heating and cooling becomes finally 100 s (273)' *• This expression fulfils the condition of being equal and opposite to the equivalence value of the surplus heat actually transferred between the two bodies. This transfer therefore is not uncompensated, but fully compensated as the second principle- requires. We thus see that the operation suggested by Wand does not give the smallest, ground for objecting to the principle. Another objection is drawn by Wand from the follow- ing considerations. He proposes the question whether the principle can be derived by mechanical reasoning from the ideas which we can form of the nature and action of heat. With this object he first applies the hypothesis main- tained by the author and others as to the molecular motion of gaseous bodies, and finds that this does actually lead to the principle in question. He then says that it ,is not sufficient to prove that one particular hypothesis leads to the principle, bat that all possible mechanical hypo- theses on the nature of heat must be shewn to lead to it. Accordingly as a second example he takes another hypo- thesis, which he conceives specially adapted to represent the phenomena of expansion, and of the increase of pressure by heat. On this hypothesis a row of elastic balls, any two of which are connected by an elastic spring, vibrate in such a way that all of them are in similar phases. From this hypothesis he arrives at an equation different from that which he has chosen as the criterion of the second principle, and then draws the conclusion, "the second principle cannot therefore be universally derived from the principles of mechanics." But upon such a conclusion the question may S60 ON THE MECHANICAL THEOET OF HEAT. be asked, -whether in reality the hypothetical motion, which he supposes, agrees with the actual motion, which we call heat, in such a way that the same equations must hold for both. So long as this is not proved the conclusion cannot be considered to be made out. Finally, Wand considers the process which Occurs in nature, when in the growth of plants, under the influence of the sun's rays of light and heat, carbonic acid and water are absorbed, and oxygen liberated ; whilst the organic substances thus formed, if afterwards burnt or serving as nourishment to animals, unite themselves again with oxygen to form car- bonic acid and water, and thereby generate heat. This transformation of the sun's heat he considers to be in flat contradiction to the second principle. To the analysis which he gives many objections might be taken; but the author considers that a process in which so much is still unknown, as that of the growth of plants under the influence of the sun, is altogether unfit to be used as a proof either for or against the principle in question. § 12. Tait's Objections. Finally the author has to mention certain objections re- cently raised against his theory by Tait ; objections which have surprised him equally by their substance and by their form. In an article which appeared in 1872 on the History of the Mechanical Theory of Heat*, the author had observed, that Tait's work, A sketch of Thermodynamics, no doubt owed its existence chiefly to the wish of claiming the Mechanical Theory of Heat as far as possible for the English nation ;-^a supposition for which the clearest grounds can be adduced. Further on in the article he had observed that Tait had ascribed to Sir William Thomson a formula due to the author, and had quoted it as given by Thomson in & paper which contained neither the formula itself nor any- thing equivalent to it. The author expected that Mr Tait, in answering this article, would specially address himself to these two points, of which the latter particularly required clearing up. A reply aj)peared indeed -f, and one written in • ^ogg. Ann., Vol. cxlv. p. 132. t Phil. Mag. , Series 4, Vol. xlui. DISCUSSIONS ON THE THEORY. 361 p. very acrimonious tone ; but to the author's surprise these two points were nowljere touched upon, a different turn being given to the whole matter. For whilst in the work previously alluded to the, author's researches on the Mechani- cal Theory of Heat, even if in his own view they were not put in their right relation to those of English writers, were, yet described at great length and with a general recognition of their merit, their correctness was here at once assailed, by the declaration that the fundamental principle, that heat cannot of itself pass from a colder to a hotter body, is untrue. To prove this two phenomena relating to electric currents are adduced. But in a rej oinder published shortly afterwards * the author was easily able to prove that these phenomena in no way contradict the principle, and that one of them is even so evidently in accordance with it, that it may serve as an example specially adapted to illustrate and establish it. As electrical phenomena are not here treated of, this is not the place to enter further into this subject. Tait further observed in his reply, that by the author's introduction of what he namdd Internal Work and Disgrega- tion, he had done a serious injury to science, offering however nothing to support this, beyond the brief remark : " In our present ignorance of the nature of matter such ideas can do only harm." What Tait has to object to the conception of internal work, it is difficult to understand. In his first paper on the Mechanical Theory of Heat the author divided the work during the body's change of condition into External and Internal Work, and shewed that those two quantities of work differed essentially in their mode of action. Since that time this distinction has been similarly made by all writers, so far as he is aware, who have treated of the Mecha- nical Theory of Heat. As regards the method (which will be described on a future occasion) of calculating the combined internal and external work, and the conception of disgregation introduced, by the author with this object, purely mechanical investiga- tions have recently led to an equation, exactly corresponding with that which the author proposed in the science of heat, and in which the disgregation makes its appearance. If * Phil. Mag., Series 4, Yol. xlhi. 36^ ON THE MECHANICAL THEORY OF HEAT. these investigations cannot yet be considered as complete, they nevertheless shew, at least in the author's opinion, that the nature of things requires the introduction of this con- ception. He therefore leaves the objections of Tait, with as much confidence as those of Holtzmann, Decher, and others, to the good judgment of the reader *. * For a rejoinder by Prof. Tait to these observations, see Sketch of Thermodynamics, 2nd edition, 1877, p. xv. Finis. APPENDIX I. ON THE THERMO-ELASTIC PROPERTIES OP SOLIDS. Sir William Thomson -was the first who examined the thermo- elastic properties of elastic solids. Instead of abstracting his investigation {Quarterii/ Mathematical Jmirnal, 1855) it may be well to present the subject as an illustration of the method of treatment by the Adiabatic Function. Consider any homogeneously-strained elastic solid. To define the state of the body as to strain six quantities must be specified, say 11, V, w, X, y, z: these are generally the extensions along three rectangular axes, and the shearing strains about them, each relative to a defined standard temperature and a state when the body is free from stress. The work done by external forces when the strains change by small variations may always be expressed in the form IJJhi+7bv+ ...)x volume of the solid, because the conditions of strain are homogeneous. U, V are the stresses in the solid : each is a function oi uv and of the temperature, and is determined when these are known.. Let 6 denote the temperature (where 6 is to be regarded merely as the name of a temperature, and the question of how temperatures are to be measured is not prejudged). Amongst other conditions under which the strains of the body may be varied, there are two which we must consider. First, suppose that the temperature is maintained constant; or that the change is efiected isothermally. Then 6 is constant. Secondly, suppose that' the variation is efiected under such con- ditions that no heat is allowed to pass into or to leave the body ; or that the change is efiected adidbatically. In the latter' case 6,u,v, are connected by a relation involving a para- meter which is always constant when heat does not pass into 364 ON THE MECHANICAL THEORY OF HEAT. or out of the solid : this parameter is called the adiabatic June-, iion. We have now fourteen quantities relating to the body, viz. six elements of strain, six of stress, the quantity 6 which defines the temperature, and the parameter ^ the constancy of which imposes the adiabatic condition. Any seven of these may be chosen as independent variables. Let the body now undergo Carnot's four operations as fol- lows : — 1°. Let the stresses and strains vary slightly under the sole condition that the temperature does not change. Let the conse- quent increase of <^ be 8i^. Heat will be absorbed or given out, and, since the variations are small, the quantity will be propor- tional to 8(^, say f{e,u, v,w, .'