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o(p 3. 6 ^ xj , j BOOK 530. W69 v. 1 c. 1 WILSON # THEORETICAL PHYSICS lllllil 3 T153 DD12fiM7fl 7 Digitized by the Internet Archive in 2011 with funding from LYRASIS members and Sloan Foundation http://www.archive.org/details/theoreticalphysi01wils THEORETICAL PHYSICS THEORETICAL PHYSICS BY W. WILSON, F.R.S. Vol. I. Mechanics and Heat. Newton — Carnot. Vol. II. Electromagnetism and Optics. Maxwell — Lo rentz . Vol. Ill, Relativity and Quantum Dynamics. Einstein — Planck. THEORETICAL PHYSICS BY W. WILSON, F.R.S. HILDRED CARLILE PROFESSOR OF PHYSICS IN THE UNIVERSITY OF LONDON, BEDFORD COLLEGE VOL. I MECHANICS AND HEAT NEWTON— CARNOT WITH EIGHTY DIAGRAMS NEW YORK E. P. BUTTON AND COMPANY INC. PUBLISHERS ■c5^3t) (fi PRINTED IN GREAT BRITAIN ^ PREFACE THE purpose of the present work is to present an account of the theoretical side of physics which, without being too elaborate and voluminous, will nevertheless be sufficiently comprehensive to be useful to teachers and students. This, the first volume, deals with mechanics and heat ; the second volume will be devoted to electromagnetism and optics and possibly the introductory part of relativity ; while the remaining volume will deal with relativity and quantum dynamics. The contents of this part are based on the notes of lectures delivered at one time or another at Bedford College and King's College (London). In selecting the subject-matter I have been influenced chiefly by its importance from the point of view of exhibiting the unity of physical theory and in a secondary degree C" by any special interest, historical or other, commending it, or ^|V^by its suitability as a means of preparing the ground for more important things to follow. The unavoidable incompleteness is compensated to some extent by the bibliographical references T' and notes appended to many of the chapters. ^ Each part of the subject is developed in a way which follows, f. broadly speaking, its historical growth, and this first volume is fCs entirely ' classical ', the dynamical part of it being based on the foundations of Newton. At the same time every opportunity • that presents itself is utilized to open the way for the description T of modern developments of physical theory which will occupy -~ later parts of the work. The methods of elementary vector and "C tensor calculus are introduced at the outset and consistently V followed, partly on account of their fitness and utility, and J^ partly as an introduction to a more complete account of tensor calculus which will have a place in a later volume. *^ Care has been taken to make the nature of fundamental ^ principles as clear as possible and everything is developed from ^ the simplest beginnings. No very serious demands are made • on the mathematical equipment of the reader. A certain ac- ^^ quaintance with the elements of the calculus and analytical y^ V vi THEORETICAL PHYSICS geometry is assumed and any mathematical methods which extend beyond this are explained as they may be required. Needless to say, I have derived much assistance from many classical works and papers ; in fact from most of those mentioned in the bibhographical appendices. I am indebted to Dr. Maud 0. Saltmarsh for reading the proofs. W. W. January i 1931 CONTENTS PREFACE GENERAL INTRODUCTION PAGE V CHAPTER I FOUNDATIONS OF EUCLIDEAN TENSOR ANALYSIS SCALAHS, VeCTOBS Aiq^D TeNSORS Scalar and Vector Products . Co-ordinate Transformations . Tensors of Higher Rank Vector and Tensor Fields 4 4 5 9 11 12 CHAPTER II THE THEOREMS OF GAUSS, GREEN AND STOKES. FOU- RIER'S EXPANSION 16 Theorem of Gauss ........ 16 Green's Theorem . . . . . . . .18 Extensions of the Theorems of Gauss and Green . 22 Theorem of Stokes ........ 23 Fourier's Expansion . . . . . . .28 Examples of Fourier Expansions ..... 32 Orthogonal Functions . . . . . . .36 CHAPTER III INTRODUCTION TO DYNAMICS 39 Force, Mass, Newton's Laws ...... 39 Work and Energy ........ 42 Centre of Mass ........ 44 Path of Projectile ........ 45 Motion of a Particle under the Influence of a Central Attracting or Repelling Force . . . .47 Angular Momentum of a System of Particles free from External Forces ....... 48 Planetary Motion ........ 50 Generalized Co-ordinates . . . . . .57 Moments and Products of Inertia ..... 60 The Momental Tensor ....... 63 Kinetic Energy of a Rigid Body . . . . .63 The PENDULuiki . . . . . . . . .65 vii VIU THEORETICAL PHYSICS CHAPTER IV DYNAMICS OF A RIGID BODY FIXED AT ONE POINT Euler's Dynamical Equations . Geometrical Exposition . Euler's Angular Co-ordinates The Top and Gyroscope . The Precession of the Equinoxes page 72 72 78 82 84 91 CHAPTER V PRINCIPLES OF DYNAMICS 92 Principle of Virtual Displacements .... 92 Principle of d'Alembert ....... 95 Generalized Co-ordinates ...... 97 Principle of Energy . . . . . . .100 Equations of Hamilton and Lagrange . . . .102 Illustrations. Cyclic Co-ordinates .... 105 Principles of Action . . . . . . .109 Jacobi's Theorem . . . . . . . .116 CHAPTER VI WAVE PROPAGATION .... Waves with Unvarying Amplitude . Waves with Varying Amplitude Plane and Spherical Waves . Phase Velocity and Group Velocity Dynamics and Geometrical Optics . 121 121 131 132 136 138 CHAPTER VII ELASTICITY 141 Homogeneous Strain ....... 141 Analysis of Strains ....... 146 Stress .......... 153 Stress Quadric. Analysis of Stresses . . . .156 Force and Stress . . . . . . . .159 Hooke's Law. Moduli of Elasticity . . . .162 Thermal Conditions. Elastic Moduli of Liquids and Gases 166 Differential Equation of Strain. Waves in Elastic Media 168 Radial Strain in a Sphere . . . . . .171 Energy in a Strained Medium . . . . .174 Equation of Continuity. Prevision of Relativity . 175 CHAPTER VIII HYDRODYNAMICS .... Equations of Euler and Lagrange Rotational and Irrotational Motion Theorem of Bernoulli The Velocity Potential . JCdstetio Energy in a Fluid 180 180 185 186 189 192 CONTENTS IX Motion of a Sphere through an Incompressible Fluid Waves in Deep Water ...... Vortex Motion ....... page 192 196 198 CHAPTER IX MOTION IN VISCOUS FLUIDS Equations of Motion in a Viscous Fluid Poiseutlle's Formula .... Motion of a Sphere through a Viscous Liquid OF Stokes ...... Formula 203 203 205 209 CHAPTER X KINETIC THEORY OF GASES .217 Foundations of the Kinetic Theory. Historical Note 217 Boyle's Law 218 Laws of Charles and Avogadro. Equipartition of Energy 221 Maxwell's Law of Distribution ..... 225 Molecular Collisions. Mean Free Path . . . 232 Viscosity. Thermal Conductivity ..... 236 Diffusion of Gases ........ 240 Theory of van der Waals ...... 244 Loschmidt's Number ....... 253 Brownian Movement . . . . . . .255 Osmotic Pressure of Suspended Particles . . . 256 CHAPTER XI STATISTICAL MECHANICS 259 Phase Space and Extension in Phase . . . .259 Canonical Distributions ....... 262 Statistical Equilibrium of Mutually Interacting Systems 263 Criteria of Maxima and Minima . . . . .267 Significance of the Modulus . . . . .267 Entropy .......... 270 The Theorem of Equipartition of Energy . . . 270 CHAPTER XII THERMODYNAMICS. FIRST LAW 272 Origin of THERMODYNA]\ncs ...... 272 Temperature ......... 272 Equations of State ........ 275 Thermodynamic Diagrams ...... 276 Work Done During Reversible Expansion . . . 278 Heat 280 First Law of Thermodynamics . . . . .280 Internal Energy of a Gas . . . . . .282 Specific Heat ......... 282 The Perfect Gas 283 Heat Supplied to a Gas During Reversible Expansion . 286 THEORETICAL PHYSICS PAGE CHAPTER XIII SECOND LAW OF THERMODYNAMICS . . . . 288 The Perpetuum Mobile op the Second Kind . . . 288 Cabnot's Cycle ........ 289 Carnot's Principle ........ 290 Kelvin's Work Scale of Temperature . . . .292 The Work Scale and the Gas Scale . . . .296 Entropy .......... 297 Entropy and the Second Law op Thermodynamics . 298 Properties of the Entropy Function. Thermodynamics AND Statistical Mechanics ..... 300 CHAPTER XIV THE APPLICATION OF THERMODYNAMICAL PRINCIPLES 304 General Formulae for Homogeneous Systems . . 304 Application to a v. d. Waals Body ..... 308 Thermodynamic Potentials ...... 309 Maxwell's Thermodynamic Relations . . . .311 The Experiments of Joule and Kelvin and the Realization OF THE Work Scale of Temperature . . . 312 Heterogeneous Systems . . . . . . .317 The Triple Point ........ 317 Latent Heat Equations . . . . . . .318 The Phase Rule . . . . . . . .320 Dilute Solutions ........ 323 INDEX OF SUBJECTS 327 INDEX OF NAMES 331 THEORETICAL PHYSICS §1. GENERAL INTRODUCTION PHYSICAL science, in the restricted sense of the term, is concerned with those aspects of natural phenomena that are regarded as fundamental. Broadly speaking it inves- tigates things with which we are brought into immediate contact through the senses, hearing, touch and sight ; and it includes, among others, the familiar sub-divisions sound, heat and light. But sense perceptions themselves hardly enter into physics. Indeed, they are deliberately excluded, as far as may be, from physical investigations. Spectra are observed photographically and the colour of a spectral line is really not a thing in which the physicist takes any interest. No great effort of imagination is needed to conceive the possibility of photometric devices whereby a completely blind observer might carry out for himself aU the observations on a spectrum which have any significance for physics. Temperature is not measured by feeling how warm a thing is, nor in acoustical investigations do we rely on the sense of hearing. In fact, the use of human senses is practically con- fined, in experimental physics, to the observation of coincidences, such, for example, as that of the top of the mercury column in a thermometer with a mark on the scale of the instrument, or that of the spider line in a telescope with a star or a spectral line, and the associated coincidence which gives the scale reading. Physical science is the cumulative result of a variety of closely correlated activities which have given us, and are adding to, our knowledge of what we shall call the Physical World, and the present treatise is an attempt to present, in outline, a con- nected account of the body of doctrine which has grown out of them. There are three well-defined periods in the development of the theoretical side of physics since the time of Galileo. The earliest of these, which we may call the ' matter and motion ' period, came to an end in 1864 when Clerk Maxwell's electro- 1 2 THEORETICAL PHYSICS magnetic theory of light appeared.^ The physicist of this period conceived the world as built up, roughly speaking, of minute particles (atoms) endowed with mass or inertia and capable of exerting forces (gravitational, electric, etc.) on one another. Their behaviour and mutual interactions were subject to certain djmamical principles, summarized in Newton's laws of motion. A phenomenon was considered to be satisfactorily accounted for when it could be represented as a mechanical process ; when it could, as it were, be reproduced by mechanical models differing merely in scale from something that might be constructed in a workshop. This mechanical physics was extraordinarily successful, and was tenaciously adhered to and defended, even so recently as the opening years of the present century, as the following quotation from the preface to the first edition of an admirable work on the theory of optics ^ will show. ' Those who believe in the possibiHty of a mechanical conception of the universe and are not wiUing to abandon the methods which from the time of GaHleo and Newton have uniformly and exclusively led to success, must look with the gravest concern on a growing school of scientific thought which rests content with equations correctly represent- ing numerical relationships between different phenomena, even though no precise meaning can be attached to the symbols used.' The second period, from 1865 till the opening years of this century, has a transitional character. Maxwell's theory (which, it may be remarked, united the previously disconnected provinces of light and electricity) led eventually to the abandonment of the effort to establish electrical phenomena on the old-fashioned ' matter and motion ' basis and placed ' electricity ' on equal terms by the side of ' matter ' as a building material for the physical world. In the 'eighties indeed the most characteristic property of matter, namely mass or inertia, was successfully accounted for in electrical terms, and attempts began to be made (with some success) to provide a purely electrical basis for theo- retical physics. Distinguishing marks of the present period of theoretical physics (since 1900) are the development of the quantum and relativity theories and the consequent overthrow of the sover- eignty of Euclidean geometry and Newtonian dynamics. These latter, however, retain their practical importance in almost undiminished measure, and it would indeed be inaccurate to ^ It is noteworthy that Maxwell wrote a little book called Matter and Motion which, though he was the inaugurator of a new epoch in physics, presents a very fair picture of, and indicates his sympathy with, the ideals and aims of the earlier period. ^ Schuster : An Introduction to the Theory of Optics (Arnold, 1904). GENERAL INTRODUCTION 3 speak of them as untrue or disproved ; but they now appear as limiting cases of the more comprehensive modem theories. The old problems of the explanation of electrical phenomena in mechanical terms, or of matter in electrical terms, have no longer any significance, and physical theory is approximating more and more to a vast and unified geometrical structure such as was not dreamt of in the philosophy of Euclid or Newton. CHAPTER I FOUNDATIONS OF EUCLIDEAN TENSOR ANALYSIS § 2. SCALABS, VeCTOES AND TeNSOES A PHYSICAL quantity which can be completely specified by a single numerical statement (the unit of measure- ment having once been chosen) is called a scalar. Examples of scalars are electric charge, mass, temperature, energy and so forth. A vector is a physical quantity associ- ated with a direction in space. For its complete specification three independent numerical statements are necessary. The typical example of a vector is a displacement. If a small body or particle is given a series of ° displacements represented by AB, BC, ^ CD, DE (Fig. 2), which are not neces- sarily co-planar, it is obvious that these bring about a result equivalent to the single displacement represented Fig. 2 by AE. The displacement AE is called the resultant of the dis- placements AB, BC, CD, DE. Any vector can be represented in magnitude and direction by a displacement. Examples of vectors are force, velocity, momentum, electric field intensity and so on. If, for instance, four forces are applied to a body (to avoid irrelevant complications we shall suppose them all to be applied at the same point in the body) and if a straight line AB (Fig. 2) be constructed having the direction of the first force and a length numerically equal to it in terms of some convenient unit, and if a second straight line BC be drawn to represent the second force in a similar way and so on ; then the four forces are equivalent to a single or resultant force which is represented in magnitude and direction by AE. The three independent numerical data which are necessary to express the vector completely may be given in various ways. We may, for example, give the absolute value of the vector, i.e. the length of the line AB or BC (Fig. 2), representing it ; in which case we have to give two additional numerical data to fix its direction §21] EUCLIDEAN TENSOR ANALYSIS 5 relative to whatever frame of reference we may have chosen. The three independent data, however they may be chosen, are called the components of the vector. It is usual, however, to restrict the use of the term ' component ' in the way indicated in the following statement : Any vector can be represented as the resultant of three vectors which are parallel respectively to the X, Y and Z axes of a system of rectangular co-ordin- ates. These three vectors are called its components in the X, Y and Z directions. Unless the contrary is stated, or implied by the context, we shall use the term ' component ' in this more restricted sense. If A represents the absolute value of a vector, we shall represent its components by A^, Ay and A^ and refer to it as the vector A, or the vector (A^, Ay, A^). The statement ' A^, Ay and A^ are the X, Y and Z components of the vector A ' may conveniently be expressed in the abbreviated form : A ^ (Ag., Ay, A^). It is clear that when a vector is represented by a line drawn from the origin, 0, of a system of rectangular co-ordinates to some point P, its components are the co-ordinates of P. Besides scalars and vectors we have still more complicated quantities, or sets of quantities, called tensors. A tensor of the second rank requires for its complete specification 9 or 3^ inde- pendent numerical data, which are not necessarily aU different. Just as a vector can be represented by a displacement, so can a tensor of the second rank be represented, in its essential pro- perties, by a pair of displacements. This will be more fully explained later. The state of stress in an elastic solid is an example of such a tensor. It has become customary in recent times to use the term ' tensor ' for all these different types of physical quantities. A scalar is a tensor of zero rank ; it requires for its specification 3° or 1 numerical datum. A vector is a tensor of the first rank, requiring for its specification 3^ independent numerical data and so on. § 2-1. Scalar and Vector Products The inner or scalar product of two vectors is defined to be a scalar quantity numerically equal to the product of their absolute values and the cosine of the angle between their direc- tions. If the absolute values are A and B, and if the included angle is d, the scalar product is AB cos 6. It is convenient to abbreviate this expression by writing it in the form (AB) or (BA). A very important instance of a scalar product is the work done by a force when its point of application is displaced. 6 THEORETICAL PHYSICS [Ch. I If, for example, A represents a force (which we shall suppose to be constant), B the displacement of the point where it is applied, and 6 the angle between their directions, the work done is expressed by AB cos 6 or briefly by (AB). To elucidate the properties of the scalar product it is con- venient to represent the vectors A and B by displacements from the origin O of rectangular co- ordinates (Fig. 2-1). Let the ter- minal points, p and q, of the dis- placements be joined by a straight line, the length of which is repre- sented by t. Then we have t^ = A^ + B^ - 2AB cos 6. Since the co-ordinates of p and q are (A^, Ay, A J and (B^, B^, B^) respec- tively, it is evident that t is the diag- onal of a parallelopiped, the edges of which are parallel to the axes X, Y and Z and equal respectively to \A^ — BJ^, \Ay — By\ and \A^ — Bj, the symbol \x \ being used to represent the absolute value of x. Therefore P = {A, - B,)^ + (A, - B^y + (A, - B,r. If we remember that A^ = AJ-{-A/ + A,^ .... (2-10) and B^=BJ + B/ + B,^ we find on equating the two expressions for t^ Fig. 2-1 ABcos0=:^A+^A+^A . . (2-11) This important result may be expressed in words as follows; — If the like components of two vectors are multiplied together, the sum of the three products thus formed is equal to the scalar product of the two vectors. When the angle between the two vectors is a right angle it is obvious that ^A + ^A + ^A = . . ._ . (2-12) and conversely, when equation (2*12) holds the directions of the two vectors must be at right angles (if we except the trivial case where one or both of the vectors are equal to zero). If we refer the vectors A and B to new rectangular co-ordinates, in which their components are a;, A^', a:, and BJ, B/, B/, the scalar product will now be ajb,'+a;bj+a:b: §21] EUCLIDEAN TENSOR ANALYSIS 7 and we must have A A + AyB, + A A = AjBj + a;b; + a:b:, since the value of the scalar product is clearly independent of the choice of co-ordinates. We have here an example of an invariant, i.e. of a quantity which has the same numerical value whatever system of co-ordinates it may be referred to. The product of the absolute values of the two vectors and the sine of the angle included between their directions is called their outer or vector product. In the case of the vectors A and B (Fig. 2-1) we have vector product = AB sin 9 We shaU usually abbreviate this expression by writing it in the form [AB]. Squaring both sides of (2-11) we have A2B2 - A2B2 sin^ d = {A,B, -F A,B^ -\- A,B,Y or, by (2-10) A^B^ sin2 d = (^,2 _j_ A/ + ^/)(5,2 + 5/ + 5^2) -(^A + ^A+^A)^ On multiplying out, we easily recognize that this last equation is equivalent to A^B^ sin^ e = {A,B, - Afi,)^ + (Afi, - A,B,)^ + (A^,-A,B,)^ . . . .(2-13) Obviously we may change the sign in any of the expressions AyB^ — Afiy, etc. on the right without affecting the equation. This ambiguity is intimately associated with a corresponding feature in rectangular axes of co-ordinates, and it now becomes necessary to give a precise specification of the type of rectangular axes we propose to use. We shall do this in the following terms : The motion of an ordinary or right-handed screw travelling along the X direction turns the Y axis towards the Z axis. In this description the letters X, Y and Z may, of course, be interchanged in a cyclic fashion. It is evident from equation (2*13) that the three quantities, ^x = ^y^z — ^ A' ^y = ^ A — ^x^z5 ^z = ^ A — ^ A can be regarded as the components of a vector the absolute value of which is AB sin d. The question arises : What is the relation between the directions of the vectors a, A and B ? The scalar product (a A) = a^^ -I- GyAy -\- a,A, = (AyB, - A,By)A, + (^A - ABMy + (^ A - A^x)A = identically. 2 8 THEORETICAL PHYSICS [Ch. I It follows that the vector o is at right-angles to A and by forming the scalar product (oB) we can show further that o is also at right angles to B. Let us turn the co-ordinate axes about the origin so that the vectors A and B lie in the XY plane in the way indicated in Fig. 2-11. The components a^ and ay will now be zero, and we see that a^ is positive, since A^By — AyB^ is obviously greater than zero. This means that when the co-ordinate axes are placed in this way relatively to the vec- tors A and B , the vector o will be in the direction of the Z axis, and so we conclude that the motion of an ordinary or right-handed screw travel - Hng in the direction of o turns the vector A towards the vector B. We shall extend the use of the notation [AB] to represent the vector product completely, i.e. both in magni- tude and direction. That is to say [AB] means the vector, the X, Y and Z components of which are respectively AyB, - Afiy, A,B, - A,B„ A^y - AyB,, and [BA] means the vector (ByA, - BAy, BA. - BJL,, B^y - ByA,), which has the opposite direction. The scalar product of any vector G and [AB] is (G[AB]) = ClAB], + Oy[AB]y + ClABl, (G[AB]) =. GMyB. - A,By) + Cy(A,B, - A^ * + CM^y - AyB,), Fig. 211 or (G[AB]) A.. A. Bx, By, B (2-14) Clearly this determinant is an invariant, since a scalar product is an invariant. If again we imagine the axes of co-ordinates to be turned about the origin till the vectors A and B lie in the XY plane, as in Fig. (2-11), the scalar product (G[AB]) becomes C^[AB]^, since the X and Y components of [AB] are both zero. Therefore (G[AB]) = G cos £ AB sin (9 . . . (2-15) where s is the angle between the directions of G and of the Z axis. If therefore e is less than -, the scalar product (G[AB]) §2-2] EUCLIDEAN TENSOR ANALYSIS 9 or the determinant (2*14) is equal to the volume of the parallelo- piped which is determined by the displacements A, B, G. We may formulate this result as follows : If the motion of an ordinary or right-handed screw travelling along the direction of C turn A towards B, then the determinant (2* 14) is equal to the volume of the parallelepiped deter- mined by the vectors A, B and G. Obviously we may interchange A, B and G in cyclic fashion in this theorem. If cos £ in (2*15) is zero, the vectors G and [AB] are at right- angles to one another ; but this means that A, B and G are in the same plane and on the other hand that the determinant (2*14) is zero. In fact, if A, B and G are all different from zero, the necessary and sufficient condition that they shall be co-planar is : = (2-16 § 2-2. CO-ORDIKATE TRANSFORMATIONS Let X, Y, Z and X', Y', Z' be two sets of rectangular axes of co-ordinates with a common origin ; and let P be any point, the co-ordinates of which are x, y, z and x' , y\ z' in the two systems respectively (Fig. 2-2). Let us further represent the cosines of the angles be- tween X' and X, Y, Z, by Z^,, \ and l^ respectively ; those be- tween Y' and X, Y, Z by m^, tYiy and m^ respectively, and so on. The problem before us is : given ic, y^ z, the co-ordinates of P in the system X, Y, Z, to find x\y\z\ its co-ordinates in the other system X', Y', Z', and vice versa. Drop a perpendicular Pm on OX', so that Om is equal to x' , the X' co-ordinate of P in the system X', Y', Z'. We may regard both OP and Om as vectors and we have clearly ?^;jJ;^;of)systemX',Y',Z'. The rule (2*11) gives us for their scalar product the alternative expressions, 10 THEORETICAL PHYSICS [Ch. I x\x + x'l^y + x'l^ (System X, Y, Z) and x'^ (System X', Y', Z'). On equating these two expressions, and dividing by the common factor x', we finally obtain x' = l^x + l^y + %z. In a similar way we may show that y' = m^x + m^y + m,z, z' = n^x -j-riyy '\-nz^. - - - - \ ) The equations of the inverse transformation are easily found to be X = l^x' + m^y' + n^z' y = l^x' +m^y' + n^z' .... (2-21) z = l^x' + my + n^z\ We may, evidently, regard l^,ly and l^ as the components of a unit vector (i.e. a vector the absolute value of which is unity) in the system X, Y, Z. A similar remark applies to (m^, m^, mj and {n^, Uy, n^). And in the system X', Y', Z' we may regard (h^ ^x. ^x). ik^ ^y' ^1/) aiid (?„ m„ rij as unit vectors. For many purposes it is convenient to represent these direc- tion cosines by a single letter, distinguishing one from another by numerical subscripts, thus : (^x5 ^y ^z) ^ (^iij ^12) Ctiajj (n^, Uy, n,) = (a. All six equations of transformation given above are con- veniently represented in the following schematic form : — . (2-22) Mathematically a vector may be defined as a set of three quantities v^hich transform according to the rules em- bodied in (2-22). There are certain important and interesting relations between the direction cosines a. For example, the sum of the squares of the a's in any horizontal row, or in any vertical column of (2*22) is equal to unity : X y z x' ail ai2 ai3 y' ^21 "22 "23 z' ttsi "32 ^33 etc. ail + aia^ _|_ a^32 ^ j^ ai2^ + a22^ + asa^ = 1 . (2-23) §2-3] EUCLIDEAN TENSOR ANALYSIS 11 The correctness of these equations is obvious, since in each case the left-hand member can be regarded as the sum of the squares of the components of a unit vector. Further, the sum of the products of corresponding a's in any two horizontal rows, or in any two vertical columns is zero, e.g., aiittis + OLiiCn^z + a3i«33 = 0, a2ia3i + a22a32 + a23a33 = ... (2-24) and so on. These equations follow since the left-hand member in each case can be regarded as the scalar product of two unit vectors which are at right angles to one another. Finally we have the relation = 1 (2-25; Ctll, «12j 0^13 0^21? Ct225 Ct23 «31j «S2j Ct33 since by (2*14) this determinant represents the volume of the paraUelopiped bounded by the three mutually perpendicular unit vectors (Z^, ly, \); (m^, m^, mj and (n^, riy, n,), § 2-3. Tensors of Higher Rank We are now able to define more precisely a tensor of higher rank than a vector. Take, for example, a tensor of the second rank, such as that which expresses the state of stress in an elastic solid. It is a set of 3^ quantities, called its components, Pxx^ Vxy^ Vxzy jPyxi jPyyi Pyz) Pzx^ Pzyj Pzz^ having the property that the values of the components p^', p^y, etc., in the system X'Y'Z' are calculated from p^^, p^y, etc., the components in the system X, Y, Z, by precisely the same rules as those for calculating AJBJ, AJBy, etc., from the pro- ducts AJB^, AJBy, etc., where A^, Ay, By, etc., are the components of two vectors. A tensor of the second rank is said to be sym- metrical when the subscripts of a component may be interchanged, e.g., when Pxy JPyx The system of stresses in an elastic solid in equilibrium consti- tutes such a symmetrical tensor. If, on the other hand, jPxy Pyx^ the tensor is said to be anti-symmetrical. Since in this case, Pxx Paxa^ the components p^, Pyy, etc., with two like subscripts wiU all three be zero. As an example of an anti-symmetrical tensor we 12 THEORETICAL PHYSICS [Ch. I may instance that formed from two vectors A and B in the following way : — AA -AA, AA - A A A A -AA AB. -AJi,, AyB, - A A. A A - ^ A. AB. -AA AA - A A. AA - ^A- Its XX, YY and ZZ components are zero and the remaining six are the components of the vectors [AB] and [BA]. In fact we may dispense with these vector products by employing this tensor. More generally, if we have n vectors and select one component of each and multiply them together, the 3^* products obtained from all the possible selections constitute a typical tensor of the nth rank, and any set of 3'^ quantities will constitute a tensor of rank n if they obey the same laws of transformation as the 3'* components of the typical tensor. § 24. Vector and Tensor Fields We shall often be concerned with regions in which electric, magnetic or gravitational forces manifest themselves. We call such regions fields of force. They are characterized in each of these examples by a vector which varies continuously from point to point in the region and which may be termed the intensity of the field. In hydrodynamics we are concerned with regions filled with a fluid, the motion of which can be described by giving its velocity at every point in the region. In all these examples we may use the general term vector field for the region in question. Or we may be concerned (e.g. when we are studying the state of stress in an elastic solid, or the Maxwell stresses in an electrostatic field) with the components of a tensor of higher rank than a vector and with the way in which they vary from one point in the field to another. In such a case we may call the region a tensor field. The description and investigation of vector or tensor fields involves the use of partial differential equations, and we shall O O O therefore study some of the features of the operations -^^ ■^, -^, ox cy cz where the round cZ's are the conventional symbols for partial differentiation, i.e. -^r- means a differentiation in which the other ox independent variables y, z and the time are kept constant. In O o o the first place we may show that -^^, ^r-, ;:-- have the same trans- Bo; 31/ dz formation properties as the X, Y, Z components of a vector. §2-4] EUCLIDEAN TENSOR ANALYSIS 13 Let ^ be any quantity which varies continuously from point to point. Then by a well-known theorem of the differential calculus, dx' dx dx' dy dx' dz dx' But by (2-22) X = a^^x' + aay + o.^^z\ y = aiao;' + aaa^/' + ctsi^', z = ttiso;' + a^sy' + cLsz^', therefore dx _ ^y _ ^^ _ and on substituting in (2 •4) we have dcf) _ ^</> I ^^ \ ^^ cx ox dy oz or, dropping </>, we have the equivalence, ^-, = ttii^ + ai2^ + ai3-- . . . (2-41) ox ox oy oz This is sufficient to establish the vectorial character of these operations. If A = {Ay., Ay, A^) is a field vector, the quantity dA^ dAy dA, dx dy dz wiU be an invariant since it has the same transformation pro- perties as a scalar product. It is called the divergence of the vector A and is written div A. Furthermore the three quantities dA^ dAy dA^^ dA^ dAy dA^ dy dz ' dz dx ' dx dy must be the X, Y and Z components of a vector, since they have the same transformation properties as the components of a vector product. This vector is called the curl of A or the rotation of A and is written curl A or rot A. It is easy to show that div curl A = 0, . . . . . (2-42) where A is any field vector. We have in fact, r) 7) r) div curl A = — {curl A}^ + {curl A}^ -|- ^{curl A}^ ox cy oz 14 THEORETICAL PHYSICS [Ch. I d* 1 A = —f—' - —A -4- —f^-^ - ^^A dx\ dy dz ) dy\ dz dx ) 'bzx dx dy d (dA^ _ dA^\ It will be seen that this is identically zero. If A = (^^, Ay, Ag) is a vector and ^ any scalar quantity, it is obvious that iA^, ^,0, A,ct>) is a vector. Similarly /dcj) dcf) 8</>\ \dx' dy' dz) is a vector. Such a vector is called the gradient of the scalar quantity and is written grad ^. We have therefore ^^^'^-(^i'%t)- ■ ■ ■ t^-^' The components of the vector curl grad ^ are all identically zero. Take the X component for example : {curl grad ^ L = ;g- (grad */» L - ^ (grad ^ }y d {dci>) d(dci>' {curl grad ^L = ^^^^^^ a.^a^ = identically. We may write this result in the form curl grad ^ = . . . . . (2-431) rl r) f) The quasi- vectorial character of the operations ^r-, -^r-, yr- dx dy dz makes it often a convenience to represent them by the symbolism used for vectors. We shall frequently denote them by the symbol V (pronounced nahla), thus V- (V. V. V.) - (4 |, I) and therefore grad (/> ^ (V.^, V,*^, V.^.) or grad </» = ^^, and div A ^ V.^4^ + V,^, + V«^. ^ (VA). The quantity dx^ "^ dy^ '^' dz^ is called the laplacian of the scalar (/> in honour of the great French mathematician Laplace. Our notation enables us to represent it by V^^* §2-4] EUCLIDEAN TENSOR ANALYSIS 15 A useful formula, frequently used in electromagnetic theory, is the following : — div [AB] = (B curl A) - (A curl B) . (2-44) This can be proved by writing out div [AB] in full. div [AB] = ^[ABL + |[AB], + 1[AB1. The first of the three terms on the right expands to ,dB dA, dB, dA, and the remaining two terms on the right give jS£ » dA,_dB,_dA, 'dy "^ "^ dy "Sy 'Bf The pair of terms, marked o, taken together make 5Jcurl^}^. Similarly, we find a pair of terms equivalent to j5Jcurl^}^, and another equivalent to J5,{curl^}„ so that six of the terms make up (B curl A). In the same way the remaining six are seen to make up - (A curl B) and thus the formula is established. An equally important formula is : curl curl A = grad div A — V^A . . (2'45) which we can likewise establish by writing out the. left-hand side, or the X component of the left-hand side, in full. (curl curl A> - ^ 1^^' - ^^4 - ^ i^^" - ^^' icurlcurl A}, -g-|^ -g-| g-|-g- — {curl curl A}. = ^{-^ + -^) - (-g^^ + ^^), and if we add and subtract -;r— ^ on the right-hand side we get {curl curl 4}. = - div A - (-^^^ + ^^ + ^), a result which may be expressed in the form (2*45). or CHAPTER II I THE THEOREMS OF GAUSS, GREEN AND STOKES. FOURIER'S EXPANSION § 3. Theorem of Gauss MAGINE a closed surface, ahc, Fig. 3, and a field vector (A^, Ay, A^) which varies continuously throughout the volume enclosed by it. We shall investigate the integral \\\ div A dxclydz (3) It is important to grasp the precise meaning of this integral. We suppose the whole volume ahc divided into small elements and each element of volume mul- tiplied by the value of div A at some point within it. The integral (3) is the limit to which the sum of all the products so formed approximates as the ele- ments of volume become indefinitely small. It is not essential that the elements of volume should be rectangular, or that the sum should be expressed by the use of the triple symbol of integra- tion. We may write (3) in the form FiQ. 3 div A dv (3-001 where dv represents an element of volume of any shape. It is convenient, however, to use the triple symbol when we wish to draw attention to the 3-dimensional character of the region over which the integration extends. From the definition of div A (§ 2-4) we have for (3) 16 § 3] THEOREMS OF GAUSS, GREEN AND STOKES 17 HI w+is' + wl**"' • ■<»»2> so that it may be treated as the sum of three integrals. We shall begin with -^dx dy dz, dx ^ ' and carry out the integration or summation over all the elements of volume from 1 to 2 (Fig. 3) in a single narrow vertical column with the uniform horizontal cross-section dy dz . For this restricted volume we have nr I -^^^ ^y ^^ = dy dz -^dx =^dydz{{A,),-(A,),} . . . .(3-003) where (A^)i and (A^)2 are the values of A^ at the terminal points 1 and 2 respectively, where the vertical column cuts the surface abc. Let the elements of area at the two ends of the vertical column be (dS)i and (dS)2. It is helpful to imagine short perpendiculars erected on the surface at the points 1 and 2 and directed outwards, each perpendicular having a length equal to the area dS of the corresponding element; (see (dS)i and (dS)2 in Fig. 3). These perpendiculars may be regarded as vectors with the absolute values (dS)i and (dS)2. Let ^i and ^2 he the angles between the directions of the vectors (dS)i and (dS)2 respectively and the X axis. We have then (dS)2 cos <^2 = dy dz, — (dS)i cos (^1 = dy dz, or (dS^)2=dydz, — (dS^)i =dydz. Substituting in (3*003) we get 2 dy dz j ^ydx = (A,d8,), + {AjiS,), 1 or, otherwise expressed, the integral \\\ — — ^ dx dy dz. dx ^ ' when extended over such a vertical column, is equal to the sum of the products A^dS^, where the surface abc is cut by the column. When the integral is extended over the whole volume abCy i.e. over all the vertical columns in it, we get the sum 18 THEORETICAL PHYSICS [Ch. II of the products AJIS^ for aU the elements of area making up the surface. Therefore ^^^^^ dx dy dz = ^^AM., where the summation on the left extends over the whole volume ahc, and that on the right over the whole surface ahc. Similarly we have, IIl^' ^-^ dy dz = ^JA,dS„ and I j I ^ tZa; c?!/ ^2 = | | ^2^^2- Adding these three equations, we get r r {AJS^ + AydSy + A,dS,} If' or [[[div A dxdydz = [f(A dS) (3.01) where (A dS) on the right hand is the scalar product of the vectors A = (A^,Ay,A^) and dS = {dS^, dSy, dS^). Equation (3'01) expresses the theorem of Gauss. § 3-1. Green's Theorem Let the vector A in (3 '01) have the form U grad V, where U and V are scalars, which, with their first and second differential quotients, are continuous functions of x, y and z in the volume abc. We thus have [jfdiv {U grad V) dxdydz = ff(C7 grad F, dS), or {{{w^W dxdydz -{- [[[(grad U, grad V) dxdydz = \\{U grad V, dS) . (3-1) Interchanging U and F in (3'1), we get I [ [ Vy^U dxdydz + {{{ (grad U, grad F) dx dy dz = \\{V grad U, dS) . . (3.11) § 31] THEOREMS OF GAUSS, GREEN AND STOKES 19 and on subtracting (3*11) from (3'1) we obtain I [ f {UyW - YTJ^'V) dx dy dz = {{(U grad V, dS) - {{{V grad U, dS) (3-12) This result, known as Green's theorem, was published in 1828 by George Green in an epoch-making work entitled An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. If we represent distances measured in the direction of an outward normal to the surface abc by the letter n, the normal component of grad V, i.e. the product of grad V and the cosine of the angle between its direction and that of dS or of the normal, is dv dn so that (3*12) may be written in the form \\i ^l-'sl-^ • ■ ■ <™' {C/y^F - Vy^U}dxdydz In this equation let the value of U at any point be equal to - where r is the distance of the point from the origin. If then the origin is outside the volume abc over which the triple integral is extended, we have from (3*13) Since V^- can be shown, as follows, to be zero. We have namely r ii) _ 1 dr dx r^' dx 2 /ar\2 1 av therefore ^^, ^,^^^ Now r^ = x^ + y^ + z^, dr therefore 2r^r- = 2x ) -^.S- • ■ ■''■»'> or dx dr _x dx T 20 THEORETICAL PHYSICS [Ch. II Further 8V CX' _ ^^ 1 dr 9V Substituting these expressions for — and ^-^^^ (3 •141) we get vx ox or Similarly and \rj 2x^ dx^ y.5 "(i) Sx^ 1 dx^ y5 ^3. -<) 32/2 1 dy^ J.5 f3' <) _ _ 3«2 1 On adding the last three equations, we find dx'' + dy'' + 3^2 _ 3(0;^ + ^2 _|_ ^2) _ 3_ _ Q^ ^5 y 3 Let us apply (3'14) to the case where the volume integration X Fig. 31 extends over a region like that indicated by the shaded part of the diagram in Fig. 3'1. This region is enclosed between the § 3-1] THEOREMS OF GAUSS, GREEN AND STOKES 21 surface abc and the surface of a sphere of small radius, R, having the origin for its centre. The surface integration is now extended over the surface ahc and over the surface of the small sphere as well. At points on this latter surface dn dr' since the direction of the outward normal is exactly opposite to that of r. Similarly 1 '© '. I -lUf-^ii; dn dr B^ Therefore the part of the surface integral of (3* 14) extended over the small sphere may be expressed as 'dS OT7" OTT" where __ and V are average values of -^ and V respectively over the surface of the sphere. This part of the surface integral is therefore equal to - 4.7zB-^ - 471 F, dV — and since -^ and V are continuous it will approach the limit - 47rFo as B approaches zero, if Vq is the value of V at the origin. We have therefore dl - F-^ \dS . . , (3-15) dn ) In this formula the surface integral is extended over the outer surface ahc of Fig. 3-1, and it is understood that the volume integral now means not merely the result of integrating over the 22 THEORETICAL PHYSICS [Ch. II shaded volume, but the limit approached by this integral when R approaches the limit zero. Imagine the surface ahc to be enlarged, so that the distance r of any point on it from the origin approaches infinity, or so that - approaches zero ; then it may happen that the surface integral also approaches zero in the limit. It is easily seen that this must happen when V diminishes in the same way as - at great dis- tances from the origin, that is to say, when the product rV never exceeds some finite number, however great r may be. For '© - -— and V_SLL are both of the order of magnitude of —, r en ^n ^ whereas the area of the surface is of the order of r"^. In such a case (3*15) becomes where the integration is extended over all space. § 3-2. Extensions of the Theorems of Gauss and Green If A, B and C are three vectors, the quantity c^A + c^ji, + CAA is the X component of a vector, since OA + C,B, + CA = (CB) is a scalar or invariant quantity. Now A^B^, A^By and A^B^ are the XX, XY and XZ components of a tensor of the second rank and it follows that ^x-^ XX I ^y-'- XV ~r ^z-^ XZ is also the X component of a vector, if G is any vector and T^y, etc. any tensor of the second rank. This will be understood when it is remembered that the components of tensors are defined mathematically by their transformation properties (see §§ 2*2, 2*3). We may similarly infer that y x^ XX ~r y y-^ XV "T" y «-* xz^ dT dT dT or ^^ _j_ XV I ^^ XZ dx dy dz is the X component of a vector and dT dT dT ^-^ yx I ^-^ yy i ^^ yz dx dy dz §3-3] THEOREMS OF GAUSS, GREEN AND STOKES 23 and -^ + -^ + -^ ox oy dz are respectively the Y and Z components of the same vector. It is usual to extend the scope of the term divergence to include this vector. Therefore div T ^ (^-^ ^^J:^ + Em,Em + ^lm +^J^, \ dx By dz ' dx dy dz dT^^ , dT^,, , dT, zx I ^^ zy + ^^ + ?).... (3-2) dx dy dz The method employed to deduce the theorem of Gauss can be applied to prove the statement : — ill ^+^"+^[c?x%(^« ^^{T^dS^ + T^dS^ + TJS,} . . (3-21 Green's theorem, and the formulae deduced from it, naturally admit of a similar extension. We can, for example, deduce the equation F^==-^^^^^'dxdydz . . . (3-22 which corresponds to (3*1 6) and in which F^ means the value of F^ at the point r = 0. The validity of this formula is subject of course to conditions strictly analogous to those which apply in the case of (3'16). § 3-3. Theorem of Stokes It has been shown already (2*42) that div curl A is identically zero and therefore III div curl A dx dy dz = 0, the integration being extended over any volume within which the vector A and its first derivatives are continuous. Now applying the theorem of Gauss we get II (curl A, dS) = . . . . (3-3) the integration being now extended over the bounding surface (abed J Fig. 3-3). Imagine the surface abed to be divided into 3 24 THEORETICAL PHYSICS [Ch. II two parts by the closed loop a^yd. Equation (3*3) may be written [[(curl A, dS) + jl (curl A, dS) = . (3-301) ab c d Where ah indicates the part of the integral over the portion ab of the surface to the left of the loop a^yd and cd the part extended over the portion to the right of the loop. Now suppose Fig. 3-3 Fig. 3-31 W the surface cd to be replaced by another surface ej with the same boundary line apyd. We shaU have [[(curl A, dS) + [[(curl A, dS) = (3-302) ah ef From (3-301) and (3-302) we have (curl A,dS) = [[(curl A4S), c d ef and therefore the value of the integral can only depend on the values of the vector A along the curve a^yd which forms the boundary of the surface cd or ef. This suggests the problem of expressing (curl A,dS) in terms of what is given for points on the boundary a^yd (Fig. 3-31) of the surface. Let us construct two sets of lines on the surface, each set containing an infinite number of lines. The first set, which we shall call the d lines, all begin at a common point (1 in Fig. 3-31) and all end at a common point (2 in Fig. 3-31). We shall suppose them to be sensibly parallel to one another in any small neigh- bourhood. The element of area between any two adjacent d lines may be called a d area. The second set of lines, which may be termed 5 lines, are so drawn as to divide the d areas § 3-3] THEOREMS OF GAUSS, GREEN AND STOKES 25 into infinitesimal parallelograms, the area of any one of which may be symbolized by dS. The increment of any quantity </>, as we travel along a d line in the direction 1 to 2 (shown in Fig. 3-31 by an arrow), from one 8 line to the next, will be represented by dcj). In the same way dcj) will represent the increment of ^ which occurs in travelling along a 5 line, in the direction indicated by the arrow, from one d line to the next. If the letter I be used for distances measured along any of these lines, an element of area dS will be equal to 61. dl. sin d (see Fig. 3-32). As usual we shall regard dS as a vector and write dS = [61, dl] We can visualize dS as a short displacement perpendicular to the surface of the element and directed away from the reader. We evidently have dS^ — dy dz — dz dyA dSy = dz dx — dx dz,> (3'31) dS^ = dxdy — dy dx.) In these equations 51 = (dx,dy,dz)\ /a.crtn dl = (dx,dy,dz)f [6 6ll) Let the X co-ordinate of the point 1 (Fig. 3-32) be x ; the X co-ordinate of the point 3 will be a; + dx, and as we pass from 3 to 4 we realize that the X co-ordinate of the point 4 must be X -}- dx -{■ d{x + dx), or x -{- dx -\- dx + ddx .... (3'312) If we travel from the point 1 to the point 4 by way of the point 2, we find the X co-ordinate of 2 to be a; -f dx and that of 4 to be X -{- dx -\- d(x -\- dx), or X -\- dx -{- dx -{- ddx .... (3*313) Both of the expressions (3'312) and (3*313) represent the X co-ordinate of the same point, and it follows that ddx = ddx (3-314) This means that the operations d and d are interchangeable, at any rate when applied to the co-ordinates. The integral (curl A,dS) over a surface bounded by 28 THEORETICAL PHYSICS [Ch. II the line a^yd may be treated as a sum of integrals, each ex- tended over a d area. A typical d area is shown in Fig. 3-33. The surface integral over it may be written J {[curl AldS, + [curl AldSy + [curl^L^^J, in which the double symbol of integration has been dropped, since we are now dealing with the sum of a singly infinite set of elements extending over the d area from 1 to 2. Suppose the integrand to be written out in fuU, using the definition of curl in § 2-4 and equations (3'31). The part of the integral involving A^ only is 2 -^((32 dx — dx dz) — -~\dx dy — dy dx) , Fig. 3-33 J [11% + 'H" dy Adding and subtracting -^dx dx, ox + -^dz\dx . dz j J we get for this part of the integral [SA^dx — dA^dx}. Now add to this result the integral 2 I d{A,dx) ■which is equal to zero, since dx vanishes at the points 1 and 2. §3-3] THEOREMS OF GAUSS, GREEN AND STOKES 27 We have therefore for the part of the integral under investigation, 2 [SA^dx + d{AJx) — dAJx] 1 2 = j [dAJx + A^ddx], 1 2 = j [dA,dx + ^,^6^a^], 1 2 1 Now this expression is the difference of two line integrals, namely 2 2 AJ.X - A^dx, LI Rl the one distinguished by the letter L being taken along the left- hand boundary of the d area and the other, marked R, along the right-hand boundary. The difference can be expressed as the line integral 1 AJx, taken right round the d area in a clockwise sense. When we take terms involving Ay and A^ into account we find the surface integral j I (curl A,dS) over the d area to be equivalent to the line integral (A^dx + Aydy + A^dz) taken round it. Otherwise expressed jj(curl A,dS) = j (A,dl), dl being a vectorial element of length along the boundary of the d area. Finally the integral (curl A,dS), when extended over the whole surface, is equivalent to the sum of all the line integrals, (A,dl), taken round all the d areas of which the surface is made up. This sum must be equal to the line integral 28 THEORETICAL PHYSICS [Ch. II taken round the boundary apyd, since along every other line numerically equal integrals are extended in opposite senses. We thus arrive at the important result known as the theorem of Stokes, ^(Adl) = [[(curl A,dS) (3-32) The line integral is extended, as the symbol ^ is meant to indicate, round the boundary a^yd of the surface over which the integral I is extended. itegral I i If we imagine a screw turning in the sense in which the line integral is taken round the boundary, the vectorial elements of area dS will be directed to the side towards which the screw is travelling. §4. Fourier's Expansion A very extensive class of functions can be represented, for a limited range of values of the independent variable, by the sum of a series of trigonometrical terms. If cj) be the independent variable and /(</>) the function, we have f((f>) == ^0 + ^1 cos <^ + ^2 cos 20 4- ^3 cos 3^ + . . . + jBi sin ^ + ^2 sin 2(/> + JSg sin 30 + . . . (4) The coefficients are defined by the equations, i^\mdr, + n A^^ =- f{r) cos ^T cZr, >^ = + 1 to + 00 . (4*01) + n B^ —- \ /(r) sin nr dr n J where n may have all integral values from 1 to oo. The sum of the series (4) will correctly represent the function /(0), subject to a qualification given below, for all values of between — 7t and + 71. The expansion is due to Jean Baptiste Fourier and wiU be found in his TMorie Analytique de la Chaleur published in 1822. Its validity was established by Lejeune Dirichlet in 1837 for aU one- valued functions of the type which can be represented graphically. A discussion of the validity of Fourier's expansion (4) is beyond the scope of this book ; but if we accept §4] THEOREMS OF GAUSS, GREEN AND STOKES 29 the validity, it is easy to prove that the coefficients are those defined by (4*01). In evaluating any integral, such as +^ /(t) cos {nr)dr, — IT we have simply to make use of the relations : COS 2 (nr) dr — 71, + ?r sin^ (nx) dx = n, + rr J COS (nr) sin (mr) dr = 0, n — m, or n j^ m, (4-02) + 77 COS (nr) cos (mr) dr =0, n ^ m, + T sin (nr) sin (mr) dr = 0, n ^ m. In Fig. 4 the abscissae are the values of the independent variable, ^, and the ordinates those of a function, /(</>), which -37r ^ ^ Fig. 4 may be given quite arbitrarily. Whether the function is periodic or not, the expansion will represent it correctly between — tz 30 THEORETICAL PHYSICS [Ch. II and + n, with the possible exception of the points — n and + 71 themselves. Outside this range of values of the independent variable, it is evident that the arbitrarily given function /(<^) cannot in general be equivalent to the sum of the series (4), since the periodic character of the cosine and sine terms neces- sitates that the values of the function in the interval — n to + n will be reproduced by the expansion in every further interval of 271, e.g. from 7i to Stt, or from — Stt to — n. This is indi- cated in the figure by the broken lines. We notice that the sum of the series (4) may approach two different limiting values from the two sides of the points —n and 4- 7i, and it can be shown that the result obtained by sub- stituting — jr or + TT f or (^ is the arithmetic mean of the two limiting values. If we wish to expand an arbitrary function \p{x) in a series of cosine and sine terms which wiU be valid for any prescribed range of values of x, we can quite simply reduce the problem to the one we have discussed by introducing a variable , 71 and we shall arrive at a result which is valid for values of x between — L and -\- L. When the expressions (4'01) for the coefficients A and B are substituted in (4), Fourier's series takes the form /(^) = — f{r)dr + - y^ /(r) cos nr cos ncjxir — IT n=+l — XT 4- - y f(r) sin nr sin n(j)dr, or W= + l — TT + 1T n=+QO +77 — 77 n = + l —77 Since cos X = cos (— x) we may evidently write (4*03 ) in the alternative form + 77 n= — 00 +77 §4] THEOREMS OF GAUSS, GREEN AND STOKES 31 and on adding (4*03 ) and (4*031) we get + 1T «= + ltO+00 +77 m4') =l\ mdr +1 2^ J fir) cos n{r - 0)rfr, — 77 n=— ItO— 00 —77 and hence n= + CO +77 M)=-^ Z j/Wcos7i(T-c^Mr. . (4-04) r by and (j) by n= — oo —IT If now in this formula we replace (7 » a 1 a where a is a positive constant, we shall have /(r) cos n(r — (b)dr =/( - ) cos -{a — w)da.-. \a/ a a Write - = A, and suppose a to be very large, approaching oo in the limit. Then - becomes dX and the summation, 2", with a respect to n becomes an integration, , with respect to X. Therefore, if we write F{a) for /(-) and F{xp) for /(- ), and observe that the limits for a = ar must be — oo and + oo, we arrive at the interesting result + 00 +00 F{^^)) = ^ [ [ F{a) cos A((7 - tp)dadX . . (4-05) — 00 —00 which is known as Fourier's Theorem. The derivation just given is not rigorous, but it shows the connexion between the theorem and Fourier's expansion. It can be proved to be valid for arbitrary functions of the type that + 00 can be exhibited graphically, provided the integral F(a)da is — 00 convergent. The arbitrarily given function F(\p) may have a finite number of discontinuities, i.e. there may be a finite number of values of ^ at each of which F(\p) has two limiting values. For these values of \p the integral in (4'05) gives the mean of 32 THEORETICAL PHYSICS [Ch. II the two limiting values and in this respect is like Fourier's expansion. § 4-1. Examples of Fouribii Expansions When/(^) in (4) is equal to ^, all the coefficients, A, vanish, since the value of the function merely changes in sign when the sign of ^ is changed. Therefore M) =4>= ^^n sin ^^. By (4-01) + 7r '■'^l\ T sin nr dr. This gives on integrating by parts, j5„ = — - cos nn, n and therefore ^ = 2{sin ^ - J sin 2^ + i sin 3(?f» h . • .