xiv CONTENTS. PAGE 11.Divergence of the Results obtained from Pambour's Assumption 251 812. Determination of the work done during one stroke,taking into consideration the imperfections already noticed 252 $18.Pressure of Steam in the Cylinder during the different Stages of the Process and corresponding Simplifications of the Equations 255 $14.Substitution of the Volume for the corresponding Tempera- ture in certain cases 257 $15.Work per Unit-weight of Steam.. 259 s16. Treatment of the Equations. 259 17.Determination ofand of the product Tg 260 dt $18.Introduction of other measures of Pressure and Heat 261 s19. Determination of the temperatures Ta and 7,·· 262 §20. Determination of c and r. 264 §21. Special Form of Equation(32)for an Engine working without expansion、· 265 s22. Numerical values of the constants 267 $23.The least possible value of V and the corresponding amount of Work 268 824. Calculation of the work for other valuea of 269 §25. Work done for a given value of V by an engine with expansion 271 省26. Summary of various cases relating to the working of the engine 273 827. Work done per unit of heat delivered from the Source of Heat 274 528. Friction 275 §29. General investigation of the action of Thermo-Dynamic Engines and of its relation to a Cyelieal Process 276 30.Equations for the work done during any cyclical process 279 831. Application of the above equations to the limiting case in which the Cyclical Process in a Steam-Engine is reversible 281 832. Another form of the last expression 282 33. Influence of the temperature of the Source of Heat 284 534. Example of the application of the Method of Subtraction .286 CHAPTER XIL. ON THE CONCENTRATION OF RAYS OF LIGHT AND HEAT,AND ON THE LIMITS OF IT8 ACTION. 1.Object of the investigation. 293 I.Reasons why the ordinary method of determining the mutual radiation of two surfaces does not extend to the present Cn80. 296 $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 8 nrface,· ·297 $4.Indeterminateness of the Formula in the case of the Concen. tration of ray8.· .300 poiredvGOOgle
XlV § 11. § 12. § 18. § 14. § 15. § 16. § 17. § 18. § 19. § 20. § 21. § 22. § 28. § 24. § 25. 011 26. § 27. § 28. § 29. § 80. § 81. § 82. § 88. § 84. CONTENTS. PAGB Divergence of the Results obtained from Pambour's Assumption 251 Determination of the work done during one stroke, taking into consideration the imperfections already noticed • • • 252 Pressure of Steam in the Cylinder during the different Stages of the Process and corresponding Simplifications of the Equations • • • • • • • • • • Substitution of the Volume for the corresponding Temperature in certain cases • • Work per Unit-weight of Steam • Treatment of the Equations Determination of 'i: and of the product Tg Introduction of other measures of Pressure and Heat Determination of the temperatures T. and Ta • Determination of c and r. . . . • . . . Special Form of Equation (82) for an Engine working without expansion • • _ • Numerical values of the constants • • • • • • The least possible value of V and the corresponding amount of Work • • . • • • . Calculation of the work for other values of V. . . • Work done for a given value of V by an engine with expansion. Summary of various cases relating to the working of the engine Work done per unit of heat delivered from the Source of Heat • Friction • • • • • , • • • • • General investigation of the action of Thermo-Dynamic Engines and of its relation to a Cyclical Process. • • • • Equations for the work done during any cyclical process. _ Application of the above equations to the limiting case in which the Cyclical Process in a Steam·Engine is reversible Another form of the last expression. • • Influence of the temperature of the Source of Heat Example of the application of the Method of Subtraction CHAPrER XIL 255 257 259 259 260 261 262 264 265 267 268 269 271 278 274 275 276 279 281 282 284 286 ON THE CONCENTRATION OF RAYS OF LIGHT AND HEAT, AND ON §l. § 2. §8. §4. THE LIMITS OF ITS ACTION. . Object of the investigation. • • • • • • • 298 I. Reasons why the ordinary method of determiniug the mutual radiation of two surfaces does not extend to the present case. • • • • • . • • • • 296 Limitation of the treatment to perfectly black bodies and to homogeneous and unpolarized rays of heat • • • • 296 Kirchhoff's Formula for the mutual radiation of two Elements of surface. • • • • • • • • • • 297 Indeterminateness of the Formula in the ease of the Concentration of rays. 800 Digitized by Coogle
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. 801 §6. The relation of corresponding elements of surface 304 7.Various fractions formed out of six quantities to express the Relations between Corresponding Elements 3C9 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 . 319 $10.Radiation 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· $11.Relations between B.D.F,and 316 310 12.Application of A and C to determine the relation between the elements of surface 318 $13.Relation between the quantities of Heat which ds.and ds radiate to each other 319 V.Relation between the Increment of Area and the Ratio of the two solid angles of an Elementary Pencil of Rays.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. 824 $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··· ·35 17.Matnal radiation of entire surfaces 8:8 $18.Consideration of various collateral circumstances. 329 s19.Summary of re8ults,·· 330 CHAPTER XIII. DISCUSSIONS ON THE MECHANICAL THEORY OF HEAT AS HERE DEVELOPED AND ON ITS FOUNDATIONS, $1.Different views of the relation between Heat and Work 332 $2.Papers on the subject by Thomson and the author ,8E3 $8. On Rankine's Paper and Thomson's Second Paper. 3e4 s4. Holtzmann'g Objections 837 55. Decher's Objections 339 $6.Fundamental principle on which the author's proof of the second main principle rests 840 $7.Zeuner's first treatment of the subject 341 s8. Zeuner's second treatment of the subject. 842