.....) S being added because the variation was per- formed under the condition that ^ was constant. By properly choosing the parameter (1); "Work done by external forces = 86 x Si^ (2). We must now examine more particularly the variations in the stresses and strains. Denote the values of U, V, ....... u,v, by different suffixes for the four operations. The work done by the external forces in these operations is respectively and the sum of these is equal to S(^Sd. Hence a variety of important relations may be obtained. Let all the strains but one be constant : then we have M,= w. du ,, «3 = -u^ du ,. M,= = «a du , -d4>'^'^' with similar equations for U,, &c. Hence the Work done in the successive operations is, , 366 ON THE MECHANICAL THEOEY OF HEAT, __ dll ,. „ dU ,. jj^^^de^u, + ^d6 ^ „ dU J. „ dU^, vi__ ,_ae. Adding these, the total "Work done becomes du dU du dU\ ( du du du dU\ joji \-d^M^re-d^)''^^^^' the differentiations being performed -when u and Zf are expressed as functions of 6, and the five other stresses. But from (2) the "Work done = d6x d<^. Hence it follows generally (using the well-known theorem as to Jacobians) that dtf) dO d d9 _^ ,„. dUd^'d^dU' ^ '' d^~ d4> ' du du de_ de4>_d d^ du 1 duV ~du^du~dU dF~~'de^~W dU dU d$ de _ d dU _ 1 _ djju dU~ dU~ du dO^ d^" 'df' du du 368 ON THE MECHANICAL THEOEY OF HEAT, These relations are true provided each of the other strains, or else its corresponding stress, is constant. Take the last of these for interpretations When 6 is con- stant we have bj (1), Heat absorbed in any change (or dq) = 6d. Hence de dU 1 dq \ ' e dU' or, by the fourth' relation, d^ dd ' 1 dq ddU Here -— is the coefficient of dilatation. This, under the condi- tions assumed, will, of course, be different according as the other stresses or other strains are maintained constant. In the case of a bar of india-rubber stretched by a variable weight, all the elements of stress but one vanish or are constant. If the stress be somewhat considerable it is found that -r^ is nega- da tive. It follows that increase of weight will liberate heat in ' the indiarrubber. But the same will not be true if the stretching weight be nil or very small, nor again if the periphery of the bar is held so that it cannot contract transversely as the weight extends it longitudinally, unless (which is Improbable) it should be found that in these cases the coefficient of dilatation is negative. APPENDIX II. ON CAPILLARITY. The equations obtained in Appendix I. may also be applied to the eqailibiium of the surface film of a liquid in contact with its owu vapour. This subject is generally known under the name of Capillarity, from its having first been studied in connection with Capillary tubes (Maxwell, On Heat, 1871, p. 263). Thus let the body considered be the film of fluid at the surface of a fluid, and let it be so small in volume that its capacity of heat in virtue of its volume may be neglected. Let T be the surface tension, and __ldg *''' dO" ^~ edS (by Equatiion (1*), Appendix I.), where q is heat absorbed. ' dT If then -j^ is negative, as- is usually the case, extension of dd surface means absorption of heat. The question of the equilibrium of vapour at a curved sur- face of liquid has been treated by Sir Wm. Thomson. The following is mainly taken from his paper {Froc. Royal Society of -Edinburgh, 1870, Vol. vii., p. 63). In a closed vessel containing only liquid and its vapour, all at one temperature, the liq'iiid' restsj with its free surface raised or depressed in capillary tubes and in the neighbourhood of the solid boundary, in permanent equilibrium according to the same law of relation- between curvature and pressure as in vessels open to the air. The permanence of this equilibrium implies physical equi- librium between' the liquid and the vapour in contact with it at all parts of its surface. But the pressure of the vapour at difierent levels difiers according to hydrostatic law. Hence the pressure of saturated vapour in contact with »■ . liquid differs c. 24 370 ON THE MECHANICAL THEORY OF HEAT. according to the curvature of the bounding surface, being less ■when the liquid is concave, and greater when it is convex. And detached portions of the liquid iu separate vessels, all enclosed in one containing vessel, cannot remain permanently with their free surfaces in any other relative positions than those they would occupy if there were hydrostatic communication of pressure between the portions of liquid in the several vessels. There must be evaporation from those surfaces which are too high, and con- densation into the liquid at those surfaces which are too low — a process which goes on until hydrostratic equilibrium, as if with free communication of pressure from vegsel to vessel, is attained. Thus, for example, if there are two large open vessels of water, one considerably abpve the other in level, and if the temperature of the surrounding matter is kept rigorously constant, the liquid in the higher vessel will gradually evaporate until it is all gone and condensed into the lower vessel. Or yre may suppose a capillary tube, with a small quantity of liquid occupying it from its bottom up to a certain level, to be placed upright in the middle of a quantity of the same liquid with a wide free surface, and con- tained in a hermetically sealed exhausted receiver. The vapour wUl gradually become condensed into the liquid in the capillary tube, until the level of the liquid in it is the same as it would be were the lower end of the tube in hydrostatic communication with the large mass of liquid. The effect would be that in a very short time liquid would visibly rise in the capillary tube, and that, pro- vided care were taken to maintain, the equality of temperature all over the surface of the hermetically sealed vessel, the liquid in the capillary tube would soon take very nearly the same level as it would have were its lower end open ; sinking to this level if the capillary tube were in the beginning filled too full, or rising to it if there is not enough of the liquid in it at first to fulfil ''the con- dition of equilibrium. The following shews precisely the relations between curva- tures, differences of level, and differences of pressure with which _ we are concerned. Let jo be the pressure of equilibrium above the curved surface in the tube, rr' the principal radii of curvature of that surface, and T the surface tension : then by the principles of Capillarity the upward drag on the surface, due to the tension, is ^ ( - + -, ) . ithin the liquid Hence the pressure within the liquid, immediately below the surface is ON CAPILLARITY. 371 Let V be the equilibrium pressure of th.e vapour at the plane sur- face of the liquid, p the density of the liquid, tr the density of the vapour, h the height at which the liquid stands in the tube above the plane surface : then clearly -h^^^y +h (1). But also IT =p + ha- (2). Therefore TT (p - 0-) =^ (p - ok of the sort in English. It eluci- dates instructively the methods of a teacher of the very highest rank. ■ We most cordially tecomviend, it to, (ill our readers J' — Mechanics' Magazine. 