} . (4-1) The sum of the series (4*1) approaches the limit tt as ^ approaches n from below, and the limit — tt as approaches — n from above. On account of the periodicity of the terms it will also approach the limit — tt as </> approaches n from fi^) -^ Fig. 4-1 above. There are therefore two limiting values of the sum of the series at n and at every point nn where n is odd. When however we substitute nn for </> the sum of the series is found to be zero, which is the arithmetic mean of — tt and + ^ (Fig. 4-1 ). §4-1] THEOREMS OF GAUSS, GREEN AND STOKES 33 If we wish to represent ip{x) — x by a trigonometrical series in an arbitrarily given interval, e.g. X we may substitute tt- for <^ in (4'1) thus nx ^( . nx 1 . ^nx , 1 . Stzx _ = 2jsm-^ --sin 2^ + _ sm — - + . . and so we get 2L( . Jtx 1 . ^Ttx . 1 . ^nx . I /A ii\ When and the function is an even one and the coefficients B vanish (Fig. 4-11). Therefore f(<t>) =Ao + ZA,, cos nct>. 1 Equations (4*01) give us or and ^0 -2' ^^ = -j — r COS nrdr ■}- \ r cos nt dr Integrating by parts, we find 2 A,::^~{cosn7t-l), 34 Therefore and THEORETICAL PHYSICS [Ch. II = 0, A,= - A,= - and so on. 2 " M) 4 4 ¥^' 4 Therefore -jcos <J^ + 3-2 cos 3^ + — cos 5^ -f . . .} (4-12) The sums of the series (4-1) and (4-12) are equal when ^ ^ <[ itt. If f{cl>) = -l, ^<0, Fig. 4-12 (Fig. 4-12) we find the expansion sin If and (Fig. 4-13) M)- <j> + -sin B<j> + -sin B<j> + . . .1 im = - 1, (4-13) ^ 2' ^>I' i'-l<^<i' we obtain 4(1 1 = -j cos — - COS 3^ + - cos 50 h (4.14) 7p 7T Fig. 4-13 §41] THEOREMS OF GAUSS, GREEN AND STOKES 35 As a further example take the case, illustrated in Fig. 4-14, where Fio. 4-14 M) =n -4>, m = - (TT + 4,), (4.15) The function is an odd one and hence the coefficients A are all zero, while Bn = -V- sm n-, so that the appropriate expansion is 4 f 1 1 /(</>) = -| sin ^ - 3-2 sin 3</) + - sin 5^ - + . As a concluding example we may take the function /((/,) = — sin (/>, —n <,(l)<0, M) = + sin 9f>, O^cj^^n, (see Fig. 4-15), for which we find the expansion 2 _ 4|cos 2(f) cos 4^ cos 6(^ ^ S rT3~" 3-5 5-7 '* /(^) (4.16) Fig. 4-15 In all the examples where points of discontinuity occur, e.g. at ^ = in (4.13) and at ^ = - ^ and ^ = + - in (4.14), it 36 THEORETICAL PHYSICS [Ch. II may be verified that at such points the sum of the series, which is of course a one-valued thing, is equal to the mean of the two limiting values of /((/>) at the point in question. We obtain interesting verijfications of the formulae given above when we substitute special numerical values for </>. For example, if we substitute the value - for ^ in (4'1) or in (4* 13), or if we substitute the value in (4*14) we get - = 1-1 + 1-1 + -... 4 3 5 7 If we put ^ = in (4'12), or ^ = - in (4«15) we obtain -^ = 1+1 + 1+1+... Both of these formulae are well known, and can easily be arrived at in other ways. It is possible, and often convenient, to give the expansion (4) the form v =+co m = Z ''^'"' • • • • (4-17) !/ = — 00 This is effected by the substitutions cos n(j) sin n(^ 2 2i ' where i is the usual abbreviation for V — 1. The series (4'1), for example, when expressed in this way, becomes ^ ^2 3 _!_ i e-i4> — ie-i20 _J_ l.Q-iS'i^ h . . . 2 3 § 4-2. Orthogonal Functions The most important property of the trigonometrical functions in a Fourier series is that which finds its expression in equations (4*02 ). An aggregate of functions like cos nr and sin nr with this property is called a system of orthogonal functions, because of the analogy between the equations (4'02) and the equations (2*23) and (2*24) which give the relations between §4-2] THEOREMS OF GAUSS, GREEN AND STOKES 37 the direction cosines of mutually perpendicular lines. The analogy becomes more obvious if we adopt the functions cos {m) , sin (nr) Vn Vn instead of cos (nr) and sin (nr), for then the non-vanishing integrals of (4*02) are When the non- vanishing integrals are thus modified so as to have the value unity, the orthogonal functions are said to be normed. For the interval — n to + jt the normed trigono- metrical orthogonal functions are cos (n^ ^^^ nm (nj,) ^ n=^\,^,...oo Vn V, 7t and to these we may add the constant ^ , since V27t — TT In terms of the normed functions, the Fourier expansion becomes ^/JLN / 1 \ I cos cf) cos 20 , + ^,E2i + ^,EL|^+ ... (4-2) the coefficients being given by "^^iHvEy^' — TT 71 = 1, 2, 3 ... 00. ^„= j/(.)j!iL|i)j^., 38 THEORETICAL PHYSICS [Ch. II Let us now introduce the notation, V2n Vn E^i^) = £2!i, EM) = !HLP, Vn V n ^ J cos (m^) „ sin(m + l)<^ Vn Vn The Fourier expansion now becomes m=Zc,EM) .... (4-21) and since the integral relations (4*02) now have the form j E^(r)EJr)dr = I, n = m, = 0, 71=^ m . . . (4-22) the coefficients c^ are given by + 77 On = ^^nr)E,(r)dr, n = 0, I, 2 . . . oo . (4-23) — TT Systems of orthogonal functions are of course not limited to trigonometrical functions. A system of orthogonal functions -^71 (^) by nieans of which an arbitrary function /(^) can be expanded in the form (4'21) is called a complete system of orthogonal functions. CHAPTER III INTRODUCTION TO DYNAMICS § 5. Force, Mass, Newton's Laws THE notion of force has its origin in the feeling of muscular effort . Quantitative estimates based immediately on the feeling or sensation of effort are, however, too rough and uncertain to serve any purposes where precision and consistency are demanded. Consequently force, like all other physical quantities, is measured by devices which entirely eliminate any dependence on the intensity of a sensation. Typical of such devices is a spring (as in a spring balance, for example). Instead of the uncertain comparison of two weights by feeling how big are the muscular efforts exerted in supporting them, it is better to use the extensions they produce in a spring as measures of their weights. The procedure in measuring temperatures is quite analogous. The sensation of warmth i' or hotness plays no part whatever in such measurements. Let us examine more closely the measurement of force by the extension of a spring. To fi:K our ideas we may continue to keep the spring balance in mind. The upper end. A, of the spring is attached to a support, which for our i Y purposes may be supposed to be rigidly fixed. A heavy Fig. 5 body is suspended at the lower end, B. The extension can be represented completely by a vertical line (XY in Fig. 5) the length of which is made equal to it. The upward and downward directions in XY have equal claims on our atten- tion, and we shall say that a force is exerted in an upward direction on the body suspended at B, and that a force is exerted in a downward direction on the support at A. Both forces are measured by the extension, XY, of the spring and are therefore numerically equal to one another. This is the (so-called) law of action and reaction, which here emerges as a necessary consequence of the measuring device and associ- ated definitions. It is implied, in what has been said, that the unit force is the force associated with the unit exten- sion, and it follows that different units of length will, in 4 39 4:0 THEORETICAL PHYSICS [Ch. Ill general, have to be used with different springs. There is no difficulty in deciding when the extensions produced in a number of springs are aU associated with the same force ; but the fol- lowing circumstance has to be considered : though different springs may agree with one another approximately, or even very closely, when used to measure forces extending over a certain limited range, and though the intervals on the evenly-divided scales attached to them may have been chosen so that all the springs are in precise agreement in the case of one particular force (the adopted unit) ; they wiU nevertheless be found to dis- agree to some extent when measuring other forces. In fact, when springs are used in the way described, each measures forces according to a scale of its own. There is a close parallel to this in the measurement of temperature by thermometers of different types. We find it necessary to define a force scale independent of the peculiarities of the particular measuring device — some- thing precisely similar is done in the measurement of temperature. We might do this by adjusting a number of weights so that they all produce the same extension in one particular spring, when hung from it separately. We should then be able to calibrate any spring by suspending the weights from it, one, two or more at a time. The following observational facts point out another way of defining a force scale (which will, in fact, amount to the same thing in the end) : In the first place, if the point of sup- port, A, referred to above, be caused to ascend with a uniform speed, the extension of the spring (after the initial oscillations have been damped out) will not be altered. This means that no force is needed to maintain the constant velocity of the body at B. In the next place, if A ascends with a constant acceleration, the extension of the spring will be increased by a definite amount, proportional (approximately) to the acceleration. In other words, the upward acceleration of the body suspended at B is approximately proportional to the resultant force (as measured by the spring) to which the acceleration is due. In consequence of these facts it is possible, and many reasons make it desirable, to define the force scale by the statement F = ma (5) where F is the resultant force, a is the acceleration, and m is a constant characteristic of the particular body, and called its mass. If we make m unity for some arbitrarily chosen body, we shall thereby fix the unit of force at the same time. It will in fact be the force which causes it to move with the unit ac- celeration. In the case of bodies made entirely of the same material, e.g. brass, the mass is found to be very nearly propor- §5] INTRODUCTION TO DYNAMICS 41 tional to the volume. This is the justification of the rather imperfect definition of the mass of a body as the quantity of material in it. Representing the velocity of the body by v, equation (5) may be written or F = _(mv), (XT/ or finaUy ^^^ (^'^^^ where M = mY ...... (5*011) is called the momentum of the body. Equation (5*01) embodies in a single statement Newton's first and second laws of motion ; the law of action and reaction being Newton's third law. In applying these laws generally, we must regard the body as very small in its dimensions, i.e. as a particle, in order to avoid the difficulty which would appear if the velocities or accelerations of its parts differed from one another ; and in dealing with classical dynamics we shall assume, as Newton appears to have done, that the mutual forces, exerted by two particles on one another, are directed along the straight line joining them. The unit of mass adopted for scientific purposes is the gram, i.e. the mass of a cubic centimetre of water at the temperature of its maximum density ; the unit of length is the centimetre, and the unit of time the mean solar second . With these funda- mental units, the unit of force fixed by (5) or (5*01), i.e. the force causing the unit rate of change of momentum, is called the dyne. It is an experimental fact of great importance for the science of physics that unsupported bodies at the same place, i.e. bodies which have been projected and are falling freely, have the same downward acceleration. This is usually represented by the letter g, and is equal to 980-6 cm. sec."^ in latitude 45°, and varies from 978 cm. sec.~2 at the equator to 983-4 cm. sec."^ at the poles. By ' freely falling ' body is to be understood of course one which is not subject to the resistance of the air or any other sort of interference. There is therefore a downward force acting on the body, equal to mg. This is its weight. At the same place, therefore, the weights of bodies are proportional to their masses ; but whereas the mass of a body is a constant characteristic of it,^ and independent of its geographical position, its weight will vary 1 This statement will be modified when the theory of relativity is described. 42 THEORETICAL PHYSICS [Ch. Ill with the latitude in consequence of the variation of g. The acceleration, g, is the weight or gravitational force jper unit mass and we shall term it shortly the intensity of gravity. This is, for several reasons, preferable to the ambiguous term ' accelera- tion due to gravity '. § 5-1. Work and Energy The scalar product (Fdl), or FJtx + F^dy + F^dz, where p = (i^^^ Fy, jPJ is any force and dl = {dx, dy, dz) is a small displacement of its point of application, is called the work done by the force during the displacement dl. And when the point of application of the force travels from any point A along some path ABC to another point C (Fig. 5-1), ^ ^^.^ — j^ the work done may be represented by f(Fdl). ABC It may happen (and this is a very important case) that the work done is independent of the path. Starting -pjf.^ 5.1 from some fixed point A, the work done will depend only on the position of C. If we represent the work by W, we have W = function {x,y,z), where x, y, z are the co-ordinates of C. And if we take some neighbouring point C with co-ordinates {x -{- dx, y -f dy, z + dz) the work done will be W -{- dW, where dW = — dx -{- — dy -f — dz . . (5*1) dx dy ^ dz ^ so that _eW ^ _dW J, _dW ^ dx' ' dy' ' dz' We may therefore say that, when the work done is indepen- dent of the path, the force is the gradient of a scalar quantity. Conversely, when the force is the gradient of a scalar quantity at all points in a certain region, the work done between two points A and C in the region is independent of the path, since the curl of a gradient is zero (§ 2-4) and therefore [|(curlF,dS) and consequently, by the theorem of Stokes, the integral ct (Fdl) taken round any closed loop ABCDA in the region will be zero. §51] INTRODUCTION TO DYNAMICS 43 If the work done depends on the path, then the integral ^ {Fdl) round a closed loop such as ABCDA will, in general, differ from zero ; and since (f)(Fdl) = [[(curl F, dS), we see that curl F cannot be everywhere zero. Therefore F cannot be represented by a gradient everywhere. When dW is expressed by formula (5*1) it is called a complete or perfect differential. If we adopt the centimetre, gram and second as funda- mental units for length, mass and time respectively, the unit of work derived from them is termed the dyne -centimetre, or more usually the erg. It is the work done by a force of one dyne when the point where it is applied moves one centimetre in the direction of the force. Let us consider the motion of a particle under the influence of a force F = {F^, Fy, F^). By equation (5) (Newton's second law) we have d'^x ^ -S=^^ (5-") dx dij dz Multiplying these equations by — , -—- and — respectively we get CtZ 0/1/ Cf/v dx d^x m d { /dx\ ^) ri dx dt dt^ 2 dt(\dtj j ""dt' dt dt^ 2 dt[\dtj J 'dt' dz d^z _md(/dz\^) _-n,dz^ '^tltt^ ~'2Jt\\dt) ) ~ 'dt'' and, on adding, we have dt[2 j ""dt ~^ 'dt "*" 'dt where v is the velocity of the particle. Therefore — V^ — — Vo^ = 2 2 ° m „ m _V2 — _ 2 2 or — v^ — — Vo^ ^{FJx+Fydy+FM = J(Fdl) (5-12) 44 THEORETICAL PHYSICS [Ch. Ill In this equation v and Vq mean the velocities of the particle at the end and at the beginning of the path over which the integration is extended. When the force is the gradient of a scalar W the integral on the right of (5" 12) has the form J 1 ao; ^ dy ^ ^ dz I = W - Tf 0, and therefore equation (5' 12) becomes ^ -W = '^-W,. . . .(5-121) If we replace W hj — V and use the letter T for , we have T + F = To + Fo . . . . (5-122) The quantity T + F remains constant. This quantity is called the energy of the particle, T being its kinetic energy, a function of its mass and velocity, and F the potential energy, a function of its position. Equation (5*122) affirms that the energy of the particle remains constant. The conception of energy will be developed more fully in subsequent chapters ; for the present we are concerned only with the two kinds of energy which have just been described, and it will be noted that energy is measured in terms of the same unit as work. We shall normally regard the centimetre, gram, and second as our fundamental units — the question as to whether three fundamental units will suffice may be deferred till later — but it is obvious that the foregoing formulae (5*11) e^ <scg. are quite independent of the particular choice we make of fundamental units. § 5-2. Centre of Mass Imagine a number of particles, the masses of which are mi, m2, mg, . . . m^, and their co-ordinates {x-^,y-^,z-^, {X2,y2,^2), (a; 3, 2/3, 23), ... (Xg, 2/s, Zg) respectively. The centre of mass (x, y, z) of the system of particles is defined by Mx = HrrigXg, My = Em,y, (5-2) Mz = Em^Zg, where M = Em^, the summation being extended over all the particles of the §5-3] INTRODUCTION TO DYNAMICS 45 system. Let F^ be the sum of all the X components of the forces exerted on the particles of the system ; we shall have Em^ =^F and two corresponding equations for Fy and F^ ; or by (5 '2) M— Lf ^^.-Fy (5-21) M^ =F dt^ '' Therefore the motion of the centre of mass is the same as it would be if aU the masses were concentrated in it and aU the forces applied there. In equations (5*21 ) we may regard F as the resultant of all the forces of external origin, since by Newton's third law, those of internal origin will annul one another. An important case is that in which the system is free from external forces. In this case we can infer from (5 •21) that the centre of mass will move with a constant velocity — which may of course be zero. It is immaterial whether the particles con- stitute a rigid body or not. § 5-3. Path of Projectile Let a particle of mass m be projected from the origin of rect- angular co-ordinates with a given initial velocity, and suppose the X axis directed vertically upwards. If g is the intensity of gravity, i.e. if g is the weight of the unit mass, the equations of motion of the particle are : dH m^ = - mg, -S=« (5-^) The two latter equations (5*3) give us y = a^t -\- /^i, z = a,t -{- ^^ (5-31) where ai, ^i, a^, P^, are constants. If we eliminate t from (5*31) we get the equation or a^z = a^y + a^p^ — aa/^j .... (5-311) 46 THEORETICAL PHYSICS [Ch. Ill which is the equation of a vertical plane. The path of the particle therefore lies in a vertical plane, and it is convenient to place our axes of co-ordinates so that this vertical plane is the XY plane, and the Z co-ordinate of the particle is permanently equal to zero. The equations of motion are now d^x _ di^ ~ ~^ Therefore ^= - igt^ + At + B) y= at+^j ... (5 ^1^) If we eliminate t from these equations we obtain the equation of the trajectory of the particle, « = - xjy^ + A(y-^) +B . . (5-32) The constants m and g, which appear already in the differential equations (5 '3) before any steps have been taken to integrate them, we shall term inherent constants. Such constants are characteristic of the system to which they belong and are not in any way at our disposal. It is otherwise with the constants of integration A, B, a and ^. If the particle is projected from the origin at the time ^ = with a velocity V and in a direction making an angle d with the horizontal axis, Y, we have from (5*3 12) B =0, and since dx , , A 5f = «' the jrefore F sin = A, F cos = a. Substituting in (5*32), we have r — - ir/ ^ -I- F sin e If ^Vcos2 + F cos 6^ or 2:i:.F2cos ^0 = -gy^ + 2F2sin d .cos e.y If in (5*321) we put a; = 0, we find or y =0 F2 sin 26 y = :: ' (5-321) §5-4] INTRODUCTION TO DYNAMICS 47 the latter of these values of y representing the horizontal distance travelled by the projectile between two instants when it is in the plane x = 0. § 5-4. Motion of a Particle fndee, the Influence of a Central Attracting or Repelling Force Let the point, towards or away from which the force on the particle is directed, be the origin, 0, of rectangular axes of co- ordinates, and let r = {x, y, z) be the displacement of the particle from the origin. Let the absolute value of the force be F. We have for the equations of motion, X d^x F. dt^ r (5-4) dh dt^ F.^. Fig. 5-4 since F is directed along the radius vector r. Multiply the first two of these equations by y and x respectively and subtract. thus obtain We mx or m dt^ d ( Jiy di my- d^x df^ 0, Consequently / dy\ d / dx\ \dt) ~ ^tVdt) = 0. dy mx-^ dt (X/X y-. where Q^ is a constant. Similarly dz dy ^ my— — ms-^ = L?,,, ^dt dt dx dz ^ mz—- — mx-- = 14. dt dt " (5.41) Evidently Q^, Qy, Q^ are the components of a vector. In fact £1 =m[rv] (5-411) (§ 2-1) where v is the velocity of the particle. The constant £1 is called the angular momentuin of the particle. Since 48 THEORETICAL PHYSICS [Ch. Ill the vector product [rv] has the absolute value rv sin 6 (Fig. 5-41), it must be equal to twice the area swept out per unit time by the radius vector r. For if in Fig. 5*41 the points 1 and 2 represent two neighbouring positions of the particle dl apart, the corre- sponding area swept out by the radius vector is J . r sin . dl, and dividing by dt, the time taken by the particle to travel dl, we see that the area swept out per unit time is in fact |r sin 6v. The angle swept out by the radius vector while the particle travels the distance dl is dl sin 6 and consequently the angular velocity of the particle is V sin 6 io = . r We may therefore write ^=mr2.to . (5-412) We may summarize as foUows : —When a particle moves under the influence of a force directed towards or away from a fixed point its angular momentum remains constant. Since £1 is equal to the vector product of r and v multiplied by the invariant factor m, it must be at right angles to the directions r and v (see § 2-1). Therefore the scalar product (Sir) is equal to zero, that is Q.^x +Q,yy +Sl,z =0 , . . . (5-42) This is the equation of a plane passing through the origin, i.e. through the attracting centre. § 5-43. Angulab Momentum of a System of Particles free FROM External Forces We may write the equations of motion of any one of the particles, which we shall distinguish by the subscript, s, in the usual way : ~dP '""S-J^ = Fsy, 'dt^ §5-43] INTRODUCTION TO DYNAMICS 49 By a procedure similar to that used to deduce equations (5*41 ) we get : ^sik^ - x,^\ = ^sF.. - ^.P., . (5-43) dt [ ^ dt ^dt Vs-iT \ — ^s-^ sy Us^ & 'dt[ 'dt ""'dt which may be expressed more briefly in the following form : -|n, = [r,FJ .... (5-431) The vector product [r^Fg] is called the moment of the force F^ with respect to the origin. If we add aU the equations (5*431) for the whole system of particles, we have |sa=S[r,FJ (5-432) the summation being a vectorial one ; i.e. Sa, ^ 0Q,,, Si3,„, Si?,,), |[r.F.] - {^(VsFs. - 2.-f»)> S(z,^,, - x,FJ, ^{x,F,„ - y.FJ-^ Equation (5*432) states that the rate of increase of the result- ant angular momentum about the origin is equal to the resultant of the moments of the forces with respect to the origin. Now there are by hypothesis no forces of external origin, and if we suppose that the mutual action between any two particles is directed along the straight line joining them, the right-hand side of (5*432) can be shown to be zero. For the moment of a force is numerically equal to the product of its absolute value and the perpendicular distance from the origin to its line of action. Therefore the resultant of the moments due to the mutual action of any two particles on one another must vanish, since the forces are equal and opposite and the perpendicular distance mentioned above is the same for both. We conclude therefore that the resultant angular mo- mentum of a system of particles remains constant in magnitude and direction provided the only forces are those due to the mutual action of the particles on one another and that the force exerted by any particle on another is directed along the straight line joining them. 50 THEORETICAL PHYSICS [Ch. Ill § 5-5. Planetary Motion We shall now study the motion of a particle under the influ- ence of a central force varying inversely as the square of the distance of the particle from a fixed point, which we shall take as the origin of a system of rectangular co-ordinates. Since, as we have already proved (§ 5-4), the particle moves in a plane, we shall place our co-ordinates so that this plane is the XY plane. The Z co-ordinate of the particle will remain constant and equal to zero. We now introduce polar co-ordinates, r and 6, defined by X = r cos 6, 2/ = r sin 6, The kinetic energy of the particle is easily calculated. Since if dl is an element of the path of the particle, dl^ = dx^ + dy^ = (dr. cos - r sin d.dd)^ + {dr.Qin 6 -\- r cos d.dd)^ = dr^ + rHd^ Therefore the kinetic energy ©■=KJ)'+Kf)' ■ ■ <'■" rlfl We may eliminate the angular velocity, — , by making use of do "="*<!) • • • • ^'-'''^ The kinetic energy may thus be expressed in the form ;m\ , /dr ^ \dt dr\^ Q' The central force is expressed by where J5 is a constant, positive or negative, according as it is a repulsion or attraction. The X. component of the force will therefore be F =?^ -■ = -£©• p Therefore -, plus a constant, is the potential energy of the §5-5] INTRODUCTION TO DYNAMICS 51 particle (see § 5-1). It is convenient to fix the arbitrary constant in such a way as to make the potential energy zero when the particle is infinitely distant from the centre of force. Repre- senting the sum of the kinetic and potential energies by E, we have . /dry ^ Q^ ^ B ^ ,- -,, In this equation Q and E are constants of integration (§§ 5-1 and 5-4). To get the equation of the path or orbit of the particle we eliminate dt in (5*51) by means of the equation of angular momentum (5*501) which may be expressed in the form, dt = -jr-dd. Equation (5*51) now becomes J^(±y+J^+B^E . . .(5-511) 2mr^\dd/ 2mr^ r We now introduce a new variable u defined by u=- (5-512) r so that dr = — —^ (5*513) Substituting in equation (5*511) we get Differentiating this equation with respect to d and dividing out the common factor 2-— we obtain dd d^u . mB ., _^, 5e-. + « = -^ • • • • (5-52) The general solution of (5-52) is ■» = - ^ + -B cos (e - ^) . . . (5-53) where R and r} are constants of integration. This equation may be expressed in the form : r^ ^^^ . . . .(5*531) 1 + £ cos (0 - ?^) ^ ^ where £ and r\ are constants of integration. The latter evidently depends on the fixed direction, that of the x axis, from which 52 THEORETICAL PHYSICS [Ch. Ill the angle 6 is measured and it is convenient to choose our system of reference so that ^ = 0. There is no loss of generality in taking £ to be positive only, since the effect of changing the sign of e is just the same as that of rotating the fixed direction, from which 6 is measured, through the angle n. Equation (5*531 ) represents a conic section ; hyperbola, parabola or ellipse. If the central force is one of repulsion, B is positive and therefore the numerator of (5*531) is negative ; and since only positive values of r have a physical significance, it follows that the de- nominator must also be negative. This is only possible when s is greater than unity. The orbit of the particle is therefore one branch of a hyperbola (represented by the full line in Fig. 5-5). Fig. 5-5 The centre of force is the focus 0. It will be seen that the path of the particle goes round the other focus, 0'. The asymptotes make angles Oq with the axis of reference (the X axis) determined by the equation . 1 cos Oq = — -. e If the central force is one of attraction (B negative), the numerator of (5*531) is now positive and consequently the de- nominator must be positive too, in order to furnish physically significant values of r. It is obvious that s is not now restricted to values greater than unity ; but if it should be greater than unity, we have again a hyperbola (Fig. 5-51 ) and the same relation, cos do = — -, £ for the directions of its asymptotes as in the case illustrated in Fig. 5- 5. In the present case, it will be observed, the orbit goes round the focus 0, the centre of force. When £ == 1, the path of the particle becomes a parabola, the attracting centre being in the focus. §5-5] INTRODUCTION TO DYNAMICS 53 The case where e is less than unity is of special interest. The orbit is now elliptic, the centre of force being in one of the foci. If a and h are the semi major and minor axes respectively of the ellipse, we have -^ =- .... (5-532) mB a The force exerted on one another by two gravitating particles is -^ (5-533) where M and m are the masses of the particles, r the distance between them and is the constant of gravitation. If the ratio — is negligibly small compared with unity, the centre of mass of the pair of particles practically coincides with the particle M O' >x Fig. 5-51 and if we use axes of co-ordinates with the origin in the centre of mass we shall have the centre of attraction fixed in the origin. The force exerted on m is — where B= - OMm .... (5-534) by (5-533). It follows from (5-532) that QM^^^Il .... (5-535) We have seen (§ 5-4) that Q is equal to the product of m and twice the area swept out by the radius vector in the unit time. There- fore Q = 2m— (5-536) T , (5-54) 54 THEORETICAL PHYSICS [Ch. Ill if T is the period of a complete revolution. From (5*535) and (5*536) we get Johannes Kepler (1571-1630) inferred from the observational data accumulated by Tycho Brahe (1546-1601) that the planets travel round the sun in elliptic orbits, the sun being in one focus of the ellipse, and that the radii vectores sweep out equal areas in equal times. These inferences, known as Kepler's first and second laws, were published in 1609. Equation 5*54 expresses Kepler's third law, namely that the cubes of the major axes of different planetary orbits are propor- tional to the squares of the corresponding periods of revolution. This was published by him a few years later. A rather important proposition connected with elliptic motion under the inverse square law of force is the following : — The energy of the particle m is completely determined by the length of the major axes of the ellipse. The energy equation (5*51) reduces to 2mr or r2-^r-ii-=0 .... (5-55) E 2mE ^ ' when the planetary body, electron or whatever it may be, is just dT at one or other end of the major axis, since in this case — = 0. dt If the corresponding values of the radius vector are r^ and r^, we have of course ri +^2 = 2a; but from (5*55) we have B and therefore ^ = 1 (5.551) which proves the proposition. We have so far supposed the attracting (or repelling) centre to be fixed in space, or at all events fixed relatively to the axes of co-ordinates. We have now to amend equation (5*51) so that it will apply more generally to two mutually attracting or repelling bodies, not subject to other forces. Let M and m be their masses and B and r their respective distances from their §5-5] INTRODUCTION TO DYNAMICS 55 common centre of mass in which we suppose the origin to be placed. The energy equation now becomes Since r and U are the distances of the masses m and M respect- ively from the centre of mass, rm = EM, m B or = __ = (7 M r and therefore (5*56) becomes 2 Therefore This equation would be identical with (5*51) if in the latter m were replaced by m' = m(l + cr) and B hj B' = ; so that the problem of the motion of the particle m is reduced to the one we have already solved. In particular we find for the energy of the system 2a 2aM-{-m where a is the semi-major axis of the ellipse in which the particle m is moving. An instructive example of a central force is that represented 7? O by —-+ — ^ where B and C are constants. The potential energy is obviously if we adopt, as usual, the value zero for the arbitrary constant involved. The energy equation is now Imi — ] + 4- h — 7. = E . . . (5*57) ^ \dtj 2mr^ ' ^ ' 2r2 ^ ' 66 THEORETICAL PHYSICS [Ch. Ill and differs from (5*5 1) only in the expression for the potential energy. The same method as that which was used in the case of the inverse square law leads to instead of (5*5 14), and to instead of (5-52). TThCJ We shall restrict our attention to the case where 1 + -— - is positive and not zero. The general solution of (5*58) may then be expressed in the form mB u = — _/'!^^ ^. + R cos -! ^ -^ "^ ^ ■?7 (5-59) where R and rj are constants of integration. We may write this equation in the form Q^ + mC mB l+.cosj^/l+^ 6 (5-591) where both e and ^ / 1 + — — may, without loss of generality, be taken to be positive. It will be seen that (5*591) resembles the equation of a conic. This resemblance is made more obvious still if we write it in the following way : Q^ +mG mB (5-592) V' 1 + £COS {(9 - 7f} where The essential difference between the orbit represented by (5-531) and that represented by (5-592) lies in the fact that whereas in the former t^ is a constant the corresponding quantity rj' in the latter is continuously increasing or decreasing during the motion, according as a / 1 mC If we give the axes of co- is less than or greater than unity, ordinates a suitably adjusted motion §6] INTRODUCTION TO DYNAMICS 57 of rotation about the Z axis we may keep y] constant and evidently the orbit referred to such moving axes will he a conic section. We may say therefore that the orbit is a conic section the major axis of which is in rotation, in the plane of the motion, in the same direction as that in which the particle is moving, or in the opposite one, according as a / 1 + greater than unity. mC is less than or § 6. Generalized Co-ordinates The n independent numerical data, q^, q^, q^, . . . q,i, which are necessary for the complete specification of the configuration of a dynamical system, are called its generalized co-ordinates, and the system is said to have n degrees of freedom. Such co- ordinates may be chosen in various ways ; for ex- ample in the case of a single particle they may be its rectangular or polar co- ordinates. A rigid body, free to move in any way, will require six co-ordin- ates ; but only three if a single point in the body is fixed. Associated with each ^ is a generalized velocity -?, and a gene- at ralized momentum usually represented by the letter p. A rigid body, the only possible motion of which is a rotation about a fixed axis, has only one q, which may conveniently be the angle between a fixed plane of reference, OA in Fig. 6, and another plane, OB, fixed in the body, the intersection of the planes coinciding with 0, the axis of rotation, which in the figure is perpendicular to the plane of the paper. The kinetic energy of any particle of mass m in the body is Fig. 6 mv^ = ^mr \dt, where r is the perpendicular distance of the particle from the axis, 0, and (/> is the angle between r and OA. If the body is rigid d(j) _ dq dt ~"di' 58 THEORETICAL PHYSICS [Ch. Ill and the kinetic energy of the particle is Since (-r-) ^^.s the same value for all particles in the body, the total kinetic energy will be T = i{Imr')(^^y (6) The quantity Zmr^ is called the moment of inertia of the body with respect to the axis in question, and it occupies in formula (6) the same position as that of the mass in the ex- pression ^mv^ for the kinetic energy of a particle. The generalized momentum corresponding to -— is defined by dt ^ dt ^ ' This definition is in accord with the use of the term angular momentum in § 5-4 (see equation 5 '412). If the axis of rotation of a rigid body pass through the origin and if x, y and z are the co-ordinates of any particle of mass m, in the body, the components of the angular momentum of the body will be ^ / dz dy\ ^ / dx dz\ ^» = M4-^J) • • • (^-^^^ see equations (5*41). Equations (5*432) also will apply to a rigid body and may be expressed in the form dt = 2{yF. - - ^F„), dt = 2(^F. - - xF,), dp, dt = ^^Fy ■ -yF,) . . . (6-02) In these equations we may, just as in § 5-43, regard the forces as of external origin, since the forces of internal origin con- tribute nothing to Z{yF^ — zFy), etc. The right-hand members of (6 '02) are the components of the applied torque or couple. § 6-1] INTRODUCTION TO DYNAMICS 59 The parallelism between equations (6*02) and those for the motion of a particle should be noticed. The former state that Rate of Increase of Angular Momentum = Applied Couple, and the latter that Rate of Increase of Momentum = Applied Force. § 6-1. Work and Eneegy Let dq = [dq^, dqy, dq^) be a small rotation of the body- about an axis through the origin. We may represent dq by a straight line from the origin, of length dq and drawn in the direction in which an ordinary screw would travel with such a rotation. If dl = [dx, dy, dz) is the consequent displacement of a particle, the distance of which from the origin is r = (x, y, z). We have, numerically, dl = r sin 6 dq, where 6 is the angle between the directions of dq and r ; and when we study the directions we find dl = [dq, r] (6-1) The work done during the displacement dl is equal to the scalar product of F, the force acting on the particle, and dl the dis- placement. Therefore the work is (dl F) = ([dq r]F). Reference to equation (2' 14) will show that this is equal to ([r F] dq). In a rigid body dq is the same for all particles, therefore the work done by all the forces acting on the body will be given by (dqr[rF]), and the rate at which work is done will be equal to This must be equal to the rate of increase of the kinetic energy of the body, so that we have In the summation we need only take account of forces of ex- ternal origin since the contribution to Z[r F] of the mutual forces exerted by the particles of the body on one another is zero. Once more there is a close parallelism with a correspond- ing result in particle djniamics. Equation (6*11) states that the rate of increase of the kinetic energy is equal to the scalar 60 THEORETICAL PHYSICS [Ch. Ill product of angular velocity and applied couple. The rate of increase of the kinetic energy of a particle is equal to the scalar product of its velocity and the force acting on it. § 6-2. Moments and Products of Inertia If Jo and I, represent respectively the moments of inertia of a rigid body with respect to any axis, 0, and an axis C, through the centre of mass of the body Y and parallel to 0, I, = Mh^+I, . (6-2) M being the mass of the body and h the perpendicular dis- tance between the two axes. To prove this it is convenient to place the co-ordinate axes so that the axes and C are in the XZ plane and perpen- dicular to the X axis (Fig. 6-2). Fig ^gT^ '^^^ contribution of any par- ticle m to Jo is mr^ _ jjiji2 _|_ ^^2 _ 2mlis cos (/> or mr 2 = mh^ + ms^ + 2hm{x — ^^o), where x is the X co-ordinate of the particle and Xq is the X co-ordinate of the centre of mass. Summing over all the particles in the body we get Jo = Mh^ + Jc + 2h{Zmx - Mxo}, and, by the definition of centre of mass, 31 Xq = Emx, therefore equation (6*2) follows. The radius of gyration of a body with respect to any axis is defined to be the positive square root of the quotient of the moment of inertia of the body with respect to that axis by the mass of the body : h M , so that I = 3Ik^ (6-21) Unless the contrary is expressly stated, or clearly implied by the context, we shall associate the radius of gyration with axes through the centre of mass of the body. Let a, p and y be the direction cosines of an axis through V 31 §6-2] INTRODUCTION TO DYNAMICS 61 the centre of mass, which we shall suppose is at the origin of rectangular co-ordinates. The contribution of any particle of mass m to the moment of inertia, /^^^, with respect to the axis (a, p, y) is mr^ sin^ 6, where r = {x, y, z) specifies the position of the particle and 6 is the angle between the directions of r and of {a, ^, y). Since r^ sin^ is the square of the vector product of the vector r and the unit vector (a, ^, y), mv^ sin^ d = m{(yy — z^Y + (s^a — xyY + {^^ — ya)^}- Therefore •^aiSv = a^Zm{y^ + z^) + P^Zm{x^ + z^) + y'^Zm{x'^ -\- y^) — 2pyZmyz — 2ayZmxz — 2apZmxy, J,^^ = Aa^ + Bp + Cy^ + 2Dpy + 2Eay + 2Fa^ . (6-22) The coefficients A = Imiy^ + z^), B = Em(x^ + z^) and C = Em{x^ + y^) are, clearly, the moments of inertia of the body with respect to the X, Y and Z axes respectively. The remaining coefficients — D= Zmyz, — E — Emxz and — F = Emxy are known as products of inertia. Consider the surface ^|2 4_ Bri^ + CC2 + 2J[)^C + 2^^f + ^F^ri =M . . (6-23) where M is the mass of the body and i, tj and C are X, Y and Z co-ordinates. Let q be the length of a radius vector from the origin to a point (I^^C) on the surface, so that ^2 _ |2 _|_ ^2 _(_ ^2, We have for a radius vector in the direction (a^y) a = ^/q, P = r]/Q, y = C/q, and therefore, dividing both sides of (6*23) by q^, M Aa^ + 5^2 ^ Cy^ ^ 2D^y + 2Eay + 2FaP = — , so that Q This means that the length of any radius vector of the surface 62 THEORETICAL PHYSICS [Ch. Ill (6*23 ) is equal to the reciprocal of the radius of gyration of the body with respect to an axis in that direction. The equation (6 '23) must be that of an ellipsoid, since in any direction whatever q wiU have a finite positive value. It is caUed the momental ellipsoid or ellipsoid of inertia. A suitable rotation of the axes about the origin will reduce the equation to AP +Brj^ -\-GC^ =M . . . (6-235) where A, B and G have not necessarily the same values as in (6-23). A, B and G (6-235) are called the principal moments of inertia of the body and the corresponding X, Y and Z directions are called the principal axes of inertia. M M In equation (6-235) let us replace |, y} and C by -—^', -—r\' A B M and -^C' respectively, so that we get the equation A '^ B '^ G M t'2 ,/2 ^'2 |^ + f. + F-.= l (6-24) /t/j A/2 A/3 in which hi, kz and k^ are the radii of gjrration about the prin- cipal axes of inertia. The surface (6-24) is called the ellipsoid of gyration. Any point (^', rj', C) on it corresponds to a point (I, 7], f ) on the momental ellipsoid, where ^ A^' M n-^n' ' (6-241) ^ G^ The length of the perpendicular, P, from the origin to the tangent plane at (|' t]' ^') is, by a well-known rule, J_ M V A^ B^ (72 or P == , ^ - -, by . . (6-241) Vr+^2_|_^2| Q and it is easily seen that its direction cosines are the same as those of the radius vector q from the origin to the point (i r] C) §6-4] INTRODUCTION TO DYNAMICS 63 on the momental ellipsoid. It foUows that the length of the perpendicular from the origin to a tangent plane of the ellipsoid of gyration is equal to the radius of gyration of the body about an axis coincident with the perpendicular. § 6-3. The Momental Tensor Equation (6*22) may be written in the form : a{Aa +F^ + Ey} + P{Fa + BS -f Dy} + y{Ea-{-D^ + Cy}^I^^^ . . . : (6-3) The right-hand member of (6*3) is an invariant, since it is the moment of inertia of the body with respect to a specified axis in it and must therefore be independent of the system of reference chosen. This suggests that the set of quantities A F E F B D E D C (6-301) are the components of a tensor of the second rank. It can easily be proved that this is the case (see § 2-3). It is called the momental tensor and is more appropriately represented in the following way : V V i. (6-302) If we make a corresponding change of notation for the direction cosines {a^y), equation (6*3), becomes : •*■ a ^x \^x ^xx i ^y ''xy ~r ^z '^xz J + tty {a^ lyy, "J" tty I yy "f tt^ '^1/3 / + «.K hx + «y hy + «. %z} • • (6-303) The moment of inertia I^ is therefore to be regarded as the scalar product of two vectors, namely the vector (a^ a^ a^) and the vector, the components of which are represented by the ex- pressions in brackets in (6*303). § 6-4. Kinetic Energy of a Rigid Body If the motion of the body is a rotation about an axis a, the angular velocity being to = -^, its kinetic energy is T = J/a<^^ and since co^ = hsa^, o)y = isiay, co^ = coa^, we obtain from (6*303) T = i(o^{oj^ i^^ + ojy i^y + CO, *^J + i^y{(^x ijx + ^y Ky + ^z \z} + i^zi^x hx + ^y hy + ^\ iz} ' ' ' i^'^) 64 THEORETICAL PHYSICS [Ch. Ill If the axes of co-ordinates coincide with the principal axes of inertia, this becomes T = H^^x' ire + <^-^/ V + ^z" hz) ; or, if we revert to the notation of (6*235 ) for the principal moments of inertia, T = |{^co,2+^co,2 + (7a),2} . . . (6-41) If in (6*4) we represent the angular velocity co by -~, we have where p^ = ij^^ + ij^ + i,§, ^^ '""dt ^ ''dt ^ "'dt' dq^ . dq, dq, ''''di ^ ^''dt ^ ''W Pz = co^y - co,^x, are the components of the angular momentum. In fact the X component of the angular momentum is and -^ = cojjc — cojz, dt dz di (see equation (6*1) for example) ; therefore p^ = Zm{co^^ — cOyXy — co^xz + co^z^} = Kxf^x + hy(^y + hz(^z' It will be observed that _dT _dT _dT ^qx Gqy dq, where q^ = co^ = _^, etc (6-42) Any motion of a rigid body can be regarded as a motion of translation, in which all the particles of the body receive equal and parallel displacements, on which is superposed a rotation about a suitably chosen axis. Let r be the distance of any particle of mass m from a point, P, on the axis of rotation. A rotation dq will give it a displacement [dq r]. The total displacement of the particle will be the vector sum of [dq r] §6-5] INTRODUCTION TO DYNAMICS 65 and the displacement of P. Let the co-ordinates of the particle be X, y and z and those of P be Xq, y^ and z^ ; then T = {x -x^, y -y^, z — z^), and dx = dxo + {dqy{z — ^o) — dq,{y — 2/0)} • (6-43) Therefore % = ^ox + '^ix • • . • (6-435) Where v is the velocity of the particle, Vq the velocity of P and ^,, = J"(z-2„)-^%-2/„) . . (6-436) The kinetic energy of the particle is = imvo^ + imvi2 -f m{vo^Vi^ + VoyV^y + Vo^v^,), and the kinetic energy of the body, T ^T, + T, + Zm{vo,v,^ + v^yV.y + ^^o^^ij . (6-44) In this equation To = iMv,^ T, == ilco^ M is the mass of the body and / is its moment of inertia with respect to the axis of rotation. If the point P is the centre of mass of the body, since, as reference to (6*436) will show, it consists of terms, each of which contains one or other of the factors Em[z — Zq), Em{y — y^), etc., all of which vanish if (a; 2/0 ^0) is the centre of mass. We thus arrive at the important result, T = lilfvo^ + i/to2 .... (6-45) where Vq is the velocity of the centre of mass and co is the angular velocity relative to the centre of mass. § 6-5. The Pendulum The pendulum is usually a rigid body mounted so that it can turn freely about a fixed horizontal axis, O, (Fig. 6-5), which we may suppose to be the Z axis of rectangular co- ordinates. The position of the pendulum is determined by the vector q ^ {(ix cLv qz), where q^ = qy = and g^ = g is the angle between the plane XZ and the pla.ne, OC, containing the centre of mass and the THEORETICAL PHYSICS [Ch. Ill axis of rotation. The positive direction of q is indicated in the figure by an arrow. The equations (6'02), when apphed to this case reduce to or yF.) if the Y axis is directed vertically downwards, since the im- pressed forces on the body are due to gravity only. The force Fy on any particle is equal to mg, therefore i^ = gEmx = Mgx^, or -/ Mgh sin Q, Fig. 6-5 where is the angle between OC and the vertical and h is the distance of the centre of mass from the axis of rota- tion. It is convenient to write the equation in the form : d^d , Mgh . , _ do If we multiply by — and integrate, we get at *( dOy di) Mgh cosd = K (6-5) (6-501) where ^ is a constant of integration. If K exceeds Mgh the kinetic energy, |7( — ) can never sink to zero, and the body win keep on rotating in the same sense round the axis with a periodically varying angular velocity. The case of interest to Mgh us is that in which K is less than There wiU then be a dd value ^0 of ^ between and n for which -— is zero and dt Mgh cos do = K, ^6-5] INTRODUCTION TO DYNAMICS and consequently , /de\ 2 3Igh , ^ (1) where 67 'de\^ Mgh , . „ . 2 , _0 _6o ' ~ 2' '^ ~ 2" The time required by the pendulum to travel from the position = to an extreme position 6 = :j- do or from 6 = ^ 0o to =0 is {dt^ IJL_ f J V Mqh J de Mgh J V sin^ £o — sin^ e | and therefore the complete period of oscillation is Mgh J Vsin2 £„ - sin^ e I T = 4 (6-51 To evaluate the integral we introduce a new variable, </>, defined by sin s = k sin 0, where /c = sin £o- On substituting in (6*51 ) we get 7r/2 ^~ ^ ~Mqh J Vl -/c^sir ^^ ' • • ^^'^^^) sin ^(/) The elliptic integral in (6*51 1) is now expanded by means of the binomial theorem, thus, 7r/2 ff/2 + 2X6'^ ™ ^+ . . .} This can be integrated term by term, by using the well-known reduction formula 7r/2 r/2 [ sin2» <j>d<t>= — ?• [ sin2^-2^ dcl>. We get, finally, 68 THEORETICAL PHYSICS [Ch. Ill When the amplitude, ^o, is small, i.e. when k is small, the period, is independent of the amplitude. This result might have been reached much more shortly by replacing sin in (6*5) by 6 when 6 is small. The equation then becomes cm . Mgh. _ ^ W^'^ I ' or, if we write co^ for the positive quantity ^, + (^^6=0 (6-522) The general solution of this equation can be put in the form l9 = ^ cos (co^ - ^) . . . . (6-523) where A and ^ are arbitrary constants. Since d will repeat its values every time cot — cj) increases by 27t ; we must have {co{t + r) - ^} - {ojt - cf>) = 27Z cor = 271, or T = — (6-524) CO This is identical with (6*521) when co is replaced by / — ~- • The type of motion defined by (6*522) is called simple harmonic motion. It has the important property that the period is independent of the amplitude. By making use of (6*2) we may give to (6*521) the form V- 2. ■^' + '' In the ideal simple pendulum, ^ = 0, A is the distance, usually represented by I, from the point of support to the bob, and therefore, for small oscillations To = 271 J - If Ti represents the still better approximation obtained by ignoring quantities of the order of ac* and higher powers of k, we have from (6*52) =r.(i + y, §6-5] or since <c = sm INTRODUCTION TO DYNAMICS «y = T„(l+isin2|), T,=To(l+j|-) . . since the difference between the squares of sin (6-526) - and ^ is of the 2 2 order of /<*• Cycloidal Pendulum. We have seen that the period of the type of pendulum we have been studying is a function of the amplitude. It was shown by Huygens {Horologium Oscil- latorium) that the motion of a particle, constrained to travel along a certain cycloid, is strictly isochronous, i.e. the period is inde- pendent of the amplitude. The equations of motion of a particle, P, constrained to travel along a curve in a vertical plane, the XY plane in Fig. 6-51, are ^-7^ = Qx. m dp Y mg + a, (6-53) Fig. 6-51 where Q^ and Qy are the components of the constraining force. If s is the distance travelled by the particle along the curve, measured from some arbitrarily chosen point, 0', the vector ds = {dx, dy) is perpendicular to the vector 0- Therefore (&x + Qydy = (6-531 Multiply the equations (6*53) by -— and ~ respectively and add, dt cit dx d^x , dy d^y dy , ^dx , ^dy dt dt' dt dt^ dt 'dt dt' or m d ( /dx\^ , /dy\^] dy 11), and tb m d ( /ds\ JJtlxdi) by equation (6*531), and therefore m d ( /ds^ 2 mg cos £ ds dt' 70 THEORETICAL PHYSICS [Ch. Ill ds if £ is the angle between dy and ds. Dividing through by m— dt we have — 2 = 9^ cos £ (6-54) The motion of the particle will be simple harmonic (see the definition 6*522), and its period consequently independent of the amplitude, if cos 8 = — as (6-541) where a is any positive constant. Equation (6*54) then becomes ^+"^^ = ^' and the period of the motion is seen to be If I be the length of the simple pendulum, the small oscillations of which have the same period, 1 and (6*541) may be written as cos £ = - ? (6-542) V This is the equation of the required curve. On differentiating it we get ds = I sin £ ds and therefore dx = I sin^ £ ds, dy = I sin s cos s ds. Consequently dx =-(1 - cos d)dd, 4 dy =— sin dd ; 4 where = 2e. On integrating we have X =l{d - sin 6) +A y =— cos -{- B . 4 . . (6-543) Let 0', from which s is measured, coincide with the origin. 0, so that X = when y = ; and suppose that the particle, §6-5] INTRODUCTION TO DYNAMICS 71 P, is moving vertically downwards when in this position, i.e. £ = = when x = and y = 0. For this position of the particle, therefore, equations (6*543) become and on substituting in (6*543) we have the familiar equations of the cycloid X =R{d - sin (9), y = B(l - COB 6) (6-55) in which R has been written for — . 