CONTENTS. xv PAGB II. Determination of corresponding pointe 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 Relations between Corresponding Elements. • • • lIC9 m. Determination of the mutual radiation when there is 110 eoncentration of rays • • • • • • • 310 § 8. Magnitude of the element of surface corresponding to dI. on a plane in a particular position • • • • • • • 310 § 9. Expressions for the quantities of heat which dI. and dB. radiate to each other. • . • • • •• 312 § 10. Radiation 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, and E • • • • • • 316 § 12. Application of .A and 0 to determine the relation between the elements of surface • . . • . . . • 818 § 13. Relation between the quantities of Heat which dI. and ds. radiate to each other • • • • . . • • 819 V. Relation between the Increment of Area and the Ratio of the two solid angles of an Elementary Pencil of Rays 821 § 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. . . • . • 824 § 15. General view of the concentration of rays. . . • • 824 § 16. Mutual radiation of an element of surface and of a finite surface through an element of an intermediate plane • 3~ /) § 17. Mutual radiation of entire surfaces • • 3~8 § 18. Consideration of various collateral circumstances 32!J § 19. Summary of results 330 CHAPTER XIII. DISCUSSIONS ON THE MECHANICAL THEORY OF HEAT AS HERE DEVELOPED AND ON ITS FOUNDATIONS. § 1. § 2. § S. § 4. § 5. § 6. § 7. § 8. Different views of the relation between Heat and Work Papers on the subject by Thomson and the author On Rankine'S Paper and Thomson's Second Paper Holtzmann's Objections Decher's Objections • • • • • • • • • Fundamental principle on which the author's proof of the second main principle rests • • Zeuner's first treatment of the subjeet Zeuner's Beeond treatment of the subject 832 8[3 81:4 B37 339 340 8U 342 Digitized by Coogle ~
xvi CONTENTS. PAGE $9.Bankine's treatment of the subject. 345 $10.Hirn's Objections. 848 S 11.Wand's Objections 353 $12.Tait's Objections. 360 APPENDIX I, Thermo-elastic Properties of Solids.... 863 n II. Capillarity 869 III Continuity of the Liquid and Gaseous states. 373 Google
xvi CONTENTS. § 9. Bankine's treatment of the subject § 10. Him's Objections. • • • § 11. Wand's Objections § 12. Tait's Objections • ApPENDIX L Thermo·elastic Properties of Solida II. Capillarity • " III Continuity of the Liquid and Gaseous states. ..... Digitized by Coogle PAGE 345 348 3.53 360 363 369 S7S
ON THE MECHANICAL THEORY OF HEAT. MATHEMATICAL INTRODUCTION. ON MECHANICAL WORK,ON ENERGY,AND ON THE TREATMENT OF NON-INTEGRABLE DIFFERENTIAL EQUATIONS. 1.Definition and Measurement of Mechanical Work. Every force tends to give motion 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 remains 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 atraight 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 the 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 positive or negative in sign,according as it tends in the C 1 pioiedvCG0Ogle
ON THE MECHANICAL THEORY OF HEAT. MATHEMATICAL INTRODUCTION. ON MECHANICAL WORK, ON ENERGY, AND ON THE TREATMENT OF NON-INTEGRABLE DIFFEREN'rIAL EQUATIONS. § 1. Definitioo and Measurement 01 Mechanical, Work . . Every force tends to give motion to the body on which it· acts; but it may be prevented from doing so by other opposing forces, 80 that equilibrium results, and the body remains 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 the work performed is represented by the product of the distance moved through, and the component of the force resolved in the direction of motion. This component of force in the line of motion may be positive or negative in sign, according as it tends in the c. 1 Digitized by Coogle
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 overcome 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 kilogrammetre.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 gravity 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 by 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 Googfe
2 ON THE MECILUfICAL 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 quantities 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 overcome 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 indifference; in applied mechanics they are generally the kilogram and the metre respectively, and then the unit of work is called a kilogrammetre. 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 gravity does not come directly into play, can also be expressed in kilogrammetres, by comparing the forces employed with the Rtandard force of gravity. § 2. Mathemattcal determination of tM 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 Digitized by Coogle