5,000. 3. 79 A SCIENTIFIC CATALOGUE. Bayma.— THE elements of MOLECULAR MECHA- NICS. By Joseph Bayma, S.J., Professor of PhUosophy, Sto.nyhurst Cmlege. Demy 8vo. cjotiij; loj. bd, \ \\ i'. '.{ v'Boole.— Worts by G. BooEe, D.C.L, F.R.S., Professor of Mathematics in the Queen's University, Ireland !-^ > • •-■ . A TREATISE ON DIFFERENTIAL EQUATIONS. Third. Edition. Edited by I. Todhuntek. Crown 8vo. cloth. 14J. "yi treatise incomparably superior to anyothef^eletnentArj) book on the subject with which we are acquainted" — Philosophical Magazine. A TREATISE ON DIFFERJENTIAL EQtJATldlSfi' Supple- mentary Volume. Edited by I. TodhunteR. Crown 8vo. cloth. 8j-. dd. THE CALCULUS OF FINITE DIFFERENCES. CrOwn 8vo. cloth. \Qs, 6d. New Edition revised. ■■■,rr r Cheyne. — AN ELEMENTARY TREATISE ON THE PLANETARY THEORY. With a Collection of Problems. By C. H. H. Cheyne, M.A., F.R.A.S. Second.Edition. Crown 8vo. cloth. 6s. 6d. ' . : ; "^ Clifford.— THE ELEMENTS OF DYNAMIC. An Intrpduction to the study of Motion and Rest in Solid and Fluid Bodies. By W. K. Clifford, F.R.S., Professor of Applied Mathematics and Mechanics at University College, London. Part I. — Kinematic. Crown 8yo. "Js. 6a. . _ ,. , "^Cuxnilling.— AN INTRODUCTION TQ THE THEORY QF ELEpTRICITY. With numerous Examples. By L'inn^us CuJiMlNG, M.A., Assistant Master at Rugby School. Crown 8vo. 8j. 6d. : ■■ Clithbertson.— EUCLIDIAN. GEOMETRY. By F. Cuth- BERTSON, M.A., Head Mathematical Master of the City of London School, Extra fcap, 8vo. , 4?. 6j/.; , . ■■ ^ '^ Ferrers.— Works by the Rev. N.M. Ferrers, M, A. , F.R. S. ,]^ellow and Tutor of Gonville and Caius College, Cambridge ; — AN ELEMENTARY TREATISE ON TRILINEAR CO-ORDI- NATES, the Method of Reciprocal Polar^, aiid the Theory 6f Projectors. Third Edition, revised. Crown 8vo. ^s. 6d. SPHERICAL HARMONICS AND SUBJECTS CONNECTED WITH THEM. Crown 8vo. Js. 6d.- MATHEMATld^. ^ Frost. — ^Works l?y Percival- Frost, M.A., late Fellbw of St. John's College, Mathematical Lecturer of King's Coll. eambridge:— THE FIRST THREE SECTIONS OF NEWTOiS^'S PRIN- ClPIA. With Notes and Illustrations. Also a Collection ,of Problems, principally intended as Examples of Newton's- Methods. Third Edition. 8vo. cloth. I2,s. AN ELEMENTARY TREATISE ON CURVE TRACING. 8vo. 12s. SOLID GEOMETRY. Being a New Editioh, revised and enlarged, of the Treatise by Frost and Wolstenholme. Vol. I. 8vo. i6j. Godfray. — Works by.IJUGH Godfray, M.A., -Mathematical Lecturer at Pembroke .College, Cambridge : — A TREATISE ON ASTRONOMY, for the Use of CoUegfes and Schools. 8vo. cloth. 12s. 6cl. '^ It is a ■/vorkmg book," says the puardian, ^'taking Astronomy in its prober place injhe Mathematical Sciences. . . . It is'a book which is not likely to be got up wninielligeutly." " AN ELEMENTARY TREATISE ON THE LUNAR THEORY, with a Brief Sketch of the Problem up to the time.flf Newton. Sepphd Edition, revised. Crovm 8vo. cloth. Jj. D(^. "As an elementary treatise and introduction to the subject, we think it may justly claim to supersede all former ones." — London, Edin- burgh, and Dublin Phil. Magazine. , , v'Green (George).— mathematica* papers of the LATE GEORGE GREEN, Fellow, of Gonville and Caius , College, Cambridge. Edited by N. M". Ferrers, M.A., Fellow q.nd Tutor of Gonville and Caius College. 8vo. I5r. "It has been for some time recognized, that Green's writings are amongst the most valuable ,pfafhematical productions we possess."— Athenseum. Hemrning.— AN ELEMENTARY TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS. For the Use of Colleges and Schools. By G. W. Hemming, M.A,, Fellow of St. John's College, Cambridge.! Second Editieny with Corrections and Additions. 8vo. cloth. 9J. Jackson.— GEOMETRICAL CONIC SECTIONS. An Ele- mentary Treatise in which the Cpnic Sections are defjiied as the Plane Sections of a Cone, and treated by the Method of Projections. By J.' StuarT Jackson, M.A., late Fellow of Gonville and Cams College. Crown 8vo. \s. 6d. A 2 SCIENTIFIC. CA TALQGUE. ^Kelland and Tait. — an introduction TQQUATERj NipNS. ,,With numerous Examples. By p. Kbslland, M.A., F.ii.S., and P. G. Tait, M. A., Professors in the department of Mathematics iii the University df Edinburgh. Crown' 8vo. 'js. 'dd. Kempe.— -HOW TO draw A- STRAIGHT LINE., A Lecture on Linkages. By A. B. Kempe, B. A. Illustrated., Grpwn 8vo. ij.e^/. ' ^ Merriman.— ELEMENTS OF THE METHOD OF LEAST SQUARES. By Mansfield Merriman, Professor of Civil and Mechanical Enginpeiiug, Lehigh Universityj Bethleh^nj, Pqnn., U.S.A. ■ Crown, 8vo. ' 7j.'6ii ";,,, ' ,/;' ,- ' ; .,-,',,' Morgan.— A COLLECTION OF PROBLEMS AND EXAM- PLES IN MATHEMATICS. With Answers. By H: "a; Morgan, M.A., Sadlerian and Mathematical Lecturer of Jesus College, Cambridge. Croym 8vo. cloth. 6s. 6d. v^Newton's Principia.— 4to. cVoth. 31s. 6^. ' If is a sufficient guarantee of the reliability of- this complete edition of yyra/iftfw'j Prilicipia that it has been printed for wtid under the care of Professor Sir William Thomson and Professor Slackbum, of Glasgoiv University. . , '<' •^Parkinson. — a treatise ON optics. By S. Parkin- son,. D.D., F.R.S., Fellow and Tutor of St. John's College, Cambridge. Third Edition, revised and enlarged. Crowii 8vo. cloth. loj. 6d. ' Phear.— ELEMENTARY HYDROSTATICS. With Numerous , I |Exan?ples. By J. B,jPhear, M.A., Fellow and late Assistant Tutor ofiClare Coll. Cambridge. Fourth Edition. Cr. 8vo. cloth. , 5^. to. Pirrie.— LESSONS ON rigid dVnAMICS. By the Rev. G. PiRRiE, M.A., Fellow and Tutor of Queen's College, Cambridge. Crown 8vo. bs. Puckle.— AN ELEMENTARY TREATISE ON CONIC SEC- TIONS AND ALGEBRAIC GEOMETRY. With numerous .Exij.inples and Hints for their Solution. . By G. Hale Puckle, M.A. Fouth Edition, enlarged. Crown 8vo. "js. 6d. V Rayleigh. — THE THEORY OF SOUND. By Lord Rayljeigh, F.R.S., formerly Fellow of Trinity College, Cambridge. 8vo. Vol. 1. izs. dd. ; Vol. II. izs. 6d. [Vol. III. in preparation. Reuleaux.-^THE KINEMATigS OF MACHINERY. Out- lines of a Theory of Machines, By Professor F. Reuleaux. Translated' aiid edited by A. B. W. KeNnedy, C.fi., Professor of Civil and Mechanical Engineering, University College, London. With 450 Illustrations. Royal 8yo.- 20J.' MA THEM A TICS. Routh.; — Works by Edward John Routh, M.A., F,R.S., late Fellow and Assistant Tutor of St. Peter's College, Cambridge; Examiner in the University of London : — , AN ELEMENTARY TREATISE ON THE DYNAMICS OF • THE SYSTEM OF RIGID BODIES. With numerous Examples. Third Edition, enlarged. 8vo. 21s. Stability of a given state of motion, parti- cularly STEADY MOTION. The Adams' Prize Essay for 1877. Svo. 8j: 6d. ' Tait and Steele — DYNAMICS OF A PAI^TICLE. With • numerous Examples. By Professor Tait and Mr. Steele. Fourth Edition, revised, Crown 8vo. lis. Thomson. — papers on electrostatics and mag- netism. By Professor Sir William Thomson, F.R.S. Svo. i&r. Todhunter Works by I. Todhuuter, M,A., F.R.S., of St. John's College, Cambridge : — , "Mr. Todhunter is chiefly known to students of mathematics as the author of a series of admirable mathematical text-books, which possess the rare qualities of being clear in style and absolutely free \ • from, mistakes, tyfografhical or other'}' — S^-today Review. nV A TREATISE ON SPHERICAL TRIGONOMETRY. New Editiaji, enlarged. Crown Svo. cloth, e^. 6d. PLANE CO-ORDINATE'GEOMETRY, as applied to the Straight Line and the Conic Sections/ With numerous Examples. New Edition. Crown Svo. ^s. 6d. vA. TREATISE ON THE DIFFERENTIAL CALCULUS. With numerous Examples. New Edition. Crown Svo. loj. 6d. vh TREATISE ON THE INTEGRAL CALCULUS AND ITS APPLICATIONS. With numerous Examples. New Edition, revbed and enlarged. Crown Svo. cloth. loj. 6d. EXAMPLES OF ANALYTICAL GEOMETRY OF THREE DIMENSIONS. New Edition, revised. Crown Svo. cloth. 