4 BIBLIOGRAPHY Galileo Galilei : Discorsi e dimostrazioni raatematiche, 1638. Galileo's dialogLies have been translated by Henry Crew and Alfonso de Salvio. (The Macmillan Co., 1914.) Newton : Philosophise natnralis principia mathematica, 1687. Ernst Mach : Die Mechanik in ihrer Entwickelung, 1883. Huygens : Horologium oscillatorium, 1673. CHAPTER IV DYNAMICS OF A RIGID BODY FIXED AT ONE POINT § 7. Euler's Dynamical Equations LET P be any vector and P^, Py and P^ its components referred to rectangular axes of co-ordinates. Let PJ, Py and P/ be the components of the same vector referred to a second set of rectangular co-ordinates, the origin, 0', of which coincides with 0, the origin of the first system. Therefore, by (2-22) Px ^^ ^iiPx ~i" (^2iPy ~r CisiPz and ^- = |{a,,P; + a,iP/ + a„P/} . . (7) We shall suppose the first set of co-ordinates to be fixed and the second set to be in motion about their common origin. The cosines an, a 21, etc., are then variable and (7) becomes dP. dPJ , dPJ . dPJ + a,^—^ + a 3r dt dt dt dt p , aaii p ,0021 I p /dosi //v.ooi "i "^ ^ dt '^ ' dt '^ ' dt ^^ ^ Now a 11, a 21 and a 31 are the co-ordinates in X\ Y\ Z' of a point on the X axis at the unit distance from the origin, and therefore — yi^, —7^ and — -^ are the components of the velocity of this point relatively to the moving axes. Therefore if co^', my and m^ are the components of the angular velocity of the fixed co-ordinates relatively to the moving co-ordinates, dt da = co^ asi — CO, ttai = co.aai — oy^a, dt da^i 21 = co/an — ctyJa^i o^x "21 — ojy «ii = cOyaii — oy^fL^ii dt "''"'' ' 72 §7] DYNAMICS OF A RIGID BODY 73 where co^ = — co^;'? ^y ^^ ~ ^y ^^^ <^2 — "~ ^z\ so that m^, (Oy and co^ are the components of the angular velocity of the moving axes X', X' , Z' relatively to the fixed axes X, X, Z. If we now substitute these expressions for --^, etc., in equations (7*001) we get, dP, dPj ^ dp; , dp; -dt ^ ""''-df + ""''-df + ""''-W + P;(a>,aii - co.a.i) . . (7-002) At an instant when the fixed and moving axes are coincident, «ii = 1, a^i = aai = and equation (7*002 ) becomes dt dt to which we may add i = :^ + p;co, - P>,. dP dP ' ^ = ^ + P' ny, - PJoy^ , . . (7-01 ) dt dt "" ' ' "^ ^ ' dP, ^ dP,' dt dt It is very easy to be misled by these equations, and we shall therefore inquire carefuUy about their significance before applying them. In arriving at the transformation (2-22) we represented the vector concerned (in the present case P) by a straight line drawn from the origin in the direction of the vector, and having a length numerically equal to it. Therefore P^, Py and P^ are the co-ordinates of the end point of the line. Equations (7*01) apply at the instant when the two co-ordinate systems coincide. Hence P„ = P/, Py = P/, P, = P/. Suppose now that P is the angular velocity of a rigid body with one point fixed in the common origin of the co-ordinate systems. Clearly the components of the angular velocity of the body have the same values in both sj^stems of co-ordinates when they happen to coincide. It is important to note this and so avoid the error of confusing the angular velocity referred to the moving axes with the angular velocity relative to the moving axes. In fact, if the moving axes were fixed in the rigid body, its angular velocity would be (CO3,, My, coj in both systems of co-ordinates ; but obviously zero relative to the moving axes ; and we note too that the rate of change of co is the same whether referred to the fixed or the moving axes, as is immediately evident on substituting CO for P in (7-01). 74 THEORETICAL PHYSICS [Ch. IV Let us now suppose the moving axes to be fixed in a rigid body and to coincide with its principal axes of inertia through the fixed point of the body (the common origin of both systems of co-ordinates) and let us further suppose P = {P^, Py, PJ to be the an2;ular momentum of the body. Then ( ~, ~—^, -— ? ) ^ -^ \ dt dt dt J becomes the torque or couple applied to the body. In what follows we shall denote this by (L, M, N), At an instant when the axes are coincident P, = P,; = Am,, p, = p; = Ceo,, where A, B and C are the principal moments of inertia of the body. Clearly dPJ _ jdoy, dt dt ' dPy' ^ -^dwy dt dt ' dPJ ^ (7^ dt dt ' On making these substitutions in equations (7*01 ) we obtain i = 4^- + (C - B)o>,oy^, N = Cf^ + (B - A)c^co, . . . (7-02) When the applied couple vanishes these equations become ,dco^ dt ^-^ = iB- (^H^. b"^^ = {G - A)co,co,, C^ = {A - B)m,cOy . . . (7-021) The equations (7*02) and (7*021) are the well known dynamical equations of Euler. On multiplying (7-02) by ca,, cOy, and w^ respectively and adding, we get §7] DYNAMICS OF A RIGID BODY 75 which states that the rate of increase of the kinetic energy of the body is equal to the rate at which the applied couple does work, a result we expect on other grounds (equation 6*11). When the couple applied to the body is zero, i.e. when L = M = N = 0, we find, by multiplying equations (7*02 1) by Aco^, BcOy and Cco^ respectively and adding or A^co^^ + B^co/ + C2ca,2 = Q^ . . . (7-04) where D^ is a> constant. This equation is also to be anticipated on other grounds, since it expresses the constancy of the angular momentum (§6). A particular solution of equations (7*021 ) is (^x — <^v — ^ '^ <^z — <^05 ^ constant. This represents a rotation with constant angular velocity about a principal axis of inertia. Suppose the body to be rotating in this way and then slightly disturbed, so that it acquires very small angular velocities co^ and cOy about the other principal axes of inertia. How will it behave if it is now left to itself ? Since CO3. and cOy are small (i.e. by comparison with coo), we shall ignore the product co^(Oy. Euler's equations now become ^W "^ ^^ ~ ^^^'''^' = . . . . (7-05) B^ + (A- C)co,co, = 0. Differentiating the former of these with respect to the time, and eliminating — -^, we obtain W + (£z1^^.lAW..,.0., (7.051) By differentiating the second of the equations (7*05 ) in a similar way we obtain ^. + (^-^^^^Wco„ = . (7-052) The constant iC-B){C-A) AB in both of these equations is positive if the moment of inertia G is either greater than A and B or smaller than A and B. In such a case 0)^ = B cos {at — (f)) o)y=S COS {at -y)) . . . (7-053) 76 THEOHETICAL PHYSICS [Ch. IV where R and >S' are small real constants, (/> and xp are constants, and coo . . (7-054) J {G~A){G-B) AB We see therefore that the motion of rotation is stable since cOg, and ojy never exceed in absolute value the small constants R and 8. It should be noticed that R, S, ^ and y) are not all independent. The reason for this is that equations (7'051) and (7'052), in the solutions of which they occur, are more general than the equa- tions (7*05) with which we are really concerned, since they are obtained from the latter by differentiation. If we abbre- viate by writing p = at — cj), q = at — ip, and substitute the solutions (7*053 ) in equations (7*05), we get sin p (G — B)Soyo and therefore cos Q aAR ' sin p _(^ - G)Rm, cos aBS sin p sin q (7-055) cos p COS q in consequence of (7-054). It follows that p and q differ by an odd multiple of — and the solutions (7-053) may consequently be put in the form 0)^ = R cos (at — cf)), C0y==8 sin [at -cf>) ... (7-056) the first of the equations (7-055) now becomes (G - B)Scoo 1 ARa whence we get ^ = ^70^ • • • ^'-'"'^ If we represent the angular velocity o> by a straight line drawn from the origin, equal in length to to, and in such a direc- tion that the co-ordinates of its end points are co^, cOy and co^ (= (Oq) respectively, we see that it describes a small cone in the body. The end point travels along the small ellipse with semi-axes R and S. § 7] DYNAMICS OF A EIGID BODY 77 The form of equations (7*021) suggests that their general solution can be expressed in terms of elliptic functions. Con- sider the integral e } dd ^^ J Vl-y^2 singer ^'<^' which belongs to the class of integrals called elliptic integrals. The upper limit, 0, is termed the amplitude of | and may be denoted by am §• Therefore sin 6 = sin am |, or, in the usual notation sin 6 = sn ^. Similarly cos 6 = cos am i = en i. The function Vl — k^ sin^ 6\ is usually called A^? /)^d = /\ am ^ = dn ^. The three functions, sn^, cn^ and dni are called elliptic func- tions. The differential quotient of sni with respect to | is d sn i _d sin 6 dO ~~dl dd~'di' i^^Go^Q.Vl -yb^sin^^l, d I dsn ^ f. J f. or — -— - = en i dn |. di Similarly d en ^ f. T ^ —nr- = — sn^dn^, d^ i^ = -k^snicn^ . . . . (7-06) a f These equations suggest, as a solution of (7*021 ), <^x = coi sn {at — (f)), cOy = (O2 c^ {Git — ^)j CO, = (Oodn {at — (j)) .... (7'07) where coi, CO2, coq, a and ^ are constants, which, as we shall see, are not all independent. Substituting in (7*021 ), we find B -G acoi = — cogCOo — aft) 2 = — Fi — coiCOq — ak^coo = — -^ — ft)ift>2 . . . (7*071) G A u G — A B A — B 78 THEORETICAL PHYSICS [Ch. IV Therefore coi^ {G - B)B m^ {C-A)A' {A-B)A m,'^ (C -B)C coo' (7-072) 2 _ (^ -B)(G-A)^_^ , and a^ = -^: — — coq Of the six constants, coi, coa, oy^, k, a and ^ therefore, three can be expressed in terms of the remaining three. These latter may be chosen arbitrarily and the solution (7'07) is therefore the general one. Let us select a>i, coq and cf) as the arbitrary constants and consider the case where coi and co^ are very small compared with (Oq. The parameter k^ will be a small quantity of the second order, by the second equation (7'072). We shall therefore ignore it. We thus get in the equations defining the elliptic functions. Therefore sn i = sin 6 = sin |, en ^ = cos 6 = cos |, dn i = I, and the solution (7-07) reduces, as of course it should, to that already found for this special case (equations 7*056 and 7*057). § 7-1. Geometuical Exposition We have in (7*07) the solution of the problem of the motion of a rigid body, one point in which is fixed, for the special case where the forces acting on the body have no resultant moment about the fixed point. A very instructive picture of the motion is provided by the geometrical method of Poinsot (Theorie nouvelle de la rotation des corps, 1851). The results we have aheady obtained indicate that the instantaneous axis of rotation wanders about in the rigid body and therefore sweeps out in it a cone {s, Fig. 7-1), having its apex at the fixed point, 0. The positions of this axis in the body at successive instants of time are represented by Oa, 06, Oc, Od, Oe, etc. The lengths of these lines may conveniently be made equal, or proportional, to the corresponding values of co at these instants. During the time required by the axis of rotation to travel from Oa to 06 the point 6 will travel in space to some point ^. That is to say, the line 06 in the body will occupy the position 0/9 at the instant when it coincides with the axis of rotation. In a succeeding interval the axis of rotation will have reached Oc, (in the body) which §7-1] DYNAMICS OF A RIGID BODY 79 wiU now have a position Oy in space, and so forth. The lines, Oa, 0^, Oy, 0^, Oe, etc., sweep out a cone, a, which is fixed in space. The motion of the body is consequently such as would result if a certain cone, s, fixed rigidly in the body, were to roll, with an appropriate angular velocity, on another cone, a, fixed in space. The cone, s, wiU cut the momental ellipsoid (which may likewise be described as fixed in the body, or rigidly attached to it) in a closed curve, as will be shown. This curve Poinsot called the polhode {noXoQ, axis ; odog, path). Its equations can be found in the following way : Using {x, y, z) in place of (^, y}, C) in the equa- tion (6*235), of the momental ellipsoid, we have for the com- ponents of the angular velocity, cj, Fig. 7-1 CO, CO ■X, COy CO -y, (^z CO —z (7-1 Q - Q Q p meaning, as in § 6*2, the radius vector from to {x, y, z). The perpendicular, p, from to the tangent plane at {x, y, z) is p = Q cos d, if 6 is the angle between p and p. Therefore p is the scalar product, (p N), of p and a unit vector N in the direction p. Consequently p = xa ^ yP -\- zy, if a, /5 and y are the components of N, or the direction cosines of p. The equation of the tangent plane at {x, y, z) is Ax^ + Byt] + CzC = M, if (I, rj, C) is any point on it. Therefore Ax VA^x^ +B^y^ + GV\ P y = p ^ By VA^x^ +B^y^ + C^z^l Cz VA^x^ + B^y^ + CV\ M (7.11) 80 THEORETICAL PHYSICS [Ch. IV But by combining (7*04) and (7*1 ), we find that A^x^ +B^y^ -\-C^z^=^^ . . . (7-12) Therefore i^ = ^ (7-121) Similarly, by combining (6*41) and (7*1) we obtain Ax'' + By^ + Cz^ = 2T^ . . . (7-13) or M=2T^ . . .(7-131) It follows that — is a constant, namely and consequently CO P (7-132) ^^V2TM\ . . _ .(7.133) a It is therefore constant and its direction cosines (7-11) are the same as those of the angular momentum £1. Consequently, it is invariable in length and direction, and the tangent plane remains fixed in space during the motion of the body. The last of the equations (7-11) gives us AH^ -{- B^y^ -^ CV = ^ . . . (7-14) This equation holds for any point (x, y, z) where the axis of rota- tion cuts the ellipsoid of inertia and it, together with the equation of the ellipsoid, Ax^ +By^ -\-Cz^=M . . . . (7-15) determines the polhode. If we multiply (7-14) by p^ and (7-15) by M and subtract, we get (p2^2 _ MA)x^ + tp2^2 _ MB)y^ + (p^C^ - MC)z^ = (7-16) which is the equation of the polhode cone s. The curve traced out on the fixed tangent plane by the instantaneous axis of rotation was called by Poinsot the her- polhode (from squelv, to crawl, like a serpent). The corre- sponding herpolhode cone is the cone a, fixed in space, on which the polhode cone rolls. We have now a very clear picture of the motion, especially if we remember (7-132) that the angular velocity about the instantaneous axis is proportional to § 7-1] DYNAMICS OF A RIGID BODY 81 p, the radius vector of the momental ellipsoid which coincides with the axis. The cone, s, fixed relatively to the ellipsoid rolls on the cone, a, in such a way that the ellipsoid is in contact with a fixed plane, the velocity of rotation at any instant being proportional to the distance, q, from the fixed point, 0, to the point of contact with the fixed plane. The semi-axes of the ellipsoid of inertia are Suppose, A>B>G, then f-'-^i In one extreme case --I (7-17) and the equation of the polhode cone s (7' 16) becomes (B^ - AB)y^ + (C2 - AC)z'' = 0. Since both terms on the left of this equation have the same sign, the only real points on it are the points y — z = 0, and the cone reduces to a straight line, or, strictly speaking, to two imaginary planes intersecting in a real line, the X axis. There is a similar M state of affairs if ^^ j^^s the other extreme value — . If however G 2 if ^' = F' the equation of the cone becomes (^2 _ AB)x'' + ((72 - 05)22 _ 0^ In this equation A'^ — AB is positive and (7^ — CB is negative. It therefore represents two real planes intersecting in the Y axis. Instead of combining equations (7*14) and (7*15) to get the equation of a cone, let us eliminate x'^. We thus obtain [B'^ - AB)y'' -\- {G'' - AG)z'' = ^ - AM . (7-18) Reference to (7*17) will show that the right-hand member of this equation is negative or, in the extreme case, zero, and since this is true likewise of the coefficients of ^/^ and s^, we conclude that the projections of the polhodes on the YZ plane are ellipses. %i^^^ 82 THEORETICAL PHYSICS The ratio of the semi-axes of any of the ellipses is lB(A \ C{A B{A~B)\ [Ch. IV (7-181 Similarly, we can show that the projections of the polhodes on the XY plane are the ellipses, (^2 _ AC)x^ + (52 - BG)y^ = ^-. MG . (7-182) and ratio of the semi-axes being in this case V A(G -A) (7-183) B{G -B) This result should be compared with (7-057). The projections on the XZ plane are the hyperbolas [A^ - AB)x^ + (C2 - GB)z^ = ^ -MB (7-184) § 7-2. Efler's Angular Co-ordinates We shall continue to use a system of axes, X', Y', Z', fixed in the body, and coincident with its principal axes of inertia. Let X, Y, Z, be another set of axes fixed in space, the Z axis being directed vertically upwards, and the two sets of axes having a common origin, 0, in the fixed point of the body. Let the angle §7-2] DYNAMICS OF A RIGID BODY 83 between Z and Z' be denoted by 0. The X'Y' plane intersects the XY plane in the line, OH, (Fig. 7-2). The angle between OH and OX is denoted by ^, and that between OX' and OH by ^. The positive directions are indicated in the figure by arrows. The position of the body, at any instant, is completely determined by the values of these three angles, called Euler's angles. The Eulerian angles are illustrated by the method of mounting an ordinary gyroscope (Fig. 7-21). There is a fixed ring, ABC. Within this is a second ring ahc pivoted at A and B so that it can turn about the vertical axis, AB. The axis AB corresponds to OZ (Fig. 7-2). Within the ring, ahc, is stiU another ring, a^y, pivoted at a and 6, so that it can turn about the horizontal axis, ah. This axis corresponds to OH (Fig. 7-2). The gyroscopic wheel, itself, is pivoted at a and ^ in the innermost ring, so that it can spin about an axis, a/5, perpen- dicular to ah. The axis, a/?, corresponds to OZ'. Let us now express the components, co^, ca,, and oi.} of the angular velocity of the body, in terms of -j-, j- and ^. It is dt dt dt dcf> clear that co^ and o^y do not depend on ^ and we must therefore az have oy. CO, = ^ COS {ZY') + ? cos (HY'). az az Obviously m^ is not identical with -^ since ^ is an angle measured az To get cOg we have to add to ~ the az from the moving line OH. angular -j- multiplied by cos (ZZ'), therefore ^ Note that cox, (Oy, coz have the same meaning as in Euler's equations. They are the components of the angular velocity referred to axes X', Y', Z' fixed in the body. 84 THEORETICAL PHYSICS [Ch. IV The direction cosines in these equations are easily seen to have the values set out in the table : X' Y' Z' z sin 6 sin ^ sin Q cos (ji cos 6 H cos (j) — sin For example, cos [ZX') is X' co-ordinate of a point on Z the unit distance from 0. The distance from of the projection of this point on the X'Y' plane is sin Q and the angle between this projection and OX' is obviously the complement of ^. Hence we get the projection on OX' by a further multiplication by sin (j). We therefore arrive at the following relationships : — dw . ^ . J . dd J 0)^ = -J- sm & sm ^ + — cos ^, Clt U/t dw . ^ , dd . , cOy = -r^ sm cos — —- sm A, " dt dt dip ^ , d(h CO, = -^ cos (9 + -^. dt dt (7.2) § 7-3. The Top and Gyroscope We shall now apply Euler's equations to the problem of the symmetrical top (or gyroscope) supposing the peg of the top (or the fixed point in the gyroscope) to be fixed in the origin. If the Z' axis is the axis of symmetry of the top, and if the distance of the centre of mass from is h, the couple exerted has always the direction OH, and is equal to mgh sin d, m being the mass of the top. We must substitute for L, M and N in Euler's equations the components of this couple along the directions Z', 7' and Z' . The table of cosines (§ 7-2) gives us L = mgh sin d cos 0, M = —mgh sin d sin 0, iV^ -0. §7-3] DYNAMICS OF A RIGID BODY 85 On substituting these values for L, M and N in Euler's equations, (7'02), we have mgh sin cos ^ = ^ -~ + (C — B)cOya}y, Cut — mgJi sin 6 sin ^ = B-yJ + (A — 0)G>/a„ = Cf^-^ + (B - A)co^m,. . (7-3) If we replace co^, cOy and co^ in (7'3) by the Eulerian expres- sions (7*2) we obtain three differential equations the solution of which gives the character of the motion of the top. Instead of proceeding in this way it is simpler to make use of the energy equation, and obtain two further equations by equating the angular momenta about the Z and Z' axes to constants. This we are at liberty to do, since the applied couple is in the direction OH, that is to say, in a direction perpendicular to Z and to Z', so that its component in either of these directions is zero. We obtain the energy equation by multiplying equations (7*3) by 0)^, (Oy, and cOg respectively and adding. In this way we get — {^Aco^^ + iBcOy^ + iOco/} = mgh sin 6(co^ cos cf) — cOy sin ^). clt If now we write A = B, on account of the symmetry, and sub- stitute for (o^ and ca^ their Eulerian values (7*2), we have l(*4^»'»(S)'+(§)"]+*M =•"'*»■» in which we have replaced co^^ by coq^, which is a constant by the third of the equations (7*3). Thus on integrating we arrive at the result "»■»©" +©"="-¥-» • <"" where a is a constant of integration. This is the energy equation. The table of direction cosines (§ 7-2) gives for the angular momentum in the Z direction, Aco^ sin 6 sin ^ + BcOy sin 6 cos ^ + Cco^ cos 6, or A sin d{(o^ sin cf) -{- cOy cos </>) + Gcoq cos d. On replacing cOg. and cOy in the usual way by the expressions in (7*2), we get for the angular momentum about the Z axis, A sin Ofsin 6 ~\ + Ocoo cos = a constant, or sin2 6>^ = a-^°cos6l . . . . (7-32) dt A ^ ' where a is a constant of integration. 86 THEORETICAL PHYSICS [Ch. IV For the third equation we have Cm^ = a constant, or, by (7-2), C\ cos 6 -^ -\- -^\ = a constant, I dt dt) or finally coseg + ^^=co„ .... (7-33) dt dt The three equations, 7'31, 7*32 and 7*33 completely describe the behaviour of the ordinary top, when its peg is prevented from wandering about, or the motion of any rigid body with axial symmetry (gyroscope), when one point on the axis is fixed in space. By eliminating -~ from (7*31) and (7'32) we arrive at the equation If Coy, J 2 /f^0x2 2mgh ^ ,^^^, We can simplify this and the remaining equations by the follow- ing abbreviations : ^ = ?^, 6=^,/. =cos9 . . (7-345) and consequently -^^rt- ■ ■ ■ (^-^4^) We have therefore, when we substitute in (7'34), '^/*\' _ ^^ _ ^^)(i _ ^2) _ (^ _ 5^)2 . (7.35) (i)=<° an equation which may be expressed in the integrated form t= f , ^/^ (7-351) J A/(a - /S/i)(l - /i^) _ (a _ 6„)2 C if c is the value of ju at the instant ^ = 0. Equations (7'32) and (7'33) take the respective forms, dip _ a -hfi n'Z(y\ dt 1-/^2 ^ ^ #_^^_M^-6^ .... (7-37) dt I — fj,^ The integral (7'351) belongs to the class of elliptic integrals, and therefore /u, or cos 6, is an elliptic function of the time and § 7-3] DYNAMICS OF A RIGID BODY 87 consequently oscillates periodically between a fixed upper limit ^oj and a fixed lower limit ^i. Otherwise expressed, the angle, 0, between the axis of symmetry and the vertical, will change in a periodic fashion between a smallest value Oq and a greatest value d 1. This motion of the axis is called nutation. The motion expressed by -~, that is to say, the motion of the axis, OH CLZ (Fig. 7-2), is called precession. We can easily learn the general character of the motion from equations (7'35), (7*36) and (7'37) without making explicit use of the properties of elliptic functions. If we denote (-r-) hy f{/x), equation (7'35) becomes f{f,) = {a-M{l-f,^)-(a-bf,)\ Since ^ is a positive constant, /(_00) = - CO, /(+ OO) = + 00, and further /(-l) = -(a + 6)^ f(+l) = -(a-b)K Therefore /(— 1) and/(+ 1) are necessarily negative (or zero), and the general character of the function f(jbi) (=(-7-) ) is that illustrated by Fig. 7-3. Only positive values of f(jLi) and values of fi between — 1 and + 1 can have any significance in the motion of the top. The significant points in the diagrams (Fig. 7-3) are therefore those in the shaded areas of a, b and c. During the motion fi varies backwards and forwards between fixed upper and lower Limits, /liq and ^j respectively and associated with this is a corresponding variation of the angle, 0. At the same time the precessional velocity, — will also vary periodically with the same period as jti (see equation (7*36)). If we restrict our attention to the case where a and b are positive, we have the following possibilities : (1) if 6 is small enough, i.e. if the top is not spinning fast (see (7*345)), -~ will remain positive (7'36) ; cit (2) if the top is spinning very fast (6 large enough) -^ may change Coo sign during the nutational motion between 61 and Oq. This will happen when fj, is equal to -. A special case, (3), is that in 7 88 THEORETICAL PHYSICS [Ch. IV a which [Xq = -. This is the case when the top is set spinning and released in such a way that initially -r^- = and -^ = 0. at at filA -I (at*/ M (a) '\|_j/"-¥ AtA f(fA (0 +1 /" Fig. 7-3 J"(/^) The three cases considered are illustrated by {a), (b) and (c) respectively in Fig. 7-31, which exhibits the curve traced out by the centre of mass on the sphere with centre, 0, and radius h. Fig. 7-31 The figure illustrates the possibilities in the case of the ordinary top, for which ju is always positive. It is instructive to consider the case where the top is not spinning and where the angular momentum about the vertical §7-3] DYNAMICS OF A RIGID BODY 89 axis, Z, is zero, so that coq and a are both zero, (7*345) and (7*32 ). We have then the equation m = {^^' = (,o.-M{^-i^ or if we substitute cos for ix, As ^ = — ~- we may express the equation in the form ■./dd\^ . mgh ^ a This equation is seen to be identical with (6*501), since A and / have the same meaning, and the difference in sign is merely due to the fact that 6 in the one equation is the supplement of 6 in the other. The motion is that of the pendu- lum. We find, just as in ^ Q'5, two sub-cases. If - exceeds —K-, i.e. if a exceeds ^, — never vanishes and body rotates con- 21 dt tinuously round a fixed horizontal axis, but with a periodically varying velocity. This is the case illustrated by (c) in Fig. 7*3, since ( y- ) = fii^) vanishes at the points /^ = + 1 and fi = — I and nowhere else between these limits. On the other hand, if a<^ /5, /(^) again vanishes at + 1 and — 1, (7*35) since a and b are both zero, but also at a point, ^ = -, between these limits. P This corresponds to the ordinary pendulum motion and is illus- trated in (6), Fig. 7-3. The significant values of (.i extend from — 1 (when the pendulum is vertical) to -. P Another interesting special case is that for which the interval /^o/^i within which /(//) is positive is contracted to a point, so that the curve for/(^) touches the [x axis as in Fig. 7-3 {d). Therefore 11 is constant during the motion and consequently so is -J-. The axis of the top or gyroscope sweeps out a circular cone in space with a constant angular velocity. Let us consider the case where [x is zero and the axis of symmetry therefore horizontal. We see (7*36) that -~ is equal to the constant a. It is an instruc- az tive exercise to determine a by means of equations (7*01). In 90 THEORETICAL PHYSICS [Ch. IV these equations we must remember that [m^, cOy, a>^) represents the angular velocity of the moving axes and not necessarily that of the gyroscope. Suppose the X' directed vertically upwards, Fig. 7-32, and fixed, while the axis of symmetry of the top or gyroscope coincides with the Z' axis. We have therefore for (ft)^, COy, coj of equations (7*01) co^ = a. a 0, CO. CO, =0. Y Let PJ, Py and PJ be the components of angular momentum relative to these axes. Then P ' Fig. 7-32 . 0, P ' =0 PJ = Ceo, where coq has the same meaning as before. dP^ dt We have further 0, dP, dt = — mc since these quantities represent the rate of change of angular momentum with respect to the fixed axes, which are momentarily coincident with the moving ones. The equations (7*0 1) are then satisfied if — mgh = — Ccjo^a, mgh or Ceo, C^o We can of course arrive at this result in a much simpler way. Let OA (Fig. 7-33) represent the angular momentum Ccoq at any instant. The applied couple will produce in a short interval dt a change of momentum dO, at right angles to OA, as shown in the figure. The angle dyj swept out during dt will therefore be or dQ Ccoq dn 'di Ccoq = dip dip dt' §7 4] DYNAMICS OF A RIGID BODY 91 But in the present case -r- is constant and equal to mgh, therefore dip _ _ mgh dt C(Oq as we have already found. § 7-4. The Precession" of the Equinoxes The earth behaves like a top. The attraction of the sun is exerted along a line which does not pass through the centre of mass of the earth except at the equinoxes. It thus gives rise to a couple tending to tilt the earth's axis about its centre of mass and make it more nearly vertical. The state of affairs is very similar to that we have just been studying. The centre of mass of the earth corresponds to the fixed point, 0, the peg of top. Consequently the earth's axis exhibits a motion of nutation and precession. The line in which the equator cuts the ecliptic corresponds to the OH in Fig. 7-2. The points where it cuts the celestial sphere are caUed the equinoctial points, from the circumstance that day and night are equal in length when the sun passes through them. In consequence of the precession the equinoctial points travel slowly round the heavens in the plane of the ecliptic in a retrograde direction, a whole revolution requiring a period of 25,800 years. Associated with this is a corresponding motion of the celestial poles which in the same period describe circles of 23° 27' in diameter round the poles of the ecliptic. BIBLIOGRAPHY EuLEE, : Mechanica, sive Motus Scientia analytice exposita, 1736. PoiNSOT : Theorie nouvelle de la Rotation des Corps, 1851. Gray : A Treatise on Physics. (Churchill.) Webster : The Dynamics of Particles and of Rigid, Elastic and Fluid Bodies, (Teubner.) Cbabtree : Spinning Tops and Gyroscopic Motion. (Longmans.) 1 CHAPTER V PRINCIPLES OF DYNAMICS § 8. Principle of Virtual Displacements N order that a particle may be in equilibrium, the resultant of all the forces acting on it must necessarily be zero. If F ^ {F^, Fy, F^) be the resultant force, F = F = F = This condition may be stated in the following alternative way : (F 61) = FJx + Fydy -}- F,dz = . . (8) where 81 = {dx, dy, dz) is an arbitrary small displacement of the particle, i.e. any small displacement we like to choose. For sup- pose we assign to dy and dz the value zero, and to dx a value different from zero. Equation (8) then becomes F^dx = 0, and hence F^ = 0. Similarly the statement (8) requires Fy and F^ to be zero. Consider any number of particles, which we may distinguish by the subscripts 1, 2, 3, ...<§,... , and let the respective forces acting on them be Fj, Fg, Fg, . . . F^ . . . Further, imagine the particles to suffer the arbitrary small displacements, {dx^, dy^, dz^), {dx^, dy^, dz^), . . . (dx„ dy„ dz^), . . . Then the condition for the equilibrium of the system of particles is ^{FJx,+FJy, + FJz,)=0 . . (8-01) the summation being extended over all the particles of the system. The arbitrary small displacement (dx, dy, dz) is called a virtual displacement and the statement (8) or (8*01) is called the principle of virtual displacements or the principle of virtual work. The utility of the principle becomes evident when we apply it to cases where the particles are subject to constraints. As an illustration consider the case of a single particle so constrained that it cannot leave some given surface. There will in general be some force, F' = (FJ, Fy, F/), normal to the surface, and 92 §8] PRINCIPLES OF DYNAMICS 93 of such a magnitude that it prevents the particle from leaving it. Let us write equation (8) in the form {F, + FJ)dx + {F, + F,')dy + {F, + F:)dz = 0, where F = (F^, Fy, F^) represents the part of the force on the particle not due to the constraint. We shall call it the impressed force. Of course the principle of virtual displacements requires that F, + F,' =F,-h Fy' =F,+F: =0; but this is not the most important, nor the most interesting inference from the equation. If we subject the virtual dis- placement 51 = (dx, dy, dz) to the condition that it has to be along the surface, we have, since F' is normal to the surface, (F'51) = FJdx + Fy'dy + F.'dz = and consequently FJx + Fydy + F,dz = ... (8-02) In this statement of the principle all reference to the force F' due to the constraint is eliminated ; but in applying it we have to remember that the virtual displacement is no longer arbitrary, and we cannot infer therefore that F^ = Fy = F^ = 0. Indeed, this would in general be untrue. Let the equation of the surface, to which any motion of the particle is confined, be cl>{x,y,z)=0 (8-03) The virtual displacement is therefore subject to the condition |.. + |., + |.. = 0. . . (8.031) Let us eliminate one of the components of 81, e.g., dx, with the help of (8*02). We can do this most conveniently by multi- plying (8*031) by a factor, A, so chosen that i^. - A^ = . . . . (8-032) and subtracting the result from (8*02). This gives The components dy and dz of the virtual displacement can be chosen arbitrarily, since whatever small values we assign to them we can always so adjust dx as to satisfy (8*031), the condition to which the displacement has to conform. Hence we infer F,-^i=0 . . . . (8-034) oz 94 THEORETICAL PHYSICS [Ch. V In order therefore that the particle may be in equilibrium the impressed force, F = {F^, Fy, F^) must satisfy the equations (8*032) and (8*034), or, what amounts to the same thing, F F F (8*035) d(f> d<f) dcf) • • ' ' dx dy dz Consider next the case where a particle is constrained to keep to a curve. Suppose the latter to be the intersection of two surfaces, ^ (x, y, z) = 0, rp{x,y,z)=:0 (8*04) The virtual displacement, 61 = {dx, dy, dz) has consequently to satisfy the conditions and we infer that OX dy dz OX dy dz (8-041) ^'-'fz (8*05) or, what amounts to the same thing, F ^ ^ ^' dx dx F ^ ^ ^' dy' dy F ^ ^ ^' dz' dz = . (8*051) The principle of virtual displacements may be illustrated by the following examples : — Let the particle be confined to a spherical surface, but other- wise perfectly free, and suppose the force impressed on it to be directed vertically downwards. It might, for instance, be its weight. Let the origin of co-ordinates be at the centre of the §8-1] PRINCIPLES OF DYNAMICS 95 sphere and the X axis have the direction of the force. Equation (8*031) becomes xdx + ydy + zdz = 0, and for (8-032) and (8-034) we have F^ = ?^x, F, = h. Now since Fy = F, = 0, F^ = 2.x, = 2y, = 2z, and as F^ is not zero, A cannot vanish, and therefore y =z = 0. Consequently a; = + r or — r, where r is the radius of the sphere, or the possible positions of equilibrium are the uppermost and lowermost points on the sphere.^ An instructive example is that of a rigid body which can turn freely about a fixed axis, which we shaU take to be the Z axis of rectangular co-ordinates. The conditions for equili- brium are expressed by equations (8-01), the forces, F, being the impressed forces : not those due to constraints, together with the equations describing the constraints. These latter are, for every particle, s, Szs = 0, where d<j) is the same for all the particles in the body. Equation (8-01) therefore becomes 6<l>-^(x^,, - yJPJ = 0. Now d^ is arbitrary, hence 2(a:,i^» - yj«) = 0. This means that the sum of the moments of all the impressed forces with respect to the Z axis is zero, a result we have already obtained by a different method (§ 6-1). § 8-1. Peinciple of d'Alembert The principle of virtual displacements is a statical one. It provides a means of investigating the conditions necessary for Fx Fv Fz ^ Alternatively, the equations (8*035) become — = — = — and as Fy = Fz^ ^it follows that 2/ = 2: = 0. 96 THEORETICAL PHYSICS [Ch. V the equilibrium of a dynamical system. Its scope can be extended, however, by a device due to d'Alembert (Traite de Dynamique, 1743), so as to furnish a wider principle which constitutes a basis for the general investigation of the behaviour of djmamical systems. Let F = (F^, Fy, FJ be the resultant force exerted on a particle of mass m, then rr£^ -F,=0, dt^ ^ ' d^Z -r, We may express these equations in the single statement {-§ - ^0 '' + ("S - ^0'^ + (™S - ^^^ = ' ^'-'^ if (dx, dy, dz) is an arbitrary small displacement, since this necessi- tates the vanishing of the coefficients of dx, by and bz. Now (8*1) can be extended to apply to a system of particles, subject possibly to constraints, in the following way : If the system should be subject to constraints, F^ will signify the force impressed on the particle, s, and will not include the force or forces due to the constraints, and the virtual displace- ments {bx'^, bys, bZg) are not all arbitrary, but subject to the equations defining the constraints. Equation (8*1 1), with the interpretation just given, expresses the principle of d'Alembert. A simple illustration of the principle is furnished by the example of the rigid body in § 8. The procedure here differs d A only in the substitution of ^^W-^ — -^sa? for the F^^ of § 8, and (tt corresponding expressions for F^y and F^^. We thus get §8-2] PHINCIPLES OF DYNAMICS 97 and therefore, on account of the arbitrariness of dcj), same thing as which is the same thing as di (see § 6). § 8-2. Generalized Co-ordinates We shall now introduce the generalized co-ordinates of § 6. The rectangular co-ordinates of any particle, s, of a system may be expressed in terms of the generalized co-ordinates, q, in the form ^s =fs tei, ^2, . . -qj, Vs = ds (^1. ^2, . . . gj, ^s = K (s'l. 9^2, .. • gj, in which /g, g^ and ^^ are given functions of the g's and the inherent constants of the dynamical system. We have in consequence dx,=^dq,-^ ^dq,^- , . . ^^^J^dq^, dqi dq^ dq^ and similar equations for dy^ and dz^. It is convenient to use the symbol Xg itself to represent the functional dependence of the co-ordinate, x^, on the g's. We therefore obtain dx, = ^dqi + ^dq^ + . . . + ^%„, ^qi ^^2 ^qn dVs = -^dq, + -^dq^ + . . . + ^dq^, agi dq^ dq^ CZ VZ cz dz,=^dq.+^dq, + ...+^dq^. . (8-2) The symbol ' d ' will be used for increments which occur during the actual motion of the system, or during any motion we may tentatively ascribe to the system in the process of dis- covering the character of the actual motion. The symbol ' 6 ' will be used for virtual displacements and the increments depend- ing on them. The components, dxi, dyi, dzi; dx^, dyz, dz^\ ... of the virtual displacements of the particles of the system are not in general all arbitrary, as they may be subject to certain constraints. On the other hand the components, dq-i_, dq^, . . . dq^^ of a virtual displacement of the system are obviously quite 98 THEORETICAL PHYSICS [Ch. V arbitrary, since the generalized co-ordinates are in fact so chosen as to be independent of one another. They thus satisfy the conditions imposed by the constraints, as it were, automatically. If we replace the (i's in (8*2) by ^'s we get a corresponding set of equations for the virtual displacement {dx^, dy^, dz^) of a particle in terms of the associated dq's. The velocity of the particle, s, is given, in terms of the gener- alized velocities, by where q means -^. There are similar equations for -|^ and dt dt dz ~. In these equations it will be observed that each differential dt cX cl/ quotient, — , ^, . . . is expressed as a function of the ^'s and constants inherent in the system (§ 5-3). (jIT uij dz By squaring -— ?, -^-, -— respectively, adding and multiplying cit at u/t by nis, the mass of the particle, we obtain twice its kinetic energy. Therefore if T represents the kinetic energy of the system 2T = Qiiqiqi + 0i2gi^2 + . . . + Qmidn + — + Q^lMl + • - + QnnMn • (8'22) in which ©12, for example, means Q,, = E^ip p + lyi p + ^N . (8-221) ' (^qi^q^ oq^dq^ oqioq^) Each Q is therefore expressed as a function of the g's and the inherent constants and it will be noticed that Q^^ = Q^^. It is convenient to abbreviate (8*22) by writing it in the form 2T = Q^M^ .... (8-222) in which the summation is indicated by the duplication or repe- tition of each of the subscripts a and p, and not by the symbol SS. a ^ We see that (8-22) can be written in the form 2^ =^igi +^92^2 + . . -Mn • • • (8-23) or briefly . 2T = pjq^ where p, = Q,4^ + ^^2^2 + . • • + Qan% • • (8*24) or i>, = Qa^q^' §8-2] PRINCIPLES OF DYNAMICS 99 The quantity p^ is the generalized momentuin correspond- ing to the co-ordinate q^. Differentiating 2T partially with respect to g„, we get (see § 6-4), dT ^« = ai: («-2«) dT If, for instance, we take ^— - , we might carry out the differentia- cq2 tion firstly along the second row of (8 '22), thus obtaining and then along the second vertical column, obtaining 612^1 + ^22^2 + Q^^iz + . . . Qn4n' The two expressions are equal to one another (since Q^^ = Q^^) and together make 2jp^. Therefore in agreement with (8*241). It is important to note that (8*22) expresses 2T as a function of the q's, q's and the inherent constants of the system. It is a quadratic function of the g's. In (8*24) each p is also expressed as a function of the g's, g's and the inherent constants. It is a linear function of the g's. From (8*24) we derive the equations qi = I^llPl + ^12P2 + . . . + BlnPn, q^ = B^^p^ + R22P2 + . . . + R2nVn, in = KiPl + ^n2P^ + . . . + B^nPn ' • (8*25) in which the R^^ sue functions of the g's and inherent constants, and B^p — B^^. If we use the symbol i Q\ for the determinant Qllj ^125 • • • Qin ^2l5 ^2 2? • • • ^2n and the symbol \Q\a^ for the determinant which is formed by omitting the row, a, and the column, ^, each of these sub-deter- minants or minors having its sign so adjusted that, for example 101 =QM\.i + QM\..+ . . . +QM\^n (8-251) then JJ„^=1^ (8-252) 100 THEORETICAL PHYSICS [Ch. V Substituting the expressions (8*25) for q^ in (8*23), we get + E21P2P1 + R22P2P2 + . . . -\-R2np2pn + + KlPnPl + I^n^PnP^ + • • • + RnnPnPn (8*26) Which expresses 2T as a function of the g^'s, the ^'s and the inherent constants. From equation (8*26) we get dT ^- = E^^pi + K2P2 + • ' ' + KnPn GPa 'ir'' ^'-''^ by a process similar to that used to derive (8*241 ). The work done, during a small displacement of a system, is and therefore, when we substitute for the dx^, dy^, dz^ the expres- sions in (8*2), we get <t>idqi + (/»2^g2 + . . . + (l>ndqn, or ct>Jq^ (8-28) in which it is easily seen that We may term ^1, (1)2, etc. the generalized forces corresponding to the co-ordinates q^, q^, etc. 6 =I^{f ^ 4- F ^' -\- F ~'\ (8-281) § 8-3. Principle op Ekeegy The use of the term ' energy ' is of comparatively recent origin ; but the conception of energy began to emerge as far back as the time of Huygens (1629-1695). In § 5-1 we deduced from Newton's laws that the increase of the kinetic energy, ^mv^, of a iDarticle is equal to the work done by, the force or forces acting on it (5*12). In any mechanical system whatever the work done by the forces of the system is equal to the corre- sponding increase of its kinetic energy. This is the principle of energy in the form (principle of vis viva) which is of peculiar importance in mechanics, i.e. in connexion with problems in which we are concerned only with movements of material masses under the influence of forces, explicitly given, or due to con- straints or analogous causes. §8-3] PRINCIPLES OF DYNAMICS 101 The further development of the conception of energy is linked up with discoveries in different directions. In many mechanical problems, some of which we have already dealt with, the work done by the forces can be equated to the decrement of a certain quantity, V (§ 5-1), a function of the co-ordinates of the system, or a function, we may say, of its configuration. Since this is equal to the increment of the kinetic energy, T, a function of the state of motion of the system and its configuration, the sum T -\- V remains unaltered. The work is done at the expense of V and results in an equal increase of T. Then the consistent failure of all attempts to devise a machine (perpetuum mobile) capable of doing work gratis, and the success, on the other hand, in devising machines capable of doing work by the consumption of coal, gas or oil, gradually produced the conviction that work can be done only at some expense ; that whenever work is done, something is necessarily consumed. This something is called energy, and we conventionally adopt the amount of work done as a measure of the energy consumed. This does not mean that the energy of a body or a system is merely its capacity for doing work. There is some reason to believe that energy has a more substantial character, more perseity than is suggested by ' capacity for doing work '. Finally the experimental work of a long line of investigators, Count Rumford, Davy, Colding, Hirn and above all. Joule, established that when heat ^ is generated by doing work, as for example in overcoming friction, and alternatively when work is done, as in the case of the steam engine, at the expense of heat, the quantity of heat (generated or consumed as the case may be ) is proportional to the work done ; the factor of proportionality (mechanical equivalent of heat) being the same, whether work is done at the expense of heat or heat produced in consequence of work done. This suggested that the heat in a body should be identified with the kinetic energy (or kinetic and potential energy) of the particles (molecules) of which it is constituted, and gave rise to the modern Principle of Conservation of Energy, according to which the energy in the world remains invariable in quantity. The constancy of ^ + F in certain mechanical systems is merely a special case therefore of the wider energy principle, and in the middle period of last century, and still later, it was generally held that, not only heat, but all other forms of energy were either kinetic energy or potential energy in the sense in which these terms are used in mechanics. The principle of conservation of energy is in excellent accord with the view, which until quite recent times was universally ^ Heat measured by the use of mercury thermometers. See § 15" 5. 102 THEORETICAL PHYSICS [Ch. V held, that physical and chemical phenomena are au fond mechanical phenomena ; and almost till the closing years of the century physical theories were held to be satisfactory or other- wise, just in proportion to the degree of success with which they furnished a mechanical picture of the Newtonian type. § 8.4. Equations o^ Hamilton and Lagrange If a function F = F{?., 2., • • • ffJ ■ • • • (8-4) exist, such that where </>!, ^2? • • • ^^ are the generalized forces (8*28), the work done by them, during a small displacement of the system, will be ^y^ ^y 1 ^y 7 ^oA^^^ - ^li - -^k^ ... - ^qn . 8-402 dV or - ^qa> This must be equal to the corresponding increase of the kinetic energy, T, Therefore dT = -^^dq^ (8-41) and in consequence of (8*4) dT = -dV or d(T -]-V) =0; so that the mechanical energy, T -\- V, remains constant. Such a system is said to be conservative. This is the exceptional case. In general T -\- V varies. This may happen in conse- quence of a complementary variation of the T -{- V oi some other system, or it may be associated with the development of heat, as when there are frictional forces, or with variations of other forms of energy. Instead of confining our attention to conservative systems, let us suppose that there is a potential energy function, F, such that the generalized forces are given by (8*401) ; but that V has the form y = F(2i, q,, . . . q, t) ... (8-42) -=tv + '> d{T + V) - -'>■ d(T + V) _ dt dv dt ■ §8-4] PRINCIPLES OF DYNAMICS i03 Equation (8'41) will stiU hold, but since we have -^-w^ = Tt • • • • («-42i) As the time does not appear explicitly in the expression for T, whether we take (8-22) or (8-26), we get from (8-421) dt dt When T is expressed as a function of the generalized mo- menta (8*26 ) we shall represent J' + F by the symbol H, so that H does not merely denote the energy, T -{- V, but it is also a functional symbol. Since V does not contain the p's it is clear that (8-423) cp^ cp^ and therefore by (8-27) dH _dT dp, dp, dH '^ - dp. dq, dH dt dp. or .^ _ ^ (8-43) It is essential that we should bear in mind that the partial dT differential quotient — , T being expressed as a fu7iction of the dT p's and q^s, is quite different from - — obtained from T expressed as a function of the q's and q's. In fact, the former differentiation is subjected to the condition that the ^'s and the remainder of the g''s are kept constant, while the latter is subjected to the condition that the g's and the remainder of the g's are kept constant. To avoid confusion let us write dT{p, q) ^DT dq ~ Dq and dTii^^dT dq dq 104 THEORETICAL PHYSICS [Ch. V We may express a small change dT in the kinetic energy of a system in the following different ways : — 2dT = pMa + q.^Pa .... (by 8-23) dT dT ^^=f>+f>- • • • («-^^^ DT Subtracting the last of these from the first, and replacing-—- by q, (8-27), we get dT==pMa--^dq^, . . . .(8-441) and the second equation (8*44) may be expressed in the form dT=pMa + ^^dq^ . . . (8-442) dT since Pa=;^—' Hence by comparison of (8*441) and (8*442) dq^ we find f: = -^ (8-45) Adding ^r— to both sides of this equation, we obtain D[T + F) _ _ d[T - V ) Dqa. dq, which we may put in the form ^^=-f (8-451) dqa ^a where X = J' — F is also a functional symbol indicating that T is expressed as a function of the g's and g's. On the left of this equation D/Dq^ has been replaced by d/dq^, since the functional symbol H akeady indicates that the T in it is a function of the g's and ^'s. From (8-41) we have dt dq^ dt therefore dT dp^ D{T + V) dq, _ dp, dt Dq, dt §8-5] PRINCIPLES OF DYNAMICS 105 and replacing ^— by ■— (8-27), we obtain \dt ~^ dqj dt ' and consequently also /dp^ _^_L\dq^ ^ \ dt dqj dt (See 8-451). This suggests, though it does not prove, the equations dt dq^ and dt dq^ (8-46) (8-461) Their validity will be established in § 8-6. The equations 8-46 together with (8-43) are known as Hamilton's canonical equations. The equations (8-461) are the equations of Lagrange and are usually written in the form dt\dqj dq^ (8-462) We may write them in this way, because V does not contain the g's and therefore dT _ d{T -V) _dL Ma Ma Ma The function L is called the Lagrangian Function. Pc § 8-5. Illustrations. Cyclic Co-ordikates As a first illustration we may take the case of the compound pendulum § 6-5. Here we have one q, which is, conveniently, the angle 0, Fig. 6-5. The energy equation is (6-501) i/C^y - Mgh cos 6> = ^ . . . (8-5) and system is conservative. The corresponding p is ^=W='dt • • • • (8-501) 106 THEORETICAL PHYSICS [Ch. V Therefore H = ^- Mgh cos d and the canonical equations are | = -|(g_M..c,..), -7- = ?r-( — ^ — -^^Q^ COS ) ; dt dp\2I ^ ) ' whence we obtain dS ^p It~I and therefore /^ = - Mgh sin e, in agreement with (6*5 ). The Lagrangian function for the pendulum is L = im + Mgh cos d, and consequently the Lagrangian equation is d^d whence /— - + Mgh sin = dt^ as before. It will be noticed that when there are n degrees of freedom there are 2n canonical equations of the first order while there are n Lagrangian equations of the second order. The case of the pendulum is merely illustrative. It is clear that nothing is gained by the equations of Hamilton or Lagrange in cases like this. Having set up the energy equation, it can only be described as a retrograde step to differentiate it. It is when we come to systems with more than one degree of freedom that the merits of the methods of Hamilton and Lagrange begin to appear. Let us turn to the case, § 5-5, of a particle moving under the influence of a central force — . We get (see 5*51) for the Hamiltonian function H=f + /^+? .... (8-51) one of the q's is the radial distance, r, and the other is the angle, d. 37 ^^ {¥^^ + ^^^ cos d) —^ {1/02 + Mgh cos 0} = 0, §8-5] PRINCIPLES OF DYNAMICS 107 In this case Hamilton's equations become dt dr\2m 2mr^ r J' dpe _ _i/^_^ JV__^^\ dt dd\2m 2mr^ r )' dt dp\2m 2mr^ r )' dt djpe\2m 2mr^ r )' On carrying out the partial differentiations we get dp, _ Po^ B dt mr^ r^' dr _ p^ dt m ' dd _ Pe dt mr^' From these four first order equations we may derive the following two second order equations : d^r pe^ , B dt^ mr^ r^ ^\mr^^\ =0 (8-52) dt\ dt) ^ ^ This example illustrates two points : (i) The two equations we have obtained are sufficient, since the object may be said to be to express r and d as functions of the time. We have already one equation, the energy equation, at the very outset, and therefore we do not need both the equations (8*52) which we have derived. Instead of employing for the final solution of the dynamical problem the equations (8*52), it is preferable to use the energy equation and one of them. The reason for this is that the energy equation has already advanced one step in the series of integrations marking the way to the final goal, the accompanying constant of integration being in fact the most important of all, namely the energy, (ii) Whenever one or more of the co-ordinates do not appear explicitly in the function H, as for example 6 in the problem of the motion of a particle under a central force, the corresponding momentum is constant. Such co-ordinates are termed cyclic co-ordinates. 108 THEORETICAL PHYSICS The Lagrangian function derived from (5-51) is B ■D or L = Imr^ + imr^S^ - -. Therefore dL '^=mr^e, ^ = -^^+^' dd "' and consequently dV ., , B i (m.^5) = 0, [Ch. V in agreement with (8'52). For another illustration we may turn to the problem of the spinning top. The energy equation (see § 7-3) gives us + iC {^ + cos dip}^ + Mgh cos d = E . . (8-53) from which we find p^ = ^^sin^ d.y) -{- C{^ + cos d.y)) cos 6, Pe = ^0, P4> = (^{^ + cos d.y)}. Hence and i: = i^(sin2 (9.^2 ^ 192) ^ ic'(^ _!_ ^os 6).y»)2 - iff^^ cos d (8-532) Whether we employ the equations of Hamilton or those of Lagrange, we find p^ = constant, p^ = constant, and these, together with the energy equation (8*53) are equiv- alent to (7-31), (7*32) and (7'33) which we have found already. The preceding examples illustrate conservative systems, in §8-6] PRINCIPLES OF DYNAMICS 109 which the potential energy, V, does not contain the time. The following example furnishes a simple illustration of a non- conservative Hamiltonian system. A particle of mass m is constrained to keep to a straight line, and subject to a restoring force proportional to its displacement from a fixed point, 0, plus a force which is a simple harmonic function of the time. Its equation of motion wiU be m— -| = — /Ltq + R C0& cot where ^, R and co are constants. In this case and ^ ^ 2m V = ~q^ — qB cos cot since F is defined to fulfil the condition The Hamiltonian function is therefore force = — 2m ^g2 — qR cos oyi, and the Lagrangian function, \mq^ — ^q^ + qR cos oyt. § 8-6. Principles of Action If we have to deal with a system of not more than two degrees of freedom we may repre- sent its configuration and be- haviour graphically, by rectan- gular axes of co-ordinates using lengths measured from the origin along two of the axes to repre- sent the values of the g's and a length measured along the re- maining axis to represent the corresponding time (Fig. 8-6). The motion of the system will be completely represented by a line such as (1, 2) in the diagram. We shall use the methods and the language which are appropriate for this graphical repre- sentation for systems of any number of degrees of freedom. The principle of d'Alembert (8-11), if applied to the type Fig. 8-6 110 THEORETICAL PHYSICS [Ch. V of djmamical system dealt with in §§8-4 and 8-5, will take the form In this equation the summations are sufficiently indicated by the repetitions of s and a. The s summation extends over all the particles of the system. The dx^, dy^, . . . dq^ . . . repre- sent virtual displacements. Our purpose is to investigate the actual motion of the system (represented by the path (1, 2) in Fig. 8-6) by studying its relation to motions represented by neighbouring paths (such as that shown in the figure by a broken line). These lines are comparable with the d lines of Fig. 3-31 in the proof of the theorem of Stokes, and we may conveniently suppose them to be drawn on a surface. It is helpful to regard the virtual displacements, dq^, as given by the intersections of this surface by a family of surfaces, f(qi,q., . . .q,,t)^c . . . . (8-61) These are quite arbitrarily chosen surfaces on account of the arbitrariness of the virtual displacements, dq^. If we pick out one of them by giving the constant C any value A, a neighbouring surface will be one for which C = K+dL The lines of intersection of this family of surfaces with the surface on which the d lines lie we shall naturally call 6 lines, as in § 3-3. The symbol 6 will therefore represent an increment incurred in passing along a 5 line from the d line representing the actual motion to the neighbouring d line ; while the symbol d will represent an increment incurred in passing along a d line from a surface C = A to a neighbouring surface C = A + (iA, i.e. from one 5 line to the next. We shall now make use of a device, already employed in previous investigations, namely that embodied in the formula db diah) . da a—- = \ ^ — 6 -— . dt dt dt Substituting for a and 6 respectively, OX, and m,— ^, 'dt we get and similar expressions for (J IJ (J z in.-~^dy, and m,——^dz,. ' dt^ ^' 'dt^ ' §8-6] PRINCIPLES OF DYNAMICS Equation (8*6 ) thus becomes Ill dt\ 'dt ' ^ 'dt dV '' 'dt ^' 'dt ' dy.d ,^ , , dz.d,^ , 'dt dr ^'^ dt dV '^ 0, or, (§ 8-2), dV dql dq^ = (8-612) We have seen (3'314) that ddx = ddx, etc., and we may show in a similar way that ddt = ddt ; but it does not follow, for d /dx\ ^mple, that -r{^^) = ^(t")' ^® ^^^^ ^^ ^^^^ (^i§- ^*^1) exai or 61 /dx\ _ (^(a; + (5:r) \di) ~ /(Za;\ _ \^/ ~ /dx\ _ (i(5a; W / ~ ~di dx dt' d{t + dt) dt ddx — dx ddt dt dt + dt ddt ddx dx ddt dt~df and similar formulae for 'dy\ , . /dz^ (8-62) <i) "" <i)^ jr+^j:+d/a:+Jjt:j pc^doo Fig. 8-61 With the aid of (8*62) we may now express (8*612) in the form (XiX „ \^M.) dt m„ and therefore (§ 8-4) \dt J 'dt \dtj 'dt \dtj /dx,\^ , /dy\^ , /dz\^\ddt , 37. 0, <^w!'- ST -2T'^ + dV -^-^c dt dt and by (8-421 l^ipM.}-ST-2Tf + SV-§St = 0, 0. 112 Now THEORETICAL PHYSICS dr ^ dt dt ' [Ch. V therefore we find '^^ipM.-m-ST-2T'^ + SV-,E'^ 0, or {Pa^qa Mt Edt} - d(2T -E) - (2T - E)~ = 0. U/t If we multiply this equation by dt and integrate between the limits 1 and 2 we obtain 2 2 Edt VMa or finally 2 pMa - Edt - [ {dtd{2T -E) -\- {2T - E)5dt} = 0, d[{2T -E)dt = . . (8-63) In this equation the variations symbolized by 6 are subject to no conditions, except that they are smaU. Let us give our attention in the first place to conservative systems, i.e. systems for which dE = 0. Since the variations in (8-63) are arbitrary we may subject them to the condition SE = db constant. We then have 2 2 2 2 or pMa - Edt PaK -I 2Tdt SE^dt -{-E 1 ddt = 0, + SEit^ -t^) =6 \2Tdt (8-631) If we suppose the two paths to join at the lower limit 1 but not at the upper limit 2, we get, on dropping the index 2, 2Tdt. . . (8-632) Padq. + (t- t,)dE = d^ or, if we use the symbol A for the integral on the right, Pjqa + (i- ii)dE = dA ... (8-633) The function A is one of those to which the term action is applied and (8-633) indicates that it may be expressed as a function of the g's and E, and therefore dA Pa =. t. = dq: M dE (8-634) §8-6] PRINCIPLES OF DYNAMICS 113 If the system is strictly periodic and the range of integration, 2 , extends exactly over the period, r, of the system, 1 2 the terms | Pa^q^ I must vanish, and we find (^2 - t^)dE = SA, or if we denote this particular value of A by the letter J, If :'y (8-^35) In the next place let us suppose the two paths to be co- 2 terminous in space (not necessarily in time) so that | Pa^^a I = 0, 1 since the terminal dq's vanish. Then if the variations are subjected to the condition dE = we find {2Tdt = (8-636) 1 for systems for which dE = 0, i.e. for conservative systems. This is the principle of least action in its original form. It was first given in 1747 by de Maupertuis, a Frenchman, who was, for a time, president of the Royal Prussian Academy during the reign of Frederick the Great. He claimed for his principle a foundation in the attributes of the Deity. ' Notre principe ... est une suite necessaire de I'emploi le plus sage de cette puissance,' i.e. ' la puissance du Createur,' and the principle has turned out to be not unworthy of the claim made for it. A better name for it would be ' principle of stationary action ', 2 since the action \2Tdt is not in aU cases a minimum. \ 1 If in (8*63) we suppose the two paths to be co-terminous in space and time, i.e. the terminal variations dq^ and dt are all zero, we get 2 d [{2T -E)dt = 1 2 or [ (T - V)dt = .... (8-64) 114 THEORETICAL PHYSICS [Ch. V This form (the most important one) of the principle of action is known as Hamilton's principle. The function S = { (T - V)dt (8-65) is called Hamilton's principal function, while the function A (8*633) is called Hamilton's characteristic function. If we take the two paths to be co-terminous (in space and time) at the lower limit only we get from (8*63), dropping the upper index, 2, Pa^qa -ESt = dS (8-66) and therefore >S is a function of the g's and the time and ^-|'-^=S • • • («-^") We may use Hamilton's principle (8*64) to establish the canonical equations and the equations of Lagrange. If we express E as a function of the ^'s, q's and t it becomes 2 or si (2T - H)dt = 1 2 6 j (jp4. - H)dt = 0. 1 Since the variations d are perfectly arbitrary, it is permissible to subject them to the condition ^^ = 0. With this condition (8*62) becomes ./cZg'X _ dSq \di) ~ It or dq=^dq . . . . (8-662) We therefore find 2 J (i>A + g«^i'. - g-^^l-. - ^%)rf« = . . (8-67) I But we have proved, (8*43), that dH qa = ^, therefore (8*67) becomes 2 j (pJqa - ^f^^y^ = 0. §8-6] PRINCIPLES OF DYNAMICS or i(4^'-'af*'->' = »' by (8-662). 1 Therefore T Now the integral 115 since the paths intersect at 1 and 2, therefore 1 As the dq^ are arbitrary this result requires that ^« + ^ = 0. dt dq^ These are the canonical equations of Hamilton. Those of Lagrange follow immediately, since dH ^ _aL There is a certain function H{pa_, qa, t) which is equal to T + F or to ^, i.e. H{p:, q^, t)-E = . . . . (8-675) and if we substitute for E and the p's the expressions in (8-661) we get Hamilton's partial differential equation . (8-68) When E is constant = a say, the equation becomes as H does not contain the time explicitly ; or, since dq. dqj 'dA H(^^,q^^a .... (8-681) 116 THEORETICAL PHYSICS [Ch. V § 8-7. Jacobi's Theobem. Hamilton's principal function, S, defined by (8*65), is a function of the co-ordinates, q, and the time ; and it satisfies the partial differential equation (8*68 ). A converse proposition naturally suggests itself. Having set up the energy equation (8*675), appropriate to a dynamical problem, and derived the partial differential equation by replacing each p^ by the corre- sponding — - and ^ by — — ; let us suppose an integral S to have been found. Will the differential quotients, n^ = t^ — , be identical with the corresponding generalized momenta of the dynamical system ? This amounts to asking if it is a matter of indifference whether we use for S a solution of (8*68) or the function (8'65). We shall prove that this is the case provided we use a solu- tion which is a complete integral of (8*68). This is an integral containing as many arbitrary constants as there are independent variables. It must be distinguished from a singular integral, which is a relation between the variables involving no arbitrary constant, and moreover is not a particular case of the complete integral ; and from a general integral which involves arbitrary functions and therefore altogether transcends the other integrals in its generality. The complete integral of (8*68) wiU have 7^ -f 1 arbitrary constants, if we suppose there are n co-ordinates, q. We shall represent them by a^, a 2, a^ . . . a^, a^+i- One of them, which we shall take to be a^+i, is merely additive. If S be a complete integral, we shall prove that the equations ds ^ ds 1^- = Pi, ^- = ^1, cai cqi ds ^ ds ... (A) ... (B) 1.^^- S.^"» • • • • ('-^^ constitute a solution of the associated dynamical problem, if /5i, ^2, ' • ' Pn ^^^ arbitrary constants, and if we identify tti, 71 2, . . .71^ with the generalized momenta, pi, Pz, - - - Pn § 8-7] PRINCIPLES OF DYNAMICS 117 respectively. Since /^i, j^a, . . . /^„ are constants, we have from the equations A (8^7) dt\daj ±(^1-) 0, dtXca^J clAdaJ and consequently d^s , d^S dq, , d^S dq, , , d^S dq^ ^ + ^ — ^ TT ~r -^ — ^ T^ -r • • • "1- 7^ — ?s r^ ^5 daidt dajdqi dt da^dq^, dt ^^i^qn ^^ d^S d^S dq, d^S dq, d^S dq. da^^t da^dqi dt da^dq^ dt ^(^t^qn ^^ d'S ^ J^ dq, ^ _d^ ^ + . . . ■;■ ^'^ dq, = (8-71) dajdt da^dqi dt da^dq^ dt ^^r?9.n ^^ From the partial differential equation (8*68), which, by hypo- thesis, 8 satisfies, we get on differentiating with respect to a^ = — + ^^ ^^^ + ^^ ^'^ ■ daidt dH d^s dS \ da-^dq^ or, remembering equations (8*7 B). ^ _ d^s d^s dH d^s dH d^s dH daidt daidqidjti da-^dq^dn^ ' ' ' da^dq^dnj to which we may add similar equations derived by differentiating with respect to ag, as, . . . a^, namely ^ _ 8^^ d^S dH d^S dH d^S dH dazdt da^dqi dn^ da^idqidn^, ' ' ' da^dq^ dn^ d^s d^s dH . d^s dH . da^dt da^dqidjii da^dq^dji. + /^ 1^(8-711) da^dq^ dn^ 118 THEORETICAL PHYSICS [Ch. V If now the equations (8*71) are solved for ~, ~, . . . -f^ ^ ^ ^ dt' dt' dt and (8'711) for - — , ■;r — , . . . ^^ — , we see at once that CTZi 071 2 071^ dq^ _ dH dt dn^' dq, _ dH dt dn^ . . . . (8-712) dq^ _ dH dt dn^ ' In order to complete to show that the proof of the theorem, we have still dn^ dH dt dq^ - dn^ dH dt dq^ . . . . (8-713) By (8-7 B) we have therefore dn, dH dt dq^' djii d dS dt dt dqi dTi, _ d^s d^s dt dtdqi ' dqidq^ dq, 1 a^^ dq, 1 dt dq^dqi dt ^ ^'S dqn dq^dq^ dt' or, using (8-712), dn, _ d^S d^S dt dtdqi dq^dq^ dH d^S dH _ djii dq^dqi dn^ 4- d^S dH dqndqi dn^ On the other hand we get by differentiating (8-68) partially with respect to q^, and remembering that the partial differentia- tion of H with respect to q^ is not merely what we represent as ;^— in which the ^'s, i.e. the ^r-'s, are treated as independent dq^ dq §8-7] PRINCIPLES OF DYNAMICS 119 variables but takes account of the g's contained in the ;:--'s, ./8^\ dq.dq^ dq^ Hence by (8-7 B), d^S dH d^S dH d^S dq^dt djti dqidqi dn^ dq^dq^ + 1^^+^^ = 0.(8.715) On comparing (8-714) and (8*715) we find djii _ dH W ~ ~ dqi' and we can establish the validity of the remaining equations (8-713) in a similar way. The theorem thus proved was first given by Jacobi {Vorlesungen 11. Dynamik, No. XX). We have seen (8*65) that 8 =[[T - V)dt ={ (2T - E)dt. 1 Therefore S =A - \ Edt by (8-633). If E is constant (conservative system) S =A -E X time .... (8-72) dS „ dS dA and = _ ^ = dt dq, dq^ Let us take the constant a^ (8-7 A) to be E ; then Hamilton's differential equation (8-68) becomes '^^ ^ a, (8-73) K« '■) From (8-72) we get as _9^_^ dai dai or ^, = f£ _ «, by (8-7 B) 120 THEORETICAL PHYSICS [Ch. V and therefore Jacobi's theorem when applied to (8*73) takes the form —- = t + Pi, ^— = ^1, cai cqi dA _ ^ dA _ .... (A) . . . (B) A being a complete integral of (8'73) and the tt's being identical with the corresponding generalized momenta. BIBLIOGRAPHY Lagrange: Mecanique Analytique (Second edition, 1811). Hamilton : Phil. Trans., 1834 and 1835. Jacobi : Vorlesungen iiber Dynamik (1866). Thomson and Tait : Treatise on Natural Philosophy. H. Weber : Die Partiellen Different ialgleichungen der Mathematischen Physik, nach Riemann's Vorlesungen. (Vieweg und Sohn.) Volume I (5th edition, 1910) contains an admirable chapter on the principles of dynamics. Whittaker : Analytical Dynamics. (Cambridge.) Webster : The Dynamics of Particles and of rigid, elastic, and fluid Bodies. (Teubner, Leipzig.) RouTH : Dynamics of a system of rigid Bodies. (Macmillan.) CHAPTER VI WAVE PROPAGATION § 9. Waves with Unvarying Amplitude A SIMPLE example of wave motion can be exhibited on a long cord stretched between two fixed points. If one end of the cord be given a sudden jerk and then left fixed the resulting deformation will travel along it towards the other end. Such a deformation is propagated without change of shape, to a first approximation at any rate, and with a constant velocity. Suppose the undisturbed cord to coincide with the X axis, and the disturbance to be travelling in that direction. Let ip (Fig. 9) represent the ordinates, or displacements, which constitute the ^ oc V^Ji X ^u i HO' Fig. 9 deformation, and which we shaU suppose are aU in the same plane. The shape of the disturbance may be represented by V=/(f) (9) where the abscissa, |, corresponding to the ordinate ^, is measured from a point, 0', which travels with the disturbance, and where, for convenience, we are taking its positive direction to be opposite to that of the X axis, since the successive displacements, ip, will then reach an observer at some fixed point on the X axis in the order of increasing values of |. The fxuiction / is quite arbitrary, depending on the initial disturbance. If x be used to represent the distance, measured in the X direction, of the ordinate ^ from some fixed origin, 0, X = (00') - I, and if we measure the time from the instant when 0' coincides with 0, so that (00') = ut, u being the velocity of propagation, then ^ = ut — X and y)^f{ut-x) ... . . (9-01) 121 122 THEORETICAL PHYSICS [Ch. VI A special and very important case of (9*01) is that in which/ is a simple harmonic function ; for example ip = A Go^ a (ut — x) . . . (9*011) where A and a are constants. If we define another constant co by ft) = au, we may give (9*011) the form y) = A COB coft-^ . . . (9-012) so that at a fixed point on the cord -^ = ^ cos [cot — const.) . . . (9-013) The period of vibration, r, wiU be 27r T = — , ft) since the values of y) will be repeated if t is increased by any 271 integral multiple of — . CO At a given time the values of ip at various points, x, will be expressed by ^ - ^ cos (const. - ^) . . (9-014) and it will be seen that the values of ip repeat themselves over intervals, A, where _ 271U ft) The distance, A, is called the wave length. We see that A = ur, and we may express (9-012) in the form w = A cos 27t( - — ^ ), I I /\ ■ ■ • ■ <^-'''' or ip = A COB 27i(- — yi A is called the amplitude, and the argument of the cosine is called the phase. It is clear that we may add any constant to the phase, since it would merely amount to the same thing as a change in the zero from which a; or Ms measured. It is an essential feature of wave equations that the dependent variable, ip, is a function of more than one independent variable. In the example just given there are two such variables, x and t. If we wish to eliminate the particular function, /, in (9-01) for §9] WAVE PROPAGATION 123 example, we shall have to differentiate with respect to these independent variables, and so we shall obtain a partial differential equation, which, since it does not contain the particular function, /, will include every kind of disturbance travelling along the cord with a constant velocity u, and without change of shape. We shall use the abbreviations ^^'f) - r and ^'•^(^' - r Differentiating (9*0 1) partially with respect to t and x, we get 11= <■ (9-03) and dx ^ ' Therefore ?^ + > = o dt ^ dx For a given value of the constant u this equation wiU not include among its solutions any representing a propagation in the nega- tive direction of X. To get a differential equation which includes both directions of propagation we may either multiply (9*03) by the conjugate equation 1-1=° "•«"' thus obtaining ©'=-(1)" <«♦' or we may form the second differential quotients, w nT, dx^ which give the equation This latter is in fact the equation we arrive at on applying the principles of mechanics to the motion of a stretched cord, provided we restrict our attention to small displacements. Let the stretching force be F and the mass of the cord per unit length be m and consider a short element of the cord (ah) 124 THEORETICAL PHYSICS [Ch. VI (Fig. 9-01) of length I. At the end a there wiU be a force with a downward component equal to dV If the slope -^ is small we may take this downward component to be dx At the other end, b, of the element there will be a force the upward component of which is dx dx\ ox J and consequently the component in an upward direction of the resultant force on the element will be Fig. 9-01 This must be equal to the mass of the element multiplied by its vertical acceleration, namely and on equating the two expressions we get d'^y) _ F d^yj dt^ m dx This equation becomes identical with (9'05) if , (9-051) u ±J-\ (9-052) and (9*01) is one of its solutions. We learn therefore that a transverse wave is propagated along the cord with the velocity given by (9*052), provided the slope, ^, is everywhere small. §9] WAVE PROPAGATION 125 It is instructive to study the transverse motions of a stretched cord in some detail. Confining our attention to motions in one plane, we may represent the arbitrarily given initial configuration by ip = ip^=f[x), and the initial velocities at different points on the cord by I =(!).='« If the ends of the cord be fixed and if the distance between them be L, the functions f{x) and F{x) will both be zero for a; = and X = L, and moreover ip and -^ will be zero at all times dt at the points x = and x = L. We are given then Wo == f(oc) ^ = O^i for all values of t 9^ ^when X = and consequently -^ = oj or when x = L. We shall term the equations (9*06) the boundary conditions. Whatever form of solution we adopt, it must not only satisfy the differential equation (9*05) or (9*051), but must also conform to the boundary conditions. Such a solution is the following : ^ = X + lit f = if{x + ut) + 4/(» - M«) + i- j F{i)di . (9-07) It satisfies the differential equation, because it is a sum of functions oi x -\- ut and x — ut each of which separately satisfies it, and it is a property of linear differential equations, i.e. equations in which powers of the differential quotients higher than the first, or products of the differential quotients, are absent, that the sum of two or more solutions is itself a solution of such an equation. It also satisfies the boundary conditions, since if we give t the value zero the limits of the integral in (9*07) become equal to one another and it therefore vanishes, while the rest of the expression becomes At the same time 126 THEORETICAL PHYSICS [Ch. VI To show this let us put the integral in the form ^=x+ut [ F{i)di = B(x + ut) - E{x - ut), where R has the property dR{^) d^ We easily find that ^^ = y\x + ut)-ir(x-ut) = -F(l). 1 f dR(x + ut) dB{x - uty 2u( d{x + ut) d(x — ut) j or 1^ = |{f (^ + ^t) -fix -ut)} + i{F{x + ut) + F(x - ut)} = F(x) when t = 0. The solution (9*07) is usually ascribed to d'Alembert. His contribution to the subject however consisted in showing that any solution of (9*05) must be contained in the expression ip =f{x + ut) + </>(a; — ut), [Memoir es de Vacademie de Berlin, 1747). It was actually Euler who first gave the solution in the form (9*07 ). Fig. 9-02 As a simple illustration of the application of d'Alembert' s solution (or Euler's solution) let us take the case of a long cord in which displacements are produced at some instant, which we may take to be zero, over a short or limited part of the cord (a b c. Fig. 9-02). And let us further suppose that at this instant the velocities are zero. We have therefore fo=f{x) where / describes the shape of the curve a b c (Fig. 9-02) and m F{x) = 0. Therefore f{x) differs from zero for values of x between a and c §9] WAVE PROPAGATION 127 (Fig. 9-02) and is zero for all other values of x, while F{x) is zero for all values of x. Equation (9'07) now becomes r = i/(^ + ^^) + i/(^ — ^^) which shows that the deformation a h c splits up into two portions, a' h' c' and a" b" c", differing from the initial deformation in having their corresponding ordinates half the original height. These are propagated in opposite directions with the velocity u. In using d'Alembert's solution (9*07) we are confronted with the difficulty that while f{x) and F{x) are defined for values of x between and L, nothing seems to be laid down for the behaviour of these functions outside the range of values to L. Yet we need to know how they behave for any real value of the independent variable, since in (9*07) the values of the indepen- dent variable in the function, /, are x -\- ut and x — ut and they also range between these limits in the integral F(^)di. The answer to the question thus raised is contained in the last of the conditions (9*06) ; but we shaU defer it until we have studied an entirely different solution of the differential equation (9*05 ), and the problem of the vibrating cord, given in 1753 by Daniel Bernoulli. Bernoulli's method consists in finding particular solutions of the differential equation, each of which is a product of a function of X only and a function of t only. Thus ipz = X2T2, W,=X,T, (9-08) where Xg is a function of x^ only and T^ is a function of t only. Substituting any one of these in the differential equation we have and on dividing by the product X^T^, 1 d^Tg _ u^ d^X T^ dt 2 X, dx^ . To satisfy this equation we must equate both sides to the same constant. Therefore 1 d^T, dt^ = ms, u^ ^s d^X, dt^ = mg, 128 THEORETICAL PHYSICS [Ch. VI where m^ is any constant. For a reason which will become obvious as we proceed, we chose solutions for which m^ is real and negative. We shall therefore write m^ = — 0)^2 where co^ is real. Consequently we find T^ = A^ cos cOgt + J5g sin co^t V and X': = M. cos — x -\- N. sin — x. u u Ag, Bg, Mg and N^ are constants of integration and we may without any loss of generality take co^ to be positive. A solution of (9*05) is therefore % = (A, cos a),t + B^ sin co,t)(M^ cos ^x + N^ sin ^x) (9-081) and we can make it satisfy the last of the conditions (9*06 ), namely ^ = at aU times when a; = or a; = L, if we make Mg = and — ^ = -=-, 5 being a positive integer. Equation (9*081 ) thus becomes % = (Ag cos (o^t + Bg sin co^t) sin ^x . . (9-082) u in which ^^iVg and B^N^ (of 9-081) have been denoted by A^ and Bg. In consequence of the property of linear differential equations, which has been described above in connexion with d'Alembert's solution, or ^{^s cos ^s^ + ^s sin o^st) sin -^x . . . (9-09) u is also a solution of the differential equation and it satisfies the conditions at the ends of the cord. We shall suppose the sum- mation to extend over all positive integral values of s. Since nu Ly m we have for the corresponding period, "^si — — )' §9] WAVE PROPAGATION 129 and for the frequency _ 1 l~F so that Bernoulli's solution represents the state of motion of the cord as a superposition of simple harmonic vibrations, the frequencies of which are integral multiples of a fundamental frequency It is an interesting historical fact, with which Bernoulli was doubtless acquainted, that Dr. Brook Taylor (Methodus Incre- mentorum, 1715) found that a stretched cord could vibrate according to the law A ^ . CO w = A cos cot sm ~x u S7Z F \ , . STl — / — Usm— i ' = A cos -=-^ — \t sin -^x, L\J m I L where s is any positive integer. Bernoulli was led to the more general expression (9*09) by the physical observation that the fundamental note and its harmonics may be heard simultaneously when a cord is vibrating. The problem of determining the coefficients A^ and B^ so as to satisfy the initial conditions was not solved till the year 1807 when Fourier showed how an arbitrary function may be expanded as a sum of cosine and sine terms. If in (9*09) we make ^ = we have f{x) = EAg sin — X, 111 and we can determine the coefficients A^ by the methods of § 4, since f{x) is given between the limits x = (} and x = L. Similarly if we differentiate ip partially with respect to t we obtain ^ = Z{ — cOg^s sin cOgf + oyfi. cos cof,) sin ~x, 01 u and on making f = 0, (¥)o = ^(^)=^«^^^«i'^S«'' from which Fourier's method enables us to determine the coJB^ and hence the coefficients B^ themselves. The difficulty which appeared in connexion with d'Alembert's solution does not arise at aU in the Bernoulli-Fourier solution 130 THEORETICAL PHYSICS [Ch. VI of the problem. If in the Fourier expansions we substitute values of x outside the limits to L, we find f{L-x) = -f{L+x), .... (9-091) F(x) = -F{-x), F{L -x) == -F(L+ X), This suggests that in d'Alembert's solution we should adopt y}(x) = —yj( - x), \p{L — x) = — ip{L + x), \dt/x \dtj \dt/L-x \dt / L+x (9-092) dxp\ If we do this and imagine the cord extended (Fig. 9'03) both ways beyond the points and L to — L and 2L, it is obvious that the points and L on the cord must remain undisplaced and the motion of the part between and L will be precisely the same as if these two points had been fixed. As an illustration suppose f{x) =sx, <x < L/2, f(x) =s{L -x), 2 - ^ - ^' where e is a small positive constant (see Fig. 9-03), and assume the initial velocities to be zero, i.e. F(x) = 0. ^ The appropriate Fourier expansion (see § 4-1) is easily found to be „, , 4:sL/ .71 I . Znx , 1 . 5jt •)■ Therefore A,. = 0, A.= 4:SL 2^2' S^Tt §91] WAVE PROPAGATION 131 and so on. The coefficients B^ are all zero, and we have • 2 — — u L' 0)3 __S7C u L' cOs _ 571 ^ u L ' therefore 4sL( nu^ . n 1 ^^"^j^ • 3jr w — \ cos -^^ sm— a; — — cos 3-:=^^ sm -^x ^ 7i^\ L L 3^ L L + i^cos S^t sin ^o; - + . . .1 . . (9-092) O^ Jb Li j and the motion is a superposition of simple harmonic vibrations the frequencies of which are odd multiples of the fundamental 1 j~^ frequency —f. — • The absence of even multiples is due of course to the special choice of initial conditions. § 9-1. Waves with Varying Amplitude The type of wave represented by equation (9*01), which we may term a one-dimensional wave, since there is only one spacial independent variable involved in its description, is propagated without change in shape or magnitude. We shall now study two other types of one-dimensional wave. These are also propagated without change in shape ; but they become more and more reduced in magnitude the further they travel. If the values of ^ at a given position, x, are plotted against the time, the shape of the graph is the same for all positions, x, but the bigger x is, the smaller is the biggest of the ordinates ip. The first of these is represented by w = -f(ut-x) (9-1) X If we slightly extend the use of the term amplitude, we may say that the amplitude of this wave is inversely proportional to the distance it has travelled from the origin, x= 0, Writing the equation in the form xip = f(ut — x), and referring to (9*01) and (9*05), we see that the correspond- ing partial differential equation is q^)^^.8^) .... (9.101) ei+s)' - ■ ■ <'-<'^' 132 THEORETICAL PHYSICS [Ch. VI This is equivalent to The other tjrpe is one in which the amplitude varies exponen- tiall}^ It is represented by the equation y) = e-<^ f{ut - x) . , . . (9-11) where a is a positive constant. On differentiating we get and on eliminating /' and f by means of u dt and I^ = e— /", we find for the corresponding differential equation This type of differential equation will be encountered in studying the propagation of an electrical disturbance along a cable. § 9-2. Plane and Spherical Waves The equation (9*01) will also describe a wave propagated in the direction X in a medium, if x, y and z are the rectangular co-ordinates of a point in the medium. Such a wave is called a plane wave since ip has the same value at all points in any plane x = const. We can easily modify the equation so that it will represent a plane wave travelling in any direction in the medium. For this purpose we introduce new axes of co-ordinates X', Y\ Z' with the same origin as Z, 7, Z (§ 2:2), so that ^ =f{ut- (Ix' + my' + nz')} . . . (9-2) where I, m and n are the cosines of the angles between the direc- tion of propagation, X, and the axes X', Y', Z' respectively. A plane Ix' -\- my' + nz' = const., at all points in which ip has the same value at a given time is called a wave front. In general we shall use N to represent the direction of propagation, or a normal to the wave front, §9-2] WAVE PROPAGATION 133 and we may drop the dashes in (9'2). We can eliminate the particular function, /, by means of and so obtain a^2 Vao;^ dy or ^^V^ 2V-72 dt (9-21) since l^ -{- m^ -{- n^ == 1. This last equation is of course much more general than the primitive (9*2) from which it has been derived. The following important example will illustrate this. We may suppose ip to be a quantity which is determined by r the distance from the origin, so that ip = function (r). We then have dip _ dip dr dx dr dx and since r'^ = x^ -{- y^ -\- z^, dv we have 2r-- = 2x, therefore dx dr _x dx r' and consequently ^ =-^.-. dx dr r Differentiating again with respect to x we get d'^ip _ x^ d^ip I dip x^ dip dx^ f2 g^2 ^9^ ^3 9^' a ^ip a ^w and there are similar expressions for -—-^ and ^-^. Adding all three equations we find 2 _ d^ip ,2 dip dr^ r dr' Consequently (9*21) becomes a^-^ a^+r"a^^ • • • • (^22) 134 THEORETICAL PHYSICS [Ch. VI and reference to (9* 102) and the equations immediately preced- ing it, shows that a solution of (9*22) is yj=}:f(ut-r) .... (9-221) This represents a spherical wave propagated with the velocity u and having an amplitude inversely proportional to the distance from the origin. Except in the case of the transverse wave along a cord we have left the character of the dependent variable, ip, undefined. It may be a scalar or a vector quantity. In the latter case we have three similar equations associated with the three axes X, Y, Z respectively. Under this heading we may usefully study a more general type of equation which we shall meet when investigating the propagation of electromagnetic disturb- ances, and of the strain produced in an elastic medium. This equation has the form ^=^V>«+-Bl(div.j>) . . . (9-23) and there are of course two others similarly related to the Y and Z axes. If div <]> = we may, provided B is not infinite in such a case, satisfy the equations (9*23) by 4» =f{ut - (Ix -\-my + nz)}, I, m and n being constants and u being equal to VA \ ; so that Wx = «/. % = rL where a, /5 and y, which are the cosines of the angles between the direction of ^ and those of the X, Y and Z axes respectively, are also constants. We easily find that div t|> = — (aZ + /5m + yn)f, and in order that this may vanish, without involving the simul- taneous vanishing of /', it is necessary that al + i^^ + yTi = 0, i.e. the scalar product of the vectors (a, ^, y) and (I, m, n) must be zero. This means that the two vectors, one in the direction of i]> and the other in the direction, N, along which the wave travels, are at right angles to one another. Such a wave is called a transverse wave. Waves in which the displacements are in the line of propagation are known as longitudinal waves. §9-2] WAVE PROPAGATION 135 Turning to the case where div t]> is different from zero, let us differentiate the equations (9*23) with respect to x, y, and z respectively and add. We thus get a^ (divvl>) or, if we write = A\7^ (div ^) + 5V' (div t|>) D = divtp, ^^^ ={A+ B)yW .... (9-24) so that the scalar quantity, D, is propagated with the velocity VA -\- B\. Consider now any point on the wave front at some instant, and for convenience imagine the axes placed so that the point is on or near the X axis, and so that the direction of propagation is that of the X axis. We may consider any suffi- ciently restricted part of the wave front in this neighbourhood to be plane, therefore (see the beginning of § 9-2) differential quotients of the components of 4* with respect to y and z are zero in such a neighbourhood and D or div vb reduces to -^, dx or to -~-^, if n represents distances measured along the direction of propagation. In (9*24) therefore we are concerned only with displacements in the direction of propagation and the equation represents a longitudinal wave. When we differentiate the first of the equations (9*23) with respect to y and subtract the result from that due to differen- tiating the second one with respect to x, we get dtAdx dy J ^ \dx dy )' or ^ = ^ "^'(^^^ .... (9-25) if we represent curl ij^ by o. And we have, of course, two further equations containing a^ and Oy. Once again let us imagine the axes moved so that some arbitrarily selected point on a wave front is travelling along the X axis at a given instant. Then in its neighbourhood differential quotients of the components of v|> with respect to y and z must be zero, and we are left with -^ and -^ only, since —^ does dx ex dx not occur in o = curl 4». The equations (9*25) involve there- 10 136 THEORETICAL PHYSICS [Ch. VI fore only displacements in directions perpendicular to that of propagation and the equation represents a transverse wave travelling with the velocity 'VA |. § 9-3. Phase Velocity and Group Velocity The differential equations in the foregoing paragraphs, e.g. (9'21) and (9*23), represent wave propagations having the characteristic feature that the velocity of propagation is inde- pendent of the form of the disturbance or deformation which is being propagated. The velocity of a small transverse disturb- ance produced in a stretched cord, for instance, in no way depends on the function /(§ 9) which describes its shape. Con- sider now a simple harmonic wave such as that represented by (9'02) which travels with the velocity u = X/r. It may happen that when t is given some other value t' the velocity u' = A'/t' differs from A/r. This is the case with light waves in material media. There is no unique velocity of propagation for a luminous disturbance. A question both of practical and theoretical im- portance is the propagation of a group of superposed simple harmonic waves having a narrow range of periods extending from r to T + A'^^ and a corresponding range of wave lengths from A to A + A-^- Let us first consider two superposed waves of the same amplitude. The resultant disturbance may be expressed thus y)=Acos 2n(i - ?) + ^ cos 27i(-^ - |-,) . . (9-3) where we have written t' for r -\- /\t: and X' for A + A^- This is equivalent to ip = 2A cos 27ih( \t - l(- - j\x\ cos 271 If now x' — t( = /\r) and X' — X( = A A) are both very small, then ^ = 2^ cos2jrjiA(-)^ -4a(^)^| cos27r|--|| . (9-301 (t X or w = A' cos 2ti\ - — - where A' = 2A cos 27e|ia(-)^ - iA(^)^| . (9-302) 9-3] WAVE PROPAGATION 137 If we plot the values of ip at some given instant against x we shaU get a curve like that in Fig. 9-3. We shall refer to the full line as the wave outline. A crest, a, of the wave outline will travel in the X direction with the horizontal velocity u — A/t, since it is a point where the phase retains the same value, and therefore '^K -:)}=«' or dx . , dt ' The velocity u = l/i is called the phase velocity. It should be noted that the crest, a, will become a trough of the wave outline if it passes the point c where the variable amplitude Fig. 9-3 A' (9*302) changes sign. In fact the point, a, will in general travel along the curve represented by the broken line. On the other hand a point, 6, on a crest of the broken line will travel with the velocity A V = © A (i) (9-31 because it is a point where the amplitude A' remains unchanged and for which therefore or .{2.(A(i>-Ag»j=0 This velocity is called the group velocity. We may obviously regard the group velocity as the velocity of propagation of a maximum amplitude and it is clear that, if we have not merely two but any number of simple harmonic 138 THEORETICAL PHYSICS [Ch. VI waves superposed on one another they will have a definite group velocity provided the extreme range of periods A^ is small. § 9-4. Dynamics and Geometrical Optics Hamilton's principal function, S, (8*65) plays a part in dynamics like that of the phase in v/ave propagation. The resemblance between the roles of the two functions— we might almost say their identity — has been so fruitful and suggestive in the recent development of quantum djniamics, that it will be well to study it briefly here. To begin with we have S or f (2T - E)dt, S=^ (Ma - E)dt, and consequently S = (Padqa — Edt). The simplest case is that in which there is only one degree of freedom and where the potential energy is constant, e.g. a single particle not under the influence of forces, or a body rotating about a fixed axis with no impressed couple acting on it ; so that the energy may be regarded as a function of p only, and during the motion jp will remain constant. In such a case S =pq - Et, or S = px — Et, if, for the present purpose, we use x instead of q for the positional co-ordinate. On the other hand the phase, in the case of a plane sinusoidal wave (see 9*02), may be put in the form ^-Ki-5' so that we may think of S, or rather, the product of S and a constant of suitable dimensions, as the phase in a plane sinusoidal wave travelling in the X direction, thus kS = <!>, or kS = 2n (x __ t\ and therefore Kp = — -, r §9-4] WAVE PROPAGATION 139 where ac is a constant of suitable dimensions. It is usual to represent — by Ji, so that h E = - . (9-4) S ='6-3 The phase velocity of the wave will evidently be u=^~ (9-41) In classical djmamics there is nothing which enables us to assign a determinate value to k or li, and moreover the energy, E, involves an arbitrary constant so that u is an arbitrary velocity. Consider now a small change A^ in E and the corresponding small change Isp in p. Suppose them to be produced by a force, F, in the case of the particle, or a couple, jP, in the case of the rotating body, acting for a short interval of time A*^, during which it travels (or rotates) the distance (or angle) A^- Then we have /\E = FAx, AP = FAt, -, .1 Ax AE and consequently — = , ^ ^ At AP or t; = A? (9-42) Ap This result (9*42) is obviously a special case of the more general equations (8*43) given above. It thus appears that the velocity, V, of the particle is identical with the group velocity of the corresponding ' mechanical wave ' . Unlike the phase velocity this is something quite definite. The analogy between classical dynamics and wave propagation extends still further. There is a complete correspondence between the principle of least action of Maupertuis (8*636) and Fermat's principle in optics. This will be fully explained later. It will suffice at this stage to say that Fermat's principle is the basis of geometrical optics, i.e. of optical phenomena in which the wave length of the light is very short in comparison with the dimensions of the optical apparatus, apertures, lenses, etc. In these phenomena the absolute value of the wave length is not 140 THEORETICAL PHYSICS ^ [Ch. VI of importance, a circumstance which corresponds to the fact, pointed out above, that classical dynamics does not contain anything that enables us to assign a value to the constant h. Now classical djniamics becomes inadequate when applied to very small systems (electrons, atoms, etc.) and the analogy between it and geometrical optics suggested to Schroedinger that this inadequacy may be of the same kind as that of the principles of geometrical optics when the dimensions of the apparatus or apertures are very small. We shall refer to this assumption as Schroedinger' s Principle and leave a more complete study of its consequences till a later stage. It wiU be recollected that the phase velocity, E u = —J P of the ' mechanical wave ' of classical dynamics is indeterminate on account of the presence in E of an arbitrary constant. Let us briefly study the consequences of the relativistic hypothesis that the energy of a particle is proportional to its mass, i.e. E = mc^ (9-43) where c is a universal constant with the dimensions of a velocity. We shall have from (9-42) therefore or 2i= — m A(mv) A-" (■-S) j2\ i and hence mf 1 — — i = constant. ('-»■ This constant is obviously equal to the mass of the particle when its velocity is zero, and if we denote it by mo we have m = mo(^l-J)~* .... (9-44) for the law of variation of mass with velocity. Equation (9*44) shows that c is upper limit of velocity for a particle, since ii v = c the mass m becomes infinite. It has received a beautiful experimental confirmation by Bucherer who found c to have the same value as the velocity of radiation in empty space. CHAPTER VII ELASTICITY § 9-5. Homogeneous Strain THERE is overwhelming evidence for the view that all material media have a granular constitution. They are made of molecules, atoms, electrons and, for anything we know, still smaller particles, which we may be able to recognize in the future. Now when we speak of a volume element, dx dy dz, in a medium, as for example in the theorem of Gauss in § 3, we have in mind a small volume which in the end approaches the limit zero, or to be more precise, dx, dy and dz separately approach the limit zero. We shall, however, make negligible errors when we are concerned with large volumes, or distances, if we suppose dx, dy and dz to approach some very small limit differing from zero. When this small limit is large compared with the distances separating the particles of which the medium is constituted we shall speak of the medium as continiwus. Let [x, y, z) be the co-ordinates of a point (e.g. +he middle point) in a volume element of a continuous medium when in its undisplaced or undeformed condition, and let (a, /5, y) be a displacement (which we shall usually take to be small) of the medium which, in its undisplaced condition, is at the point {x, y, z) ', then a, p and y will be functions of x, y and z and the time, t, or a = a(a;, y, z, t), P^P{x,y,z,t) (9-5) y = y{x, y, ^, 0- When we are dealing with static conditions we may omit the reference to the time, and equations (9*5) become a = a{x, y, z), P=P{x,y,z) (9-501) y = y{^, y, ^j). In consequence of this displacement, a particle of^the medium, 141 142 THEORETICAL PHYSICS [Ch. VII originally at (x, y, z), will have moved to a neighbouring point (I, r], C), such that i = X -{- a, v = y + ^, (9-51) If {Xi, yi, Zi), («!, Pi, yi) and (|i, r]i, Ci) refer to a neighbouring particle, we shall have a^- a = —{Xj^ -x) + —(2/1 -y) + -^(z^ - 2;) . (9-52) Now it follows from (9-51) that ii — i = Xi — X -{- ai — a, and we have therefore ii- i = x,-x+ pjx,- X) + p(y,- y) + g^(^i- z) (9-521) and corresponding expressions for 77 1 — ?^ and f 1 — C- In these equations, x-i_ — x, yi — y and z-s_ — z are the X, Y and Z components of a vector r which specifies the position of one particle, relatively to that of the other, before displace- ment has occurred. Let p be the corresponding vector after displacement. We have therefore ^cc — *^1 '^J ^y = yi- y, r^ = Zi — z, . . . . . (9-522) From (9-521) and (9-522) we get /^ , da\ . da . da «.='l+'.('+l)+'l' ^. = '4 +'.| +<'+!) ■ . (9-523) It may happen that the displacements (a, p, y) merely move the medium, or the body which it constitutes, as a whole, i.e. as if it were rigid ; but in general the change will consist of such a motion of the body, as a whole, together with some deformation or strain. Instead of considering the point (x, y, z) and one neighbour- ing point {Xi, 2/1, Zi), let us consider three neighbouring points § 9-5] ELASTICITY 143 which we shall distinguish by the subscripts 1, 2, and 3. We shall now have three vectors, r, namely : Ti = {Xj_ - x,yi- y, Si - z), ra = {x^ -x,y^~ y, z^ - z), . . . (9-53) Ts = (Xs -x,y^~ y, Ss - z), in the undisplaced or undeformed state of the medium, which, after displacement become pi = (li — I, ?yi — '^, ?i — f), P2^ (I2-I, ^2-^, C2- f), . . .(9-531) p3 = (1^3 — i, rjs — ?7, Cs — C). The vectors r will determine a parallelopiped the volume of which is (§ 2-1) ^1x5 ^lyj 'Is (9-532) (9-533) ' 2xi ' 2j/J '2% /^ M fUt ' 3a!J ' 3|/j ' 3z After displacement this volume will become Qlx^ Qly^ Qlz Qsxf Qsy, Qsz If we substitute the expressions in (9-523) for the ^'s in (9-533) we get -(' + S) + ^'-S + '4? '4x + '^^{' + 1) + ^--' dy ^8^^ da X ^Q^ *"'% "^ '''''^■' ""^^ da '■dz- dx ■ ''" dy dy ''dy V^dy)^''^d-z' ^2.^ + r + r, '''dx ^"'d^^ '' ('40 aa\ dx) da da dp \''X^^)^''^^''^' ''' 2-^..(.H-|)+,.«, dy 'dy dy^ ■^^8S + ''="8^ + H^+s)' which is equal to the product ' Ijc? ' ij/' ' iz /!< /!• (1» 2a;5 /5 ' 22; 3a;J ' 3y5 ' 3s 1 +£?, 3«' 8a;' da ^ ^di dy' dy' da dz' ^^, 1 + - dz' dz dy dx dy dy dy (9-534) (9-535) 144 THEORETICAL PHYSICS [Ch. VII as can easily be verified by applying the rule for multipljdng determinants. f)rr If the differential quotients -^, etc., are very small, so that we may neglect products of two or more of them by comparison with the differential quotients themselves, (9*535) becomes ix) ' Ij/J Iz ' 2a;) I 2yi ' 2z ^3x5 ^%5 ^32 X 9a dx dy or /volume afterX _ /originalx \^ displacement/ "~ \volumey and consequently X (1 + div (a, p, y)) . (9-536) - . , ^ > Increment in volume dlv (a, (i, y) = pgj, ^^^ ^^j^^g (9-54) (9-55) If the body is merely displaced like a rigid body, and not strained, this divergence will be zero ; but the converse proposition will not in general be true. We shall call div (a, p, y) the dilatation of the medium at {x, y, z). It is evident from its physical meaning (9*54) that it is an invariant. The set of nine quantities da da da dx' dy dz dl d_l dj dx dy dz' dy dy dy dx dy dz* constitutes a tensor of the second rank (§2-3). It is convenient to call it the displacement tensor, since in general it specifies what may be described as a pure strain superposed on a dis- placement of the body as a whole. In equations (9*521) let us suppose the origin of the co- ordinates to be shifted to the particle {x, y, z) so that X = y = z = and suppose the particle to remain at the origin so that I = ^ = C = 0. Then li =Xill da\ , da , da d_a dx) dy dz- § 9-5] ELASTICITY 145 and there are two corresponding expressions for yji and Cil or, dropping the subscript, 1, f. /, , da\ , da . da We shall now consider a strain or set of displacements with the property that the components, — , etc., of the displacement ex tensor are constants. We may therefore write (9*56) in the form f] = ?.2iX + A222/ + -^232;, . . . (9'561) C = Agio; + A322/ + -^332;, where the coefficients, X, are constants. It is clear that, on solving (9-561) for x, y and s, we shall get equations of the form y = /^2il + /^22^ + i^asC, . . . (9-562) 2; = ^3i| + ^32^ + /^SsC, where the coefficients, ^, are likewise constants. Consider now two parallel planes, in the undisplaced medium represented by Ax ^ By -^Cz -\-D =0, Ax -\-By -\-Cz-\-D^ = () . . . (9-57) After displacement the particles in these planes will be situated in loci, the equations of which we shaU obtain by substituting for X, y and z the expressions (9*562). Obviously we shaU again obtain linear equations and it will be seen that, in both, the coefficients of |, r^ and C are the same, i.e. the equations have the form A^ -\- Mri-\-N^ -\- Q =0, A^ -\-Mri-\-Nl:-{- Q^ = , . .(9-571) where A, M, N, Q and Qi are constants. Expressed in words : particles, which before displacement or strain lie in parallel planes, will lie in parallel planes after displace- ment. It follows, since planes intersect in straight lines, that particles, which in the unstrained condition of the medium lie in parallel straight lines, will also be found to be in 146 THEORETICAL PHYSICS [Ch. VII parallel straight lines in the strained condition of the medium. Such a strain is called a homogeneous strain. § 9-6. Ais-ALYSis OF Strains It is clear that a homogeneous strain, as just defined, includes not merely a strain in the stricter sense of the term, i.e. a pure strain, but also, in general, a displacement of the medium or body as a whole. Let us examine what happens to the portion of the continuous medium within the sphere a;2 + 2/2 + 2;2 = i?2 (9.5) when subjected to a homogeneous strain, supposing the central point to continue undisplaced, a supposition which does not really entail any loss in generality, since we may, if we desire, imagine the medium to be given a subsequent translation as a whole. On substituting for x, y and z the expressions (9'562), we obtain an equation like a|2 + 6^2 _|_ cj2 _!_ 2/9/C + 2^|C + ^Un = ^' . (9-601) where a, b, c, etc., are constants formed from the constants jLi in (9*562). This must represent an ellipsoid, since the radii vectores p = {i, r], C) are necessarily positive and finite in all directions ; and we may, by altering the directions of the co- ordinate axes, give the equation the simpler form aoP + Kf]^ + Co;2 = E^ . . . (9-602) We conclude therefore that a pure strain (if it is homogeneous) consists in extensions parallel to three lines at right angles to one another. These three mutually perpendicular lines are called the principal axes of the strain and the ellipsoid (9-601) or (9-602) is called the strain ellipsoid. It is perhaps needless to remark that the term extension is used algebraically to include contraction. It will be observed that, when the co-ordinate axes are parallel to the principal axes of strain, equations (9-56) or (9-561) take the form : < ■+s> ^ = 2/(1 + 1), . . . . (9-603) <'+! )■ § 9-6] ELASTICITY 147 the ^, — , ^, etc., vanishing. Similarly equations (9*562) ox oy cy become ox - *^ .... (9-604) (' 4) 0?/, SO that the equation (9*602) of the strain ellipsoid is J^2 ..2 ^2 + , ^ c... . + . . .. = ^' • (9-605) ('43" (-I)" ('40 The components of the tensor 9*55) do not, in general, all vanish even when the medium is not strained at all in the stricter sense of the term. They vanish for a pure translation, since each of the components a, ^ and y has the same value at all points {x, y, z). Consider now a very small pure rotation, for convenience about an axis through the origin, and represented in magnitude and direction by The consequent displacement of a particle, the original position of which is determined by r = {x, y, z), is (see equation 6*1) ^=q,x-q^z, (9-61) y = ^xy - c[yX. The q^,, qy and q^ have of course the same values for all particles and are therefore independent of x, y and z. We have consequently da „ da da & = "' dy=-^"dz=^- dx ^" dy ' dz ^"' dy dy dy „ 148 THEORETICAL PHYSICS [Ch. VII In this case therefore the tensor (9*55) becomes 0, — qz, <lv q.. 0, -g, . . . . (9-62) - qy^ qx^ 0, all the components being constants. We notice the following relations between them : ^+^^ = 0, dy dx ' dz dy ' and also that the components of the small rotation q = [q;^,qy, q^) can be expressed in terms of those of the displacement tensor in the following way : '■ S ij- We see that ;^, ^, ^ and the three quantities represented by ox cy cz the expressions (9 '621) are unaffected by any small displacement of the body as a whole, and therefore their values are determined by the nature of the strain only. This suggests that we should seek to describe a pure strain in terms of these six quantities. It is easy to do this. The first of the equations (9 '52 3) may be written ". = '■(■ +l)+'.»(r;4f) +'■*(£ + !) , 1 /da 8/S\ , 1 /da dy\ or, by (9-622), «■='•(■+ 1) +'■*(!+ 1) +<M) + qy'^z - qz^v The last part of this expression merely represents a contribution due to the rotation of the body as a whole (9-61). The rest is §9-6] ELASTICITY 149 quite independent of any displacement of the body as a whole and we may therefore describe a pure strain by the equations da , ,/da , dp\ , ,/9a , dy\ We shall speak of the sjrmmetrical tensor da ,(da dl\ i/3a , 3y\ dx Ady^ dxj' ^\dz ^ dx)' ^(dj_^da\^d^^ ,/dJ_^dY\ ^\dx dy/' dy ^\dz dyj' as the strain tensor and represent it by ^xx> ^xyi ^xz) °vx^ "yyy ^yzi Szx, ^zy, ^zz (9-641) If we write |, r] and C for the components oi p; x, y and z for those of r and {a, p, y) for the difference of these two vectors, i.e. (a, ^, y) = (q^ - r^, Qy - r^, q, - r,), then equations (9*63) assume the more compact form a = xSrf^ + ySj^y -\- zSg.^, P = XSyy, + ySyy ^ ^J^yg , y = xs,^ + ys,y +ZS,, .... (9-65) If M represent the scalar product of (a, ^, y) and r = (x, y, s), we have + Sy^yx + Syyy^ + Sy,yz + s,^zx + s,yzy + s,,z^ = M ... (9-66) In Fig. 9-6 the vector (a, /?, y) is represented by (ab) and the scalar product, M, is therefore equal to the product of r and (ac), or the product of r and the component of the displacement (a, p, y) in the direction of r. The quotient of (ac), by r is called the elongation in the direction of r. The elongation is therefore equal to (ac) ^{ac)r^M 150 THEORETICAL PHYSICS [Ch. VII Obviously the elongations in the directions of the co-ordinate axes are —, J- and ;^. The elongations in the directions of dx cy dz the principal axes of the strain are called the principal elongations. If we introduce the principal axes of strain as co-ordinate axes, (9-66) becomes 8,,x^ + S,,y^ + S,,z^ = M . . (9-662) in which we have used S^^, Syy and S^^ to represent the values which s^^, Syy and s^.