4J. vA TREATISE ON ANALYTICAL STATICS. With numerous Examples. ' New Edition, revised and enlarged. Crown Svo. cloth. lOJ. 6d. VA HISTORY OF THE MATHEMATICAL THEORY OF PROBABILITY, from the Time of Pascal to that of Laplace. Svo. iSj. vliESEARCHES IN THE CALCULUS OF VARIATIONS, Principally on the Theory of Discontinuous Solutions ; An Essay to which the Adams' Prize was awarded in the University of Cambridge in 1871. Svo. 6s. SCIENTIFIC^ CA TALOGUE. Todhunter — cpntmued. ,.,..r >i/AHrSTORY; OF THE MATHEMATICAL THEORIES'" OP : ' ATTRACTION, and the Figure of the Earth, from the time of Newton to that of Laplace. Two vols. 8vo. '24r. " Probably no man in England ii so qualified to do justice to the ■theme as Mr.' Tblthkliiter. ' ' To all mathematicians these volumes ■will be deeply interesting, and to all succeeding investigators ^ of the highest practical tttUity. "— Athensurii. ' ' ' ' "^AN ELEMENTARY'TKEATISE ON LAPLACE'S, LAME'S, AND BESSEL'S FUNCTIONS. Crown 8vo. lor. '6^. Wilson (W. P.)— A TREATISE ON DYNAMICS. ' By W. V. Wilson, M. A., Fellow of St. John's College, Cambridge, and Professor of Mathematics in Queen's Collegfe, Belfast. 8vo. gj. 6d. J';' Wolstenhblme. — MATHEMATICAL PROBLEMS, oii Sub- jects included in the First and Second Divisions Of the Schedule of Subjects for the Cambridge Mathematical Tripos Examination. Devised and arranged by Joseph Wolstenholme, late Fellow of Christ's College, sometime' Fellow of St. John's College, and ■ Professor of Mathematics in the Royal Indian Engineering College. New Edition, greatly enlarged. Svo. iSj. Young SIMPLE PRACTICAL METHODS OF CALCU- 'LATING STRAINS ON GIRDERS, ARCHES, AND TRUSSES. With a Supplementary Essay on Economy in suspen- sion jBridges. By E. W. Young, Associate of King's College, London, and Member pi the Institution of Civil Engineers. Svo. 7^. dd. PHYSICAL SCIENCE. PHYSICAL SCIENCE. Airy (G. B.)— popular astronomy. With illustrations. By Sir G. B. AlRY, K.C.B., Astronomer Royal. New Ifdition. iSipo. cloth. 4P. 6a?. ' - Bastian. — Works, by H. Charlton Bastian, M.D., F,R.S., Professor of Pathological Anatomy in University College, London, &c. : — THE BEGINNINGS OF LIFE : Being some Account of the Nature,. Modes of Origin, and Traiisfornnations of Lower Organisms, In Two Volufties. With upwards of lOO Illustrations. Crown 8vo. 28.r. "/if is a book that cannot be ignored, and must inevitably lead lo renewed discussions and repeated observations, and through these to the establishment of truth." — A. R. Wallace in Nature, EVOLUTION AND THE ORIGIN OF LIFE. Crown 8vo. "Abounds in information oj interest to the student of biological science."- -Daily Nev/s. . ,■ ' ■ Blake.— ASTRONOMICAL MYTHS. Based on Flammarion's "The Heavens." By John F. Blake. With numerous Illustra- tions. Crown 8vo. ^. Blanford (H. F.)— RUDIMENTS OF PHYSICAL QEO- GRAPHY FOR THE USE OF INDIAN SCHOOLS. By H. F. Blanford, F.G.S. With numerous Illustrations and Glossary of Technical Terms employed. New Edition. Globe 8vo. 2s. 6d. Blanford ("W. T.)— GEOLOGY and ZOOLOGY OF ABYSSINIA. By W. T. Blanford. 8vo. 21s. Bosanquet— AN ELEMENTARY TREATISE ON MUSICAL INTERVALS AND TEMPERAMENT, With an .Account of an Enharmonic Harmonium exhibited in the Loan Collection of Scientific Instruments, South Kensington, 1876; also of an Enhar- monic Organ exhibited to the Musical Association of London, May, 1875. By R. H. Bosanquet, Fellow of St. John's College, Oxford. 8vo. 6s. Codl : ITS HISTORY AND ITS USES. By Professors GREfek, MiALL, -Thorpe, Ruckek, and Marshall, of the Yorkshire College, Leeds. With Illustrations. 8yo. 12s. 6d. . , ' ' It furnishes a very com^ehMsive treatise on the whole subject of Coal 'from the gealogicttlpchemical,- mechanical, andindMstrig4 points 'of viewt .co^cltiMn-g it/itfia chapter on the important topic known as the ' Coal Question: "-Daily News. SCIENTIFIC CATALOGUE. Cooke (Josiah P., Jun.)— FIRST PRINCIPLES OF CHEMICAL PHILOSOPPIY. By Josiah P. Cooke, Jun., Ervine Professor of Chemistry and Mineralogy in Harvard College. Third Edition, revised and corrected. Crown 8vo. I2J-. "-'ly Cooke (M. C.)— HANDBOOK OF BRITISH FUNGI, with full descriptions of all the Species, and Illustrations of the Genera. By M. C. Cooke, M.A. Two vols, crown 8 vo. 2'+?.' " Will maintain its fldce as the standard English hook, on the . , subject of which it treats, for many years to come. " — Stand,ar4.. Dawkins,— CAVE-HUNTING : Researches on the Evijience of Caves respecting the Early Inhabitants of Europe. By W. Boyd Dawkins, F.R.S., &c.. Professor of Geology at Owens College, Manchester. With Coloured Plate and WCodcuts. 8vo.< zu. " The'mass of infdrmatioH he has brought together, with the judicious use he has made of his materials, •will be found to' invest his book with much of new and singular value." — Saturday Review. Dawson (J. W.)— ACADIAN GEOLOGY. The Geologic Structure, Organic Remains, and Mineral Resources of; Nova Scotia, New Branswick, and Prince Edward Island. By John William Dawson, M:A.', LL^.D., F.R.S., F.G.S., Principal and Vice-Chancellor of M'Gill College and University, Montreal, &c. ; ^ith a Geological Map and numerous Illustrations, Third Edition, with Supplement. ;8vo. 2 1 j. i Supplement, separately,- za'. 6i^. " The book tuill doubtless fnd a place, in. the library, not 'dnly of the scientific geologist, but fUsp- of all who are desirous of the in- dustrial progress and commercial prosperity of the Acadian i>ro- vinces." — Mining Journal. , , .,, , .. Fleischer.— A SYSTEM OF .VOLUMETRIC ANALYSIS. By Dr. E. Fleischer. Translated fro;m the Second German Edition by M.' M. Pattison Muir, with; Notes and Additions. Illustrated; Crpwii 8vo. "js. 6d. . ■ Forbes.--THE transit of VENUS. By George Forbes, B.A,, Professor of Natural Philosophy in the Andersonian Univer- sity of Glasgow. With numerous Illustrations. Crown 8vo. y. 6d. Foster and Balfour. — elements of embryology By Michael Foster, M.D., F.R.S., andF. M, Balfour, M.A., Fellow of Trinity College, Cambridge. With numerous Illustra- tions. Parfl. Crown 8vo. Js. 6d. Gdlton Works by Francis Galton, F.R.S. :— METEOROGRAPHICA, or Methods of' Mapping the Weather. Illustrated by upwards of 6oo Printed Lithbgrapnic Diagrams. 4to. gs. PHYSICAL SCIENCE. Gallon — continued. ■ , - »» HEREDITARY (iENTOS ; Ajx Inquiry intq its Laws and Con- sequences. Demy 8vo. i2j. , ; The Times calls it " a most able and most interesting book." ENGLISH MEN OF SCIENCE; THEIR NATURE AND NURTURE. ,, 8vo. is. ed " Tlie book iseirtainly one of very great interest." — Naturei' Geikie. — Works by ARCHIBALD Geikie, LL.D., F.R.S., MurcHson Professor of Geplogy and Mineralogy at Edinburgh :— ELEMENTARY LESSONS IN PHYSICAL GEOGRAPHY. With numerous lUustrations. 'Fcap. 8vo. 4f. 6d. Questions, is. 6d. FIELD GEOLOGY. ■ With illlustrations. Crown 8vo. [Shortly. PRIMER OF GEOLOGY. Illustrated. 1811I0. is. PRIMER OF PHYSICAL GEOGRAPHY. Illustrated. iSmo. is. Gordon. — AN ELEMENTARY BOOK ON HEAT. By J. E. H. Gordon, B.A., Gonville and Caius College, Cambridge. Crown 8vo. zj.' ' , '"Guillemin.— THE FORCES, OF NATURE: A Popular Intro- ' duction to the Study "of Physical Phenomena. By AM^ofe GuiLLEMiN. Translated from the French by Mrs. Norman ■ LofeKYfiR ; and Edited, with Additions and Notes, by J. Norman Lockyer, F.R.S. Illustrated by Coloured' Plates, and 455 Wood- cuts. Third ajid, cheaper Edition. Royal 8vo. 2i,s. "translator and Editor have done justice .to their trust. The text has all iJie force and flow ' of original writing, combining faithfulness to the authors meaning with purity and independence in regard to idiom ; while the historical precision and .accuracy pervading the' work throughout, speak of the watchful editorial supervisionwhich has been given to every scientific ddail, 'N(^Hng can well exceed the cifnrness,'av4 delicacy of the illustrative, wood- cuts. Altogether, the work may be said to'hflvef no parallel, either ' in point of fulness or attraction, as, a popular, manual of physical science." — Saturday Review. .-,■> THE APPLICATIONS OF PHYSICAL FORCES. . By A. Guillemin. Translated from the French by Mrs. Lockyer, and Edited with Notes and Additiunsi by, Ji N. Lockyer,' F.R.S. With Coloured Plates and numerous Illustrafions. Chea,i)pr ■ ■' Edition.'' Imperial 8vo. clotli, extra gilt 36^. ; '"' ' Alsoin'Eigbte^n Monthly Parts, price is. each. Part I. in Novfember, 1878. ' ■' . ^ , '■'^A book ■uihichr vie can heartily recommend, both on account of the width and . soundness of its contents, and also- because of the excel- lence of its. print. Us illustrations,,, amd external \appecirance." — Westminster'.Review. < '• i ' SCIENTIFIC CATALOGUE. Hanbury. — SCIENCE PAPERS: chiefly .Pharmacological ajid Botanical. . By DANIEL Hanbury,,, F^.R.S. Edited, with Memoir^ by J. Ince, F.L.S., and Portiait engraved by C. H. Jeens. 8vo. I4f. Henslow.— THE THEORY OF EVOLUTION OF LIVltSTG THINGS, and Application of. the- Principles of Evolution to Reli^on considered as Illustrative of the Wisdom and Benefi- cence of the Almighty. By the Rev. George Henslow, M.A,jF.L.S. CrownSvo, , .6j. ; ■ Hooker.— Works by Sir J. D. HOOKER, K.C.S.I., C.B., F.R.S., M.D., D.C.L. :— . THE STUDENT'S FLORA QF THE BRITISH ISLANDS. Second Edition, revised and improved. Globe 8vo. los. bd. The object of this work is to. supply students and field-botanists with a fuller account of the Plants of the British Islands than the manuals hithertq in. use aim at giving. " Certainly the fullest arid vipst acetfrdte manual of the kind that has yet appeared. Dr. Hooker has shown his characteristic indiistry and ability in t/jie care and skill which he has thrown into the characters of the ptanis. ThiSi are top, great extent, origiffal, and are really admirable ,fhr their (oit^i'ftanah of clearness, brevity, and completeness. "-^Pall liiall Gazette. 'V ', . PRiMER (i)F BOTANY. With Illustrations., i8mq. is. New Edition, revised and corrected. Hooker and Ball.— jpuRNAL OF A TOUR IN MARdCCO AND THE GREAT ATLAS. By Sir J. D. HookER, K.C.S.L, C.B., F.RiS., &c., and John Ball, F.R.S. With. Appendices, including a Sketch of the Geology of Marocco. By G; MAW, F.Li.S., F.G.S. With Map and Illustrations. " 8vo. 2W. Huxley and Martin.— a COURSE OF PRACTICAL IN- STRUCTION IN ELEMENTARY BIDLOGY; By T. H. HtrXLEY, LL.D., Sec. R.S., assisted by H. N. MarTiN, B.A., M.B., D.Sc., Fellow of Christ's College, Cambridge. Crown 8vo. 6j. " This is the most thoroughly valuable hook to teachers OMd' students of biology which has ever appeared in the English tongue" — London Quarterly. Review. ' .; Huxley (Professor).— lay SERMONS, ADDRESSES, AND REVIEWS,,. By T. H, Huxley, LL.D,, F.R.S. New and Cheaper Edition. Crown 8vo. Is. 6d. FoU7 teen Discourses on the following suijeetst^-\ i ) On the Advisable- ness of Improving Natural ICmrwledge;: — (2) EmanHpation — Slack and White : — (3) A Liberal Education, and where to find it; — (4) Scientific Education: — (5) On -ithe Educational' Value of PHYSICAL SCIENCE. Huxley (Prp,fessor)-Tr-coi the Natural History Sciences: — (6) On the Study of Zoology: — (7) On the Physical Basis of Li/e:^S) The Scientific Aspects of Positivism: — (9) On a Piece of Chalk: — (10) Geological Contem- poraneity and. Persistent Types, of Life: — (11) Geological ■Reform: — (i2) The Ori^n of Species.: — (13) Criticisms on the '" Origin of Species:" — {14) On Descartes' '' Liiscourse touching tjie Method of . using Onis Season rightly and i>f seeking Scientific Truth." ' : ESSAYS SELECTED FROM "LAY SERMONS, AD- DRESSES, AND REVIEWS." Second Edition. Crown 8vo. is. CRITIQUES AND ADDRESSES. 8vo. ' ios,.6d. ' Contents: — i. AdmiHisiraiive Nihilism. 2. The School Boards : what they can do, and what they may do. 3. On Medical Edu- cation. 4. Yeast. 5. On- the Formation of Coal. 6. On Coral and Coral Reefs. 7. On the Methods and Results of Ethnology. 8. On some Fixed Points in British Ethnology. 9. Pala;ontology and the Doctrine of Evolution. 10. Biogenesis and Abiogenesis. , II. Mr. Darwin's Critics. 12. The Genealo0t of Animals. 13. Bishop Berkeley on the Metaphysics of Sensahoti. LESSONS IN ELEMENTARY PHYSIOLOGY. With numerous Illustrations. New Edition. Fcap. 8vo. 4s. 6d. ■ ■ ■ , "Pure gold- throughout:"— GmsxAisxi.. " Unquestionably the (ledrest and most ipmplete elementary treatise on this subject that we possess in any language." — Westminster Review. AMERICAN ADDRESSES: with a Lecture on the Study o^ . Biology. 8vo. 6/. (>d, PHYSIOGRAPHY: An Introduction to the Study pf Nature. With •■ Coloured Plates and numeroils Woodcuts. New Edition. Crown 8vo. 7j. (>d. Jellet (John H., B.D.) — a TREATISE ON THE ■ THEORY OF FRICTION. By John H. Jellet, B.D., Senior Fellow of Trinity College, Dublin ; President of the Royal Irish Academy. 8vo. • 8f. (sd. Jones.— THE OWENS COLLEGE JUNIOR COURSE OF PRACTICAL CHEMISTRY. By Francis Jones, Chemical Master in the Grammar School,- Manchester. With Preface by Professor Roscoe. New Edition. i8ra.o. with Illustr^ttipns.; 2s. dd. Kingsley.— GLAUCUS : OR, the wonders OF THE '.SHORE. By Charles Kingsley,= Canon '■ o5 "Westminster. ' New Edition, with numerous Colouired Plates.'. Crown Svo* 6s. Lahgdon.— THE APPLiGATiON of' eLectrjgity *ro RAILWAY WORKING,, ' ByW. E. Langdon, lyiember of the Society of "Telegraph Engineers. With numerous Ill,us|trations. Extira fcap. 8vo. 4/. 6(/. ' . . , .12 . SCIENTIFIC CATALOGUE. Lockyer (J. N.) — Worksby J. Norman Lockyer, F.R.S.— ELEMENTARY LESSONS IN ASTRONOMY. With nu- merous Illustrations. New Edition. i8m6; 5j. 6d. ' , ''The book is full, clear, sound, and worthy of attention, not only as a popular exposition, but as a scientific 'Index.' " — Athenaeum. " The most fascinating of elementary books on the .Sciences." — '. - Nonconformist. ' ''■ •^THE SPECTROSCOiPE AND ITS APPLICATIONS. By J. ,1 ■ '. Norman Lockyer, F.R.S. With Coloured' 'Plate and numerous , Illustrations. Second Edition. Crown 8vo. 2^- ^^• ■^CONTRIBUTIONS TO SOLAR PHYSICS. . By J. Norman Lockyer, F. R. S. I, A Popular Account of Inquiries into the Physical Constitutiqn ,of the Sun, with especial reference to Recent ^jSpe.ctroscopic Researches. II. Communicatioiis to the Royal Society of London and the French Academy of Sciences, with Notes. Illustrated by 7 Colc^ur§d Lithographic Plates ^and 1 75 Woodcuts. Royal 8yo. cloth, extra gilt, price SW. dd. " The book may be taken as an authentic exposition of the present state of science in connection with the. importcmt subject of spectro- scope analysis. ... Even, the unscientific public may derive much information from it." — Daily News. PRIMER OF ASTRONOMY. With Illustrations.. iSmo. - u. Lockyer and Seabroke.^STAR-GAZlNG : PAST AND PRESENT. An Introduction to Instrumental Astronomy. By J. N- Lockyer, F.R.S. ; Expanded- from Shorthand' Notes of a Course of Royal Institution Lectures with, the assistance of iG. M. , Seabroke, F.R.A.S. Withnumerous.IUustiations. RoyalSyoj 2ij. " '" A ' book, of great inieresi and utility to the astronomical student." ' ' — Athenssum. • ■ ' ■ Lubbock.-^Works by Sir John Lubbock, M.P.jF.R.is.,D.QX.: THE; ORIGIN AND METAMORPHOSES OiF INSECTS. :With Numerous Illustrations. . Second Edition: Crown 8vo. 3J. bd. "As a summary of th'e phenomena Of insect ihetamorphoses Ms little book is of great value, and will be read with interest and profit by all students of natural history. . The, i/uhole chapter on the ' 'origin of insects is most interesting and Valuable'. The ijliisira- tions are numerous and good." — Westminster Review. ON BRITISH WILD FLOWERS CONSIDERED IN RELA- TION TO INSECTS. With Numerous IIlustrati6ns. Second ,E^j;ion. Crown 8yo. i^.dd., i ... Macmillan (Rev. Hugh). — For other Works by the same Author, see Theological Catalogue. .' ■ . . . SOUDAYS ON HIGH LANDS ; or, J^amblps. and^Ineid^ts in " ; . sedrcH 'of Alpine Plants. ' Globe 8vo, cloth, (is. ' 7 •:''' ■-■"■' ' '"■One of the most charmirig' books ^ of its kind' ever written." — Litetary Churchman. "'Mrl' Maciriiilan's glo^oing pictures of. Scandinavian scenery." — Saturday Review. PHYSICAL SCIENCE. 13 Macmillan (Rev. Hugh)— B(,«;j««^ar, ;.' FIRST FORMS ©F VEGETATION. Second Edition, _ corrected and enlarged, with Coloured Frontispiece and numerous lUiistra- tions. Globe 8v(5. 6j. . . > The first edition of this book was published, under the name of "Footnotes from the Page of Nature; or. First Forms afWegeta- tion. Probably the best popular guide to the .studyi of mosses, , .