^ respectively assume when these axes are used {s^y, s^^, etc., of course vanish ; see the remark after equation 9*603). If S^^, Syy and S^^ in (9-662) are all positive, M must be positive and the locus of all points {x, y, z) for which M has the same positive value is an ellipsoid. AU particles, which in their undis- placed condition lie on this ellipsoid, experience an elongation equal to M/r^ (by equation 9-661). The radial elongation is positive in all directions and inversely proportional to the square of the radius vector r. On the other hand if S^^., Syy and S^^ are aU negative the radial elongation will be negative in all directions. When S^^, Syy and S^^ have not all Fig. 9-6 the same sign, the locus of the points (x, y, z) for which M has the same value will be an hyperboloid. This hyperboloid and its con- jugate, obtained by giving M the same numerical value, but with the opposite sign, will represent the elongation of the medium in all directions ; and here it should be remarked that it is of no consequence (so far as the elongation is concerned) what the absolute value of M may be, since the elongation in a given direction is the same for aU particles when the strain is homogeneous and pure. To see that this is the case divide both sides of (9*662) by r^. We obtain SrJ^ -f SyyTn^ -{- Sg^n^ = elongation ; therefore for given values of I, m and n, i.e. for a given direction the elongation is constant. We may therefore just as well assign to M the absolute value 1, and the locus (and its conjugate, if it has one) is called the elongation quadric. The direction cosines of the normal at a point on the surface § 9-6] ELASTICITY 151 (9'66) or (9*662) are proportional to the components a, ^ and y of the corresponding displacement. The conjugated elongation quadrics, or the two families of surfaces, (9-66) or (9'662), obtained by assigning to M every positive and negative value, are separated by the asymptotic cone ^..x^ + S^^jV^ + S,,z^ = . . . (9-67) for all radial directions along which the elongation is zero. This is a special case of the cone of constant elongation for which where K is the constant elongation. Substituting in (9*662 ) we find for the equation of this cone, since r^ = x'^ +2/^ + 2;^, {S^ - K)x^ + {S^^ - K)y^ + {S,, - K)z^ = . (9-671) When the principal elongations are all equal we have a uniform dilatation. It will be remembered that the terms dilatation, elongation, etc., are used in an algebraical way ; for instance a uniform contraction will be treated as a dilatation by using a negative sign. Another simple type of strain is the simple shear, for which one of the principal elongations is zero, while the remaining two are numerically equal ; but have opposite signs. For example -^ = while -^ = — ^. It will dz ex dy be noticed that there is no change in volume since div (a, p, y) is zero. We shall use the term shear for any pure strain not associated with a change in volume. Any pure strain may be regarded as a superposition on one another of a uniform dilatation and simple shears ; for any pure strain consists in three elonga- tions — , 7/- and TT^ in the directions of its principal axes and ex dy dz ^_^^x{^_a,djdy\/da_dj\ ,/da _ dy\ dx ^\dx "^ a^/ dzj "^ ^\dx dy) "^ \dx dz)' dy \dy dx) "^ ^\dx ~^ dy~^ dz) "^ Ady dz)' dy _ (dy _ da\ Jdy _ d^\ ^da ,dB dy\ ^ ,g di-'KFz d~x)^'\dz d~y)^\dx^dy^dz) ' ^^ ^^^ The strain therefore consists of a uniform dilatation which may be regarded as due to three principal elongations each equal to il^ +^+^); a simple shear associated with the X and Y axes consisting in an X elongation of if z— — zf- 1 and a Y ^ ^ ^\dx dy) 11 152 THEORETICAL PHYSICS 'dp da' [Ch. VII elongation oi U^ "~ ^ ) ^^^ *^^ other simple shears associated with the YZ and ZX pairs of axes respectively. It follows too that any shear can be regarded as a superposition of a number of simple shears. Let abed, Fig. 9-61, represent a cubical portion of the medium, each side of which is taken, for convenience, to be 2 cm. in length, and let it be given numerically equal elonga- tions in the X and Y directions, the former positive and the a. p I } f /\ f A9\ 9 s ■n. c ^ d' \s'y \y d r i c Fig. 9-61 latter negative. Since elongation means increase in length per unit length, the block will be stretched so that da ^ = W) = {g^') = e, say. while dy («/) = {^g) = e. Its dimensions in the Z direction are unaffected. Consider the portion of the cube cut out by four planes parallel to Z and bisecting ah, be, cd and da along the lines p, q, r and s. This portion of the block becomes, on shearing, p', q', r' , s' . It is easy to see that the small angle, e, between pq and p'q^ is equal to the elongation, e. It is in fact equal to -^^^ divided by one half of pq or ^ / 4 V2. V2 / ' If the sheared block be turned, so as to bring the face r's' into §9-7] ELASTICITY 153 coincidence with rs, the faces p's and q'r will be inclined to ps and qr by an amount = 2£ = 2e, (9-69) (see Fig. 9-62). The angle ^ = 2e is usually taken as a measure of the simple shear. If the sides, jps and sr, of the unsheared Fig. 9-62 Fig. 9-63 block are parallel to the co-ordinate axes, it is evident that (j> = Apsp' + Arsr' (Fig. 9-63), (9-691 or <^ = ?? + ?^ dy dx The physical meanings of aU the components of the strain tensor are now evident. § 9-7. Stress A condition of strain may be set up in a medium in various ways ; for example by gravity or by electric and magnetic fields. Every material medium is normally slightly strained by reason of its weight. The insulating medium, glass, mica or ebonite. F^ A ^-^ >F Fig. between the plates of a condenser is in a state of strain when the condenser is charged. Weight and electric or magnetic forces are examples of impressed forces which bring about a condition of strain in material media. Correlated with the strain at any point in a medium we have a corresponding state of stress, which is evoked (in accordance with Newton's third law) by the impressed forces producing the strain. To fix our ideas, suppose a cylindrical rod (Fig. 9-7) to be strained by 154 THEORETICAL PHYSICS [Ch. VII numerically equal forces, F, applied at its ends and stretching it along its axis. It is evident that the material to the left of any cross-section, A, will experience a force, F, directed to the right while a numerically equal force, in the opposite sense, will be exerted on the material to the right of the section. The term stress in its widest sense is applied to forces of this kind. It is clear that, in order to specify completely the state of stress at any point in a material medium, we must be able to express the magnitude and direction of the force per unit area on any small area in the neighbourhood of the point, for any orientation of this area. Consider a small element of area, dS, (Fig. 9-71) in a con- tinuous medium. It will be helpful to follow our usual practice and regard it as a vector. We shall imagine an arrow drawn perpendicular to dS and having a length numerically equal to it. The components of dS, namely dS^^, dSy and dS^, will be equal to the projec- tions of dS on the YZ, ZX and XY planes respectively, provided these are furnished with appropriate signs. If f be the force exerted by the medium, situated on the Fig. 9-71 side of dS to which the arrow is directed, on that situated on the other side, we may express its X component in the form /. = «.»dS (9-7) SO that t^n is the X component of the force on dS, reckoned per unit area.i Sometimes it will be convenient to use the alternative definition, f^ = -p^^dS (9-701) or fj = p^JS, in which f ' = — f is the force exerted on the medium on the same side of dS as that to which the arrow (Fig. 9-71) is directed. By definition, therefore, P.n= -t.n (9-702) The component, f^, can be expressed as the sum of three terms, in the following way : Let dS be the face, abc, of a tetra- hedron oabc (Fig. 9-72). The components of dS are dS^ in the direction X, equal to the area obc ; and dSy in the direction Y, equal to the area oca ; and dS^ in the direction Z, equal to the ^ The plan is adopted here of using the first subscript, in this case x, to indicate the component of the force, and the second subscript, 7i, to indicate the direction of the vector dS. §9-7] ELASTICITY 155 area oab. The X component of the force on the face obc of the tetrahedron will be denoted by p^dS^ or ^xx^^x in accordance with the definitions and notation in (9*701) and (9-702). Similarly the X components of the forces on the faces oca and oab of the tetrahedron will be and respectively. Therefore the total value of the X component of the force, due to stress, on the tetrahedron is /. - {t^dS, + t^^dSy + t,,dS,) . . . (9-71) To this we have to add a force equal to the volume of the tetra- hedron multiplied by R^, the X component of R, the impressed force, or so-caUed body force, reckoned per unit volume. This is a force of external origin, due to gravitation or other causes, and it will become negligible in comparison with the forces over the surface of the tetrahedron as the dimensions of the latter approach the limit zero. This becomes evident when we reflect that dividing the lengths of the edges of the tetrahedron by n reduces the area of any face to Fig. 9-72 n- of its original area, while the volume becomes — of the original volume. The expression, (9*71), therefore represents in the limit the X component of the resultant force on the tetrahedron. It must therefore be equal to the mass of the tetrahedron multiplied by the X com- ponent of its acceleration. But for finite accelerations this product must also be negligible for the same reason which led us to neglect the body force. Consequently we have to equate the expression (9*71) to zero, and remembering that we have similar equations associated with the Y and Z axes, we arrive at the result Jx fv CdS = tJS^, + LAS, + tJ8,. . «„dS (9-72) 156 THEORETICAL PHYSICS [Ch. VII The quantities ^xx^ '^xy^ ^xzi ^j/aj' ^w ^i/a' t.x, %y, i.. (9-721) constitute a tensor of the second rank, as the form of the equations (9-72) suggests. We shall refer to it as the stress tensor. The component t. xyi for example, means the X component of the force per unit area on a (small) sur- ^ccT/ face perpendicular to the Y axis (i.e. its vector arrow is in the direction of the Y axis), and, further, it is the ^ force exerted on the medium O situated on the side a (Fig. Fig. 9-73 9-73) by the medium situ- ated on the side 6. The tensor character of (9*721) can be demonstrated in a simple way. Let us write the first equation (9*72) in the form Jx ^^ ^xx^x ~r *xy^y ~^ ^xz^zi where 8^, Sy and ^S^^ are of course small, and are the components of a vector S. Therefore in any small neighbourhood f^ is a (linear) function of 8^, 8y and 8^, fx ^fx{^x> ^y, ^z)> ana g^ - t,y. Now, if 8y is the Y component of a vector, the operation ^r^ transforms according to the rule for the Y component of a vector (see equation (2*41), and as f^ is the X component of a vector, it follows that ~§- transforms according to the same rule dby as the product, ajby, where a and b are two vectors. Thus ^, or t^y, is the XY component of a tensor of the second rank, dby according to the definition of § 2-3. § 9-8. Stress Quadric. Analysis of Stresses Imagine a vector, r = (x, y, z), parallel to the vector dS. We shall think of it as a line drawn in the direction of the arrow § 9-8] ELASTICITY 157 associated with dS. Let us further suppose the origin of co- ordinates to be situated in dS. We have then i = ^ (^-8) and two similar equations associated with the Y and Z axes respectively. We now form the scalar product, (fr), using the equations (9*72). fx^ +fyy +fz^ = txx^(^^x + txMSy + t^,xd^, + %^zd8^ + t,yZdSy + t,,zdS,. In this equation let us replace the left-hand member by fjr, where /„ is the component of f normal to the surface dS, i.e. its component in the direction of the vector dS or the vector r. On the right-hand side of the equation we replace dS^, dSy and dS^ by -dS, -dS and -dS respectively (equations 9*8). In this r r r way we get + iyS^ + iyyV'' + hzV^ ^%,zx+t,yZy-\-t,,z^}dS. Therefore if t^^ is the tension normal to dS, i.e. fJdS, we have + Kxy^ + tyyy^ + iyzy^ -^ t,^zx ^ t,yzy -{- t,,z\ . . . (9-81) In this equation, t^, t^y, etc., are the components of the stress at the origin. They are therefore constants, i.e. not functions of X, y and z in the equation (9'81). If now we replace ^^^r^ by a constant, M, which may conveniently have the numerical, or absolute value 1, we obtain ^axc*^ "T" ''xy'^y "T ''xz'^^ -f t^^x 4- ty^y^ -f- ty,yz -{- %^zx -\- %yzy -{- t,,z^ = M . . . (9-82) which is the equation of a quadric surface. It is called the stress quadric. Obviously a suitable rotation of the co-ordinate axes reduces (9-82) to the simpler form T^x^-{-Tyyy^ + T,,z^ = M , . . (9-821) The new co-ordinate axes are naturally termed the principal axes of the stress, and T^, T^y and T^^, the values which t^, tyy and t^^ assume for these special co-ordinates, may be caUed the principal tensions or stresses {T^y, T^^, Ty^, etc., are of course zero). The tension t^n or T^^ (normal to the surface 158 THEORETICAL PHYSICS [Ch. VII element dS) is equal to M/r'^ and therefore the quadric has the property, that the normal tension in any direction is inversely proportional to the square of the radius vector of the quadric in that direction ; and this applies also to the normal pressure jPnn or P^^ = (— t^n^ — T^J. AU that has been said about the relationship between the strain quadric and the radial elongation applies, mutatis mutandis, to the stress quadric and the normal tension. If the quadric is an hyperboloid, there will be a con- jugate hyperboloid, obtained by changing the sign of M, and an asymptotic cone, analogous to (9*67), separating them and representing the directions along which the normal tension (or pressure) vanishes. There will also be a cone of constant normal tension analogous to (9*671 ). It appears then that any state of stress can be regarded as due to three principal tensions, T^^, Tyy, T^^ (or pressures P^^, Pyy, PgJ i^ directions perpendicular to one another. When the principal stresses are equal to one another (T^^ = Tyy = T^^) we have a uniform traction (dilating stress) or a uniform pressure. In an isotropic medium this must give rise to a uniform dilatation (or compression). A tension T^.^, normal to the YZ plane together with a numerically equal one of opposite sign normal to the ZX plane, Tyy = — T^, we shall term a simple shearing stress, since it will produce a simple shear in an isotropic medium. Obviously the two tensions produce numerically equal elongations of opposite sign normal to the YZ and ZX planes while the elongations which they would separately produce normal to the XY plane will also be numerically equal and of opposite signs, so that the resulting elongation normal to the XY plane is zero. Since any state of stress can be regarded as three tensions (positive or negative) in mutually perpendicular directions we may look upon it as a superposition of simple shearing stresses on a uniformly dilating stress. In fact ■'■ XX ^^ 3\-^ XX \ -^ yy ~T~ -^ zz) "T" 3' (-^ xx -^ yy) "T 3'V-' xx -^ zz) yy "3\ yy xx) ~r s\-^ xx '^ -^ yy \ ■'- zz) \ Zy yy 22/' r.. = i{T.. - T^) + \(T,, - T„) + \(T^ + T,^ + r,,) (9-83) We have here a complete analogy with a homogeneous pure strain (equation 9*68 ). There is an alternative way of describing a simple shearing stress. To show this let us consider an element of the medium in the form of a prism and having its axis parallel to the Z axis. We shaU suppose its cross-section to be an equilateral right- angled triangle (aoh, Fig. 9-8), the sides oa and oh being perpen- dicular to the X and Y axes respectively, and each equal in area to unity. The shearing stress may be a force T^ over the side §9-9] ELASTICITY 159 oa, parallel to the X axis and an equal force T over the side oh and in a direction opposite to that of the Y axis. Since body forces may be ignored for the reason already explained, these Y two forces will produce a result- ed ant tangential force over the side ha of the prism and equal to V2J T. But the area of ah is V2 |. Therefore the tan- gential stress is equal to T, We conclude therefore that we may describe a simple shearing O ^ stress as made up of two numer- Fig. 9-8 ically equal normal stresses per- pendicular to one another or, alternatively, as consisting of a single tangential stress at 45 degrees to the normal stresses. T 11 § 9-9. Force and Stress We shall next consider the resultant force exerted on the ^yx=t^ds Fig. 9-9 medium within a closed surface in consequence of a state of stress. Its X component is (see Fig. 9-9) or 160 (§ 9-7). THEORETICAL PHYSICS This is equivalent (§ 3-2) to dx dy dz F = ■^ X dx dy dz [Ch. VII . (9-9) where the integration is extended over the whole of the volume enclosed by the surface. Since equation (9*9) must be valid however small the enclosed volume may be, the X component of the force exerted on a volume element dx dy dz must be equal to ^-^^-\-^Adxdydz . . . (9-901) dx dy dz] and consequently the X component of the force per unit volume at any point must be equal to dx dy dz For the Y and Z components we find respectively (9-91 dx dy dz dx dy dz (9-91 ar-^ dr, dy, dz ' ^^' For brevity these expressions, which are divergences according to the extended modern use of the term, may be written as (div t), (div t), (div t), (9-911) Incidentally it may be remarked that the divergence of a vector (tensor of rank 1) is a scalar quantity (tensor of rank 0) ; the divergence of a ten- sor of rank 2 (the present oo*\dx.dy,dz instance) is a vector (ten- sor of rank 1) and quite generally the divergence of a tensor of rank n is a tensor of rank n — \. The result expressed by equations (9*91) is so important that it is worth while to arrive at it directly, without employing the theorem of Gauss. Let {x, y, z) be the co-ordinates of the central point of a volume element dx dy dz of the medium, and imagine a plane surface perpendicular to the X axis and bisecting the element, Fig. 9-91. The X com- djc Fig. 9-91 § 9-9] ELASTICITY 161 ponent of the force exerted over this plane on the part of the element to the left of it is t^ dy dz, t^ meaning the average value of t^^, over the plane in question. Therefore the X component of the force on the face dy dz on the right of the element must be L + i-^dxldydz. This is a force tending to drag the element in the X direction. In the same way it wiU be seen that a force ixx - i-^dx^dy dz, tending to drag the element to the left, is exerted over the face dy dz on the left. Consequently the resulting X com- ponent of the force on the volume element, so far as it is due to stresses on the faces perpendicular to the X axis, will be which reduces to -^^x dy dz, ox "^-dx dy dz dx in the limit when dx, dy and dz are sufficiently small. In a similar way we may show that the part of the X component of the force exerted on the element, in consequence of the stresses over the faces perpendicular to the Y axis, is -^dx dy dz, dy ^ while that due to stresses over the faces perpendicular to the Z axis is -^dx dy dz. dz ^ On adding all three together we arrive at the expression we found by the use of the theorem of Gauss. We may of course replace t^, 4, and i^, by - p^, - p^y and - p^, respectively (9*702) and thus obtain the alternative expression, ( ^Pxx I ^Pxv I ^Vxz \ (9»912) dx dy dz ) for the force per unit volume. 162 THEORETICAL PHYSICS [Ch. VII The X, y and z in the foregoing equations (9 '91), etc., refer to the actual or instantaneous positions of the parts of the medium and not to their positions in its undeformed state. We ought therefore to have used the letters |, rj and C in order to avoid confusion and possible error. If, however, as we are assuming, the differential quotients ^r-, -^r^, etc., in the strain tensor are ox cy negligible by comparison with unity, no errors will arise if we use X, y and z in sense defined in the description of strain. To show that this is the case, consider the differential quotient ^, where ^ may mean a stress t^^,, or any other function of ex (x, y, z) or (I, ri, Z). Since I = a; + a, r} =y ^- P, C = 2 + r, (equations 9*51), M = ^^ + ^^ 4- ^K dx di dx dfjdx dC dx' dcjy __ dcf)/ da\ ,dcf>dB d(j> dy dx d^\ dxj drj dx dC dx' d<f) _ d(j) d(j) da dcj) dp dcj) dy ^ dx di di dx df] dx dC dx ' and this reduces to d^ _a^ dx di da dp when — , ~, etc., are very small compared with unity. vx ox § 10. Hooke's Law — ^Moduli of Elasticity The question now arises : What is the relationship between a state of strain in a medium and the correlated stress ? Gener- ally speaking the relationships between physical quantities can be expressed by analytic functions. It is probable that this statement is strictly true when it is confined to the quantitative relationships in macroscopic phenomena. The phenomena of elasticity, with which we are now concerned, come under this heading. In fact in § 9-5 we assumed that even the volume element dx dy dz was very large when measured by the scale of the granular structure of the medium. Roughly speaking, an analytic function is one which can be expanded by Taylor's § 10] ELASTICITY 163 theorem. If 6, ^ and ip are the independent variables in such a function, any sufficiently small increments dO, dcj) and dip wiU give rise to an increment of the function equal to rl-p the differential quotients ■—, etc., being independent of the increments dO, dcj), dip. We should therefore expect a priori that the components of the stress tensor are linear functions of those of the strain tensor when these latter are small. Experi- ment shows that this is the case. We have here in fact a slight generalization of the law stated by Robert Hooke (1635-1703) in the famous anagram ce Hi n o sss tt uu{= ut Tensio sic Vis). First of aU let us consider a uniform dilatation. Of the com- ponents of the strain tensor all vanish except ^, -^, ^ and ox dy cz S = 1 = 1 = *^ ''""*""""• In an isotropic medium therefore and Hooke's law requires 4x = ^ X dilatation, where ^ is a constant, called the bulk modulus of elasticity. Therefore «„ = 3fex|| ..... . (10) when the strain is a uniform dilatation. We might of course have defined this modulus as equal to k' = Sk in the equation The definition given is the one which is universally adopted and is probably the more convenient of the two. A simple shear may be regarded as due to a tangential stress (§ 9-8). Let us suppose it to be in the XY plane ; then, in accordance with Hooke's law, we have for an isotropic medium '.-"(l+i) ■ • ■ ■ "»■'»> where ^ is the angle of shear (equations 9*69 and 9*691) and n is a constant called the simple rigidity or modulus of rigidity. 164 THEORETICAL PHYSICS [Ch. VII From (lO'Ol) we can derive another equation involving n. We have seen (§ 9-6) that a simple shear is equivalent to two numer- ically equal elongations, of opposite signs, along lines at right angles to one another and that ^ = 2e, i.e. twice the positive elongation. Furthermore instead of attributing the shear to a tangential stress, for example t^y in (lO'Ol), we may attribute it to an equal normal stress, t^,^, perpendicular to the YZ plane, and a stress, tyy = — t^y, perpendicular to ZX plane. Therefore (lO'Ol) is equivalent to t^^ = 2ne .... (10-011) . ^ da *- = 2^. From the theoretical point of view these are the simplest relations between stresses and strains. It should be observed that since the effect of a tangential stress is merely to produce a simple shear, equation (10*01) is a general expression for t^y ; on the other hand equations (10) and (10*011) are expressions for ^3, which are true in special cases only, the former for a uniform dilatation, the latter for the case of a simple shear. We have to search, therefore, for general expressions for t^, tyy and %^. The expressions (9*68) show the general strain to consist of (a), a uniform dilatation in which each axial elongation is ^/da dp dy\ (6) three simple shears, a typical one consisting of the elongation e = i,^« ^^^ and -e = j(|-|).J.toZX The dilatation (a), contributes to t^ an amount equal to ^Tc X (axial elongation) in accordance with (10) or da , dp , dy^ dy and under (6), we have a contribution to t^^ equal to 2n X ^^^" - ^P' and another equal to contribution ,/aa _ a^\ Adx dy)' 2n X il (da dy\ dx dzj* § 10] ELASTICITY 165 in consequence of (lO'Oll). Adding all three contributions to t^ we get "^ \dx '^ dy~^ dz) '^ 3 \dx dy) "^ 3 \dx dzj' This is the general expression for 4x- We may write it and the corresponding expressions for tyy and %^ in the following more compact form : '»=('+I)S+('-I)^('-t)I '.=('-i)^('-i)i+('+f)i<'»-»^' When the state of stress consists of %V ~ ^zz ~ ^} the elongations ^ and ~ become equal to one another of course, ^ dy dz and equations (10-02) become » -('-?)!+<'+ IF 3jdy' dy' Eliminating ^ we find dy and for ratio, ^ = ~ J^/ dy/ dx _ %k — "In ^ ~ 2(3F+T) The constant (1004) _ _^nk_ (10-041) is called Young's modulus of elasticity, and the ratio, s, of the lateral contraction to the longitudinal elongation is known as Poisson's ratio. Young's modulus, Y, and the modulus of rigidity, n, can easily be determined experimentally and the formula (10-041) enables us to find the bulk modulus, k, from the experimentally determined values of Y and n. 166 THEORETICAL PHYSICS [Ch. VII § 10-1. Thermal Conditions. Elastic Moduli of Liquids AND Gases It is convenient to speak of a body or a medium as elastic when there is a linear relationship between stress and strain or between a small change in the stress and the resulting deform- ation. The analysis in the foregoing paragraphs tacitly assumes that an elastic body, or a portion of an elastic medium, has a finite and determinate volume even when the stress components are all zero. It is thus restricted to solid and liquid media, the latter being media for which n = and in which there are consequently no shearing stresses (see equation 10*01) of the elastic type. In a liquid therefore the stresses are aU normal stresses. It is quite true that in actual liquids and gases we may have shearing stresses, due to viscosity ; but we are con- fining our attention at this stage to cases where such stresses may be ignored. If n is made equal to zero in (10*02) it will be seen that the state of stress in a liquid is a uniform dilating (or compressing) stress and da j^ S/5 ^ dy^ dy or, writing t for t^, we have t = kdV/V, or P = - JcdV/V .... (10*1) In these equations t is the tension at the point in question, p = — t 18 the pressure and the divergence has been replaced by its equivalent dV/V or the increment in volume per unit volume. There is clearly only one modulus in the case of a liquid, namely the bulk modulus, k. It should be noted that t may be positive as well as negative in the case of a liquid. That is to say it is possible to develop in a liquid a condition of stress giving rise to a positive dilatation. If a glass vessel with fairly strong walls and a narrow stem (after the fashion of an ordinary mercury thermometer) be nearly filled with water from which air and dissolved gases have been expelled by prolonged boiling, and if it be sealed off while the water is boiling in the upper part of the stem, so as to enclose nothing but water and water vapour, we have a state of affairs in which the closed vessel is full of (liquid) water except for a very small space at the top of the stem which contains only water vapour. By judiciously warming the vessel and contained water the latter may be caused to expand till it fills the whole vessel and presses hard against its walls without however developing a pressure big enough to t -t -t -kf^^ + ^^ + ^y^ § 10-1] ELASTICITY 167 break the vessel. If now it be allowed to cool the liquid is still firmly held to the sides of the vessel and continues to fill it ; but it is now in a state of tension. Gases differ from liquids in that a state of ^positive tension cannot be produced in them. In fact a gas is always subject to a positive pressure (negative tension) which can only approach the limit zero when the volume of the gas becomes very great. Its elastic behaviour can how- ever be brought within the scope of the preceding theory if we agree to use the term ' stress ' for any small change in the pressure of the gas. For gases therefore equation (10*1) becomes dp = -h^r . . . . (10-101) In § 10 it is implied that stress and strain mutually determine one another ; that for instance the components of the strain tensor are uniquely determined by those of the stress tensor and vice versa. Now small changes in temperature can bring about appreciable volume changes while the condition of stress is maintained constant. Such volume changes are relatively enormous in the case of gases. It is therefore important that definite thermal conditions should be laid down in dealing with elastic phenomena. Unless the contrary is stated or clearly implied we shall take the temperature to be constant without expressly mentioning this condition. That is to say we shall suppose the strain to occur under isothermal conditions. There is however one other thermal condition, or set of conditions, in which we are specially interested and which may be called adiabatic or isentropic. We shaU understand by an adiabatic strain one which is produced very slowly and in such a way that heat is prevented from entering or leaving the strained medium. The isothermal relation between the pressure and volume of a given mass of gas is approximately expressed by pv = constant (Boyle's law), and therefore dp = — p—, V so that under isothermal conditions (equation 10*101) k=p (10-11) Therefore the isothermal bulk modulus of elasticity, or briefly the isothermal elasticity of a gas is equal its pressure. The adiabatic relation between pressure and volume in the case of a given mass of a gas is approximately pyy = constant, 12 168 THEORETICAL PHYSICS [Ch. VII where y is a constant which varies from one gas to another. Consequently we have 5 dv op = — yp— and therefore (equation lO'lOl ), the adiabatic elasticity of a gas is k = yp (10-12) § 10-2. Differential Equation of Strain. Waves in Elastic Media When we equate the force per unit volume of the medium to the product of its density (mass per unit volume) and its acceleration we have the equation Here R = (jR^., By, B^) is the so-called body force per unit volume and q is the density. Substituting for t^ the expression in (10*02), for t^y the expression (10*01) and the analogous expression for t^^ we get iiC+T)^ ('-¥)!+ ('-1)11 ^j /8a . dd\] . d After a little reduction this becomes +^i"(M))4K£+2)h''.-S- n /d^a , d^a . d^a\ , /, n\ d /da , ^^ , M , z> or 'a + (^ + f)| {div (a, /?, y)) + i?, = e^^i (10-201) and we derive, of course, two similar equations from the Y and Z components of the force per unit volume. If the body force R is negligible or zero (10*201) is essentially identical with the wave equation (9*23). Instead of the vector 4* = (v^^j V'i/j %) in (9*23) we have the vector (a, /5, y) ; instead of the constant A in (9*23) we have here the constant n/q and instead of the constant B we now have ih -f- - )/^. The discussion in § 9-2 enables us to infer, therefore, that when a small strain is pro- duced in an elastic solid two waves will travel outwards from § 10-2] ELASTICITY 169 the centre of disturbance, a longitudinal (or dilatational) wave with a velocity V f » + ('+?) . (10-21 and a transverse (or distortional) wave with a velocity 10-211) In the cases of liquids and gases, for which 7t = 0, transverse waves obviously cannot be propagated, and the expression for the velocity of longitudinal waves in such media simplifies to ^ (10-212) 4- The expressions (10-21) and (10-211) can be verified by considering a plane wave travelling in the X direction. In this case the differential quotients, d/dy and d/dz, with respect to the Y and Z axes are all zero and equations (10-201) reduce to ( The longitudinal wave was one of the difficulties in the elastic solid theories of light of Fresnel, Neumann and MacCuUagh. There are no optical phenomena requiring such a wave. The difficulty was at first imperfectly met by assuming the luminifer- ous medium to be incompressible, i.e. by assuming div {a, p, y) = 0. This assumption makes h infinite, if the stresses are not zero, and hence the longitudinal wave travels with an infinite velocity. While getting rid of the longitudinal wave the assumption, div (a, p, y) =0, led to insurmountable diffi- culties in other directions. Lord Kelvin solved the difficulty (so far as the wave phenomena of light are concerned ; there are other phenomena which make the hypothesis of an elastic solid aether untenable) by the bold, but not very credible hypothesis that or , 4:71 170 THEORETICAL PHYSICS [Ch. VII This contractile aether banished the longitudinal wave by making it travel with zero velocity and it was shown by Willard Gibbs and others that it was adequate in other respects. We might be tempted to adopt the expression (10'21) for the velocity of a longitudinal disturbance along a thin rod. Closer investigation however shows that this would be an error. Let AB (Fig. 10-2) represent an element of the rod dx in length parallel to the X axis and suppose x to be the co-ordinate of the y{'^-i^^]ds^ y[^'k^dj:]dS middle point or section of the rod, C. The force exerted over the area dS of the cross-section G must be equal to dx since the tension (force per unit area) is equal to the product of Young's modulus and the elongation. Therefore the force over the section B, tending to pull the element to the right, will be equal to ■da d^a dx]dS. ^[dx'^hx^ / The force exerted over the section A, and puUing the element in the opposite direction, will be equal to The resultant force in the X direction is consequently Yy^—dx dS. dx^ This has to be equated to the product of the mass of the element and its acceleration, namely (10-22) § 10-3] ELASTICITY 171 On equating the two expressions and dividing both sides by the volume, dx dS, we get Consequently the velocity of propagation of such a disturbance along the rod is The apparent discrepancy between this result and that expressed by formula (10*21), which undoubtedly represents correctly the velocity of propagation of purely longitudinal motions in a medium, is due to the fact that the propagation along the rod consists of longitudinal displacements associated with lateral contractions which travel along with them (see equation 10*04). This explanation can be verified by considering under what circumstances the longitudinal motions in the rod would be unaccompanied by lateral motions. This would be the case if Poisson's ratio (10*04) were zero, i.e. if Sk = 2n. When this relation subsists between k and n, the velocities (10*21 ) and (10*22) are in fact identical as we should expect. § 10-3. Radial Steain in a Sphere If the parts of the elastic medium are in equilibrium, and the body forces are negligible or zero, equation (10*2) becomes n\/^a + fk |)idiv(a,A,)=0 and with it are associated two similar equations W/5 + (^ + 1)1" div (a, P,y)=0 nV'y + (^ + 1)1 div (a, p,y)=0 . (10*3) If now the strain consists in displacements w along radial lines from the origin of co-ordinates we have X a = -w, r y = ^-w, .... (10-301) r 172 THEORETICAL PHYSICS [Ch. VII where r = {x, y, z) and x, y and z are the co-ordinates (in its un- displaced condition) of the particle which suffers the displacement vV T Y X w. Then, remembering that ^ = - and therefore - — = — -^ ox r vx V T dw xdw , ^ . and TT- = - -^^5 we obtam ex r dv da dx /I x^\ x^ dw dy \Y ry r2 dr Consequently (-r-r> + PT- • • (^"-^^^^ diy {a, l^,y)=-w + ^^ . . . (10-31) r or In a similar way it is easy to show that On substituting in (10*3) we get f 2a; , 2a; cZt(; , a; d^w n\ - —w +-__ + _ I r^ r^ ar r dr^ If we now turn the axes of co-ordinates about the origin to make the X axis coincide with r or w, we shall have x = T and the r\ 7 differentiation — becomes—-, since for any function, ^ of r only ex CvJL d(f) _d(f) dr dx dr ' dx X dS dS , = -—L. = _Lj when r = X. r dr dr The equation (10'33) therefore simplifies to /, , 4:n\ id^w , 2 dw 2w] or r^^ + 2r^ - 2w; = . . (10-34) dr"" dr ^ ^ § 10-3] ELASTICITY 173 If we substitute r** for w in this equation we find that it is satisfied provided ?^ is a root of the equation n[n -I) -\-2n -2 =0 or 7^2 _|_^ _ 2 = . . (10-341) Such an equation is called an indicial equation. Its roots in the present instance are + 1 and — 2. Therefore r and r ~ ^ are particular solutions of the differential equation (10*34) and the general solution is w = Ar-^-^ (10-35) A and B being arbitrary constants. For the normal tension ^xx = ^rr along a radial line through origin let us write — p^, so that p^ is the corresponding pressure. We have now ^ = ^ and — = ;r^ = -. This latter relation follows at once from oy dz r aiv,.,,,,, =1 + 1 + 1 dw , 2w dw , dB , dy o^ -T- + — =:r- + 5^+/- dr r dr dy dz Therefore by (10-02) and on substituting for w the expression (10-35) we get -p^ = UA -^ . . . . (10-36) If we write — p^ for the tensions tyy = t^^ in directions per- pendicular to r we shall have -'-('-t")S+('+I)?+('-I)f or -p^=^kA-\-^ (10-361) If we consider a spherical portion of the medium with its centre at the origin, it is evident that the constant B must be zero, otherwise the displacement, w, as well as the pressures p^ and p^ would be infinite at the centre. In this case then p^ = p^ = — SkA, and we have a uniform pressure the corresponding dilatation being 3A. Indeed the dilatation will in any case be constant and equal to 3A, as will at once appear on substituting the expression (10-35) for w in equation (10-31). 174 THEORETICAL PHYSICS [Ch. VII We next consider a spherical shell, i.e. a portion of the medium enclosed between concentric spheres of radii r^ (inner) and r^, the common centre being at the origin, li p2^ is the pressure on the outside and pi^ on the interior, (10*36) gives the two equations '2 which enable us to determine A and B in terms of these two pressures and the elastic moduli. We find ^ ^ ri^Pir — r^^p^ ^ ^ {Plr -ff2r)^1^^2^ 4?^(r2^ — Ti^) and on substituting these expressions for A and B in (10*35), (10*36) and (10*361) we can evaluate the displacement and the pressures radial and transverse at any point in the interior of the sheU. The type of problem just solved is of practical importance, for instance in the measurement of the compressibility (i.e. the reciprocal of the bulk modulus) of liquids. § 10-4. Energy in a Strained Medium Imagine a cylindrical element of volume with its axis parallel to the X axis. Let its length be I and cross-sectional area dS and suppose the state of stress in the medium is simply a tension 4a;. Then the work done in producing a displacement a of one end of the cylinder relative to the other wiU be t^JiSda, since ^^ means the force per unit area. If the length of the cylinder be I this may be written dS,l . \ w© The volume of the element is IdS and when its dimensions are very small a _ 8a _ (;v § 10-5] ELASTICITY 175 and hence the energy of strain is per unit volume. If we use the principal axes of the strain (or stress, since we are dealing with an isotropic medium) as co-ordinate axes we find for the strain energy per unit volume Substituting e, f and g for 8^^, Syy and S^^ respectively for the sake of abbreviation and replacing T^, T^y and T^^ by the equivalent expressions in (10*02), we obtain e,t,Q J {(Le + Mf + Mg)de + (Me + Lf + Mg)df + (Me + Mf + Lg)dg}, in which M=k--. This becomes iL(e^ +P+ g^) + M{ef +fg + ge), or P(e+/ + g-)2+|{(e-/)2 + (/-g-)2 + (g,-e)2} . (10-4) This expression represents the strain energy per unit volume in terms of the principal elongations e, / and g and the moduli k and n, § 10-5. Equation of Contintjity. Prevision of Relativity It will be remembered that a distinction was made between the co-ordinates (x, y, z), which refer to the positions of portions or elements of the medium in its undisplaced or undeformed condition, and the co-ordinates (|, r], C) which refer to actual or instantaneous positions at some instant t. In the present paragraph we are concerned with the latter co-ordinates only, but we shall represent them by (x, y, z) instead of (i, rj, C)- Having made this clear, let us proceed to find an expression for the mass of the medium which passes per second through a closed 176 THEORETICAL PHYSICS [Ch. VII surface from the interior outwards. Let dS (Fig. 10- 5) represent an element of area of the closed surface, its vectorial arrow being (as usual) directed outwards. Let the direction of motion of the medium in the neighbourhood of dS at some instant t make an angle 6 with that of the vector dS and suppose its velocity to be c = {u, v, w). Obvi- ously u, V and w are functions of X, y, z and t. Construct a cylinder with its axis parallel to c and with across-sectional aread^^ = dS cos 6. It is not difficult to see that the mass of the medium passing through dS per second will be equal to that contained in a portion of the cylinder of length c. If ^ be the density of the medium this will be equal to ^c dA, = QC dS cos 6, = (^c, dS), and therefore the total mass emerging through the whole surface per second will be Fig. 10-5 If {QC, dS). By the theorem of Gauss (§ 3) this is equal to [ OX dy dw ' (10-5) But the mass leaving any element of volume dx dy dz per second must be equal to — ~ dx dy dz, ct and therefore (10*5) is equivalent to iJi ^ dx dy dz ot On equating (10*5) and (10*501) we obtain ni d{Qu) ^{qv) d{QW) dx + ~-^^^ + "-^^^ + 57 \dx dydz ^0 dy dz dt 10-501) (10-51) This result is true for any volume and therefore true when the volume is simply the element dx dy dz. We may therefore §10-5] ELASTICITY 177 drop the symbols of integration and so obtain the important result d(Qu) _^ dJQv) _^ dJQw) _j_ gg _ Q ^ ^ (10-52) dx dy dz dt This is called the equation of continuity. We shall now turn back to equation (10*2), and give our attention to the case where the body force is zero. The equation therefore becomes dx dy dz dt^ ' The X, y and z on the left are, as we have seen (§ 9-9), the in- stantaneous co-ordinates of the part of the medium considered and the t^.^, t^y, t^.^ are functions of these co-ordinates. On the d^a other hand the a, in ^^-^, on the right is regarded as a function ot of t and the co-ordinates of the medium in its undisplaced con- dition. We shall now express the acceleration in another way. It is of course equal to the increase in velocity u^ — u^ divided by the corresponding time t^, —t^, or, strictly speaking, the limit to which this ratio approaches as t^ — t^ is indefinitely decreased. Now if -i^ is a function of t and the instantaneous co-ordinates x, y, z, (^2 — '^i)/(^2 — ^i) becomes in the limit du , dudx , du dy , dudz dt dx dt dy dt dz df du , du , du , dw or -\-u^ -{- V— + w—, dt dx dy dz and consequently (10*2) may be written in the form _/^ + ^+^\ /S^^^^5!^ + ,|f + A (10.521) \dx dy dz J ^\dt dx dy dz J ^ ^ Now add to this equation which is simply the equation of continuity multiplied by u. We obtain ^Pxx , ^Pxy , ^Pxz\ __ ^{Q'^) , S(^^') , ^{QUV) . d{QUW) \ dx dy dz J dt dx dy dz or ^{Pxx + QU^) , djp^y + Quv) d{p^, + Quw) dJQu) _ ^ n 0-53) dx dy dz dt ^ ^ 178 THEORETICAL PHYSICS [Ch. VII We have so far spoken of the velocity (u, v, w) in terms which imply that every part of the medium within a sufficiently small volume element will have the velocity {u, v, w) ot a velocity differing from it infinitesimally. The medium is however granular in constitution, and the individual particles or molecules will have velocities which differ widely from one another. What then is the meaning of the velocity (u, v, w) 'i It is clear that when we associate this velocity with an element of volume dx dy dz it can only mean the velocity of its centre of mass. Let nig be the mass of a single molecule and {Ug, Vg, Wg) its velocity, and let (u/, v/, Wg) be its velocity relative to the centre of mass of an element of volume within which it is situated, then Ug = Ug + u, Vg = Vg' + V, Ws = '^s + ^• Consider now the quantity ^mgUgVg, where the summation is extended over the unit volume, i.e. it is carried out over all the particles in an element of volume and the result divided by the volume. The sum ^mgUgVg = ^mg{u; + u) (v/ + v) = YimgUg'Vg + vl^mgUg + uEmgV/ + uvXmg, This reduces to HmgUgVg = ILmgUgVg' + quv because Sm^-z^/ = llmg{Ug ^ u) — 0, and ^mgVg' = I.mg(Vg - v) = 0, by the definition of centre of mass. Considerations exactly similar to those explained above in arriving at the mass QCdA, passing per second through the area dA at right angles to the velocity c (Fig. 10-5) lead us to the conclusion that XmgUg'Vg'dSy is the X component of the momentum which crosses the boundary dSy (see Fig. 9-73) per second from the side a to the side 6. It is therefore equal to the X component of the force exerted on the medium on the side b of dSy by the medium on the side a. Consequently I^mgU/Vg' = p^, and therefore l^mgUgVg = p^y + quv. It is now evident that we may express (10*53) in the form a(Sm,^,2) d(LmgUgVg) d(LmgUgWg) d(LmgUg) _ ^ no-54) dx dy dz dt ^ ^ § 10-5] ELASTICITY 179 To this we may add two equations similarly associated with the Y and Z axes. The form of equation (10*54) suggests a four-dimensional divergence. This suggestion becomes still stronger if we multiply the last term above and below by a constant c with the dimensions of a velocity — we need not at present inquire whether any physical significance can be attached to c — and use the letter I for the distance ct. We thus obtain ^ ^ o o To this equation we may of course add o 7^ 7\ ^ /-\ ^ ^ ^ and the following fourth equation is suggested : 7\ ^ ^ + |(Sm3c2) = . (10-55) Now this fourth equation (10*55) is one we have already derived. It is in fact the equation of continuity since QV = Sm^Vg, QW = ^nigWg. Equations (10*55) give us a prevision of the restricted or, as Einstein prefers to caU it, the special theory of relativity, which draws space and time together into one continuum of a Euclidean character. BIBLIOGRAPHY Love : Mathematical Theory of Elasticity. See also the works of Thom- son and Tait, Webster and Gray mentioned above (pp. 91, 120). CHAPTER VIII HYDRODYNAMICS § 10-6. Equations of Eijlek, akd Lagrange WE now turn our attention to media for which n = 0. In such media the tangential stresses, such as t^ = - Pxy, are zero and p^ = p^^ = p,, = p (§ 10). Consequently the equations (10*2) take the form -1+*.=%?. • • • ■ <■»•'■> If the body force R is derivable from a potential, as is the case for example when it is due to gravity, R = - e grad V ^^ = - ^&r' ^v = - Q^> in which q is the density of the medium ; and if in (10*6) we replace — by du du du du dt dx dy dz as in equation (10*521 ) we get I dj:) dV _du ^'^1 ^u du Q dx dx dt dx dy dz and two similar equations associated with the Y and Z axes 180 § 10-6] HYDRODYlSrAMICS 181 respectively. The density, q, is a function of the pressure, p and therefore we may put — in the form dll and so we get the Q equations dn dV _ du ^'^1 ^'^ _i_ ^'^ dx dx dt dx dy dz' dn dV _dv dv dv dv dy dy dt dx dy dz' dn dV dw , dw , dw , dw ,^txms dz z dt dx dy dz These are Euler's hydrodynamical equations. It is import- ant to have clear notions about the meanings of the variables, more especially the independent variables, which appear in these and other hydrodynamical equations. We must regard x, y, and z as the co-ordinates of the centre, or centre of mass, of a small volume element at the instant t. For the sake of brevity we shall say particle instead of centre of mass of a small volume element. With this explanation we may describe x, y, and z as the co-ordinates of a particle of the medium at the instant t, or as the instantaneous co-ordinates of the particle. It is how- ever sometimes convenient to use as independent variables the co-ordinates (Xq, yo, Zq) and the time, t, where (o^o 2/o ^o) give the position of the particle at some earlier instant, ^o ; and we shall have to be on our guard against the error due to attaching the same meaning to d/dt in the two cases. If ip is some function of the four independent variables, we shall adopt for partial differentiation with respect to t the following notation : ~ when ip = function {x, y, z, t), dt and -—■ when %p = function (xq, yo, Zq, t) . (10'701) The differential quotient =p therefore means a partial differen- ut tiation of yj with respect to t when Xq, yo and Zq are not varied. It therefore means the limiting value of tz ti where ip2 and ipi are the values of ip for the same particle (xq, yo and Zq being unvaried) at the times t^ and ti respectively. But we have already seen (§ 10-5) that this limiting value is dt dx dy dw 182 THEORETICAL PHYSICS [Ch. VIII Therefore Dy) Dt dy) ~dt ' ox dy dw (10-702) Suppose for example that y) = X. Our formula becomes Dx Dt dx dt , dx dx dy dx But x y z and i t are independent variables ; therefore dx dt dx dy dz and dx dx 1, consequently Dx Dt u, as is otherwise evident from the definition of — given above. Obviously Euler's equations may be expressed in the form dx diJ dz If we multiply these equations by ^— , ^ and ^— respectively CXq OXq vXq — X, y and z can be regarded as functions of Xq, yo, Zq and t — and add, we obtain -(a> + < + !<" + '-'I. ^li'^+O _ Du dx Dv dy Dw dz Dt dxo Dt dxo Dt dxo ' _ d jj ^ _ Du dx Dv dy Dw dz dxQ Dt dxQ Dt dxo Dt dx^ to which we may of course add d iTj , TT\ — ^^ ^^ Dv dy Dw dz ~d^} "^ ^^ ~DtWoDtd^oWd^: ^^(77+7)==^^^^ +^J^-^+^f^. (10.71) dzo Dt dzQ Dt dzQ Dt dz^ These are the hydrodynamical equations of Lagrange. § 10-6] HYDRODYNAMICS 183 The complete expression for ■— being dip _dyj dx dip dy dip dz dip dt dxo dx dxo dy dxo dz dxo dt dx^ it seems as if we ought to have written the left-hand side of (for example) the first equation (10'71) as _ djn + V) d(n + V) dt_ dxQ dt dxo' But we have to remember that t and Xq are independent variables and therefore dxo Such an expression as y- ^— may be written in the form '0/ or, since t and Xq are independent, we may interchange — and ^— , Ut CXq Dt\ dxj Dt\dxJ 'Q are thus obtaining Du dx _ D / dx\ _ d /Dx\ Di'dxo~ Dt\dxo) ^\Dt) ' Du dx _ D / dx \ ^ /I 2\ '''' DtWo~ DtVd^J ~ W^"" ^' similarly __ _^ = ( t;^ ) _ (i^ax ^ Dt dxo Dt\ dxJ dxo^ ^' , DW dz D / dz\ 9 /I ox ^',r,n^1^ and -^=7- ?^r- = i=r('^^^—] — ^r-ii'^) - (10'711) Dt dxo Dt\ dxJ 8a;o Appljdng this result in equation (10'71) we get _ ^ /Tj j^ ys _D / dx dy ^^\ _ ^ a 2\ dxQ Dt\ dxQ dxQ dxJ dx^ ^ or dXfi ^ Dt\ dxo dxo dxJ If we multiply both sides of this last equation by Dt and integrate between the limits t^ and t [x^ y^ Zq being kept constant of course, so that the function i7 + F — Jc^ refers all the time to the same material in the course of its motion) we obtain J dxQ dxQ CXq 184 THEORETICAL PHYSICS [Ch. VIII since ^« = land^° = ^-^=0. If we represent the integral on the left by x> this becomes d dx , dy , dz CXq CXq CXq CXq and we may similarly derive d dx , dy , dz ^2/0 5?/o dyo dyo d dx , dy , dz ,^r.m^^ These are Weber's hydrodynamical equations. Let {Xq + Sxq, Va + %o, 2;o + ^2;o) be the position of a particle in the neighbour- hood of {xq, 2/o) ^o) at the same time t^, and multiply Weber's equations by dx^, dyo and ^^o respectively and add. We thus find - ((S - ^>^» + (a| - ^»)^^» + (S - «'»)^^"} = ?^5a; + vdy + wSz . . (10-721) If at the time ^o a velocity potential ^o exists, i.e. if everjrwhere Uo = dxo Vo = Wo = equation (10*721 ) becomes (X + ^o)^2/o + udx + vdy + wdz ; la^o^^ "^ ^'^^^' '^wJ'^ "^ ^°^^^' ^ ^J"^ "^ ^''"^^^ or smce fee = |-"& + |-% + ~'dz, ^ dx dy " dz ' 8zo = pdx + p8y + pdz, OX cy CZ = udx + vdy + wdz. 10-8] HYDRODYNAMICS [ence in such a case dx dy 185 «,= _i(,+^,) =-2 . . (10-73) This means that if at any instant ^o a velocity potential ^o exists there will always be a velocity potential ^ = ;^ + 0o. § 10-8. Rotational and Irrotational Motion In equations (9*622) a smaU rotation of the medium is represented by {q^, qy, gj and the corresponding small displace- ments hj (a, p, y). If both sides of the equations are divided by the short interval of time during which the rotation and dis- placements are effected they become o ^^ dv 2(o = " dy dz' du dw 2co = - " dz ~di'' 2., - ^"^ du ^""'^dx' 'dy . , . . (10-8) where (co^, co^, coj is the angular velocity of the medium. We may also arrive at this result in the following way. By the theorem of Stokes (T) {udx + vdy + wdz) = l [(curl c, dS), the integration on the left extending round any closed loop in the medium at the same instant, t. Now suppose a motion of pure rotation with an angular velocity lo to exist in the neigh- bourhood of some point {x, y, z) and imagine the closed loop to be a small circle of radius r having its axis coincident with the axis of rotation. The line integral on the left becomes while the surface integral on the right becomes curlc.Trr^. Therefore 2jrr c = curl c.yrr^, 2- = curl c. T 186 THEORETICAL PHYSICS [Ch. VIII But c/r is the angular velocity. Therefore 2<o = curl c. This is equivalent to (10'8). When to = we speak of the motion as irrotational. It should be observed that a mass of fluid may be revolving about some axis while its motion is nevertheless irrotational in the sense defined above. This is illustrated in Fig. 10-8. The case {a) represents irrotational motion, the true rotational motion Fig. 10-8 being shown in (6). In the former case curl c is zero notwith- standing the fact that the fluid may be said to revolve about an axis 0. Clearly the motion will be irrotational if a velocity potential exists, since the curl of a gradient is zero (2*431). § 10-9. Theorem of Bernoulli Turning to the first of Euler's equations (10'7), let us subtract from it the identical equation or 5-4^ == %- + '^o- + ^Q-- dx ax ax ax , We thus get - |.(i7 + F + 4c2) = J - 2voy, + 2wco.^, and, in a similar way, we find - ^(77 + F + ic2) = I - 2wco, + 2uco,, - |(i7 + F + ic2) = I? - 2uco, + 2voj,, (10-9; § 10-9] HYDRODYNAMICS 187 If the motion is stationary, i.e. if the state of affairs at any point (x, y, z) remains unchanged during the motion, all the differential quotients d/dt are zero, hence - |.(i7 + F + 4c2) = 2wayy - 2voj,, - |.