^jlichetts, and fungi ever iDritten. fts practical value as a help to ihe student and collector cannot be exaggo-ated." —iHwas^eisXH^ Examiner. Mansfield (C. B.) — Works by the late C.iB. Mansfield :— A THEORY OF SALTS. A Treatise on the Constitution of Bipolar (two-membered) Chemical Compounds. Crown 8vo. 14J. AERIAL NAVIGATION. the Problem, with Hints 'for its Solution. , Edited by R. B,. Mansfield. With a Preface by J. M. Ludlow. With Illustrations. Crown 8vo. loi. (id. Mayer. — SOUND ; a Series of Simple, Entertaining, and In- expensive Experiment^ in the Phenomena of Sound, for the Use of Students of every age. By A. M. Mayer, Professor of Physics in the Stevens Institute of Technology, &c. With numerous Illus- trations. Crown 8vo. 35^. bd. 'Mayer and Barnard. — light, a Series of Simple, Enter- taining, and Useful Experiments in the phenomena of Light, for the use of Students of ev.ery age. By A. M. Mayer and C. Barnard. With Illustrations. Crown 8yo. 2J. (>d. Miall.— STUDIES IN COMPARATIVE, AN ATOMY. No. i, The Skull of the Crocodile. A M^^Vial for Students. By L. C. MiALL, Professor of Biology in Yorkshiire College. 8vo. 2J. M. No. 2, The Anatomy of the Indian Elephant. By L. C. MlALt and F. Greenwood. With Plates. 5^. Miller.— THE romance of ASTRONOMY. By R. I^lley Miller, M.A., Fellow and Assistant Tutor of St. Peters Col- lege,"Cambridge. Second Edition, revised and enlarged. Crown 8vo. 4-r. 6(1?. Mivart ( St. George).^Works by St. George Mivart, F,R.S. &c.. Lecturer in Comparative Anatomy at St. Mary's Hospital: — ON THE GENESIS OF SPECIES., Crown i8vp^, Second Edition to which; notes have been added in reference and, reply to Darwin's "Descent of Man." With numerous Illustrations, pp. XV' 2Q6 Q-^' <•>««« -work in the English language lias this great-controversy been treated at once with the same broad and vigorous p-asp of ■ facts, ntid the same liberal and candid ^CTi^y wo'iild exaggerate our sense of the moral and theological value- of the work," — British Quarterly Rfeview. ' ' "^ Newcomb. — POPULAR ASTRONOMY. By Simon New- comb, LL.D., Professor U.S. Naval Observatory. With 112 Engravings and Five Maps of the Stars. 8vo. iSj. "As affording a thoroughly reliable foundation for more advanced reading. Professor Neiucomb's ' Popular Astronotiiy ' is deserving ' of strong recommendation." — Nature. Oliver.— Wprks by Daniel Oliver, F.R.S., F.L.S., PrcifeSsor of Botany in Uiiiversity College, London, and Keeper of the HerEa- rium and Library of the Royal Gardfens, Kew : — LESSQNS IN ELEMENTARY BOTANY. With, nearly Two Hundred Illustrations. New Edition, Foap. 8vo. 4^. 6d. This book is designed to teaci the. eleihents of Botany on Professor Henslov^ s pla,n of selected Types and by the use of Schedules. The earlier chapters, embracing thi elements of Structural and Physio- logical Botany, introduce us to the methodical stiidy of the Ordinal Types. The mv.clud.ing chapter ^ g,re e^f;4kd, " Hirw to D/y PHYSICAL SCIENCE. 15 OWVQV— continued. Plants " and" How to Describe Plants. " A valuable Qlossary is apjiende^io the volume. In the preparation of this work free use has been made, of the n^anuscripi materials of the late Professor ftenslaw, FIRST BOOK OF INDIAN BOTANY, With numerous Illustrations. Extra fcap. 8yp. ds. 6d. '' It contains a well-digested sumik&ry of all essential knowledge -. , , P^aining^ to Indian Botany, wrought out in accordance with the .' best 'princijiles of scientific arrangement." — Allen's lildiali Mail. Pennington.— NOTES ON THE BARROWS AND BONE CAVES OF DERBYSHIRE. With, an account o? a Descent -into Elden Hole. By RooKE Pennington, B.A., LL.B., F.G.S. 8V0. f>s. ■ Penrose (F, C.)— on a method of predicting by GRAPHICAL CONSTRUCTION, OCCULTATIONS OF STARS BY THE MO.ON, AND SOLAR ECLIPSES FDR . ANY GIVEN PLACE. Together with more rigorous methods for the Accurate Calculation of Longitude. By F. C. Penrose, F.R.A.S: With Charts, Tables^ &c. 4to. 12s. Perry. — AN ELEMENTARY TREATISE ON STEAM. By John Perry, B.E., Professor of Engineering, Imperial College of Engineering, Yedo. With numerous Woodcuts, Numerical Ex- amples, and Exercises. i8mo. 4r. dd. > ^'Mr. Perry has ^in this compact little volume brougkp together an immense amount of information, new told, regarding steam and its application, not the least of its merits being that it is suited to the capacities alike of the tyro in engineeiing science or the better grade of artisan." — Iron. ^ Pickering.— ELEMENTS OF PHYSICAL MANIPXJLATION. By E. C. Pickering, Thayer Professor of Physics in the Massa- ■ ichiisetts Institute of Technology, Part I., medium 8vo. \Qs. 6d. . Part II.,,., los. 6d. "When finished' 'Physical Manipulation' will no doubt be con- sidered the best and most complete 'text-boo^ en the subject of which it treats." — Nature. ..... ■.;[-■ ■■, wv ■ ■ , Prestwich THE PAST AND FUTUllE OF GEOLOGY. An Inaugural Lecture, by J. P?iESTWiGH, M.A., F.R.g., &c., Professor of Geology, Oxford. 8yo. zs. Radcliffe.^PROTEUS : OR UNITY IN NATURE. By. C. B. RadcLiffe, M.D., Author of "Vital Motion as a mode of Physical Motion. Second Edition. 8vo. 7j, dd. i6 SCIEI^TIFIC CAT4^0GUE. Rendu.— THE THEORY OF THE GLACIERS OF SAVOY. By M. LE Chanoine Rendu. Translated by A. Weus, Q.C, late President of the Alpine Club. To which are added, the Original Meniciir and Supplementary Articles byPrbfessoirs Tait and Rus- KiN. Edited with Introductory remarks by George Forbes, B.A., Professor of Natural Philosophy in the Andersonian Universityj Glasgow. ,'8vo.. "js.Cd. ■. , i .'./] ,'■ ■ '■.■,■. . \-\ ' Roscpe. — ^yorks by Henry E.'RpscoE, F.R.S., Professor of Chemistry in Owens College, Manchester : — LESSONS' IN ELEMENTARY CHEMISTRY, INORGANIC AND ORGANIC. With numerous Illustrations and Chromq- litho of the' Solar- Spectrum, and of the AlkaMs and Alkaline Earths.' New Edition. Fcap.'SvOi' ^s.Sii. ' CHEMICAL '^liOBLEMS,' adapted to the, atove by Professor Thorpe. Fifth Edition, with Key. 2s. " We i^ffhesitcfiingly^ pronounce it , the,, best of all our elementary treatises on Chemistry." — Medical Times. \ ' , ■ PRIMER OF CHEMISTRY. Illustrated. iSma; is.. Roscoe and Schorlemmpr.— A TREATISE ON CHE- MISTRY. .By PRaFESSQES RpscoE and Schorlemmer'. Vol. I., The Non-metaUic Elements. With numerous Illustrations and Portrait of Dalton. Medium 8vo. 21s. Vol. II., Metals. Parti. With illustrations. 8vo. t^s. "Regarded as a treatise oh the Non-rnetallii Elements,' there can bci no doubt that this volume is inc'omparahlv the most satisfactory one of which we are in possession'' — Spectator. " It would be difficult to praise the work too highly. All the merits wltich we noticed in the first volume are conspicuous in the second. The arrangement is clear and scientific ; the facts gained by modern research are fairly represented dhd judiciously selected; and the style throughout is singulgfly luijid." — Lanceti ■-.,'. [Metals,. Partll. in the Press. ^ Rumford (Count).— THE LIFE AND COMPLETE WORKS OF BENJAMIN THOMPSON, COUNT RUMFORD. With Notices of his Daughter. By George Ellis. With Portrait. Five Vols. 8vo. 4/. "i+r. 6(f. Schorlemmer.— A MANUAL OF THE CHEMISTRY OF ' THE CARBON COMPOUNDS OR ORGANIC CHEMISTRY. By C. Schorlemmer, F.R.S., Lecturer in Organic Chemistry iu Owens College, Manchester. &vo. i+r. ., -. . "Zt appears Jo us to beas^ complete ,fl, manual, of the metamorphoses, of . cfirbon as could be at /present produced, ctiid it must ptoVe eminently useful to the chemical student.'' — Athenieum. PHYSICAL SCIENCE. 17 Shann. — AN ELEMENTARY TREATISE ON HEAT, IN RELATION TO STEAM AND THE STEAM ENGINE. By G.. Shann, M.A. With Illustrations. Crown 8vo, 4r. 6d. Smith.-— HISTORIA FILICUM : An Exposition of the Nature, Number, and Organography of Ferns, and Review of the Prin- ciples upon -which Genera are founded, and the Systeias of Classifi- cation of the principal Authors, with a new GeneralArrangement, &c. By J. Smith, A.L.S., ex-Curator of the Royal Botanic Garden, Kew. With Thirty Lithographic Plates by W. H. Fitch, F.L.S. Crown 8vo. 12s. 6d. " No one anxious to work up a thorough knowledge of ferns can afford to do mthout it." — Gardener's Chronicle. "^ South Kensington Science Lectures.