(i7 + F + 4c2) = 2u(o, - 2w(o^, dy - 1(77 + F + ic2) = 2vcD^ - 2uo>^ . (10-901) If the motion is irrotational as well, co^ = My = co^ = and consequently 77 + F 4- Jc2 = constant . . . (10-91) This result, known as Bernoulli's theorem, was given by Daniel Bernoulli in his Hydrodynamica (1738). Even if the motion is rotational Bernoulli's theorem will still be true for aU points on the same stream line, that is to say 77 + F + Jc^ will remain constant along the path of a particle of the fluid, pro- vided of course the motion is stationary. This can be shown in the following way : The equations of a stream line are obviously dx : dy : dz = u: V : w, or dx = Au, dy = Av, dz = Aw (10-902) Multiply equations (10-901) respectively by dx, dy and dz and make use of (10-902). We find — d(n + F + Jc2) = 2A {uwcOy — uvco^ + VUCO^ — VWCO^ ~\- WVCOg, — WUCOy }. The right-hand side is identically zero and therefore 77 + F 4- Jc2 = constant. If the fluid be incompressible {q = constant) and F the gravitational potential, Bernoulli's theorem assumes the familiar form -+ 9^^ + ic2 = constant . . . (10-91) Q h being the height of a point in the fluid measured from a fixed plane of reference, p the pressure at that point and c the velocity of the fluid. The formula of Torricelli for the velocity with which a liquid emerges, under gravity, through a small orifice in a wide jar is an immediate consequence of Bernoulli's theorem. Let the , 188 THEORETICAL PHYSICS [Ch. VIII fixed plane of reference be that in which the orifice, A, is situated (Fig. 10-9 (a)) and consider first a point in the surface B of the liquid, the area of which we shall suppose is very great com- pared with the cross-section of the orifice. The velocity at B is practically zero, therefore the constant quantity of (10*91) reduces to where B is the atmospheric pressure. At the orifice on the other hand h is zero, the pressure is again atmospheric and the liquid has some velocity c, therefore we have for the same constant On equating the two expressions we find This result can only be approximately true for any other than an ideal liquid, since the hydrodynamical equations from which it Fig. 10-9 has been derived entirely ignore viscosity. It will be observed that the deduction is only valid for points at the surface of the emerg- ing jet, since it is only there we may assume the pressure to be atmospheric. If the orifice is very small the velocity wiU be practically the same all round the jet, whether the orifice is in the side of the vessel or at the bottom, as shown in Fig. 10-9. The stream lines converge in the neighbourhood of the orifice (see Fig. 10-9 (6)) until at a place C, a short distance outside the vessel, the cross-sectional area of the jet is reduced, as will be proved, to half that of the orifice. This is the vena contracta. § 11] HYDRODYNAMICS 189 Here obviously the stream lines become parallel and conse- quently the surfaces of constant potential, indicated in the figure by broken lines, are also parallel to one another and perpen- dicular to the axis of the jet. Thus the potential gradient, and therefore the velocity, will be constant in the vena contracta. If A be the cross-sectional area of the orifice and A' that of the vena contracta, the force exerted at the orifice on the emerging liquid must be equal to Aggh. On the other hand it must be equal to the rate at which momentum passes through A and therefore through A', if we neglect momentum produced outside the orifice by gravity. But the volume of liquid passing through A' per second is A'c and the momentum per unit volume is ^c. Hence the momentum passing through A' per second is A'qc^, Consequently Aggh = A'gc^ or c2 = _- gli. On comparing this with Torricelli's formula we see that A = 2A\ § 11. The Velocity Potential When a velocity potential exists, i.e. when the velocity is a gradient, it is evident that the motion of the fluid is irrotational, since 2(o is equal to the curl of the velocity, and the curl of a gradient is zero (2*431 ). Conversely if the motion of the fluid is everywhere irrotational, i.e. if to is everywhere zero, the line integral ^(udx -f vdy + wdz), taking round a closed loop in the fluid at some definite instant t will be zero by the theorem of Stokes. Hence the integral c I {udx -f vdy + wdz) from a point A to another point C is A independent of the path or ' (udx + vdy + wdz) = (udx + vdy + wdz), ABC ADO (see § 5-1 and Fig. 5-1). Whence it follows that c = (u, v, w) is a gradient ; in other words a velocity potential exists. This type of field vector is called a lamellar vector and the corre- sponding fluid motion is called lamellar motiono 190 THEORETICAL PHYSICS [Ch. VIII If the fluid is incompressible {q constant) the equation of continuity ( 10*52) becomes du dv dw _ dx dy dz or div c = (11) and consequently, by (10*73), This is Laplace's equation. We shall see later that it is also the equation for the potential in an electrostatic field, in regions where there is no charge, when </> is the potential ; and therefore problems of lamellar flow in an incompressible fluid are mathe- matically identical with electrostatic problems in regions free from electric charges. For the sake of argument let us imagine fluid to be created at one point (which we may conveniently take to be the origin of rectangular co-ordinates) at a constant rate of Q cubic centi- metres per second. Since q is constant Q cubic centimetres will then pass per second through any spherical surface of radius r with its centre at the origin. This of course will be true of any surface completely enclosing the origin, on the assumption we are making that the density q is constant. Hence the volume passing through any square centimetre of the spherical surface per second will be Q/4:7tr^. This means 47rr2 We have therefore an inverse square law for c, which, being the negative gradient of the velocity potential, corresponds to the field intensity in electrostatics. The point source of the fluid corresponds exactly to a point charge of electricity equal to Q in suitable units. Obviously the velocity potential is --^, so that - is a solution of (11*01). This has already been proved r jn § 3-1. Applying the method of § 3-1 to find the values of n for which r^ is a solution of (11*01), we get V 2(rn) ^ |3^ + {n - 2)n}r''-^ Therefore n^ -\- n = 0, or 7^ = — 1 or 0, and the corresponding solutions are </> = constant, ^ , constant ,^^ ^^ ^ v and ^= (11*02) § 11] HYDRODYNAMICS 191 We can derive an unlimited number of solutions from (11*02) by differentiation. We have and, in consequence of the independence of the variables, x, y, z, v-(i).o. Therefore if </>i is a solution of V^<^ = 0? so is ^. Hence among the particular solutions of ^/^cf) = are included the following : 5a + 6 + c/_\ (11-03) dafdy^dz' where a, h and c are any positive integers. Instead of imagining a point source, let us suppose the fluid to be created at constant rate over an extended region. This means that we are giving up within this region the equation div c = 0, and therefore also V^^ = 0- If s is the volume of fluid created per second in one cubic centi- metre. The volume Q created per second in a sphere of radius r wiU be ^ 4:71 ^ Q = .^r'. If we assume radial symmetry the velocity at points on the surface of the sphere will be Q s c = = — r 47rr2 3 ' or c = f^x, ^y, ^z\ = {u, v, w) and therefore J. du , dv , dw div c = — + — + — ex dy oz or div c = e, or otherwise expressed \J^= -e (11-04) This result, known as Poisson's equation, will still hold even if the assumption of radial symmetry is dropped since any additional velocity, c', which we may imagine to be superposed on that possessing radial symmetry, is bound to conform to div c' = 192 THEORETICAL PHYSICS [Ch. VIII § 11-1. Kinetic Energy in a Fluid The kinetic energy in a given volume of the fluid will be J ^c^ dx dy dz, the integration being extended over the whole volume in question. If the fluid is incompressible (q constant), and if a velocity- potential exists, this becomes ^ = k [ [ [ (^rad cl>)^dx dy dz. If we write Z7 = F = ^ in equation (3-1) we find T= - iQ\\\^'^ ^<l>dxdy dz + ^{[{cl, ^rad <!>, dS), and since VV = ^ T=|fj(^grad^, dS) . . . (IM) the integration extending over the bounding surface or surfaces. Or we may express it in the form ^=l\\€'' ("-^5) where (§3-1) ^^ means differentiation in the direction of the dn outward normal to the surface. § 11-2. Motion of a Sphere through an Incompressible Fluid We shall now study the steady irrotational motion in an incom- pressible fluid through which a sphere is moving with a constant velocity. Let the centre of the sphere travel along the Z axis with the velocity Co, so that Cq = (0, 0, Cq). We may obviously take the velocity of the fluid at points very far away from the sphere to be zero. If we now imagine a velocity — Co super- posed on the sphere and fluid, the former will remain at rest, and we may choose that its centre is at the origin of co-ordinates ; while the distant parts of the fluid wiU have the velocity u — 0, V = 0, w = — Co (11*2) We now inquire about the velocity potential. Apart from a constant it must have the value cj^^coz (11-201) § 11-2] HYDRODYNAMICS 193 at distant points, in order to give the velocity (11*2). Near the surface of the sphere <f) must conform to the condition ^ = .... . (11-202) dr on substituting R, the radius of the sphere, for r, since the component of the fluid velocity in directions normal to the surface of the sphere must necessarily be zero at the surface. In addition to the conditions (11*201) and (11*202) ^ must of course satisfy equation (11-01). The particular solution ^ =- does not help us, because of its radial symmetry. We want a solution with the axial type of symmetry and this at once suggests ^ = i(i) .... (11.203) where A is some constant. The sum of the particular solutions (11*201) and (11*203) will also be a solution of Laplace's equation. We therefore try ^ = ^"^ + Kt) or (l> = CqZ — —- (11*21) This satisfies Laplace's equation and the condition (11*2) for the motion of the distant parts of the fluid, and we have still to inquire if the remaining condition (11*202) dci> dr = r=R can be satisfied by giving A a suitable value. We can put <^ in the form , z A z 4> =Cor-- --, or = [c^v --^ cos (9, where is the angle between the Z axis and the radial line from the centre of the sphere through the point {x, y, z). The difler. entiation — means a differentiation subiect to the condition that or cos d is not changed ; therefore a*^ / , 2A\ . ^ = ( Co + -X- ) cos 0, or \ r^ J 194 THEORETICAL PHYSICS [Ch. VIII and at the surface of the sphere this becomes r = R dr This will vanish if A = - 0« + ^) 2A\ . cos d. CoB^ 2 ' so that the appropriate expression for ^ is ^ = «««+l^'« .... (11-22) This is the velocity potential for the case where the sphere is at rest with its centre at the origin, and the distant parts of the fluid have the velocity (11*2). If we now superpose on the whole system the velocity ^ = 0, V = 0, W = Cq, we have our original problem again, and (/> becomes ^='# (11-23) It is important to note that this expression represents the velocity potential only at the instant (^ = 0) when the centre of the sphere is at the origin of co-ordinates. We are now able to work out an expression for the kinetic energy Tp of the fluid. Equation (11*15) gives us The integration has to be extended over the boundary of the fluid. We may think of the fluid as bounded by the sphere of radius R with its centre at the origin, and by a sphere of infinite radius with its centre at the origin. As the differentiation d/dn is in the direc- tion of the normal outwards from the fluid, it will be equivalent to d/d r on the large sphere and to — d/dr on the surface of radius B. In either differentiation cos = - is constant. It is convenient r to use as variables r, 6 and cj), instead of x, y and z, where is the angle between any plane containing the Z axis and some fixed plane containing the Z axis, for example the XZ plane. We may suppose ^ measured in the direction of rotation of a § 11-2] HYDRODYNAMICS 195 screw which is travelling along the Z axis in a positive sense. The co-ordinates r, and ^ are known as polar co-ordinates and are related to x, y and z in the following way : z = r cos d, X = r ^ixid cos (f), y = r sin d sin (f), . . . . (11'241) If on a sphere of radius r we vary d and keep cj) constant we shall have a great circle. Collectively these circles will be like circles of longitude, and will have a common diameter contained in the Z axis. On the other hand if we vary and keep 6 constant we shall get a circle (hke a circle of latitude) with a radius r sin 6. A small element of area, dS, bounded by 6, 6 -\- dd, (f) and (f) -\- d(f) will be equal to dS = rdd .r sin 6 d(f) or dS = r^ sin 6 dd dct> . . . (11-242) Introducing the new co-ordinates in (11*24) we get Tf = hW'-^ COS e^l^-g. COS ey sin dddd^. 0=0 = The integration with respect to ^ is clearly equivalent to multipli- cation by 27t. We have therefore TT IT T, = .^^^^ fcos2 d sin Odd - ^^^^' fcos2 6 sin Odd, 2r^ J 2r^ J T — R r = Qo The second integral is obviously equal to zero, and we have ^ _ QTzR^Co^ ^f 3— . Writmg m = — — q, o we find Tf = JmCo^, where m evidently means the mass of the fluid filling a volume equal to that of the sphere of radius B. If M be the mass of the sphere itself, the total kinetic energy of the moving sphere and fluid will be T = iMco^ + Jmco^ or T = i(M -{- im)Co^ .... (11-25) Briefly we may say that the presence of the fluid has the same effect on the motion of the sphere as if its proper mass were increased by an amount equal to one-half the mass of the fluid flUing a volume equal to that of the sphere. 196 THEORETICAL PHYSICS [Ch. VIII § 11-3. Waves in Deep Water Let us think of the liquid as resting on a horizontal surface coincident with the XY plane, the Z axis being directed up- wards. We shall inquire about the velocity of harmonic waves (if such waves be possible) travelling in the X direction. Differ- ential quotients with respect to the Y direction are therefore zero and Laplace's equation becomes S+S=« <"•" A suitable equation for such a wave is ^ = A C0& a(x — at) .... (11*31) where a (= 27r/A, see § 9) and a are constants, and A depends on z only. Substituting in Laplace's equation, we find d^A — a^A cos a(x — at) + -^-^ cos a{x — at) = 0, and consequently from which we derive A = Aoe-^ + BoB-^' . . . (11-311) Aq and Bq being constants. So that (11*31) now becomes cl> = {A^e^^ + B^e-^^) cos a(x - at) . (11*312) The vertical component of the velocity of the water must be zero at the plane on which it rests, i.e. ©,..=»■ Therefore 0(^0^"^ — B^e-'^^) cos a[x —at) must be zero when z = 0. It follows consequently that Aq = Bq and the expression for becomes <j) = AQ{e°-^ ■\- e-""^) QO^ a[x - at) . (11*32) It is convenient to make use of the hydrodjraamical equations in the form (10*9), remembering however that co^ = co^ = co^ = and replacing u, v and w by the corresponding gradients of the velocity potential. These equations are therefore -|(i7+F + 4c^) = dtdx' _l(;7+F + ic^) = dtdy' - 1(77 + F + Jc^) = d^ dtdz (11*33) § 11-3] HYDRODYNAMICS 197 Multiplying by dx, dy and dz respectively and adding we get and therefore 77 + F + Ac^ - ?^ + C. ot Replacing 77 by - and V by the gravitational potential, gz, we obtain ^+3» + Jc2=^ + (7 . . . (11-331) Q 01 It will be noted that the process of integration out of which C has arisen leaves open the two possibilities, namely that C is a constant or depends on the time only. We may now introduce certain approximations, if we agree that the waves are to be restricted to small amplitudes and velocities. On the principle of neglecting squares and products of small quantities we shall cut out the term Jc^ in ( 11*331), and since the pressure on the surface of the liquid must be everywhere constant, we have there Dt In consequence we get from (11*331) g^^=^(^l\+^ , . . (11.332) ^Dt Dt\dt) ^ Dt ^ ^ This must hold at any rate in the immediate neighbourhood of the liquid surface .^ Now Dz d(f> Dt dz ^"""^ Dt\dt) 'W^ \x\di) + %\dt) + "^dzKdt) which becomes, in consequence of small amplitudes and velocities, Dt\dt) ~~ W Equation (11*332) thus takes the form ^dz dt^ "^ Dt ' 1 Consequently in the equations which follow z means the whole depth of the water. 198 THEORETICAL PHYSICS [Ch. VIII If we now substitute in it the expression (1 1'32) for <j) we arrive at — agAoie""^— e-'^^) cos a{x — at) + aVAoie"^ + e-*^) cos a{x ,. dC since C is either constant or depends on t only. Or more shortly -J- = K cos a(x — at), at The last equation indicates that dC/dt is zero, since if it varied with t it would necessarily vary with x also, and this latter possibility has already been excluded. Hence K = a^^Aoie'^' + e'""^) - agAoie'^^ - e-°-^) = 0, and or g e°^ — e~"-^ a 6°-'^ + e~'^^ / 27r 27r \ 271/ 2n e + e ^•) (11-34) z . In deep water therefore, where - is very great, the expression for the velocity approximates to V 27t (11-35) § 11-4. Vortex Motion A line drawn in the fluid, so that its tangent at every point on it at a given instant coincides with the direction of co at that point, is called a vortex line. The equations of a vortex line will therefore be dx dy dz ft). ft)., ft). (114) if dx, by and dz represent a short arc measured in the direction of to. If a vortex line be drawn through every point on a closed loop the resulting set of lines constitutes a vortex tube and it will be convenient to use the term vortex filament for a vortex tube of very small or infinitesimal cross-sectional dimensions. The following simple example will serve as an illustration. Imagine a fluid in revolution about an axis passing through the origin and suppose the angular velocity q of any particle about the axis to be a function of its perpendicular distance from it. § 11-4] HYDRODYNAMICS 199 We have for c the velocity of a fluid particle (see for example equations 9*61), c = [qr], where r is the vectorial distance of the particle from the origin. Therefore u==q,z- q,y, v = q,x - q^z, Now suppose the axis of revolution to coincide with the Z axis, The last equations become Hence U V w = - q^y, = 0. 2co, __ dv du dx dy =2,+.| dy or if ^2 = a; 2+1/2. 2.. = 2^.+^7^+^ Q dQ Q Therefore de' 2C0, = -^^ + # or 2co = ^^-^4-f . . . . (11-41) which becomes, if co is constant, q = to+4 . . . . (11-411) A being a constant of integration. Let the constant rotation oj be equal to tOo when q :^ Qq, and zero when q^ Qq. When therefore Q<^ Qo the constant A must be zero, otherwise q would be infinite in the axis, where ^ = 0. Consequently q = coo (11-412) within the cylinder of radius ^o and the fluid within the cylinder will turn about Z like a rigid body. Outside this cylinder o) = and therefore . A and if we wish to avoid velocity discontinuities we must have, when ^ = ^0 A q^ =(Oo = — - 14 200 THEORETICAL PHYSICS [Ch. VIII Hence outside the cylinder A = ^o^tOo and q=^^" .... (11-413) The vortical region is within the cylinder of radius ^o- It is only here that the rotation co is different from zero. Outside the cylinder it is true that the fluid is travelling round Z, but there is no rotation in the sense in which we are employing the term. The distinction between the two types of motion is illus- trated by Fig. 10-8, (a) showing the irrotational motion and (b) the rotational motion. The irrotational motion round the Z axis is like that of a man who walks round a tree while all the time facing north, whereas he would exemplify the rotational type of motion if he were to face steadily in the direction in which he is travelling while going round the tree. With the help of one or another of the hydrodynamical equations given above we can easily deduce some interesting properties of vortices. Starting with the equation (10*721) in which it will be remembered dxo, dy^ and dzQ represent the vectorial separation of two neighbouring fluid particles at the same instant, Iq, while dx, dy and dz represent the separation of the same particles at some later instant, t ; we integrate round a closed loop and thus get §{uQdxQ + v^dy^ + Wodzo) = j>{udx -f vdy + wdz) . (11*42) since the integral (f ( ~^dXo + J^dvo -{- ^^sioj obviously vanishes. ^V^^o dyo "^ dzo / Either integral is called the circulation round the loop over which it is extended, and since both of the loops thread together the same chain of fluid particles the theorem (11*42) affirms that the circulation round a loop connecting a chain of fluid particles remains unchanged in the course of the motion of the particles. The theorem of Stokes enables us to express (11*42) in stiU another way, namely [ [(curl Co, dSo) = [ j(curl c, dS), or, as curl c = 2to, {{(coodSo) = \UcodS) . . (11*421) This means that the integral j | (<odS) extended over a surface will remain unchanged as the surface is carried along by the motion of the fluid. § 11-4] HYDRODYNAMICS 201 The statement (11»42) includes the special case that if the circulation, at some instant, taken round any closed loop what- ever, is zero, then it will always be zero, and this means that a velocity potential exists (§11). It follows from (11 '421) that if to is zero in any portion of the fluid at any instant, it must always be zero in that portion of the fluid. Since co is the curl of a vector (co = J curl c), div to must be zero, by equation (2*42). Therefore the integral III div to dxdydz, extended over the fluid contained at a given instant within a closed surface must be zero also, and by the theorem of Gauss ff (to dS) = .... (11-43) when the surface integral is extended over the closed surface and dS has the direction of the outward normal. If now the Fig. 11-4 closed surface be part of a vortex tube bounded by two cross- sectional surfaces A and B (Fig. 11-4), the part of (11*43) which extends round the tube must be zero because to is parallel to the side of the tube, and therefore perpendicular to the direction of the vector dS. We are consequently left with the surface integral over the cross-sectional faces A and B, and so f [(todS) + f f(todS) =0 (11-431) A B The vector dS, having in (11-431) the direction ot tho outward normal, will have its vectorial arrow passing through A in the same sense as that of to. At the surface B the two directions have opposite senses. If we agree to reverse the sense of dS on the surface B, so that the vectorial arrows of to and dS cross both surfaces A and B in the same sense, equation (11-431) becomes [[(todS) = rr(todS) . . (11-432) 202 THEORETICAL PHYSICS [Ch. VIII We shall call co the vortex intensity and the integral 11 (to dS) the vortex flux across the area A. We have thus A learned two things about the vortex flux ; firstly that the flux across an area A remains unchanged as it is carried along by the motion of the fluid (11'421) and, secondly, that the flux through A is equal to the flux through any other section of the same vortex tube (1 1*432 ). The method by which (1 1*432) was established clearly demonstrates that a vortex tube must either extend to the boundaries of the fluid or, failing that, it must run into itself and constitute a vortex ring. There is one other feature of vortices which the same method demon- strates. Consider any surface made up of vortex luies. It may constitute a sort of longitudinal section of a vortex, or it may be a surface enclosing a vortex tube. In either case the integral (to dS) over the surface, or over any part of it, must be zero, since in the surface the vectors to and dS are perpendicular to one another ; and it will remain zero as the surface is carried along by the motion of the fluid. If two such surfaces intersect, they must do so in a vortex line, and conse- quently they will continue to intersect in a vortex line as they are carried along by the motion of the fluid. In other words, if a chain of fluid particles lies along a vortex line at any instant it will always lie on a vortex line. Consequently too the particles which are on the boundary of a vortex at any instant will always continue on its boundary. Vortices have therefore a quality of permanence. They cannot be created or destroyed. It must be remembered, however, in connexion with this last statement, that we have assumed no viscosity and also that the body force in the fluid has a potential. These assumptions however are not necessary for the validity of (11*432) which depends on the fact that the divergence of a curl is identically zero. The analogy between the lamellar (i.e. potential) flow in an incompressible fluid and an electrostatic field free from charges has already been pointed out. There is also a close analogy between the rotational fluid motion we have just been studying and the magnetic field due to a current in a wire. The analogy is very close indeed when the fluid is incompressible. The flux in a vortex or vortex ring is analogous to the current, the vortex intensity corresponding to current density, while c = {u, v, w) corresponds to the magnetic field intensity. If suitable units for current and field intensity are used the correspondence is exact. CHAPTER IX MOTION IN VISCOUS FLUIDS § 11-5. Equations of Motion in a Viscous Fluid WE shall use the equations (10*2) as a starting point for developing those of a viscous medium. As ex- plamed m § 10-5, we may replace q—, q^ and q^^ by ^— -, Q—- and q—- respectively; and if we further suppose JJt ct JJt dV the body force to have a potential, so that E^^ = — g—-, for ex example, the equations will assume the form dx ~^ dy ^ dz ^dx ^Dt' a^ a^ a^, _ aF _ Dv dx ^ dy "^ dz ^dy ^Dt The components t^, t^y, etc., of the stress tensor now include additional terms due to the friction between one part of the medium and another. We must therefore write / = /' -4- f" f — f -L t" and corresponding equations for the remaining components. In these equations f^^, f^, etc., mean the part of the stress asso- ciated with strain, i.e. the elastic part of the stress ; while '^"xx> ^"xy^ ^t<^-j represent the part of the tensor evoked by the friction between the parts of the medium. In a fluid medium, to which we now confine our attention, where p is the pressure and t'^y, t\^, t'y^, etc., are all zero. Now the part of the stress tensor due to viscosity or internal friction, 203 204 THEORETICAL PHYSICS [Ch. IX namely 1"^^., t'\y, t'\^, etc., is related to the velocity gradients V-, -K-^ ^^ ^^c., in precisely the same way as the elastic part dx dy oz of the tensor, namely t'^^, t'^, t\^, etc., is related to the dis- placement tensor ^^, ^r-, ^, etc. Therefore we have ox oy oz "'■■-= ('•+¥)£ +('•-¥)! +('•-¥)£■ dy \ Z Jdz ^J!L\^^ 4- (h' - i™^?!' -. (k' A- !^^?J. ^"^^='^'(1 + 1) ("-^^^ and so on, (lO'Ol and 10*02) ¥ and n' being constants. We obtain consequently for a viscous fluid, - I + . V% + (t + jp)5i«l.v c) - sj- . fjy, We shall assume h' to be zero, and replace n' by // so that we get -| + ''V-+f|:«"vc)-4|^.g' . . (n-52.. and two similar equations for y- and -j—. If the medium is incompressible div c = and these equations become : dp , ^2 dV Du dp , ^2 8F i)v The constant // is called the coefficient of viscosity, or briefly the viscosity of the fluid. § 11-6] MOTION IN VISCOUS FLUIDS 205 § 11-6. Poisetjille's Formula We shall now apply these equations to the problem of the steady flow of a liquid (incompressible fluid) along a horizontal tube of small radius, B. The axis of the tube may be taken to coincide with the Z axis of rectangular co-ordinates, the direction of flow being that of the axis, and we may drop the dV potential terms ^^—5 etc. In addition to the equations of motion (1 1*522), the following conditions have to be satisfied: u = V = 0, div c == ^ =0, dz w = function (r) c=Owhenr = i?. . . (11*6) In these equations, r is the perpendicular distance of any point in the fluid from the axis, and ip may mean any quantity asso- ciated with the motion of the fluid. The last of the statements (11*6) afiirms that the liquid is at rest at the wall of the tube. Experiment indicates that this is at all events very near the truth. In consequence of the conditions (11*6) the equations of motion become ox -^ = 0, (11-61) - Il + fc\7'w = 0. It follows at once that p is constant over any cross-section of the tube, and is consequently a function of z only ; while T.^ dw X dw Now =_ ox r or since w is a> function of the single variable, r, and hence d^w _ x^ d^w I dw x^ dw dx^ r^ dr^ r dr r^ dr ' d^w _ y^ d^w I dw _^y^ dw dy^ r^ dr'^ r dr r^ dr ' 206 THEORETICAL PHYSICS [Ch. IX and, on adding, since x^ -\- y^ — r^, dr^ r dr or, finaUy ^^u, = ll(r^) . . . (ll-eil) Thus the last of the equations (11*61) becomes f = /f l.(r^\ .... (11.62) dz r dr\ dr J the straight d of ordinary differentiation having been introduced to mark the fact that the left-hand member of the equation is a function of z only, and the right-hand member a function of r only. It follows that dz t<L(r^^\=G . . . (11-621) where (r is a constant. From the second of these equations we get d / dw^ 'i,- and consequently dw Or^ , . where ^ is a constant of integration. This holds for all values of r from zero to R, and on substituting the particular value 0, we see that A must be zero. The equation therefore becomes dw Or^ >''W = -2-' dw Or ^dr 2 which on integration gives us Gr^ , 7? fxw = —- -f B. Since by hypothesis w = a.t the wall of the tube, we must have 4 and therefore, on subtraction, w = —(r^-R^) .... (11-63) 4f^ § 11-6] MOTION IN VISCOUS FLUIDS 207 We shaU now deduce an expression for the volume of liquid passing any cross-section of the tube per second. The area of the part of the cross-section bounded by the circles of radii r and r + cZr is 2nrdr and the volume of liquid passing per second through this is 2nwrdr, w being the velocity at the distance r from the axis. Hence if Q be the volume flowing through the whole cross-section per second, R I* Q = 27t\ wrdr. R G Consequently Q = 27i [^(r^ - Rh)dr, or Q = -^ ..... (11-64) From the first of the equations (11'621) we have Q^ P^ -i>i .... (11-641) i where pi and ^2 ^^^ the pressures at two cross-sections separated by the distance I, the liquid flowing from 1 to 2. On substi- tuting in (11*64) we obtain the weU-known formula of Poiseuille Q = ili:zJP^ .... {11-65) The foregoing theory of the flow of a viscous liquid through a narrow tube constitutes the basis of a method of measuring the coefficients of viscosity — or the viscosities, as we say for brevity — of liquids. If the tube is not horizontal we shall have in place of the gradient dz ~ " of (11*621) another constant, namely ^ = ^(p + eF) (11-66) as is evident from equations (11*522). If for example the tube be vertical, and the liquid flowing down it {Z axis directed A .. . ^ i ^ 208 THEORETICAL PHYSICS [Ch. IX downwards), the gravitational force per unit mass is g and hence V = — gz -\- constant. We may as well take the con- dV stant to be zero — V only appearing in ^ — and we find oz so that instead of (11'65) we shall have If the apparatus be arranged — as it sometimes is — after the manner illustrated in Fig. 11-6, it is not permissible to take the pressure difference be- tween such a point as A and the point B at the end of the tube as .B equivalent to jpi—jpz. If the velocity is practi- cally zero at ^, we have Fig. 11-6 to subtract from the pressure at A the amount J^c^ to get px, the pressure just inside the tube, in accordance with Bernoulli's theorem. The formula (11'65) applies, as we have seen, to the case of an incompressible fluid ; in practice it applies to liquids. We can however very easily modify it to obtain a formula applicable to gases — or, to be precise, to fluids obeying Boyle's law. The gradient — is no longer zero and consequently w is a function of r and z. Apart from this the conditions (11 '6) continue in force. Instead of (11*62) we now find dp __ jLi d / dw\ 4 d^w since \J^w must include the term -^r— , and we have the term cz^ ^ — (div c) of (11*521). In consequence of Boyle's law w will 3 cz vary inversely as the pressure at all points at the same distance from the axis. Hence if the pressure gradient is everywhere small, ^— will also be small and r—- negligible. We may there- dz oz^ § 11-7] MOTION IN VISCOUS FLUIDS 209 fore adopt the equations (11*621), provided of course we do not lose sight of the fact that dz is no longer a constant, but a function of z, that is to say it varies from one cross-section to another. The volume — call it W — flowing per second through the cross-section at z will there- fore be W = _^^ dz Sfi ' where -f- has the value appropriate to that particular cross- dz section. On multiplying both sides by p, the pressure, we get ^ dz SjLi ' Now, in consequence of Boyle's law pW has the same value for aU cross-sections. Let us call it Q. It represents the quantity of gas passing through the tube, or past any cross- section, per second ; the unit used being that quantity of the gas for which the product of pressure and volume is unity. Therefore d{p^) nR^ Q = - dz 16/j,' Q^dl^l^ .... (11.67) § 11-7. Motion of a Sphere through a Viscous Liquid. Formula of Stokes The special problem to which we now give our attention is that of determining the force required to keep a sphere in motion with a constant velocity, through an infinitely extended mass of liquid (incompressible fluid). We shall represent the constant velocity by Co and suppose, in the first instance, the centre of the sphere to be travelling along the Z axis in the positive direc- tion. The problem is equivalent to that which arises if we imagine a uniform velocity = (0, 0, — Cq) superposed on the whole system of fluid and sphere. So that the sphere is now at rest — and we shall suppose its centre to be at the origin of our system of co-ordinates — and the infinitely distant parts of the liquid have the velocity (0, 0, — Co). Obviously we may take the pressure to be constant at distant points and it will be immaterial what value we assign to this constant. It is 210 THEORETICAL PHYSICS [Ch. IX convenient to take it to be zero. We are not concerned with any body forces and the equations of motion of the liquid (11-522) become dp , „„ Du -I +"'% = , 55. . . . ,„,, Let R be the radius of the sphere. The conditions to be satisfied are the following : u = V = w = 0, when r = R, div c = 0, c = (0, 0, — Co), when r = oo (11*71) -;^ = for all quantities, ip, associated with the motion ; and we shall impose the restriction that u, v and w are every- where small. This last condition justifies us in ignoring Du Dv T Dw TO. -=—, -.=— and -.=r-. In tact Dt' Dt Dt Du _du ^^ _i ^'^1 ?'^ Dt dt dx dy dz' Du or 77"=^+ sum of products of small quantities taken two at a time. The ignoration of the accelerations simplifies the equations of motion to ||==A*V^^ (11-72) By differentiating with respect to x, y and z respectively and adding, we obtain d^ d^ d^ ^ /a^t ,dvdw\ dx^ "^ dy^ "^ dz^ ^^ \dx '^ dy~^ dz /' or V¥ = /^V^ (div c) Consequently \j^p = o (11*721) § 11-7] MOTION IN VISCOUS FLUIDS 211 We naturally think at first of a velocity potential ; but a little reflexion will show that a velocity potential cannot exist. Consider, for example, the state of affairs at a point on the X axis close to the spherical surface. Here quite obviously s-^"- while !=«■ and hence du dw dz dx differs from zero at such a point ; and this is incompatible with the existence of a velocity potential. The kind of symmetry which the motion possesses leads us to suspect that it is the Z component of the velocity, w, which makes a velocity potential an impossibility, and we endeavour therefore to satisfy the conditions of the problem by dx' dS dy t^= -|| + ^i. . . . . (11-73) Substitution in (11*72) leads to dp ^v-72JL 5 - - 'a,'* while substitution in the divergence equation, div c = 0, leads to ^' = V2^ .... (11-732) We proceed further by adopting the simplest method of satisfy- ing (11-731) and (11-732), namely p = - pS7^4>, SJhjo^^O (11-74) 212 THEORETICAL PHYSICS [Ch. IX The pressure, p, has to satisfy (11'721) and we shall try the solution p = constant x ^(- )> (see 11 '03), where r is the radial distance from the centre of the sphere. This expression has the sort of axial symmetry characteristic of the motion. Let us write it in the form P-$ .... (11-741) We shaU see, as we proceed, that it is the right expression for p if we assign a suitable value to the constant, A. It now follows from equations (11*732) and (11'74) that w^=— (11-742) IJiT We reject the possible additive constant, since it may be con- sidered to be included in the constant velocity, — Co, of the distant parts of the liquid. Turning to the function ^, the problem of § 1 1-2 suggests putting ^ = ^^ + 5^ + ^! • • • • (11-75) The constants a and h, like the constant A above, have still to be determined ; as also has the character of the function ^i. We have therefore since the first two terms in (11*75) contribute zero to \/^<j) (see § 11-2). Consequently p = -/.V^i • • (11-751) or -^=- /"V'-^i- It is easy to verify that this is satisfied by We obtain, in fact, from this expression for ^i a^^i _ _ Az_ ZAzx^ 'dy^ ~ 2^^ 2iLir^ ' § 11-7] MOTION IN VISCOUS FLUIDS 213 aVi _ _ ^^ , 3^g^ _ Az dz^ 2/Ar^ 2iLcr^ fxr^' and on adding these together or V^^i^-^, /^ so that (11*751) is satisfied. In virtue of (11'752) the expression for ^ becomes '^ = «^+^-S + #^ • • • (11-76) 2r3 2/ir and we find for u, v and w (11'73 and 11*742) _ nhR^ A\zx /3bR^ , A\zy ,.. _,,, What we have succeeded in doing so far amounts to finding expressions for p, u, v and w which satisfy the equations of motion. We have now to investigate whether we can satisfy the conditions (11*71) by assigning suitable values to the con- stants A, a and b. Now at infinity w = — Cq and in consequence a = Cq, and the first two equations (11*761) will conform to the condition u = V = w = for r = i?if A = - SpibB, while the last of the equations (11*761) will conform to this condition if, in addition to assigning the values just mentioned to a and A, we put b = — -^. Therefore a = Co b = -j (11-762) 214 THEORETICAL PHYSICS [Ch. IX On substituting these values for A, a and b we get ^ = <'«*-^' + ^ (11-77) P=^^ (11-78) -O-J)^^ (11-781) We have thus succeeded in satisfying the equations of motion and the boundary conditions as well. To get the force exerted by the liquid on the spherical surface, we turn back to (9*72) which gives us expressions for the force exerted over an element of surface dS. Since the resultant force is obviously along the line of the Z axis, we only need This will represent the Z component of the force on an element dS of the surface of the sphere if the vectorial arrow of dS be directed away from the centre of the sphere. Consider in the first place an element dS in the plane Y = 0. dSy = and /, = t,,dS, + t,JS^, or /, = (%, cos 6 + tzx sin 0)dS . . (11-782) (Fig. 11-7). For any other element of area within the zone bounded by the angles d and 6 -{- dd f, = rds, where F has the same value as 4z cos 6 + t^^ sin 6 in (11*782). We therefore get for dF = S/^, the Z component of the force exerted on the zonal surface, the expression dF = {%, cos d + t,^ sin d)27zR^ sin Odd ; so that F= [ 27ri22 sin 0cZ0(^,, cos + 4^ sin (9) . . . (11-79) Now , /,, , ^n'\dw , /,, 2n'\/du , dv\ § 11-7] MOTION IN VISCOUS FLUIDS by § 11-5; or since div c = ; and . jZw . du\ Replacing n' by /x we get 4. = - i> + 2/^-^, /dw , du\ 215 (11-791) Fig. 11-7 On carr3dng out the differentiations ^r-, -^ and -^ and sub- dz ex oz stituting the special value E for r, we get, since — = cos 0, /dw\ \dz /r = R /dw\ \dx ) r \dzJr=.R 3Co 2R (cos d sin2 6), . = -S(^^^ 3co 2i2 sin d cos^ 0. Furthermore we get from (11-78) SjuCq cos d P = 2i^ 15 216 THEORETICAL PHYSIOS [Ch. IX Thus t„=-?^ cos 6 - ^ cos e sin2 6, t^^ == ^ (sin 6 cos2 d - sin3 (9). 2jti On substituting these expressions for t^^ and t^^ in ( 11*79) we get F = - 67tRfiCo { sin 6 d6{i cos^ |9 + cos^ d sin^ (9 whence -icos2 0sin2 + isin4 6}, F == - Q7tR/j,Co .... (11-792) This means that the force is numerically equal to QjcRjuCq and is in the direction in which the liquid is flowing. Finally let us superpose on the whole system the velocity Cq. The liquid will now be at rest at infinitely distant points and the sphere will be travelling in the Z direction with the velocity Cq. It will experience a resisting force equal to GtijuBCq. This is the celebrated formula of Stokes. In deducing it we have assumed u, V and w to be everjnvhere smaU and div c to be zero. If u, V and w are everywhere small div c will be a small quantity of the second order and may be ignored. This justifies the use of the formula for the slow motion of a sphere through a gas. A further assumption, which has been tacitly made, is that of the continuous character of the medium through which the sphere is moving. The formula begins to be inaccurate, as the experiments of Millikan have shown, when the radius of the sphere approaches in magnitude the mean distance between the molecules or particles of which the medium is constituted. BIBLIOGRAPHY Stokes : Mathematical and Physical Papers, Vol. Ill, p. 55. LoRENTZ : Lectures on Theoretical Physics, Vol. I. (Macmillan.) CHAPTER X KINETIC THEORY OF GASES § 11-8. Foundations of the Kinetic Theory — Historical Note THE kinetic theory (whether applied to gases or to other states) aims at interpreting thermal phenomena in mechanical terms. It assumes that matter in bulk is constituted of innumerable small particles or dynamical systems (molecules) and identifies heat with their kinetic energy .^ The picture which the theory gives us 6f a gas is that of an enormously large number of very minute particles flying about in a chaotic manner in the containing vessel, their collisions with the wall of the vessel giving rise to the pressure characteristic of gases. Thermal conductivity and viscosity are explained by the collisions between the individual particles. The velocities of translation account for the laws of diffusion of gases and liquids. The assumption of forces of cohesion between the particles or mole- cules together with the fact that, though very minute, they have an appreciable proper volume of their own, as distinct from the space they may be said to occupy, renders some account of the liquid and vapour states and the transition from one to the other, as well as of the phenomena of surface tension. The utility of the theory is limited, roughly speaking, to gases ; but here it has achieved a great measure of success. Daniell Bernoulli appears to have been the first to make progress worthy of mention in the development of the theory. He succeeded in accounting for Boyle's law (Hydrodynamica, 1738) ; but nothing further of any consequence was accomplished for about a century, by which time Bernoulli's contribution had been forgotten. In 1845 Waterston submitted a paper on the subject to the Royal Society. Unfortunately there were certain errors in it, and in consequence it was not published at the time. It contained among other things the theorem of equi- ^ Strictly speaking, heat is identified with the mechanical energy, kinetic and potential, of the molecules. In the case of a gas however this is prac- tically equivalent to identifying heat and kinetic energy. 217 218 THEORETICAL PHYSICS [Ch. X partition of energy and an explanation of Avogadro's law, and was eventually published in 1892 because of its historical interest. The further development of the theory is largely due to Clausius (1857) and especially to Clerk Maxwell (1859) and Ludwig Boltzmann (1868). Clerk Maxwell's great contribution was the law of distribution of velocities, while that of Boltzmann con- sisted in expressing the thermodynamic concept of entropy in terms of the probability of the state of an assemblage of mole- cules or dynamical systems, and this aspect of the theory is the main feature of the great work on statistical mechanics by WiUard Gibbs (1901). § 11-9. Boyle's Law We shall begin with the simplest possible assumptions about the constitution of a gas, namely that it consists of minute and perfectly elastic particles, so small that we may regard their proper volume (i.e. X^nr^, if they are spheres and r is the radius of a sphere, or N^nr^, if all have the same radius and N is the total number of them) as a negligible fraction of the volume occupied by the gas. Let us further suppose these particles or molecules to be flying about in the containing vessel with very high velo- cities, so that we may neglect gravity. Their Fig. 11-9 kinetic energy will be maintained by impacts with the wall of the vessel, since we identify it with the heat energy of the gas, and this will be maintained if the temperature of the wall is kept up. We shall provi- sionally make the further assumption that the wall of the containing vessel is perfectly smooth and elastic, though this is formally inconsistent with the implied hypothesis that the wall of the vessel is itself constituted of molecules. Finally let us suppose that there are no inter molecular forces. It is not difficult to obtain an expression for the pressure exerted by the gas on the wall of the vessel, assuming it to be due of course to bombardment by gas molecules. It is best to begin by calculating the part of the pressure due to the mole- cules which have velocities of the same absolute value c. These velocities may be supposed to be uniformly distributed as re- gards directions. This means that, if we draw a line to represent in magnitude and direction the velocity of every molecule in the unit volume, all the lines being drawn from the same point, O (Fig. 11-9), their extremities will be uniformly distributed over a sphere of radius c. It is to be understood that an element § 11-9] KINETIC THEORY OF GASES 219 of the surface of this sphere, notwithstanding its minuteness, nevertheless embraces an enormous number of these terminal points. If n be the number of the molecules per unit volume, the number of them whose representative points lie on the unit area of the sphere will be n 47rc2 and the number on a small area da will be nda 4:710' But da/c^ is the solid angle subtended at the centre of the sphere by da, and consequently the number of molecules whose direc- tions of motion lie within the limits of a solid angle, dSl, will be ""^^ (11-9) n = 4:7Z n' (11-902) Interior If we take for d£t the solid angle contained between the polar angles 6, + d%, ^ and </> -\- d(l) (§ 11-2) we shaU have dSl = sin e dO d(l> . . . (11-901) and we may express (11-9) in the form , _ '?^ sin 6 dd defy 4:71 The number of the molecules, travelling in directions included within dSl, which strike a small element dS of the wall of the containing vessel in the time dt, will be the same as the number of them contained in the cylinder CBED (Fig. 11-91) the volume of which is c dt dS cos 6, and therefore equal to n' cdtdS cos 6 . (11-903) Each molecule has the momentum mc with a normal component mccos 6, and on collision with dS this will be reversed, so that if we confine our attention to this component — and we may do so since it is the force normal to dS that we are investigating — the change of momentum which a single molecule suffers on colHsion will be 2mc cos 6, Fig. 11-91 220 THEORETICAL PHYSICS [Ch. X and consequently in the time dt the total change of momentum due to collisions with dS will be 2mn'c^dSdtGo^^d . . . (11-904) Substituting in this the expression (11*902) for n', we have ^^' dS dt cos2 e sin d dO d6. The change of momentum per unit time, or the contribution of these collisions to the force on dS, is therefore *^' dS cos2 Q sin d dd dcf>, In and we obtain for the force due to all the molecules with the velocity c, mnc dS f f cos2 (9 sin (9 d^ d^, 271 -3-dS. The kinetic energy of a single molecule is — —, and that of the n molecules in the unit volume, — - — ; so that their contribu- tion to the force on dS is ^KdS, where K is the kinetic energy per unit volume. It is now evident that, whatever may be the law of distri- bution of velocities, the total force on dS will be given by f (Z, + iT, + . . .)dS, where Ki, K^, K^, etc., are the kinetic energies per unit volume of the molecules with the velocities Ci, Cg, Cg, etc., respectively. We have consequently for the pressure p = ^ (kinetic energy per unit volume), . (11*91) or pv = f (total kinetic energy in the gas) . (11*911) The factitious assumption about the nature of the wall of the containing vessel is not necessary. If we consider, in the first place, only the molecules travelling up to the wall, and work out by the method just described the sum of the components of their momenta perpendicular to dS, we shall find for their contribution to the pressure — whatever happens to them on colliding with the wall — p = i (kinetic energy per unit volume), § 12] KINETIC THEORY OF GASES 221 and the assumption that the directions of the velocities, c, are uniformly distributed leads to the same contribution from the momenta leaving the wall. The two contributions combined yield the amount expressed by (11-91). The identification of heat and kinetic energy will make the right-hand side of (11*911) constant so long as the temperature is constant, and we thus have an explanation of Boyle's law. A very little reflexion will show that (11*91) is true for a mixture of gases. Therefore in such a case p=^ULi+L,-\-L,-\- . . .), where L^, L^, L^ . . . represent the kinetic energy per unit volume of the constituent gases of the mixture and consequently P =i>i+i?2+i?3+ (11-912) In this formula ^2 =1-^2, P^=^L„ .... (11*913) etc., represent the contributions of the constituent gases 1, 2, 3 . . . respectively to the total pressure. In other words, p^, P2, Pz ' ' . are the partial pressures of the constituent gases, and (11*912) asserts that the total pressure is equal to the sum of the partial pressures. This is equivalent to the state- ment that the total pressure is equal to the sum of the pressures which each constituent gas would exert if it alone were occupy- ing the volume of the mixture. This is Dalton's law of partial pressures. Since (11*91) is equivalent to where q is the density of the gas, we can easily find ^c^, or the root of the mean square of the velocity of translation of the molecules of any gas. For hydrogen, oxygen and nitrogen at normal temperature and pressure we find 1'844 x 10^, 4-61 x 10* and 4-92 x 10* cm. sec.~i respectively. § 12. Laws of Chaeles and Avogadeo — Equipartition OF Energy It will have been noted that the kinetic energy referred to in (11*911) is that of translation only. The molecules may however have kinetic energy of rotation as weU. To begin with let us imagine them to be perfectly smooth spheres. Any force exerted on such a molecule must be normal to its surface, and 222 THEORETICAL PHYSICS [Ch. X consequently passes through its centre and therefore, if it is uniform, through its centre of mass. It follows that its kinetic energy of rotation cannot undergo any change. A gas consti- tuted of such molecules will behave in precisely the same way whether they have rotational kinetic energy or not. It can have no observational consequences. Each of such molecules may be regarded as a dynamical system with three q's and the corresponding ^'s, the former being the co-ordinates of the centre of the molecule and the latter the associated momenta. Such molecules have virtually only three degrees of freedom. In the next place let us suppose the molecules to be perfectly smooth and uniform ellipsoids of revolution. The forces ex- perienced by them in collisions must again be normal to the ellipsoidal surfaces. They will therefore always pass through the axis of revolution, but not necessarily through the centre or centre of mass of the ellipsoid. What has been said about the rotational kinetic energy of the spherical molecules applies to the kinetic energy of rotation about the axes of revolution of the ellipsoidal molecules, but not to the rotational kinetic energy about other axes. The ellipsoidal molecule consequently has virtually five q's or degrees of freedom ; three to fix the position of its centre of mass, and two to fix the direction of its axis of revolution. A sixth q representing angular displacements about this axis is not associated with any observable conse- quences and is for us virtually non-existent. Quite obviously the kinetic energy of translation wiU vary greatly from one molecule to another and the average kinetic energy of translation, reckoned for a volume so minute that it contains only one or two molecules, will likewise differ very much in different parts of the gas, and at different times. For the present we shall make the following assumption, leaving the discussion of its validity till a later stage. When the tempera- ture of the wall of the containing vessel is kept constant the gas reaches in time a final state which we shall describe as one of statistical equilibrium, and when this has been attained the average kinetic energy of translation calculated for the mole- cules in any volume at a given instant approaches a limiting value as the volume taken is made sufficiently big. We assume further that this limiting value is already practically reached while the volume in question is still so small a fraction of the space occupied by the gas that it may be treated as an element dx dy dz. For brevity we may say that statistical equilibrium is associated with a uniform distribution of translational kinetic energy among the molecules. From another point of view, statistical equi- librium is associated with uniform temperature throughout the §12] KINETIC THEORY OF GASES 223 gas and we shall define a scale of temperature by the statement K =aT (12) where K means the average translational kinetic energy of a molecule, and a is a constant, the same for aU gases .^ This definition leaves the unit of temperature difference still unde- fined. It foUows from (11-911) that pv = tNaT, or, if we write k for !«, pv =NkT (12-01) If two different gases occupy equal volumes at the same pressure and temperature it follows, since k is a. universal constant, that both contain the same number of molecules. This is Avogadro's law. Furthermore if pv has the same value for a number of different gases all at the same temperature, e.g. that of melting ice, it will necessarily have the same value for all the gases at any other temperature, e.g. that of saturated steam at normal pressure. This is the law of Charles. Equation (12-01) really unites in one statement the laws of Boyle, Charles and Avogadro. The unit of temperature difference is usually fixed by making the difference between the temperature of saturated steam under normal pressure, and that of melting ice, 100. If therefore we write (12-01) in the form pv=BT (12-011) the unit of temperature difference is fixed by i? = Ml__(^» . . . (12.012) where (pv)i and {pv)o mean the values of pv at the tempera- tures named above respectively. The definition of temperature adopted above is justified by its consequences ; but it involves an assumption which it is desirable we should be able to deduce as a consequence of the statistical equilibrium of a large number of dynamical systems, namely that the average kinetic energy of translation of all molecules, however much these may differ from one another, is the same when statistical equilibrium has been set up. We shaU later establish a theorem which contains this as a special case, namely that the average kinetic energy per degree of free- dom is the same for all molecules in statistical equilibrium. This is called the theorem of equipartition of energy. Since the average kinetic energy of translation of a molecule is ^kT, and this is distributed over three degrees of freedom, it follows 1 This assumption will be justified later. 224 THEORETICAL PHYSICS [Ch. X that the average kinetic energy associated with any one q or degree of freedom is \kT per molecule. The average kinetic energy of a molecule with v degrees of freedom is consequently -^jT. If we assume the potential energy of the molecules of a gas to be an invariable quantity, we find for the energy of a gram of the gas E = l-NkT + constant .... (12-02) where iV is the number of molecules in a gram. The specific heat of the gas at constant volume is therefore or c, = jJ? (12-021) where R is the gas constant for a gram. If the volume of the gas is changed, a certain amount of work, positive or negative according as the volume increases or diminishes, will be done by the pressure. The force exerted on an element dS of the wall of the vessel will be pdS, and if dS is displaced a distance dl the work done will be ^(dS dl) ; and consequently during a small expansion of the vessel the work done will be ^S(dS dl), the summation being extended over all the elements dS which make up the surface of the containing wall. The work done may therefore be expressed in the form pdv. This gives us, for any change in volume from an initial value Vi to a final value v^, the expression W = [pdv .... (12-022) which becomes if the pressure be kept constant during the expansion, or by (12*011). Consequently the work done is equal to R, if the § 12-1] KINETIC THEORY OF GASES 225 associated rise in temperature amounts to one degree. A word of warning is needed here. Equation (12'011) was deduced on the assumption that the volume occupied by the gas was not varying. Consequently the last result will only be valid if the expansion is taking place very slowly, so that the pressure, p, is not sensibly different from the pressure that would exist if the volume were not changing at all. Such an expansion is called a reversible expansion. Reversible processes will be discussed in detail in the chapters devoted to thermodjmamics. We see now that the specific heat of the gas at constant pressure will exceed that at constant volume by the amount R. Therefore c, =|i2 + i2 . . . . (12-023) and consequently the ratio r = -^ = 1 + - . . . . (12-03) This formula is in good accord with the experimental values for gases the chemical properties of which indicate a relatively simple molecular constitution. The sma^llest possible number of degrees of freedom is three giving y = If, a value found ex- perimentally for mercury vapour, helium and argon. The value y = If is found for gases like hydrogen, oxygen and nitrogen, the chemical behaviour of which shows them to consist of mole- cules having two atoms, while the ratio y is found to approach more closely still to unity with increasing complexity of mole° cular structure. § 12-1. Maxwell's Law of Disteibution We shall now inquire about the distribution of velocities among the molecules of a gas of the type described in § 11-9. Repre- senting the velocity of an individual molecule by c = {u, v, w) we have to try to answer the question: Among the N mole- cules constituting a gas, how many have velocities lying between the limits c = (^, v, w) and c + dc = (^ + du, V -\- dv, w -\- dw) 'i All the molecules with the same absolute velocity c will be assumed to be uniformly distributed as regards direction (§ 11-9). Let us represent the velocities of the individual molecules by points on a diagram (Fig. 12-1), the X, Y and Z coordinates of any one of these points being numerically equal to the u, V and w respectively of the molecule which the point represents. Imagine two infinite and parallel planes perpen- dicular to the X axis and cutting it at u and u -}- du. The 226 THEORETICAL PHYSICS [Ch. X number of representative points between the two planes may be expressed in the form Nf(u)du, f(u) being an unknown function of u. If we construct two further parallel planes, v and v + dv, similarly related to the Y axis, and if we make the very reasonable assumption that the number of molecules with Y components of velocity lying between v and v -]- dv is quite independent of their X com- ponents of velocity, we may write for the number of molecules with representative points in the region [J, bounded by the four planes N'f{v)dv, where N' is the number of representative points between the Y U-^dxr XT c ) « iL^du ^ Fig. 12-1 planes u and u -f du. Therefore the number of representative points in this region is Nf{u)f(v)du dv. Finally we may imagine a third pair of planes perpendicular to the Z axis at w and w + dw, and we find for the number of representative points in the small volume du dv dw enclosed by the three pairs of planes Nf(u)f{v)f{w)dudvdw . . . (12-1) Our problem is to find out, if possible, the character of the function/. The product Nf(u)f{v)f(w) in (12'1) is the number of points per unit volume at {u, v, w) in the space of Fig. 12-1 which represent molecules. At all points on the surface of a § 12-1] KINETIC THEORY OF GASES 227 sphere of radius c, and having its centre at the origin, this product must have the same value, i.e. fMMfM = const., and, of course, ^2 _|_ ^2 _|_ ^2 ^ c2 = const. If therefore {u, v, w) and {u + du, v -{- dv, w -{- dw) are neigh- bouring points on the spherical surface, f{u)f{v)f{w)du -\-f(u)f(v)f(w)dv +f{u)f(v)f{w)dw = 0, and vdu + vdv + wdw = 0. In the former of these equations f(u) is an abbreviation for •% . We may write these equations in the form fM f{v) f{w) udu + vdv + 'i^dw = . . (12'11) Multiply the second equation by ^ and subtract. The result will be and if we choose A so that we shall be left with In this last equation it is evident that dv and dw are arbitrary and their coefficients must consequently be zero. In this way we get the three equations iP^ = ^y (12-12) The factor A must be a constant because the first equation represents it as a function of u only, the second one as a function 228 THEORETICAL PHYSICS [Ch. X of V only and so on. The equations (12* 12) are equivalent to — {log f(u)} = Aw, ~ {log f(w)} = 2.W. We thus get ^ogf{u) =-u^ + const., or, if we replace - by — a, f(u) =Ae-<^^' . . . . . (12-13) The following definite integrals find frequent application in the kinetic theory. If e be a positive constant and n a positive integer, 00 00 00 J,„ = '\^x^''e~">'dx = il±^l^j(!^IliU-(^) (12-131) The last two, J^^ and J^^, are derived by successive differen- tiation from J 1 and J 2 respectively. The constant, ^, in (12*13) can be expressed in terms of the constant a. The expression +00 N [f{u)du —00 must be equal to the number of molecules in the gas, i.e. equal to N. Therefore + 00 A [ e-'^'^'du = 1, §12-1] KINETIC THEORY OF GASES 229 and by the second equation (12'131) A jr*a-* = 1 or A == a^Tz-^ .... (12-14) If we use rectangular co-ordinates, the number of molecules in the element of volume du dv dw of the representative space of Fig. 12-1 is NA^e--(^'+^'+^'Hu dv dw . . . (12-15) or NA^e-'^^'c^dc sm 6 dd dct> . . (12-151) if we use polar co-ordinates c, 6 and ^. The average kinetic energy of translation, K, of a molecule is evidently given by NK = NA^ { [ f Jmc^e— c^c^c^c sin 6 dO dc[>, 00 or K = 27imA^ j cH-^-'^'dc, and by (12-131) K = 2nmA^^n^a-s or K = 2nmam~i.l7i^a~^, and therefore K = ima-i .... (12-152) Another way of expressing this is to say that Average kinetic energy per molecule If we take the average kinetic energy of translation of a mole- cule to be |A;T (see § 12 ; the constant a of equation (12) must not be confused with the constant a of equation 12-152) we get from (12-152) and therefore « = 2& (^2-154) We thus find for the number of molecules per unit volume of the representative space g . . . . (12-16) V m^^ This is Maxwell's law of distribution of velocities. Starting from (12-151), we can easily find c the average 230 THEORETICAL PHYSICS [Ch. X of the absolute values of the molecular velocities. This is obviously given by 00 77 277 00 or c = 4:71 A^ I c^e-'^^'dc 47iA^ A^ f f ce— *^'c26?c sin 6 dd d<j> 00 1 00 [ c^e-^'^'dic^) 2 I By (12-131) this is c = 271 A^a-^ and since A = a^7t~^ c = 27ia^7c~^a~^ or c = 27i~^a~^ Therefore {cY = -_ (12-17) But we have seen that a7c a or fmc^ = I—, a and therefore c2 = -i .... (12-171) 2a so that (12-17 and 12-171) Q77. c^=^(c)^ . . . (12-172) That is to say the average of the squares of the velocities is equal to the product of — and the square of the average velocity. The averages just calculated are those of quantities associated with the molecules occupying some definite volume at a given instant of time. There are certain other averages of interest § 12-1] KINETIC THEORY OF GASES 231 and importance, for instance the average kinetic energy of translation of the molecules passing per unit time (or during a given time) through an element of area dS from one side to the other. This will obviously be greater than ^hT/2 because the energies of the faster moving molecules will appear more frequently in the sum from which the average is computed. The number of molecules per unit volume, the velocities of which lie between the limits c and c + dc, in absolute value and between d, (f> and d -{- d 6, cf) -{- d cj) in. direction (Fig. 11-91) is n' = nA^e-^^'Mc sin d dd dcjy, n being the total number of molecules per unit volume. The number passing through dS (Fig. 11*91) in the time dtiB, (11*903) n'c dt dS cos or nA^cH-'^'^'dc sm 6 cos 6 dd.dcf). dS dt. The translational kinetic energy transported by them is got by multiplying this expression by Jmc^. Therefore the number passing through the unit area per second is 00 nnA^ { c^e-'^^'dc, .... (12*18) the integration with respect to the variables d and ^ extending from to n/2 and to 2n respectively. Writing c^ = a;, this becomes 00 "^ [xe-'^Hx .... (12-181) For the kinetic energy passing through the unit area per second we get in a similar way ^^nA^ f ^2^-a.dx . . . (12-182) On evaluating the integral in (12*181) we get for the number of molecules passing through the unit area from one side to the other per unit time 2 since A = a^7i~^ ; and on substituting -^ for a we finally obtain n 16 /— (12*19) V 2nm 232 THEORETICAL PHYSICS [Ch. X The average kinetic energy of translation of these molecules is given by dividing (12-182) by (12-181). This yields K' =ma-^ .... (12-191) On comparing this with K (12-152), we see that K' =\k (12-192) § 12-2. Molecular Collisions — Mean Free Path Let us assume the molecules to be spheres, each having a dia- meter, a, very small compared with the average distance travelled by any molecule between consecutive collisions in which it is involved. This average distance is called the mean free path and we shall define it precisely as the quotient of the sum of the lengths of all the free paths completed during a given interval of time and the number of these paths. The given interval of time is understood to be so long that the quotient of total distance and number of paths is independent of its duration. There are of course several alternative definitions. If we take a given instant of time and consider the distance traversed by a molecule between this instant and the instant of its next collision, the average of these distances for all the molecules is the mean free path as defined by Tait. The former definition gives us, as we shall see, V2 I na^n whereas Tait's definition leads to . _ -677 . . . Am . Tia^n In each case n is the number of molecules per unit volume. We shall begin the attack on the problem of calculating the mean free path by considering the mean of the free paths, des- cribed in a given time, by a molecule moving with a speed which is very high compared with that of the vast majority of the remaining molecules. In this calculation we may suppose the remaining molecules to be at rest. Let the velocity of the moving molecule be c and consider a cylinder the axis of which is the path of the centre of the moving molecule, and the section of which is a circle of radius a. Collisions will occur between the moving molecule and all those which have their centres within this cylinder. The length of the cylinder described per unit time will be c and its volume na^c. Hence the number of collisions per unit time will be no^nc. Dividing the total distance. § 12-2] KINETIC THEORY OF GASES 233 c, which the molecule has travelled, by the number of collisions, we get Ao=-V ..... (12-2) This will at all events give the order of magnitude of the mean free path. We can easily see that the exact expression for the mean free path, calculated in accordance with the definition we have laid down, must represent a number between that just given (12*2) and zero ; since a very slowly moving molecule must, so long as it is moving slowly, describe very short free paths. Let r be the velocity of a molecule, B, relative to another molecule, A, the absolute velocities of A and B being Ci and Cg respectively. If these two velocities be represented diagrammaticaUy by lines of length Ci and Cg, as in Fig. 12-2, it wiU be obvious that r = {ca^ + Ci2 - 2C2C1 cos d}\ since the distance between B and A will be shortened by this amount dur- ing one second. The average value of r fig. 12-2 for a single molecule B with a velocity c 2 and a large number of molecules. A, each with the velocity Ci, and uniformly distributed in direction, will be f = i [ sin dd{c^^ + Ci2 - 2C2C1 cos df, The successive steps in the evaluation of this integral may be written down without detailed explanation as follows : f = j 2 sin -cos 2^(2)1(^2 - Ci)2 + ^c^c^ sin^-j , 1 1 r = dy{{c Ci)2 + 4c2Ci2/P, and therefore r = r = Sc,^ + c,' 3c < 3Ci' + c,^ Sc >Ci, <C1 (12-21 234 THEORETICAL PHYSICS [Ch. X Clausius obtained an approximate expression for the mean free path by assuming all the molecules to have the same velocity, c, and to be uniformly distributed in direction. With this assumption (12*21) becomes f = ^c (12-211) o which represents the average velocity of any one molecule relative to aU the others. To get the number of collisions ex- perienced by a particular molecule during the unit time we may suppose it to be moving with the velocity r and all the other molecules to be at rest. The method by which (12-2 ) was reached now gives us for the number of collisions per unit time nahir, while the actual distance travelled by the moving molecule is c. We get therefore for this approximation to the mean free path A - ^ by (12-211). In arriving at (12-2) and (12-22) it has been tacitly assumed that the cylinder of volume naH or no'^ is straight. Actually it has a more or less sharp bend at each collision. It is obvious however that this will not cause the expressions na'^c or ttctV to be in error, since the space swept out will be equal to the sum of the volumes of a large number of cylinders of cross- section 71(7 2 ; the sum of their lengths being c or f as the case may be. The way leading to an exact expression for the mean free path, according to the definition we have adopted, is now clearly indicated. From (12-21), and Maxwell's law of distribution, we get for the average velocity of a molecule with the absolute velocity Cg, relative to all the other molecules, the expression CO + 471^3 [ ^^i'+^^' c,%-^^'(^Ci (12-23) Therefore the number of collisions made by it in the unit time will be nna^ (12-231) §12-2] KINETIC THEORY OF GASES 235 where f is given by ( 12*23). The number of collisions made by aU the molecules N will consequently be 00 or, written out in full, 00 2 C - U/lyg 2 CO 00 Cj or V = l^7i^ahiNA^{C -{-D} . . (12-232) In the integrals C and D the integration with respect to Ci has to be carried out first. If in C the integration with respect to Cz were carried out first, it would have to be written in the form 00 00 q since Ca ^ Ci. It is obvious now that C = D and (12*232) becomes V = Z27i^aHNA^D . . . (12-233) On evaluation D becomes and for the number of collisions which N molecules experience in the unit time, we find V = 327c^aHNA^7t^2-^a-s or since A = a%~*, V = 2V2'\7i^a^nNa-i ... . (12-234) The total length of the paths is Nc=N2n-^a-i . . . (12-235) by (12-17), and therefore on dividing (12-235) by (12-234) we find for the mean free path A=--i (12-24) V 2 I na^ 236 THEORETICAL PHYSICS [Ch. X The following table gives the mean free path in centimetres for a number of gases at normal temperature and pressure. Gas A X 10^ Hydrogen Oxygen Nitrogen Carbon dioxide 1-78 102 0-95 0-65 § 12-3. Viscosity — Thermal Conductivity When a gas is in motion as a whole, we have to distinguish between the velocity of its motion, i.e. the stream velocity, s, and the velocity of agitation, c, of an individual molecule. Let us represent the components of the stream velocity by u\ v' and w' and those of the velocity of agitation of a single molecule by u, V and w as before ; so that s = {u', v\ w'), c = (u, V, w) (12'3) Associated with the flow of a gas in a given direction will be a stream momentum Sms = (Lmu', T,mv', Hmw') . . (12*301) where m is the mass of a single molecule. When the stream velocity varies from point to point, frictional or viscous forces wiU be exerted by one part of the gas on another (§ 11-5). These forces find their expression in terms of a tensor f'^, f'yy, t'\y, etc. Imagine the Z axis to be directed upwards and the Y and Z components of the stream velocity to be zero, so that everywhere s ^ « 0, 0), and let u' be a function of z only, so that the stream velocity has the same value at all points in the same horizontal plane. The tensor of § 11-5 now simplifies to the single component ^'V,, and by (U-Sl) or r„ = ^J' (12-31) This is the force per unit area exerted in the X direction over any horizontal plane by the more rapidly flowing medium above, on the less rapidly flowing medium below. The kinetic theory § 12-3] KINETIC THEORY OF GASES 237 explains this viscous force in the following way : The molecules above the given horizontal plane have a greater stream momen- tum, mu' , than have those below it. Approximately equal numbers wiU cross the unit area of the plane in both directions in a given time and the lower portion of the medium will there- fore gain stream momentum at the expense of the upper portion. The rate of gain of momentum will be a measure of the force exerted on the gas as whole below the given plane and tending to increase its velocity of flow, or conversely it will measure the force hindering the flow of the gas above the plane. To get an expression for the viscosity, /^, we first find the amount of stream momentum passing upwards through an element of area dS (AB in Fig. 11-91, the normal, N, having the direction of Z). We shall simplify the calculation by assum- ing that all the molecules have the same absolute velocity of agitation which we take to be the average of the actual velocities of agitation, or c. We may use the expression (1 1*903) for the number of molecules passing through dS in the time dt in directions included within the solid angle dQ{— sin 6 dd d(f>). z - M This has to be multiplied by the stream momentum per molecule. Each molecule on passing dS will have travelled, on the average, a distance I Fig. 12-3 (equal to the mean free path according to one of the possible definitions) since its last colli- sion, and we may take it to have the stream momentum appropriate to the place of its last collision. If dS be in the plane z=M = const., each of these molecules starts from the plane 2; = if — Z cos (Fig. 12-3) and if the stream momentum in the plane 2; = if be mu\ each molecule in question will have the stream momentum mL-? cos 61^1 . . . (12-311) Leaving the first term of this expression on one side for the time being, it becomes , ^ du' — ml cos d -— . dz If we multiply (11*903) by this we get an expression which differs from the corresponding one in § 11-9 only in having — mZ— - replacing 2mc, dz Lcosd 238 THEORETICAL PHYSICS [Ch. X so that we have instead of (1 1*904 ) — ml--—-n'cdSdt cos^ 6, dz or - "^ ^ dS dt 8m 6 cos^ddddS . (12-312) 4:71 dz Integrating, we get mncl du' j c. 7^ — — r— ao di. 6 dz Therefore the stream momentum carried upwards through dS in the time dt is P-— ^dS6^^ . . . (12-313) Q dz ^ ' where P is the contribution (whatever it may be) due to the term we have left on one side. The corresponding calculation for the stream momentum carried downwards obviously gives P+— ^dSd:^ . . . (12-314) Q dz ^ ' Subtracting (12-313) from (12-314) we get for the net gain of stream momentum by the medium below the plane z = M\ reckoned per unit area per second, „ _ mncl du' ^ .« - -3- -^. Consequently II = -^ ...... (12-32) The I in this formula will not be very different from X^ of (12-22), so that we obtain as an approximate expression for the viscosity 11=^ (12-321) or, by (12-22) [ji = — , .... (12-322) A rigorous calculation of /jl for spherical molecules leads to [ji = -350 . . . pcX . . . (12-323) 1 where X = V2 na^n and consequently a = — ^= . . . (12-324) §12-3] KINETIC THEORY OF GASES 239 It wiU be seen that the rigorous formulae differ only very little from the approximate expressions. The viscosity, according to (12*322) or (12'324), is equal to c multiplied by a constant which depends only on the mass and size of the molecules. The theory indicates therefore that it is proportional to the square root of the absolute temperature and quite independent of the pressure of the gas. This relationship was discovered by Clerk MaxweU. Subsequent experiment fully confirmed the independence of the viscosity of the pressure, but it was found to vary more rapidly with the temperature than is indicated by the theory. The discrepancy suggests that the molecular diameter a depends on the velocity of agitation, that is on the temperature. Sutherland ^ has derived the formula /To + C\ /T \ 3/2 in which (7 is a constant characteristic of the gas. This accords well with experimental results. When the temperature varies from point to point in a gas or any other medium heat flows from places at higher to places at lower temperature. Let us suppose the temperature to have the same value at all points in any plane z — const. There will be a consequent flow of heat in a direction perpendicular to these planes if the temperature varies with z. The thermal conductivity, K, is defined by Q^^^ (12-33) OjZ where Q is the quantity of heat flowing through the unit area per unit time in the direction of decreasing values of z. The kinetic theory identifies heat with the kinetic energy of the mole- cules and the problem of finding an expression for the con- ductivity of a gas is seen to be mathematically identical with the foregoing problem of viscosity. The kinetic energy per unit mass of the gas is therefore c^T where % is the specific heat of the gas at constant volume, and therefore the average kinetic energy per molecule may be expressed in the form mc^T. This is the quantity transported by a molecule. The identity of the present problem and that of viscosity becomes still more obvious if we represent the temperature by u' instead of T, and the quantity of heat transported through the unit area per unit time by t^^ instead of Q. The constant K will then occupy the place of ^ in formula (12*31). The quantity 1 Sutherland : Phil, Mag,, 36, p. 507 (1893). 240 THEORETICAL PHYSICS [Ch. X transported by a single molecule is then m%u' instead of mu' as in the viscosity problem. Consequently we find K=ixo, . . . . (12-331) i.e. the thermal conductivity is equal to the product of the viscosity and the specific heat at constant volume. Experiment confirms the proportionality of thermal conductivity and vis- cosity indicated by (12*331), and that the two quantities vary in the same way with temperature ; but here the agreement ends. It is found in fact that K = a[x% (12-332) where a is a constant in the neighbourhood of 2-5 for monatomic gases, such as helium and argon, 1-9 for diatomic gases, such as oxygen, hydrogen and nitrogen, and still smaller for more com- plex molecules. The discrepancy between the theory given above and experiment is mainly due to the assumption made about the distribution of velocities. A more rigorous theory based on a suitable modification of Maxwell's law of distribution — the existence of the temperature gradient obviously puts the law in error — does in fact yield a = 2-5, 1-9 and 1-75 for mole- cules with one, two and three atoms respectively. § 12-4. Diffusion of Gases The phenomenon of the diffusion of one gas into another is analogous to that of the conduction of heat and the definition of the coefficient of diffusion, or the diffusivity as it is usually termed, of a gas A into another B is similar to that of thermal conductivity. Instead of a temperature gradient, we now have a concentration gradient ; and instead of considering a transport of heat or kinetic energy we have now to study the passage of the molecules of one gas into the other. Let n^ and n^, be the numbers of molecules per unit volume of two gases occupying the same enclosure, and take the case where 7^l and n^ are functions of one co-ordinate, z, only, just as in the problems of viscosity and conductivity we supposed the stream velocity or the temperature to be functions of z only. If -D12 represent the diffusivity of gas 1 into gas 2, we have dz ' where G-^ is the number of molecules of gas 1 which pass through the unit area perpendicular to the Z axis per second in the direction of increasing z. Similarly dz ' 6, = -D,,- ©2 = — 2)21- § 12-4] KINETIC THEORY OF GASES 241 Following the method of calculating the viscosity in § 12-3, we bear in mind that the molecules passing through the element of area dS, in the sense of increasing z for example, and in a direction inclined at an angle 6 to the normal (i.e. to Z), have their last collision at the average distance I cos 6 below dS ; so that in evaluating the number passing through dS we must take the concentration appropriate to z — M — I cos 6 (§ 12-3). For the number of molecules of gas 1 passing through dS in the time dt and travelling in directions included in the solid angle dQ = sin. 6 dd dcf), we easily find < rii — li cos 6--^] -—dS dt cos 6, dz J 4t7z Ui meaning the concentration in the plane z = M, which con- tains dS. As in the viscosity problem the first term will con- tribute nothing to the end result, and we are left with _ cA drn ^^ ^^ g^ Q ^^g2 Q dQ ^^ 4:71 dz which takes the place of (12'312) in the viscosity problem. On integrating we get _cAdn,^^ 6 dz and therefore the number of molecules passing upwards (i.e. in the direction of increasing z) through dS in the time dt will be p _cj^dn, ^^ ^^^ 6 dz Similarly the number passing downwards will be p c£,dn, ^^ ^^^ 6 dz Therefore the net number passing upwards through the unit area per second is The corresponding quantity for the other constituent is .... (12-401) C 2" 2 ^"^ ! 3 dz In arriving at these formulae one important circumstance has been neglected. We have tacitly assumed that while gas 1 is diffusing, gas 2 is quiescent. Since however 7^l + 7^2 remains constant, there must be in general some motion of the gas as a whole. Let us suppose the velocity of this motion (in the 242 THEORETICAL PHYSICS [Ch. X Z direction) to be w' . Then our element dB must be travelling relatively to a fixed element dS^ with the velocity w' . The expressions (12*4) and (12'401) must therefore be amended as follows : Q^= - w'n^ _^2dp _ _ (12-402) 3 dz Now since Ui -\- n^ is constant it foUows that driz _ drii dz dz ' and Consequently = - w'{nj_ + n^) - J(^i^i - ^2^2)-^' and therefore ■{Cili C202)- 3(^1 +^2) dz On substituting this expression for — w' in (12*402) we get 0^=- 3(?ii + 7I2) dz and ^ _ n,cJ,-\-n,c,h dp Therefore ^'^-^" — 3(^7+1^" ■ • ■ ^ ' If the temperature be kept constant, the ratios w 2/(^1 + ^2) and ?^ 1/(^1 -\- n^ will remain constant as also Ci and c^ ; but Zi and 1 2 wiU vary inversely as the pressure. So that the diffus- ivity is inversely proportional to the pressure at constant tem- perature. When the pressure is kept constant, the ratios ^2/(^1 + ^2) ^.nd ?^ 1/(^1 + ^2) will again remain constant ; but Zi and 1 2 will be proportional to the temperature while Ci and C2 are proportional to the square root of the temperature. We conclude therefore that at constant pressure the diffusivity is proportional to T^l'^, Combining both conclusions we may say that the diffusivity is proportional to Jf3/2 Experiment confirms this result so far as the dependence on the pressure is concerned ; but the diffusivity is found to vary § 12-4] KINETIC THEORY OF GASES 243 more rapidly with temperature than the 3/2 law indicates. It will be recollected that a similar relation between experiment and theory was pointed out in connexion with viscosity. A phenomenon of great interest is the diffusion of a gas through minute apertures in a membrane, or in the wall of the containing vessel. This must be distinguished from the streaming or effusion of a gas through apertures which, though small in the ordinary sense, are nevertheless wide enough to permit the simultaneous egress of enormous numbers of molecules. If such an aperture is very short in comparison with its breadth, the velocity in the emerging stream of gas is given approximately by Bernoulli's theorem according to which Pi-P2 = 4/0^2^ - ipv^^, Pi and P2 a^nd Vi and V2 are the pressures and stream velocities at the points 1 and 2 respectively. So that if ^ 1 is the pressure in the interior of a large vessel where the velocity Vi is practically zero, and if ^pa is the pressure just outside the aperture, we have Pi -P2 ==ipv^ or -.. _ . -J- P for the velocity reached in the aperture. This result forms the basis of a simple method of comparing the densities of gases devised by Bunsen. If the aperture is in the nature of a long channel, the streaming through it of the gas is governed approxi- mately by the formula of Poiseuille. In neither of these cases is there any separation in the case of a mixture of gases. The partial pressures of the gases play no part in the phenomena ; but only the total pressure. It is different with true diffusion which depends on the motions of the individual molecules and therefore does not become evident tiU the openings are so minute that only one or two molecules are passing through them at any one instant. If we have a number of gases (between which we distinguish by sub- scripts 1, 2, 3, . , . s, . . .) contained in two vessels separated by a partition in which are such minute apertures, and if n'g and n'g represent the numbers of molecules per unit volume of the gas, 5, in the two vessels respectively ; it is clear that the number of molecules of the gas s, which leave the first vessel per unit time wiU (other things being equal) be proportional to n'g, and the number leaving the second vessel to n'^. This is an immediate consequence of (12*19). Other things being equal therefore, the rate of diffusion of a gas (expressed by the number of molecules diffusing in the unit time) is propor- tional to the difference of its partial pressures on the two sides 244 THEORETICAL PHYSICS [Ch. X of the membrane. On the other hand the rates of diffusion of different gases under like conditions are proportional to their mean velocities Ci, Cs, Cs, . . . , and g m iCi^ = m^c^^ and therefore Ci^ _ ma Cg^ mi or by (12.172) %- J'^}' it follows that the rates of diffusion are inversely proportional to the square roots of the masses of the molecules and therefore inversely proportional to the square roots of the densities (measured under like conditions of pressure and temperature) of the gases. These deductions are identical with the experi- mental result known as Graham's law. A membrane or waU which permits only one gas in a mixture to diffuse through it is called semi -permeable. For example palladium at a suitable temperature allows hydrogen to diffuse through it ; but not other gases. The picture which the kinetic theory gives us of this state of affairs is that of a partition with apertures so small that the molecules of only one gas are small enough to enter them. Imagine a palladium tube (maintained at a sufficiently high temperature) containing within it, say, nitrogen and surrounded on the outside by hydrogen kept at constant pressure. The latter gas will continue to diffuse in- wards until its partial pressure inside is equal to its pressure outside. The excess of the total pressure inside over that outside will therefore be equal to the partial pressure of the non-diffusing gas, or the pressure it would exert if it occupied the palladium vessel alone. Similar phenomena are associated with diffusion in liquids through semi-permeable membranes (made by deposit- ing copper ferrocyanide inside the wall of a vessel of unglazed earthenware). If such a vessel contains an aqueous solution of a crystalline body, sugar for example, and is surrounded by pure water, only the latter diffuses and the excess of the pressure inside the semi-permeable vessel over that outside, when equi- librium has ultimately been reached, is naturally associated with the dissolved sugar and is called its osmotic pressure. § 12-5. Theory of van der Waals We have so far supposed the dimensions of the individual molecules to be so smaU that their total proper volume is a negligible fraction of that of the containing vessel (§ 11-9). Let us now examine some of the consequences which ensue when § 12-5] KINETIC THEORY OF GASES 245 this fraction is not negligible. The centres of any two molecules cannot approach nearer to one another than a distance a equal to the diameter of a molecule. Imagine a sphere of radius a described round the centre of each molecule in the gas. We shall caU such a sphere (after Boltzmann) the covering sphere of the molecule. The sum of the volumes of the covering spheres 4 win therefore be -na^N, or 8v, where N is the total number of o molecules and v their total proper volume. Since the centres of a pair of molecules may be separated by as short a distance as a, some of the covering spheres wiU overlap ; but this over- lapping volume will be small by comparison with v and we shall neglect it. The part of the total volume V in which it is possible for the centre of a given molecule to be situated may consequently be taken to be V - -nam. 3 Let us reconsider, in the light of this result, the deduction of the expression ( 1 1 '9 1 ) for the pressure of the gas. The centres of the molecules on coUiding with AB (Fig. 12-5) will reach a plane CD, separ- ated from AB by the distance cr/2. Let us replace the cylin- der BCDE of Fig. 11-91 by the cyhnder DCEF of Fig. 12-5, with a perpendicular distance cdt cos between its end faces. This cylinder plays exactly the same part in the calculation as the former one, and has the same volume cdtdS cos B. We have to re-calculate n' in (11'903). have been written —cdtdS cos 6 471 V where N is the total number of molecules, of velocity c. The total number of molecules per unit volume of the space available N for their centres is now seen to be not -=y, Fig. 12-5 This formula might (12.5; but N V - -nam 3 (12-501) 246 THEORETICAL PHYSICS [Ch. X If now the cylinder DCEF (Fig. 12-5) were in the interior of the gas, the space within it available for the centres of molecules would be V - %tGm cdtdScosO 1 . . (12-502) If we take dt to be very smaU indeed, the cylinder wiU be so narrow that the centres of the molecules, whose covering spheres penetrate the cylinder, will, except for a negligible fraction, lie outside it. We should say that half of these centres were outside EF and the remaining half outside CD (Fig. 12-5). Since how- ever the distance between CD and the wall AB of the vessel is actually only -, no covering spheres of molecules penetrate it from that side at all. The last expression (12*502) must therefore be amended as foUows : cdtdScosd- ^^^ . . (12-503) To get the number of the N molecules which are in the cylinder DCEF and are moving in the directions included within the limits of the solid angle dD we must therefore multiply together -— and the expressions (12*501) and (12*503). We thus obtain 4:71 N 1 - 3-3^ which has to take the place of (12*5). The total proper volume of the molecules is SO that we get dQN\ Vj 4v^ c dt dS cos 6, 47rF A _ 8v\ or, neglecting {v/YY and higher powers, ^-M—cdtdSGO^Q .... (12*51) 4:71 V — 4V In recalculating the pressure therefore, we learn that the in- § 12-5] KINETIC THEORY OF GASES 247 fluence of the size of the molecules is precisely that which might be brought about by a reduction in volume equal to 4v. If we write, as is usual, b =4.v (12-52) we must amend (12*011) to read p(V -b) =RT (12-53) A further amendment due, like that just described, to J. D. van der Waals, is based on the supposition that the molecules exert attractive forces on one another which however are only appreciable when the separation of the molecules does not exceed a certain quite small distance R. Any molecule in the interior of the gas will therefore be under the influence of those situated in the sphere of radius R described about this molecule as centre. We may therefore suppose the resultant force exerted on it to be practically zero. It is different in the case of a molecule quite close to the boundary. The attracting molecules are all, or mostly, on one side of it instead of being uniformly distributed in a spherical region round about it. Over the whole boundary of the gas there will be a layer of molecules, extending to a depth R, which experience resultant forces in the direction of the interior of the gas. This will give rise to a pressure over and above that applied through the wall of the vessel or enclosure containing the gas. Since the number of molecules in this layer is practically proportional to the density of the gas, and the same is likewise true of the number of mole- cules attracting them, it is evident that the additional pressure may be taken as proportional to the square of the density or as inversely proportional to the square of the volume of the gas. We have therefore to amend equation (12-53) by adding to p Si term a/V^, where a is a suitable small constant. In this way we obtain the improved gas equation of van der Waals, (p+^^{V-b)=BT . . . (12-54) which may also be written in the form V^ -(b-\-—\v^-h-V -—=0 . (12-541) \ p / p p The isothermals (constant temperature curves) according to (12*54) or (12-541) are diagrammatically illustrated in Fig. 12-51. The arrow indicates the order of increasing temperature. The portions of these isothermals which slope downwards from left to right, for example in the isothermal ACEG the portions ABC and EFG, correspond moderately closely with experimentally 17 248 THEORETICAL PHYSICS [Ch. X found isothermals (if suitable values are given to a, h and R) the former representing states in which the whole of the sub- stance is in the liquid phase, and the latter those in which the substance is wholly vapour. Those states corresponding to portions of the isothermals, like CDE, which slope upwards from left to right are unstable (which explains why we do not observe them). For consider the state of affairs at such a point as H. A slight increase in the pressure wiU cause a diminution in volume and, as the slope of the curve indicates, a lower pressure than the original one is now necessary (at ir Fig. 12-51 constant temperature) for equilibrium. The actual pressure is therefore operating so as to remove the substance more and more from the state of equilibrium. It should be observed that in the deduction of van der Waals' equation, the whole of the substance is supposed to be in the same state at the same instant. Suppose it were possible for the whole of the substance to be in the state, H, at some instant. A slight local increment in pressure beyond HK, which is necessary for equilibrium, would result in that part of the substance changing to the condition corresponding to some point on ABC. Similarly a local diminu- § 12-5] KINETIC THEORY OF GASES 249 tion in pressure, however slight, would result in the substance in that locality changing to the condition represented by some point on EFG. Even supposing therefore the possibility of the whole of the substance being momentarily in the state repre- sented by H, it would immediately break up into two states (liquid and vapour). The equilibrium at the boundary between the two phases is obviously independent of the relative quantities of the substance in these phases. The equilibrium pressure is therefore determined solely by the temperature. Consequently the portion of an isothermal in which liquid and vapour states coexist will be a horizontal straight line. Thermodjmamical reasons will be given in a subsequent chapter (§ 17-4) which indicate that the situation of this horizontal line (BE in Eig. 12-51) is such as to make the areas BCD and DEE equal to one another. The states EF (supersaturated vapour) and BC (super- heated liquid) can of course be produced experimentally. Indeed this fact led James Thomson to suggest that the isothermals have the shape indicated by ACEG (Fig. 12-51) before v. d. Waals developed his theory. The maxima and minima of the v. d. Waals isothermals are located on a curve CPE, shown in the figure by a broken line. The isothermal passing through the summit, P, of this curve, and all those corresponding to higher temperatures, have no portions which slope upwards to the right, and we conclude that there is only one state for the range of temperatures beginning with that of the isothermal through P and extending upwards. This is in accordance with the fact, revealed by the experiments of Andrews, that it is impossible to liquefy a gas unless the tem- perature is reduced below a certain critical temperature characteristic of the particular gas,^ and which according to the theory of v. d. Waals is the temperature corresponding to the isothermal through the point P. The term gas state, in its narrower sense, applies to the substance when its temper- ature exceeds the critical value. Let us now pick out any isothermal, ACEG for example, and differentiate its equation with respect to v. We obtain The maximum E, and the minimum C, therefore conform to ^+f.=^a(^-*)- • • • (12-55) This must be the equation of the curve CPE. It will be noticed ^ This was suggested still earlier by Faraday. 250 THEORETICAL PHYSICS [Ch. X that it cuts the axis, p = 0, Sbt V == 2b and F = oo . The location of the critical point P is obtained by differentiating (12'55) and putting -^ = 0. We thus get dp _ 2a _ _ 6a -TT v\ ,^(^ dV ~ W ~ ~ T^^ ~ ^^V^ and therefore, if F^ is the critical volume, or F, = 36 (12-56) We may find the critical pressure by substituting 36 (12*56) for F in equation (12*55). This gives P, = A-o (12-561) ±-0 2762 ^ ' Finally we get an expression for the critical temperature by substituting the values (12-56) and (12-561) for the volume and pressure respectively in van der Waals' equation (12-54). This will be seen to give T, = -^ .... (12-562) ' 21Rb ^ ' It is instructive to express the pressure, volume and tem- perature, in van der Waals' equation, in terms of the correspond- ing critical values as units. Representing them by n, co and t, we have therefore p V T On substituting in the equation of van der Waals we find an equation from which the constants, which distinguish one gas from another, have disappeared. The quantities n, co and x are termed the reduced pressure, volume and temperature respectively. A number of gases for which the reduced pressure, volume and temperature are respectively equal, i.e. for all of which the pressures, n, are equal and all of which occupy equal volumes co at the same temperature, r, are said to be in corre- sponding states and equations (12-57) express the theorem of §12-5] KINETIC THEORY OF GASES 251 corresponding states from the point of view of the theory of van der Waals. The existence of a critical temperature, pressure and volume for gases is of course an experimental fact, and the theorem of corresponding states, in its widest sense, states that a relation f{7t, o),r)=0 exists, in which / is the same function for all gases. It is very- doubtful whether the theorem is accurately true ; but in the form (12*57) it represents at least a fair approximation to the truth. Any horizontal line cuts an isothermal (Fig. 12-51) in one point or three points, as is otherwise obvious from the fact that for a given pressure and temperature ( 12*541 ) is a cubic equation for V, and has therefore one real root, or three real roots. We may regard the critical point, P, as a point where three real roots have coincident values. For this point therefore (12*541 ) becomes Hence 73 - 3F2F, + 3FF,2 - F,3 3n = 6 + ^;^ 3F,2 = ^ Pc and These equations furnish an alternative way of arriving at the critical values (12*56), (12*561) and (12*562). We shall now consider briefly the deviations from Boyle's law in the light of v. d. Waals' theory. For this purpose v. d. Waals' equation may be expressed in terms of yy(= pF),^and T. In (12*54) or (12*541) therefore we replace F by -^ and so obtain 'tf - {RT + hp)ri^ + apri - abp^ = . . (12*58) If we plot f] against p (for constant temperatures) we get ap- proximately horizontal straight Hnes (isothermals) in accordance with the approximate validity of Boyle's law. Differentiate (12*58) with respect to p twice over, keeping T constant, and then equate ^ to 0. We thus obtain the equations dp and rj^ -% -\-2ap =0 . . . (12*581) ^pr]^ -2{RT + hp)ri + ap] = 2ab (12*582) 252 THEORETICAL PHYSICS [Ch. X The former of these equations gives the positions of the minima (or maxima, if they are maxima). They are seen to lie on a parabola ((12'581), represented by the broken line in Fig. 12-52). The latter equation shows that the corresponding values of — ?? are positive (if we assume a to be positive ; i.e. if we suppose the intermolecular forces to be attractive), as we easily find by ignoring hp and ap, since an approximate estimate will suffice in order to find the sign of -—^2. Thus dp 2ab or dh] ^ dp^ ~ 3^2 _ 2BT'r]' d^rj _ 2ab dp' ~ BW^' since r] = BT approximately ; and its positive character is obvious. Consequently the values of rj( = pV) on the locus (12*581) are minima. vi'pv) Fig. 12-52 Qualitatively the agreement between v. d. Waals' theory and the observed deviations of Boyle's law is very good. The minima are in fact observed at low temperatures to move in the direction of increasing p as we pass to higher temperatures (see the minima below A in Fig. 12-52) ; while at higher temperatures they behave in the contrary way. dp On differentiating (12'581) with respect to ?;, and making -j- = 0, we find *? = nTj which is the value of rj for the point A (Fig. 12*52). § 12-6] KINETIC THEORY OF GASES 253 At the point B we find (by making p = m (12*581)) »? = i- By sub- stitution in (12*581) or in v. d. Waals' equation, and remembering that rj = pVf we find the corresponding values of V and T. These are given in the subjoined table : 7] P F T A a 26 a 862 46 9a IQRb B a b 00 a Rb The theory of v. d. Waals is not however good enough quantitatively for these numerical values to be of importance. The extent of its failure can be shown very clearly by comparing the value the ratio {pV)p=o {pV)p=pa at the critical temperature with the observed value. Using equation (12*57), it becomes for very large volumes, 8 since t = 1 ; and at the critical point n = I and co = I ; hence TtCO = 1. The ratio is consequently 2f; whereas the observed value is found to be in the neighbourhood of 3-76.^ § 12-6. Loschmidt's Number It is usual to speak of the number of molecules in a gram molecule of a gas as Loschmidt's or Avogadro's number. It was first estimated by Loschmidt in 1865. The terms atomic weight and molecular weight ^ were introduced by chemists, at a time when the absolute masses of atoms and molecules were not yet known, to represent the masses of atoms and molecules in terms of the mass of a hydrogen atom as a unit. The atomic weight of hydrogen was therefore originally unity, and its mole- cular weight was taken to be 2 on the ground of chemical evidence interpreted in the light of Avogadro's hypothesis. For example the combining volumes of hydrogen and oxygen are in the ratio ^ For an account of various alternative gas equations of Clausius, Dieterici, Callendar and others, see Ferguson, Mecfianical Properties of Fluids. (Blaclde & Sons.) 2 ' Atomic weight ' and ' molecular weight ' have the sanction of long estabhshed custom ; but quite obviously ' atomic mass ' and ' molecular mass ' are the appropriate terms. 264 THEORETICAL PHYSICS [Ch. X of two to one, and the volume of the water vapour produced is found to be the same as that of the hydrogen, when measured under like conditions of temperature and pressure. Now assum- ing Avogadro's hjrpothesis, we have in the unit volumes of hydrogen, oxygen and water vapour (at the same temperature and pressure) equal numbers of molecules, say n. Therefore the reaction may be represented in the following way : where M^, Mq and Mj^tateb represent the molecules of hydrogen, oxygen and water (in water vapour) respectively. Consequently i.e. two molecules of hydrogen and one of oxygen produce two molecules of water. The simplest constitution of water consistent with the chemical evidence is H2O. Therefore 2Ms + Mo = 2Hfi, and consequently M^ = H^ Later this assumed constitution for hydrogen and oxygen was confirmed by physical observations, for example by determin- ations of the ratio of the specific heats at constant pressure and constant volume. A gram molecule of any substance is by definition a quantity of the substance the mass of which in grams is equal to its molecular weight. More recently atomic and molecular weights have been readjusted on the basis of = 16. This makes H = l-OOS.i The kinetic theory furnishes us with a means of estimating the absolute mass of a molecule, or, what amounts to the same thing, the number of molecules in a gram molecule. For this purpose we may use the following equations : -„ SET . , , . ,., ^,., (12-322) and we might add V = =j-Tr , CL^UiVeiJ LCXit tU mc • • = :-=<• M = Nm, . (12-172) ^ Quite recent experimental investigations of the relative masses of the atoms of isotopes have led to a further very minute readjustment. § 12-7] KINETIC THEORY OF GASES 255 but some of the assumptions underlying these formulae, for instance that of spherical molecules, are so rough that we may- just as well assume c^ = (c)^. The symbols have the meanings : M = mass of a gram molecule ; B = gas constant for a gram molecule ; c = velocity of a molecule, the bar indicating averages ; fi = viscosity ; m = mass of a molecule ; V = total proper volume of the molecules ; 6 = V. d. Waals' constant ; N = the number of molecules in a gram molecule (Lo- schmidt's number). We have in these four equations four unknown quantities, namely c, c, m and N ; the other quantities being given by experimental observations. As an illustration let us take the case of hydrogen. R == 8-315 X 10^ ergs per °C. (the same approximately for all gases). M = 2-016 gram. T = 273, if we chose the temperature of melting ice. fi = 86 X 10"^ gram per cm. per sec. b = 19-7 c.c. for a gram molecule and hence v = 4-925. When we substitute these data we find a = 2-74 X 10-8 cm., A^ = 4-6 X 1023. Obviously these numbers cannot be regarded as expressing any- thing better than the order of magnitude of a and A^. § 12-7. Beownian Movement In 1827 the botanist Robert Brown observed that the poUen grains of clarkia pulchella, when suspended in water, were in a constant state of agitation. Further investigation has shown that the phenomenon is not peculiar to poUen grains, and is not confined to particles which are living organisms. It can in fact be observed with smaU particles of any kind suspended in a liquid or gas. It is independent of the chemical constitution of the particles and is not due to external vibrations, or to motions in the suspending fiuid due to temperature inequalities. When every precaution has been taken to get rid of such disturbances it stiU persists. In the words of Perrin, ' II est eternel et spontane.' These characteristics of the Brownian movement led Christian Wiener in 1863 to the conclusion that it was due to the motion 256 THEORETICAL PHYSICS [Ch. X of agitation of the molecules of tlie suspending medium. The movement is more violent in the case of smaU particles than in the case of larger ones as Brown himself observed ; a fact which supports the conclusion that Wiener arrived at. § 12-8. Osmotic Presstjee of Suspended Particles Imagine a large number of smaU particles, all having the same mass, suspended in a fluid of smaller density. Let n be the number of them per unit volume at a height z from the base of the containing vessel when statistical equilibrium has been attained, and m' the excess of the mass of a single particle over that of an equal volume of the suspending fluid. If p be the (osmotic) pressure due to the particles, we have by (10'6) dp , — -^ = nmq, dz or dp = — nm'gdz .... (12*8) and according to the kinetic theory p = nhT, and therefore _^ = — —^dz, p kT dn nh'Oj Hence logl-.f*. ..... ,m.) where tIq is the number of particles per unit volume in a horizontal plane z = M, and n the number per unit volume in a plane z = M -{- h. Perrin verified this formula experimentally by directly count- ing the numbers of small equal spherules of gamboge suspended at various heights in water in a small vessel which was placed under a microscope. He determined the size and mass of the spherules by various methods ; e.g. by measuriQg the length of a row of them and counting the number in the row ; by weighiug a known or estimated number of them ; and by measuring the rate of faU of a spherule through the water and applying Stokes' law (11*792). The data which he thus obtained enable k to be found and hence also Loschmidt's number iV, since Nk = R, where B{= 8-315 x 10^ ergs per ° C.) is the gas constant for a gram molecule. In this way Perrin found for N numbers vary- ing from 6-5 x 10^3 to 7-2 x lO^s. § 12-8] KINETIC THEORY OF GASES 257 Perrin carried out a great variety of experiments which not only settled any question as to the nature of the Brownian move- ment, but constituted most important tests of the kinetic theory of gases. Only one other of these investigations will be dealt with here. It is based on a formula deduced by Einstein. The equations of motion of a single spherule may be written in the form : m— - = — S-r- + X, dt^ dt ' m^. = ->S^ + Z; . . . . (12-82) d^ _ "df^ ~ 'dt where ( >S^, S~, /8^ j represents the resistance of the fluid to the motion of the spherule, and (X, Y, Z) is the force due to bombardments by the fluid molecules. By Stokes' law (11*792) S = OtTtrfx where r is the radius of a spherule, and [jl is the viscosity of the fluid. Multiplying the first of the equations (12*82) by x we have dH ci dx , ^ mx-j- = — Sx— + xX, dt^ dt and therefore d / dx\ /dx\^ S d(x^) , „ Consequently f^^i.