— Vol. I.— Con- taining Lectures by Captain Abney, F.RS., Professor S.tokes, Professor. Kennedy, F. J. BrAmwell, F.R.S., Professor G. Forbes, iH. C. Sorby, F.R.S., J. T. Bottomley, F.R.S.E., , S. H. ViNESj B. Sc, and Professor Carey Foster. Crown 8vo. 6s. [Vol. II. nearly ready. SpottiSWOode.— POLARIZATION OF LIGHT. By W. Spottiswoode, President of the Royal Society. With numerous Illustrations. Second Edition! ,Cr. Svbl.SJ. id. (Nature Series.) " The illustrations are exceedingly well adapted to' assist in making the text comprehensible.'" ^-K^enxura. "A clear, trustworthy manual." — Standard. Stewart (B.) — Works by Balfour Stewart, F.R.S., Professor of Natural Philosophy in Owens College, Manchester : — LESSONS IN ELEMENTARY PIIYSICS. With numerous Illnstrations and Chromolithos of the Spectra of the Sun, Stars, and Nebulae. New Edition. Fcap. 8vo. 4?. 6d. The: Educational Times calls this the beaii-idial of a scientific text- book, clear, accurate, and thorough.'^ PRIMER OF PHYSICS. With Illustrations. New Edition, with Questions. iSmo.- \s. Stewart and Tait.— the UNSEEN UNIVERSE: or. Physical Speculations on a Future State, By Balfour Stewart, F.R.S., and P. G. Tait, M.A. Sixth Edition. Crown 8vb. 6j-. " The book is one which well deserves the attention of thoughtful ana religious readers. . . . It is a petfectly sober inquiry, on scientific grounds, into the posiiUlilies of a fttture existence." — Guardian. -i Tait.— LECTURES ON SOME RECENT ADVANCES IN PHYSICAL SCIENCE. By P. G. Tait, M.A., Professor of Philosbphy in the University of Edinburgh. Second edition, revised and enlarged, with the Lecture on Force delivered before the British Association. Crown 8vo. gj. B SCIENTIFIC CATALOGUE. Tanner.— FIRST principles of agriculture;., By Henry Tanner, F.C.S., Professor of Agricultutal Science, University College, Aberystwith, Examiner in the Principles of Agriculture under the, Government Department of Science. iSmo. IS. ' ' ' ' ' Taylor. — sound and MUSIC : A Non-Mathematical Trea tise on the Physical Constitution of Musical Sounds and Harniony including the Chief Acoustical Discoveries of Pirofessor Helm holtz. By Sedley Taylor, M.A., late Fellow of Trinity Col ledge, Cambridge. Large crovpn 8vo. %s. dd. "In no previbus scientific treatise do we remember so exhaustive and so richly illustrated a description of forms of vibration and oi wave-motion in fluids." — Musical Standard. ■ ■ ' '• Thomson. — Works by Sir Wyville Thomson, K.C.B., F.R.S. THE DEPTHS OF THE SEA : An Account of the General Results of the Dredging Cruises of H.M.SS. "Porcupine" and " Lightning " during the Summers of 1868-69 and 70, under the scientific direction of Dr. Carpenter, F.R.S., J. Gwyn Jeffreys,; F.R.S., and Sir Wyville Thomson, F.R.S. With nearly 100 Illustrations and 8 coloured Maps and Plans. Second Edition. Royal 8vo. cloth, gilt. '3,u. 6d. The Athenseum says : " The book is full of interesting matter, and is written by a master of the art of popular exposition. It is excellently illustrated, both coloured maps and ivoodcuts possessing high merit.. Those who have already become interested in dredging operations will of course make a point of reading this work ; those who wish to be pleasantly introduced to the subject, an ... written in a clear style, and may be easily understood by,, even those who are not versed in such discussions!' — British Quarterly Review- A HANDBOOK OF MORAL PHILOSOPHY. New, Edition. Crown 8vo. 6^. . . ' _ ., ,.■'•' ■ "It is, we/eel convinced, the best handbook on the subject, intellectually and morally, and does inHnite credit to its author. " — Standard, "A- compact and useful work, going over a great deal of ground in a manner adapted to suggest and facilitate further study. , . . His book Tjoill be an assistance to many students outside his awn University of Edinburgh. — Guardian. THE RELATIONS OF MIND AND BRAIN. INearly ready. Fiske.— OUTLINES OF COSMIC PHILOSOPHY, BASED ON THE DOCTRINE OF EVOLUTION, WITH CRITI- CISMS ON THE POSITIVE PHILOSOPHY. By John FiSKE, M.A., LL.B., formerly Lecturer on Philosophy at Harvard University. 2 vols. 8vo. ajj. " The work constitutes a very -ejfecHve encyclopcedia of the evolution- ary philosophyi and is well worth the study of all who wish to see at once the entire scope and purport of the scientific dogmatism of the day." — Saturday Review. Herbert.— THE realistic assumptions of modern SCIENCE EXAMINED. By T. M. Herbert, M.A., late Professor of Philosophy, &c., in the Lancashire Independent College, Msinchester. 8vo. 14J. Jardine.— THE elements of THE psychology of COGNITION. By Robert Jardine, B.D., D'.Sc, Principal of the General Assembly's College, Calcutta; and Fellow of the Uni- versity of Calcutta, Crown Svo. 6j. 6 //. ffistory's Scientific Tretensioiis. JII. Damd Hume as a MetaphymeHn. IV. HuxkyismU V. Recent I'hase of Scientific Atheism. VI. Limits of Demonstrable Theism. Thring (E., M. A.)— THOUGHTS ON LIFE-SCIENCE, By :Edward Thring, M. a. (Benjamin Place), Head Master of Uppingham School. „ New Edition, enlarged and revised. ■ Crown 8vo. "js. dd. Venn.— THE LOGIC OF CHANCE : An Es^ay on the Founda- tions and Province of the Theory of Probability, with especial reterence to its logical bearings, and its apphcation to ^Mdtal and Social Science. By John Venn, M.A., Fellow and Lecturer of Gonville and Caius College, Cambridge. Second Edition, re- written and greatly enlarged. Crown 8vo. lOr. 6d. " One of the most thoughtful and philosophical treatises on any sub- ject connected with logic and evidence which has been prudu^d in this or any other country for many years." — Mill's Logic, vol. ii. p. 77. Seventh Edition. , ., , , ,' - - ,V}:'''.:A SCIENCE PRIMERS. 27 SCIENCE PRIMERS FOR ELEMENTARY SCHOOLS. Under the joint Editorship of Professors Huxley, Roscoe, and Balfour Stewart. . Chemistry — By H. E. Roscob, F.R.S., Professor of Chemistiy in Owens College, Manchester. With numerous Illustrations. l8mo. IJP. New Edition. With Questiohs. Physics.— By Balfour Stewart, F.R.S., Professor of Natural Philosophy in Owens College, Manchester. With numer- ous Illustrations. i8mo. ^ \s. NeW Edition. With Questions. Physical Geography. _By Archibald Geikik, F.R.S., Murchison Professor of Geology and Mineralogy at Edinburgh. With numerous Illustrations. New Edition with Questions. l8mp. \s. Geology By Professor Geikie, F.R.S. With numerous Illus- trations. New Edition. l8mo. cloth, is. Physiology By Michael Foster, M.D., F.R.S. With numerous Illustrations. New Edition. i8mo. \s. Astronomy — By J, Norman Lockyer, F.R.S. With numerous Illustrations. New Edition. i8mo. \s. Botany — ^By Sir J. D. Hooker; K.C.S.I., C.B., F.R.S. With numerous Illustrations. New Edition. iSmo. is. Logic — By Professor Stanley' Jevons, F.R.S. New Edition. i8mo. IS. Political Economy.— By Professor Stanley Jevons, F.R.S. l8mo. IS. In preparation: — INTRODUCTORY. JBy Pcofessor Huxley. &c. &c. ELEMENTARY SCIENCE CLASS-BOOKS. Astronomy By the Astronomer Royal. POPULAR AS- TRONOMY. With Illustrations. By Sir G. B. Airy,, K.C.B., Astronomer Royal. New Edition. i8mo. 41. dd. Astronomy.— ELEMENTARY LESSONS IN ASTRONOMY. With Coloured. Diagram of the Spectra of the Sun, Stars, and Nebulse, and numerous Illustrations. By J. Norman Logkyer, F.R.S. New Edition. Fcap. 8vo. is. dd. 28 SCIENTIFIC CATALOGUE. Elementary Science Class-books — continued. QUESTIONS ON LOGKYER'S ELEMENTARY LESSONS IN ASTRONOMY. For the Use of Schools. By John Forbes Robertson. i8mo, cloth limp. is. Bd. Physiology.—LESSONS in elementary physiology. With numerous Hlustrations. By T. H. Huxley, F.R.S., Pro- fessor of Natural History in the Rpyal School of Mines. NeWj Edition. Fcap. 8vo. 4s. 6ii. ' QUESTIONS OIJ HUXLEY'S PHYSIOLOGY FOR SCHOOLS. By T. Alcock, M.D. iSmo. is. (>d. Botany — LESSONS IN ELEMENTARY BOTANY. ByD. Oliver, F.R.S., F.L.S., Professor of Botany in University College, London. With nearly Two flundred Illustrations. New, Edition. Fcap. 8vo. d^. dd. ' ■ Chemistry — LESSONS IN ELEMENTARY CHEMISTRY, INORGANIC AND ORGANIC. By Henry, E. Roscoe, F.R.S., Professor of Chemistry in Owens College, Manchester, With numerous Illustrations and Chromo-Litho ^of • the Solar Spectrum, and of the Alkalies and Alkaline Earths. New Edition. Fcap. 8vo. 4J-. 6d. ^ '■ A SERIES OF CHEMICAL ^PROBLEMS, prepared with Special Reference to the above, by T. E. Thorpe, Ph.D., Professor of Chemistry in the Yorfehire College of Science, Leeds. Adapted for the preparation of Students for the Government, Science, and Society;of: Arts Examinations. With a Preface by Professor RoscoE. Fifth Edition, with Key. i8mo. 3J. , Political Economy — POLITICAL ECONOMY FOR BE- GINNERS. By MiLLicENT G. Fawcett. New. Edition. i8mo. 2s.6d. . . • ■ I r- Logic — ELEMENTARY LESSONS IN LOGIC ; Deductive and Inductive, with {opious Questions and Examples, and a Vocatulary of Logical Terms. By W. Stanley Jevons, M.A., Professor of Political Economy in University College, London. New Edition. Fcap. 8yo. 3^. 6d. Physics.-^LESSONS IN ELEMENTARY PHYSICS,, ; By Balfour Stevstart, F.R.S., Professor of NaturaL Philosophy ih Owens College, Manchester. With, numerous Illustrations aji^ Chromo-Litho of the Spectra 'of the Sun, Stars,',and Nebula;. Ne'w ■Edition. Fcap. 8vo. 41. 6d. Pi-actical Chemistry.—fHiE OWEJSfS cbLLEGEjUNIQR COURSE OF PRACTICAL CHEMISTRY. ' By Franci? , Jones, Chemical Mastfer in the Grammar School, Manch^ter. With 'Preface by Professor RoscoE, and,. Illustrations. • New Edition. i8mo. ds. 6d. ■ ' SCIENCE CLASS-BOOKS. 29 Elementary Science Cl^ss-books — continued, Anattomy.^LESSONS IN ELEMENTARY ANATOMY. By St. George Mivart, F.R.S., Lecturer in Comparative Anatomy at St. Mary's Hospital. With upwards of 400 Illustrations. Fcap. 8y6. 6j. 6(1. Mechanics — an elementary treatise. By A. B. W. Kennedy, C.E., Professor of Applied Mechanics in University College, London. With Illustrations. [In preparation. iSteam an elementary treatise. By John Perry, 1 /Professor- sf Engineering, Imperial College of Engineering, Yedo. With numerous Woodcuts and Numerical Examples and Exercises. l8mo. 4T. 6d. Physical Geography. _ ELEMENTARY LESSONS IN PHYSICAL GEOGRAPHY. By A. Geikie, F.R.S., Murchi- son Professor of Geology, &c., Edinburgh. With numerous .Illustrations., Fcap. 8vo. 4^.61^. • QUESTIONS ON THE. SAME. u. 6d. Geography.— CLASS-BOOK OF 'GEOGRAPHY. By C. B. ClarkEj M.A.. F.R.G.S, Fcap. 8vo. 2s.6d. Natural Philosophy.-NATURAL PHILOSOPHY FOR BEGINNERS. By I. Todhunter, M.A., F.R.S. Part I. The Properties of Solid and Fluid Bpdies. igmo. y. 6d. Part II. Sound, iight, and Heat. iSmo. 3j. 6d. Sound AN ELEMENTARY TREATISE. By W. H. Stone, M.D., F.R.S. With Illustrations. l8mo. [In ihi Press. Others in Preparation. MANUALS FOR STUDENTS. Crown 8vo. Dyer and Vines.— THE STRUCTURE OF PLANTS. By Professor Thiselton Dyer, F.R.S., assisted by Sydney ViNfeSi B.Sc, Fellow and Lecturer of Christ's College, Cambridge. With numerous Illustrations, ;• [In preparation. Fawcett.— A manual of political ECOIJOMY. By Professor Fawcett, M.P. New Edition, revised and enlarged. Crown Svo. i2J-. 611?. Fleischer.— A SYSTEM OF volumetric analysis. Translated, with. Notes and Additions, from, the second German Edition, by M. M. PaTtison Muik, F.R.S.E. . With Illustra- tions. Crown 8vo. 7^. ()d.. 30 SCIENTIFIC CATALOGUE. Manuals for St\xA^n\.s— continued. Flower (W. H.)_aN INTRODUCTION TO THE OSTE- OLOGY OF THE MAMMALIA. Being the Substance of the Course of Lectures delivered at the Royal College of Surgeons of England in 1870. By Professor W. H. Flower, F.R.S., F.R.C.S. With numerous Illustrations. New Edition, enlarged. Crown 8vo. los. 6d. ■ ' ' Foster and Balfour._THE elements of ■ EMBRY- OLOGY. By Michael :Foster, M.D., F.R.S., and F. M. Balfour, M.A. Part I. crown 8vo. ^s. 6il. Foster and Langley._A COURSE OF ELEMENTARY PRACTICAL PHYSIOLOGY. By Michael Foster, M.D., F.R.S., and J. N. Langley, B.A. New Edition; Crown 8vo. 6s. Hooker (Dr.)_THE STUDENT'S FLORA OF THE BRITISH ISLANDS. By Sir J. D. Hooker, K.C.S.I., C.B., F.R.S., M.D., D.C.L. New Edition, revised. Globe Svo. los. 6d, , Huxley — PHYSIOGRAPHY. An Introduction to the Study of Nature. By Professor HuxLEV, F.R.S. With numerous Illustrations, and Coloured Plates. New Edition. Crown Svo. 7^. 6d. ' 3 ' Huxley and Martin._A COURSE OF practical IN- ■ STRUCTION IN ELEMENTARY BIOLOGY. By Professor Huxley, F.R.S., assisted by H. N. Martin, M.B., D.Sc. New Edition, revised. Crown Svo. 6s. Huxley and Parker — ELEMENTARY BIOLOGY. PART II. By Professor Huxley, F.R.S. , assisted by — Parker. With Illustrations. ■ , \ln> prepdi^atioH. Jevons.— THE PRINCIPLES OF SCIENCE. A Treatise on Logic and Scientific Method. By Professor W. Stanley Jevons, LL.D., F.R.S., New and Revised Edition. Crown Svo. X2s.6d. Oliver (Professor)._FlRST BOOK OF INDIAN BOTANY. By Professor Daniel Oliver, F.R.S., F.L.S., Keeper of the Herbarium and Library of the Royal Gardens, Kew. With numerous Illustrations. Extra fcap. Svo. 6s. 6d. ■ Parker and Bettany._THE MORPHOLOGY OF, THE SKULL. By Professor Parker and G. T. Bettany. , Illus- trated. Crown Svo. los. 6d. • Tait— AN ELEMENTARY TREATISE , ON HEAT., By Pr«. fessor Tait, F.R.S.E. Illustrated.,'., ^In the Press. Thomson.— ZOOLOGY. By Sir C. Wyville Thomson, F.R.S. Illustrated. \In preparation^ Tyler and Lankester.— ANTHROPOLOGY. By E/B. Tylor, M.A., F.R.S., and Professor E. Ray Lankester, M.A., F.R.S. Illustrated. [In preparation. Other volumes of these Manuals will follow. NATURE SERIES. -^THE SPECTROSCOPE AND ITS APPLICATIONS. By J. N. LOCKYER, F.R>S. With Illustrations. Second Edition. Crown 8vo. • 3J. 6.^ President of the lloyal Society. Illustrated. Second Edition. Crown ON BRITISH WILD FLOWERS CONSIDERED IN relation to insects. By Sis JOHN LUBBOCK, M.P., F.R.S. Illustrated.' Second Edition. Crown 8vo. j[s. 6d. THE SCIENCE OF WEIGHING AND MEASURING. By H. W. CHISHOLM, Warden of the Standards. Illustrated. Crown Svo. 4J. ^d. HOW TO i)RAW A STRAIGHT LINE: A Lecture on Linkages. By A. B. KEMPE, B.A. Illustrated. Crown Svo. if. 6d. •'LIGHT : A Series of Simple, Entertaining and Useful J Experiments in the Phenomena of Light for the Use of Students of every Age. By ALFRED M. MAYER and CHARLES JBARNARD. With Illustrations. Crown Svo. 2J.' 6d. "" SOUND : A Series of Simple, Entertaining and Inex- pensive Experiments in the Phenomena of Sound, for the Use of Students of every Age. By A. M. MAYER, Professor of Physics in the Stevens Institute of Technology, &c. With numerous Illustrations. Crown Svo^ 3f . 6d. (Others to follow.) MACMILLAN AND CO., LONDON. Pttbhshed- every Thttrsddy, frice fyd.j Monthly PaHs ■is. and 2J. td., Half- Yearly Volumes, f^s.' nature: AJV ILLUSTRATED TOliRNAL OF SCIENCE. Nature expounds in a popular and yet authentic manner, the Grand Results of Scientific Research^ discussing the most recent scientific discoveries, and pointing out the bearing of Science upon civilisation and progress, and its claims to a more general recognition, as well as to a higher place in the educational system of , the country. It contains original articles . on all subjects within the domain of Science ; Reviews setting forth the nature and value of recent Scientific Works ; Correspondence Columns,' forming a medium of Scientific discussion and of intercom- munication among the most distmguished men of Science ; Serial Columns, giving the gist of the most important papers appearing in Scientific Journals, both Home and Foreign ; Transactions of the principal Scientific Societies and Academies of the World, Notes, &c. In Schools where Science is included in the regular course of studies, this paper will be most acceptable, as it tells what is. doing in Science . all , over the world, is popular without lowering the standard of Science, and by it a vast amount of information is brought within a small compass, and students are. directei^ to the best sources for what they need. The various questions connected with Science teaching in schools are also fully discussed, and the best methods of teaching are indicated., " '■'■