^) - .(|)^ = - f -|^ + -^ (12-821) If now x^ he the average value oi x^ for a large number of the spherules, which are of course supposed to be exactly alike, we get from (12*821) md^ 8 d^^) /12.822^ (clx\ — j is two-thirds of the average kinetic energy of translation of a spherule, and as the value of X at any given place is just as likely to be positive as negative, xX = 0. If we abbreviate by writing dx^ _ m ,de hT 2 dt dp. S + —8 = dt m 258 THEORETICAL PHYSICS [Ch. X (12-822) becomes S or -^-e=?^ ... (12-823) dt m m This may be written and therefore s-^=Ae-^i . . . . (12-83) where ^ is a constant of integration. If t is sufficiently long, the right-hand side of this equation is not sensibly different from zero, and we have d^^ 2kT and consequently dt S ' — „ 2kT S or x^ = ^r ..... (12-84) _ Snrju where a; 2 is the mean value of the square of the displacement in the X direction during a sufficiently long period of time t. This is Einstein's formula. Perrin measured x^, by means of a vertical microscope capable of motion in a horizontal plane, the individual measure- ments of a; 2 being made on the same granule, thus eliminating the errors, due to slight differences in size, which might have resulted from observations on different granules. He thus deduced values for Loschmidt's number between 5-5 x lO^^ and 8 X 1023, his mean value being 6-88 x lO^s. The importance of these results does not lie of course in the precision of the numerical results, but in the test they furnished of the essential soundness of the kinetic theory. BIBLIOGRAPHY Clerk Maxwell: Scientific Papers. L. BoLTZMANN : Vorlesungen iiber Gastheorie. (Barth, Leipzig.) Jeans : Dynaimcal Theory of Gases. (Cambridge.) G. Jager : Die Fortschritte der Kinetischen Gastheorie. (Vieweg & Sohn.) Jean Perrin : Les Atomes. (Felix Alcan.) A. Einstein : The Brownian Movement. (Methuen.) Mecklenburg : Die experimentelle Grundlegung der Atomistik. (Fischer, Jena.) CHAPTER XI STATISTICAL MECHANICS § 12-9. Phase Space and Extension in Phase IMAGINE a very large number of Hamiltonian systems (i.e. dynamical systems subject to the canonical equations (8*43 and 8*46) of Hamilton) all exactly alike and having each n degrees of freedom. For simplicity we shall suppose they do not interact with one another at all. Let the number of them which have q^ between q^ and q^ + dqi, ^n 55 qn 5) <ln -{-dq^, Vi 55 Vi 55 Pi -\-djp^ Vn 55 Vn 55 Pn + #«5 be pdq4q^ . . . dqjp^djp^ . . • #n • • (12-9) The density, p, may be regarded as a function oi q^, q^, . . . qn, Pi, P2 • • • Pn' I^ ^^^ ^^ convenient, sometimes, to replace qi, q2 • • ' qn^y ^15 I2, . . . in respectively and p^, p^, , . . p^ by in+v ^n+25 ' ' ' i2n'y SO that (12-9) may be written pdi.di, . . . diji,^, . . . di^, , (12-901) p being a function of li, I25 • • • l2n- ^or illustration consider the case where each system has only one degree of freedom and consequently /> is a function of q and the associated p. We may represent the state of the assemblage of systems on a plane, using q and p (or |i and I2) as rectangular co-ordinates, and the number of systems for which q lies between q and q + dq, and p between p and p -\- dp is pdqdp, or pdiidi^- The language and symbolism appropriate for this graphical representation of the distribution of the systems may profitably be extended to an assemblage of systems each of which has n 259 260 THEORETICAL PHYSICS [Ch. XI degrees of freedom, although we may not be able to visualize 2n axes of coordinates. We shaU term the space of such a diagram the representative space or phase space, and p the density of the distribution of the systems in phase. The equation of continuity (10*52) suggests that This is easily established in the following way : Consider the plane (or boundary), l^ for example, of the elementary region included between li and li + dii, S 2 J5 S 2 r ^S 2j Obviously the number of systems which cross this boundary and enter the element in the time dt is expressed by p^^dtd^^di^ . . . di,_^di,^^ . . . cZ|2^, in which product the differential dig is missing, its place being taken by i/it. Some of these may of course cross one or more of the 2n — I remaining boundaries ; but the number of them doing so will be a small quantity of still higher order, and need not be further regarded. The number of systems leaving the element through the boundary |g + d^^ will clearly be {pl I ^(pl) fit dtdiidiz . . . di,_T^di,^i . . . di 2ft* On subtraction we find for the excess of the number of systems leaving the element of volume of the representative space in the time dt over that entering it ^-^^tdi.di, . . . di, . . . di^; and when we take account of the remaining boundaries we get the result r This must equal -dt d^idiz • • • ^li -^£dtd$,di, . . . di, § 12-9] STATISTICAL MECHANICS 261 and equation (12*91 ) results from equating the two expressions. Since this equation may be written in the form +4:+!:^ ■■•+!:)=»■ we may use the symbolism of § 10-7 (see equations 10*701 and 10*702) and write In this equation ,— represents the rate of change of p as we follow the motion along a stream line in the representative space. In the earlier notation (12*911) becomes Now it follows from the canonical equations '^ = ¥; dH dH ^^ ^ ~ ^* (where H is the energy of any one of the systems in the element dqi . . . dq^dpi . . . dp^ ot d^i . . . d^^n) that ^« + ?2f = § = (12*92) for every s. Therefore This result is known as Liouville's theorem. We can express it in an alternative way. If A-^ be the number of systems in the element AI1AI2 . • • Ahn^ AN = QAhAh . . . Af2n, or briefly Ai^^ = ^A^; and D(AN) _ Dq Dim —Df- - ^^Dt + ^-DT' 262 THEORETICAL PHYSICS [Ch. XI If we confine the equation to the same systems Dt and by Liouville's theorem = 0, hence —^ = 0, and Q = constant ; ^^=». and AliA^2 • • • Al27i = constant. If therefore we follow the motion of N systems in the representative space, the volume, {dq^ . . . dq.dp, . , . dp^, . . (12-921) which they occupy in it will remain constant. In the language of Willard Gibbs their extension in phase remains constant. § 13. Canonical Disteibtjtions If the number of systems per unit volume at every point in the representative space is constant, i.e. if dt everjnvhere, we have statistical equilibrium. The condition for statistical equilibrium is therefore (12*92) s=2n r. s=l ^^ s=l -^* -^ * This condition will be satisfied by Q=f{H) (13-02) where / is any function, and H is the energy of a system ; for if we represent -^ by /', and r- = 1^/' = iJ'' dp, dp-' § 13-1] STATISTICAL MECHANICS 263 and consequently is^ + Ps~^j^ = ( -isi>s + i>As)f = 0. The particular case p=Ae~^ (I3'0S) where G and A are constants, is of great interest, and is naturally suggested as a generalization of the Maxwellian law of distribu- tion, § 12-1. The constant A can be expressed in terms of O and the inherent constants of the individual systems constituting the assemblage by substituting the expression ( 13*03) for p in j . . . I pd^, . . . dhn = N . . (13-04) where N is the total number of systems in the assemblage. A distribution defined by (13*03) is called by Willard Gibbs a canonical distribution, the constant, 0, being the modulus of the distribution. § 13-1. Statistical EQuiLiBiinjM of Mutually Interacting Systems We have been studying a type of assemblage, the individual members of which are conservative systems, and do not inter- act with one another at all, and in which therefore the energy is distributed in such a way that a definite portion of it is assigned to each system. No actual assemblage can be strictly of this type. There must be some interaction between individual systems, and consequently some of the energy must be ths mutual energy of groups of two or more systems. In what follows we shall take this interaction into account ; but we shall restrict our attention to cases where the mutual energy is a negligible fraction of the whole energy of the assemblage. Let the total number of systems forming the assemblage be N, and imagine the phase space to be divided into very small and equal elements A<^i, A<^2, Acog .... We may denote the number of systems in the elements Acoj, ACO2, A^Oa • • • hy Ni, N^, N^, . . . respectively and the energy of each system in these elements by El, E2, E3, . . . respectively, the total energy being E. We have therefore N = Z^=' 18 s-l. 2. 3, . . . E= 2] E,N, . . . . . (13-i: s-1. 2, 3, . . . 264 THEORETICAL PHYSICS [Ch. XI It is convenient to write f =^s and E for the average energy of a system, so that equations (13'1) become S = l, 2, 3, . . . E= ^EJ, . . . . (13-101) s=l, 2. 3. , . . Among the various distributions of the systems in the phase space, the only one which can be permanently in statistical equilibrium is that which has the greatest probability. In order to find a starting point for attacking the problem of determining the relative probabilities of different distributions, let us consider the following illustration : Imagine a large number of similar baUs to be projected, so that they fall into one or other of three receptacles, A, B and C. It may happen that they distribute themselves equally among the three receptacles, and hence the probability that any one of the balls is in the receptacle A is the same as the probability of its being in B, or in C. This is often expressed in the form : the a priori probability that a given ball is in the receptacle A is the same as the a priori prob- ability of its being in B, or in C. The term a priori is used be- cause the probability in question is one of the premisses from which we start out when we wish to find the probability of a given distribution of some definite number of balls in the three receptacles ; e.g. a total number of 6 balls, 3 in A, 2 in B and 1 in C. If in the projection of the balls, one of the receptacles is favoured in some way, so that when a large number of them is projected, twice as many fall into B as into either A or C, the a priori probabilities of a particular ball being in ^, -S or C are as 1 : 2 : 1. In the former case A, B and C are said to have equal weights, in the latter their weights are 1, 2 and 1 respec- tively. If the weights (or a priori probabilities) associated with the receptacles are all equal, the probability of a given distri- bution among them of a definite number of balls is equal to the number of ways (or complexions) in which this distribution can be made, divided by the sum of the numbers of complexions of all possible distributions. Taking the example of two recep- tacles A and B and a distribution in which 4 balls are in A and 2 in B, out of a total of 6 balls ; the number of complexions is 6! 4! 21' § 13-1] STATISTICAL MECHANICS 265 while the sum of the numbers of complexions of all possible distributions of 6 baUs between the two receptacles is 1 +? -L^ 4- ^-^'^ , 6.5.4.3 6.5.4.3.2 1 1.2 "^ 1.2.3 1.2.3.4 1.2.3.4.5 ' This is the sum of the coefficients in the expansion of (a + 6)«, and is therefore equal to 26. Hence the probability required is _^2- 4! 2! More generally if N be the total number of balls, distributed among n receptacles, so that there are Ni, N2, Ng, . . . N^ baUs respectively in the receptacles 1, 2, 3, . . , n ; the proba- bility of the distribution will be N^N^Jl...Nr ' ■ ■ ■ ^''"'^ In these examples we have tacitly adopted the usual conven- tion that certainty is represented by unity. It is more con- venient however for the purposes we have in view to use the total number — n^ in (13'11) — of the complexions of all the possible distributions, as representing certainty ; in which case (13*11) is replaced by til (IVM) N,\N,\N,\ , . . NJ ' ' ' ' ^ "^ ^) Adopting this convention, and assuming that the a priori probabilities associated with all the elements A^Ou ACO2, Aw 3, ... of the phase space are equal ; the probability, P, of the distribution in which Ni systems are in the element A^i, •^ * 2 55 J5 35 55 JJ A<^2j -^3 55 35 53 55 55 L\(^ 3} and so on, is clearly TV' P = — . . . (13-13) We assume iVi, Nz, N^, . . .to be individually very large numbers, and we may in consequence make use of Stirling's theorem, namely n\ = V27in I e-%% .... (13-14) where nis a, large integer, strictly speaking an infinite integer. 266 THEORETICAL PHYSICS [Ch. XI It follows that log nl = n log n . . . . . . (13'141) and hence logP^A^logiV^- ^ N,logN„ s = 1.2.3. ... or f = logP = -N2^fJogf,. . . (13-15) s=l, 2. 3. . . . The most probable distribution is that for which P, and consequently ip, has the biggest value, subject to the conditions (13*101). The maximum value of yj is therefore determined by dy,= -N ^ {\ogl + l)df, = 0, s=l. 2. 3. . . . the dfs being subject to the limitations imposed by 3-1. 2. 3, . . . and SE = 2^ E,df, = 0, s = l. 2. 3, . . . which merely express the fact that the total number of systems, and the total energy remain constant. These equations are equivalent to s-1. 2. 3. . .". X ^sSfs =0, s = l, 2, 3. . . . 2J ,5/, =0. ..... (13-16) s = l,2, 3, ... Hence it follows that the most probable distribution is given by log/3 +^^, +a = . . . . (13-17) where a and /5 are constants, and consequently f,=Be-^^^ .... (13-171) in which 5 is a constant. This is identical with the canonical distribution already described, since B can be put in the form B = A Aco, or B = Adq^dqz . . . dq^dpidp^ . . . dp^ . (13-172) where ^ is a constant, and hence P = l (13-18) § 13-3] STATISTICAL MECHANICS 267 The constant B can of course be expressed in terms of /3 (or 0), since S/, = 1 = 5Se-^^^ . . . (13-185) The maximum value of ip is obtained by substituting the ex- pression (13*171) for /s in (13*15). We thus have ?» = - ^ >J -Be-^^'- (log B - liE,), s = l. 2, 3. . . . or y)^ = - NlogB + ^E (13-19) in consequence of (13*185) and (13*101). § 13-2. Criteria of Maxima and Minima We have tacitly assumed that ipm is a maximura ; but the foregoing argument does not distinguish between a maximum and a minimum. To settle this question we expand dip, the small increment of ip due to small increments dfg. Since xp = -NZf.logfs, we have dip = - N^lil + Sf,) log (/, + Sf,) - f, log /J , dip = - Nu[{f, + (5/j|log/, + log (l + ^^) I -/, log/,], [ore Sy, = - Nz[sf, log/, + if, + a/,) log (l + j)]. Now when dip = ip - ip-rr,, this reduces to which is essentially negative whatever the df^ may be, provided they are small enough. Hence ipm is a maximum. § 13-3. Significance of the Modulus Let us now consider a small increment d\p.^ due to a s^nall change dE in the energy of the whole assemblage. The values of B and /5, which for a given value of E are constants, will now experience increments dB and d^, and we have from (13*19) drp^ = - n"^ 4- pdE + Edp . . . (13*30) or Therefore 268 THEORETICAL PHYSICS [Ch. XI Differentiating (13* 185) we find =dB ^ e-^^s - B^ E.e-^^^dp S = l. 2. 3, . . . S-1. 2. 3, . . . ^ dB E.^ and consequently, on substituting in (13*30), dxp^ = pdE, or (13-31) The d\p^ in this equation must be sharply distinguished from dy). The former represents the small increment of ip corresponding to the increment dE of the energy of the assemblage when statistical equilibrium is practically established. The latter means a small change in ip occurring while E remains constant, and it can only differ from zero so long as statistical equilibrium (or, strictly speaking, the most probable state) has not been reached. We now turn to the problem of the statistical equilibrium of two assemblages, which can interchange energy with one another, but are otherwise isolated ; i.e. their combined energy is a constant quantity. We shall distinguish them by the letters A and B ; so that E = E^ -\- E^. It is easy to see that where P is the probability of a state of the combined assemblages, while P^ and P^ are the probabilities of the associated states of the individual assemblages A and B respectively. Consequently The condition for statistical equilibrium of the combined systems is dip = 0, subject to SE = (13-32) Now since the individual systems, A and B, are themselves in statistical equilibrium any small changes in ip^ or ip^ must be due to transfer of energy from A to B or B to A, and are there- fore properly represented by dy)j^ and dip^. Consequently dy) = dy)j_ + dy)s, and dE =dli^ -\-dEs (13-33) § 13-3] STATISTICAL MECHANICS 269 The conditions for statistical equilibrium are therefore dE^ +dEs=0; . . . . (13-34) and, by (13'31). J dEj, dV>B = -^• ^B On substituting for dy)j^ and df^ in equations (13-34) we get dE^_dE^_ ^ Ob ' whence 0^ = 0^ (13-35) This then is the condition that the two assemblages may be in statistical equilibrium with one another. Any interaction between two assemblages which have not yet reached statistical equilibrium must be such that dy) or ^+^ (13-36) Oa Ob is a positive quantity, because it is bound to have such a character as to bring about a condition which is more probable.^ Therefore dE^ dE^ . —^ — -jY 1^ positive. If now dEA must be negative ; i.e. energy must flow from the assemblage which has the greater modulus, S, It is now clear that S plays the part of temperature, and we have reached the stage when we may claim to have given an explanation of the more obvious features of thermal phenomena in mechanical terms. Reference to §§12 and 12-1, and more especially to equations (12), (12-01) and (12-16) will indicate that we must identify S with hT. For the thermal equilibrium of two assemblages (two gases for example) Oa = O^. by (13-35),^ and the physical meaning of temperature necessitates that ^ Strictly speaking, we may only equate dy)j^ to -^— when the as- semblage A is itself in. statistical equilibrium, so that the expression ( 13*36) may only be employed for dyj when statistical equilibrium has nearly been attained. It will however suffice for the present purpose if we suppose that this is the case. 270 THEORETICAL PHYSICS consequently or '^A — ^Bi [Ch. XI and the assumption of the universal character of the constant a (or fA;) in equation (12) is now justified. § 13-4. Entropy In the chapters on thermodynamics we shall meet with a quantity, </>, first introduced by Clausius and known as entropy. We shall see that when a system is nearly in thermal equilibrium # = f. where dQ is the heat communicated to the system and d(j) is the corresponding increase in its entropy. If we compare this relation with , dE dip = e we see at once that hip-=^ (13-4) In consequence of this relationship h is often called the entropy constant. It is also known as Boltzmann's constant. § 13-5. The Theorem of Equipartition of Energy The general expression (8*26) for the kinetic energy of a Hamiltonian system simplifies in many cases to a sum involving squares of momenta, but not their products. When this happens, the energy of the system takes the form E = V -{- a,p,^ + a,p,^ + . . . + a,p,^ + • . . (13-5) where ai, ^2, . . . a^, . . . are either constants or functions of the q's only. Examples are : a particle, a rigid body or also a system consisting of two mutually gravitating bodies. It is convenient to term a^p^^, a^p^^, • • • (^sPs^^ • • • ^^^-j ^^^ kinetic energies associated with the co-ordinates 1, 2, . . , s, . . . etc., respectively. We can now establish that, in any assemblage of this kind, the average kinetic energy (of a system) associated with any co-ordinate, s, is the same for all the co- kT ordinates and equal to - or to — . The number of systems in § 13-5] STATISTICAL MECHANICS 271 the element dq^dq^ . . . dq^dp^dp^ • • • dp^ i^aay be expressed in the form : NAe ® dq^dq^ . . . dqj.p^ . . . dp^, where N is the total number of systems in the assemblage. The total kinetic energy associated with the co-ordinate, 5, in this element of the phase space is NAa,p,H © dq^dq^ . . . dq^dp^ . . . dp^. The average kinetic energy (in an element of volume dq-^dq^, . . . dq^) associated with s is consequently r -^ r dq^dq^, . . dq^ a,p,H ® dp, dqidqz . . . dq^ e ® dp, . . \p, ' ''dp, or _a^ \ e ® dp. (13-51) (13-511) Both integrals in this expression have of course the same limits — p, may range from to + oo or from — oo to + oo — in either case we get from (12-131) for the average kinetic energy as stated above. This is the theorem of equipartition of energy on which the proofs of the laws of Avogadro and Charles in §12 were based. BIBLIOGRAPHY WiLLARD GiBBS : Elementary Principles in Statistical Mechanics. See also references at the end of the preceding chapter. CHAPTER XII THERMODYNAMICS. FIRST LAW § 15. Origin of Thermodynamics THERMODYNAMICS, as we understand the term, owes its origin to the Frenchman Sadi Carnot who published in 1824 a treatise entitled ' Reflexions sur la Puissance Motrice du Feu et sur les Machines propres a developper cette Puissance.' This work, one of the most important and remark- able in the whole range of physical science, was entirely ignored for more than twenty years, when its merits were recognized by Sir William Thomson, afterwards Lord Kelvin. Classical thermodjmamics is based on two main principles, the first and second laws of thermodynamics. The first law, which is simply the principle of conservation of energy as applied to thermal phenomena, is commonly ascribed to Julius Robert Mayer, who, in 1842, evaluated the so-called mechanical equivalent of heat from the values of the specific heats of air at constant pressure and constant volume. In justice to Carnot it should be said that a precise and clear statement of the first law was found, after his death, in the manuscript notes which he left, and also a calculation of the mechanical equivalent of heat. The value which he found was 0-37 kilogram-metres per gram-calorie. The second law was also discovered by Carnot, and is contained in the treatise mentioned above. While classical thermodjniamics is based on the two laws already mentioned, a ' third law of thermodynamics ' has been added in recent times by the German physical chemist, W. Nernst. § 15«L Temperature We may define the term ' temperature of a body ' in a rough way as its hotness expressed on a numerical scale. The term ' hotness ' has reference to the sensation we experience in touch- ing a hot body. Such sensations do not enable us to construct a scale of temperature with precision, and we have therefore to make use of appropriate physical quantities for this purpose. 272 §15-1] THERMODYNAMICS. FIRST LAW 273 Of these physical quantities, one which is very commonly used is the volume of a fixed quantity of some liquid, usually mercury. We assume that the reader is familiar with the mercury ther- mometer. An arbitrary scale, for example a millimetre scale, marked on the stem of such a thermometer defines a scale of temperature as far as the divisions extend. If we place the thermometer in water contained in a beaker, the mercury will expand, or contract, according as it happens to be initially colder or hotter than the water, until a state of equilibrium (thermal equilibrium) is established, when the top of the mercury column is at some definite mark on the arbitrary scale. If we make the water progressively hotter (in the sense that it actually feels hotter), we find as an experimental fact, that the mercury column rises in the stem of the thermometer. Another import- ant fact of experience is the following : if we place two bodies, having very different temperatures, in contact ; for example if we surround some hot liquid contained in a copper vessel by cold water contained in a larger beaker, we find that ultimately a state of thermal equilibrium is set up, in which both the liquid in the copper vessel and the surrounding water have the same temperature. This is the case whether we judge the temperature by the sensations experienced on immersing the hand in the liquids or by noting the position of the top of the mercury column on the stem of the thermometer. We see that the readings of a mercury thermometer follow, as far as we can judge, the much rougher indications of our sensations of warmth or coldness. We may continue to adhere to the definition of tem- perature given above, namely, * the hotness of a body expressed on a numerical scale ' provided that the numerical scale is defined by some physical quantity, as for example the volume of a definite quantity of mercury in thermal equilibrium with the body, the temperature of which is being expressed. There are many other physical quantities which may be employed for defining scales of temperature and for temperature measurement, e.g. the electrical resistance of a piece of platinum wire, or the electromotive force in a thermo-couple ; but whatever physical quantity be used, it must express the temperature in a way that is unambiguous over the range of temperatures that are being measured. A water thermometer, for example, would not be a suitable instrument for temperatures immediately above that of melting ice, since, as it is gradually heated up the liquid column descends at first, reaches a minimum position, and then rises ; so that there are definite positions on the stem of such a thermometer each of which corresponds to two different tem- peratures. 274 THEORETICAL PHYSICS [Ch. XII § 15-15. Scales of Temperature. It is usual to subject scales of temperature to the condition that the difference in temperature of a mixture of ice and water in equilibrium under the normal pressure, and saturated water vapour under the normal pressure shall be 100.^ These two temperatures have been found to be invariable. This means of course — taking the case of ice and water in equilibrium under normal pressure for instance — that the indication of the ther- mometric device, whether it functions in terms of the volume of a definite mass of liquid, the resistance of a piece of platinum wire or in any other way, is always the same, once thermal equilibrium with the mixture has been established. If some physical quantity x, which may be the volume of a definite quantity of mercury, the pressure of a definite quantity of some gas at constant volume, the electrical resistance of a piece of platinum wire, or any other appropriate quantity, is used for thermometric purposes and if Xq and x^ represent the values corresponding to the temperature of the ice and water under normal pressure (melting ice) and the saturated steam respectively, then x^ — Xq represents a difference of 100°. A difference of 1° is defined by X 1 Xq 100 In the case of the Centigrade scale the temperature of the melting ice is marked 0°, and on this scale the value x would therefore represent the temperature, [Xi Xq) t = (X — Xo) ^ 100 or t = 100 ^__J^ ...... (15-15) X 1 Xq It is important to notice that different physical properties x define different scales of temperature. The readings of a gas thermometer for example do not agree with those of a platinum resistance thermometer. We shall see later that the second Law of Thermodynamics provides us with a means of defining scales of temperature which are independent of the physical property used in the experimental measurement. Meanwhile it may be noted that the product of the pressure and volume of a definite quantity of any gas is very nearly constant if the temperature (as indicated by a mercury thermometer for instance) is kept constant, i.e. the product is independent of the individual values oi p or V (Boyle's law). The product pv is a quantity which ^ This is merely the definition of an arbitrary unit of temperature. § 15-2] THERMODYNAMICS. FIRST LAW 275 increases continuously as the gas is heated and therefore is suitable for defining a scale of temperature, and it has the special merit, that it is the same function of the temperature (whatever arbitrary scale we may have adopted) for all gases, at any rate approximately (law of Charles). This means that if we take fixed quantities of different gases, such that pv has the same value for all of them at 0° C, it will have approximately the same value for all of them at any other temperature (§ 12). We have therefore pv=m (15-16) where t is the temperature on some definite but arbitrary scale, and / is the same function, approximately, for all gases. It is found that all gases approximate more and more closely in their behaviour to the laws of Boyle and Charles as their tem- peratures are raised, provided that the pressure is not unduly raised. We use the term perfect gas, or ideal gas for a hypo- thetical body which obeys these laws exactly and has certain other properties, to be detailed later, which actual gases approach under the conditions just mentioned. These facts suggest the use of a perfect gas to define a scale of temperature. The Centi- grade gas scale would then be expressed by the formula t =. im^l^^^^^^Mh .... (15.17) (P^)i - {l>v)o It is more convenient to define a gas scale by giving equation (15-16) the form pv = ET' (15-171) where t has been replaced by T' and i? is a constant, the value of which is chosen so that {pv), - (pv), = lOOR. The zero of temperature on this scale is called the absolute zero, and the constant B is the gas constant. § 15-2. Equations of State The equation connecting the pressure, volume and temperature of a definite mass of any substance is called its equation of state. The statements (15-16) and (15-171) are appropriate equations of state for an ideal gas. Other equations have been proposed, to which the behaviour of actual gases conforms more closely, for example the equation of van der Waals, (?' + J)(^-^)=-^2^' • • • (^5-2) where a, b and E are constants characteristic of the particular gas. 276 THEORETICAL PHYSICS [Ch. XII § 15-3. Theemodynamic Diageams It is convenient to represent the relation between the pressure and volume of a substance, or between the pressure and tem- perature, or any other pair of variables, graphically. The most important of these diagrams is that representing the relation between pressure and volume. These relations are determined by the equation of state of the substance, and the conditions to which it is subjected. For example if we take hydrogen gas, the equation of state of which is fairly accurately expressed by (15*171), and subject it to the condition of constant tem- perature, the graphical representation of the relation between p and V will be a rectangular hyperbola (see Fig. 15-3). It should be noted that when we speak of the pressure of a sub- stance we mean the pressure measured while it is in equilibrium. This is the sense in which the term pressure is used in the equation of state. It is very important to re- member that when a gas or vapour is expanding rapidly, for example, in a cylinder closed by a piston, the actual pressure exerted on the walls of the cylinder or on the piston will differ from that which would be exerted if the gas or vapour were in equili- brium, e.g. if the piston were not in motion, or if it were moving very slowly. In what follows, the term ' pressure ' will, unless the contrary is expressly stated, always be used to mean the pressure measured under conditions in which the substance is in equilibrium or expanding with extreme slowness. Any process which takes place under con- ditions which differ only slightly (infinitesimally) from those of equilibrium is termed a reversible process. Such a process is in fact reversible in the literal sense of the term. If for example a gas were expanding in the way mentioned above, the process differing only infinitesimally from a succession of states of equi- librium, it is obvious from the equation of state that an in- finitesimal increase of the pressure would cause it to reverse. It is not however the reversibility (in the literal sense of this word) which is the essential feature of reversible processes from the point of view of thermodynamics ; it is the succession of equilibrium states which is the important characteristic of them. The curve representing the relation between the pressure and § 15-3] THERMODYNAMICS. FIRST LAW 277 volume of a substance during a reversible change at constant temperature is called an isothermal. There is another relation between the pressure and volume of a substance with which we are much concerned in thermo- dynamics, namely the relation which subsists between these variables during a reversible change, which is subject to the condition that heat is not allowed to enter or leave the substance. The curve representing such a relation is called an adiabatic and such a change is called an adiabatic change. The term ' adiabatic ' is often employed rather loosely and carelessly to mean any process subject to the condition that heat is prevented from entering or leaving the substance. There are many very different processes which might be termed ' adiabatic ' in this wider sense. For example we might subject a gas to the con- dition that heat is not allowed to enter or leave it and allow it to double its volume in the following different ways : (a) by expanding into a previously exhausted space, (6) by expand- ing reversibly. In the former process, experiment shows that its temperature is only very , slightly altered, in the latter the gas is very appreciably cooled. In this treatise the term ' adia- batic ' will be used, unless the contrary is clearly indicated, for T a process subject to the two Fig. 15-31 conditions, (i ) not ransf er of heat, (ii) reversibility. The latter condition means that the process takes place in such a way that the substance remains practically in a state of equilibrium. There are other ways of representing the states of a sub- stance graphically. We may, for instance, represent the relation between pressure and temperature under the condition of con- stant volume. Such curves are called isochores. Or we may represent the relation between volume and temperature under the condition of constant pressure and we have the curves known as isopiestics. A very important example of a pressure-tempera- ture diagram is that representing the equilibrium between different phases of a substance, i.e. between its solid, liquid and vapour states, or between the phases of a system with more than one constituent, e.g. water and common salt. The phases in this case would include ice, water vapour, the solution of the salt in water, and so on. The equilibrium between the different phases of water is illustrated in Fig. 15-31. 278 THEORETICAL PHYSICS [Ch. XII When the substance is in a state represented by any point on the line (OA), the liquid and its vapour are in equilibrium, i.e. neither evaporation nor condensation goes on. For such states both phases may exist simultaneously. If however the pressure, at some given temperature, is raised above the value corresponding to a point on (OA), the equilibrium state will be one in which only the liquid phase can exist ; if the pressure is less than the value corresponding to a point on (OA), then only the vapour phase will be possible. Similar remarks apply to the curves (OB) and (00). The point, 0, represents a pressure and temperature at which all three phases can co-exist. § 15-4. Work Done During Reversible Expansion Let us imagine the substance to be contained in a cylinder (Fig. 15-4) closed by a piston. The pressure, ^3, is, by definition, the force per unit area ; so that if A represents the area of the piston, pA will be the force exerted on it during a reversible change. During any very small expansion the pressure and therefore the force, pA, exerted on the piston will be sensibly constant, and the work Fig. 15-4 done will be equal to pAs, if s re- presents the distance the piston travels. The product. As, is the corresponding increase in volume, so that during a small reversible expansion (§ 12) dW=pdv (15-4) where dW is the work done by the substance, and dv is the corresponding small increase in volume. We see, therefore, that the work done during a reversible expansion from an initial volume i;i to a final volume V2 is expressed by the formula W {j^dv .... (15-401) ^1 This work is obviously represented on the pv diagram by the area enclosed between the perpendiculars erected at Vi and V2. During an isothermal expansion for instance it is represented by the shaded area in Fig. 15-3. In the special case of the isothermal expansion of a gas, we , . ^T' have, smce p = , V w = Rr[- J V § 15-4] THERMODYNAMICS. FIRST LAW 279 or W = ET' log ^ (15-41) or, since in this case PlVl =P2V2, W Rriog^ . . . (15-411) The formulae are, of course, only approximately true for actual gases. If we deal with a gram-molecule of a gas and use absolute units, e.g. if we measure pressures in dynes per square centi- metre and volumes in cubic centimetres, the constant R has the same value, nearly, for all gases, namely R = 8-315 X 107 ergs per °C., so that the work of expansion in such a case is given by W = 8-315 X 10^^' W !^^ If we use the practical unit of work, the joule, we have obviously to give R the value 8-315 joules per degree.^ Finally we may sometimes find it convenient to express the work in terms of the equivalent number of gram calories, in which case R will be approximately 1-98 calories per degree. When a substance is made to pass reversibly through a suc- cession of states represented by a Fig. 1541 closed curve on the pv diagram, it follows from (15-401) that the net amount of work done by the substance against the external pressure, or done on it by the external pressure, according as the closed curve is described in a clockwise or counter clockwise sense, is equal to the area within the closed curve. Suppose the substance to start from the condition represented by the point A (Fig. 15-41) and to travel along the path ACB to B. The work done by it is repre- sented by the area bounded by ACB and by the perpendiculars AM and BN. If it is now caused to pass along the curve BDA to its original state A, the work done on it will be represented by the area bounded by the curve ADB and the perpendiculars AM and BN. Therefore the excess of work done by the substance over that done on it is represented by the area of the loop. 19 * Since the joule is equal to 10' ergs. 280 THEORETICAL PHYSICS [Ch. XII § 15-5. Heat The meanings of the terms temperature and scale of tempera- ture have already been explained, and we have now to distinguish between the notion of temperature and that of heat, or quantity of heat. If a piece of some metal, initially at 100° C, be dropped into a cavity in a block of ice at 0° C, thermal equilibrium will be established when the metal has cooled down to 0° C, and a definite quantity of the ice will be melted during the process. We may define heat by using the amount of ice melted to measure the quantity of heat lost by the metal. Such a calorimeter, consisting of a block of ice with a cavity in it, covered by an ice lid to prevent heat from the room melting ice within the cavity, was used by Joseph Black (1728-1799) for measuring quantities of heat, and was one of the earliest, if not the earliest, forms of calorimeter. The unit of heat, called the calorie, may be defined as the quantity of heat necessary to raise a gram of water 1° C. in temperature. The calorie so defined is not a unique quantity, since experiment shows that the quantity of heat necessary to raise a gram of water from 0° C. to 1° C, for example, is not quite the same as that needed to raise it, say, from 20° C. to 21° C. The term ' calorie ' is used for any of a number of units of heat, most of them differing very little from one another. The 15° calorie is the quantity of heat needed to raise a gram of water from 14 J° C. to 15j° C. ; the mean calorie raises 0-01 gram of water from 0° C. to 100° C. ; the zero calorie raises a gram of water from 0° C. to 1° C. and so on. All these units differ only slightly from one another. § 15-6. FiBST Law of Thermodynamics It has already been pointed out that Carnot himself arrived at the great generaHzation known as the Principle of Conserva- tion of Energy. The following passage was found after his death, in 1832, among his unpublished manuscripts : La chaleur n'est autre chose que la puissance motrice [ou plutot que le mouve- ment] qui a change de forme. [C'est un mouvement dans les particules du corps.] Partout oil il y a destruction de puissance motrice, il y a, en meme temps, production de chaleur en quantite precisement proportionelle a la quantite de puissance motrice detruite. Eeciproquement, ou il y a destruction de la chaleur, il y a production de puissance motrice. Ou peut done poser en these generate que la puissance motrice est en quantite invariable dans la nature, qu'elle n'est jamais^ a proprement parler, ni produite, ni detruite. §15-6] THERMODYNAMICS. FIRST LAW 281 This is a clear statement of the energy principle and Carnot's puissance motrice is simply what we now-a-days call energy. It is true that, since the advent of the theory of relativity, we have come to regard energy as something having a more ' sub- stantial' character than the mere capacity for doing work, or puissance motrice ; but we are not at present concerned with this. The general adoption of the principle of energy came about in consequence of the experimental work of J. P. Joule, a Man- chester brewer,^ who carried out a series of classical experiments between 1840 and 1850. He determined, in various ways, the amount of work which must be done to generate a unit of heat and his results differ only slightly from the best modern measure- ments, which yield the mean result that one 15° calorie is equiv- alent to 4-188 X 10'^ ergs. The work of Joule received im- portant confirmation a little later by G. A. Hirn, an engineer of Colmar in Alsace, who, among other researches of interest and importance, carried out experiments on a steam engine of a converse type to those of Joule. That is to say he measured the heat used up to do work and his results showed that the mechanical equivalent is just the same as when work is done to generate heat. The principle of conservation of energy viewed from the stand- point of Joule or Hirn, is the deliverance of an extensive series of careful experiments. It is therefore a physical law which (like that of Boyle for example) might conceivably, when the accuracy of temperature measuring devices is sufficiently im- proved, turn out to be an approximation only. The experiments can scarcely assure us of its exact validity. Nevertheless we have gradually, and perhaps uncritically, developed a belief in its perfect exactitude. Indeed if future experiments should reveal that in certain circumstances more heat is generated, for example, than the work done would require, we should hardly doubt the principle of conservation, but rather infer from such experiences a previously unsuspected source of energy. If dQ represent a small quantity of heat communicated to a system and dW the excess of the work done by the system ^ The untenability of the old caloric theory was demonstrated before the close of the eighteenth century by Count Rumford's famous experi- ments on the boring of cannon at Munich, and by Sir Humphry Davy's experiments in which heat was generated by friction between blocks of ice. The former indeed furnished a rough estimate of the mechanical equivalent of heat. RuMFOBD : * An Enquiry concerning the source of the heat which is excited by friction.' Trans. Roy. Soc, Jan. 25th, 1798. Davy : Collected works. 282 THEORETICAL PHYSICS [Ch. XII over that done on it, then we have for the gain in energy of the system dU=:dQ-dW . . . . (15-6) The letter U represents what is called the internal or intrinsic energy of the system. We are concerned, for the present, with systems, the equations of state of which are relations between pressure, volume and temperature ; that is to say with systems the state of which is fixed by the values of any two of these variables ; so that the internal energy of such systems is a function of the pressure and volume, or of the temperature and volume or of the pressure and temperature. § 15-7. Internal Energy of a Gas Experiments carried out by Gay-Lussac as long ago as 1807 indicated that the internal energy of a gas is determined solely by its temperature. Very similar experiments were carried out by Joule independently and much later. He allowed air, con- tained in a copper vessel under a considerable pressure, to ex- pand into a similar, previously exhausted vessel. The vessels were immersed in water, and Joule found no appreciable change in the temperature of the latter on stirring it after the expansion ; though he observed very marked temperature changes when the vessels were immersed in water in separate containers, the water surrounding the vessel out of which the air was expanding being cooled, and that surrounding the other vessel being heated. It is easy to see that the interpretation of these experiments is that given above. For no heat is communicated to or abstracted from the air during the experiments and no external work is done. Therefore by (15'6), the change in the internal energy is zero ; and since the temperature of the air as a whole is not affected we see that the internal energy is the same for different volumes at the same temperature. A more sensitive method of investigating the dependence of the internal energy of a gas on its volume was suggested by Lord Kelvin, and carried out by him in collaboration with Joule. The results and the theory of their experiments will be dealt with in some detail later ; it will suffice to state here that the internal energy of an actual gas does vary slightly with its volume. § 15-8. Specific Heat If, when a small quantity of heat dQ is communicated to a gram of a substance, there is a rise in temperature dt, we define dt § 15-9] THERMODYNAMICS. FIRST LAW 283 to be the specific heat of the substance. It is clear that this ratio will depend on the conditions under which the heat is com- municated, since we can alter the temperature of the substance quite appreciably without communicating or withdrawing heat at all ; but merely by compressing it, or allowing it to expand. We are chiefly concerned with the specific heat measured under the conditions of constant pressure (and reversible expansion) or of constant volume. If we use the gas scale of temperature, the specific heats of a gas are approximately constants. The specific heat of a gas at constant volume, for example, is nearly independent of the temperature and volume of the gas. This is sometimes called the law of Clausius. § 15-9. The Perfect Gas Actual gases, we have seen, conform approximately to three laws, namely : i. The law of Boyle, ii. The law of Joule, which may be expressed in the form (a=» "'•" iii. The law of Clausius. We shall use the term perfect gas or ideal gas for a hypo- thetical gas which obeys these laws exactly. We shall now apply the first law to a perfect gas. For a reversible process equation (15*6) becomes dQ=dU-i-pdv . . . . (15-91) since the work done, dW, is now expressed by pdv. It must be remembered that in equations (15*6) and (15*91) the heat supplied, the internal energy and the work done are all expressed in terms of the same unit-— which may for example be the erg. We shall often have occasion to make use of the formula dz=^dx+^dy (15-92) dx dy where 2 is a function of the independent variables x and y, and where the round 3's are used to indicate partial differentiation. dz In obtaining the coefficient ^ for example, the other independent dx variable, y, is kept constant during the differentiation. Since 284 THEORETICAL PHYSICS [Ch. XII the internal energy, U, of a system, is a function of the tempera- ture and volume we have, by ( 15*92) where the suffixes are used to indicate the variable which is kept constant during the differentiation. Equation (15'91) now becomes This formula is quite general. It applies to a reversible expansion of any substance. Applied to a perfect gas it takes the special form dQ = ^^AT' +pdv . . . (15-941) in consequence of the law of Joule (15*9). If we are dealing with a gram of the gas, (15*941) obviously becomes dQ = c^dT' -h pdv . . . (15*942) and if the heat dQ is communicated under the condition of con- stant pressure, pclv = RdT and therefore dQ^ = c^dT^' + RdT'^ 4TJ const, vresswe or \ c^ =c, -{- R (15*95) Expressed in words, this formula states that the excess of the specific heat of a gas at constant pressure over that at constant volume is equal to the gas constant for a gram of the gas. If the specific heats are expressed in calories per gram per degree the formula becomes c =%+j . . . . (15-951) where J is the number of ergs equivalent to one calorie, i.e. the mechanical equivalent of heat. This formula in fact furnishes us with a means of determining J. If we take one gram of air (which approximates very closely to a perfect gas) we have approximately E = 29 X 10^ ergs per degree, Cj, = -239 cal. per gram per degree, and c^ = -169 „ „ §15-9] THERMODYNAMICS. FIRST LAW 285 from which we get, by substituting in ( 15*9 51), J = 4-14 X 10^ ergs per cal. This is the method of determining J which was employed by Mayer in 1842, and still earlier by Carnot. Equation (1 5*942), which governs any reversible change in a perfect gas, will, when applied to an adiabatic change, take the form = c^dT + pdv or, since we have under all circumstances, RT' V = . V = CAT + RT'-.. V If we divide both sides of this equation by c^T' and make use of equation ( 15*95), we get, where y is employed for the ratio, c^/c^, of the specific heats at constant pressure and constant volume. When we integrate this equation we obtain O=log^ + (y-l)log^, where Tq' and Vq represent the initial temperature and volume and T' and v the final temperature and volume. This result may obviously be written in the form, log T' -\-iy- 1) log V = log To' + (r - 1) log v„ or in the equivalent forms log T' + (y - I) log V = constant . . . (15-96) ^V-i = constant . . (15-961) pvy = constant . . . (15-97) TV~^ = constant . . . (15-98) the two latter equations being obtained by eliminating T' and V respectively in (15-961) by the substitutions T' = pv/R and V = RT/p. The constant y, as defined above, is the ratio Cj,/c^,. Reference to equations (10-11) and (10-12) shows that it is also equal to the ratio, e^/e^,, of the adiabatic elasticity of the gas to its iso- thermal elasticity. This equality is the basis of the method of Clement and Desormes for determining the ratio of the specific hearts of a gas and of the method of obtaining it from the measured 286 THEORETICAL PHYSICS [Ch. XII velocity of sound in the gas. By (10*21) the velocity of sound waves in a gas is since n is zero ; and the compressions and rarefactions in sound waves of audible frequency in gases are practically adiabatic, so that h = yp. Hence u or u ^ VyBT\ . § 16. Heat Supplied to a Gas During Reversible Expansion We have seen that when we subject a gas to the condition of constant temperature, the relation between its pressure and volume is expressed by pv — constant. If it is subjected to adiabatic conditions, the relation is expressed by pvy = constant. More generally any condition to which the behaviour of the gas is subjected will make its pressure some function of its volume, i> =/(*') (16) We can deduce an expression for the heat supplied to the gas during a reversible expansion under the condition expressed by (16). From the equation of state of the gas we have ^^, ^ pdv + vdp ^ R and when we substitute this expression for dT' in equation (15-942) we get, ,^ pdv + vdp . J dQ = c,^ — -^ + pdv, V y or dQ = ' dp + — - — pdv. y - 1 y - 1 We now eliminate dp from this last equation by means of (16). We have dp = -±i-idv, dv or dp =f(v),dv, § 16] THERMODYNAMICS. FIRST LAW 287 and therefore dQ = &dv + -T-pdv. y — I y — r In the special case where /> f(v) = -, or pv' = c, c and s being constants, we have f(v) =IL^=-?l and therefore dQ='^^-^pdv (16-01) y —I or (Heat supplied) = ^^ X (work done) . . . (16*02) 7 — 1 When the expansion is isothermal, s = l, and we see that the heat supplied is equal to the work done, as indeed is otherwise evident from the fact that during an isothermal expansion the internal energy of a gas does not alter. If on the other hand we put s =y we have a further verification of our formula, since it correctly states that in this case the heat supplied is zero. BIBLIOGRAPHY RuMFORD : An inquiry concerning the source of the heat which is excited by friction. Trans. Roy. Soc. 1798. H. V. Helmholtz : Ueber die Erhaltung der Kraft. (Berlin, 1847.) J. P. Joule : On the mechanical equivalent of heat. Scientific Papers, Vol. I. Joule draws attention on p. 299 to Rumford's estimate of the mechanical equivalent. J. R. Mayer : Die Mechanik der Warme. (Stuttgart, 1867.) E. Mach : Principien der Warmelehre. (Leipzig, 1900.) M. Planck : Thermodynamik. (Leipzig.) Das Prinzip der Erhaltung der Energie. (Leipzig.) The last named work of Planck contains a very full history of the development of the energy principle and numerous references. CHAPTER XIII SECOND LAW OF THERMODYNAMICS § 16-1. The Perpetuum Mobile of the Second Kind IN the treatise referred to in § 15, Carnot makes the state- ment : ' La production de la puissance motrice est done due, dans les machines a vapeur, non a une consommation reelle du calorique, mais a son transport d'un corps chaud a un corps froid, . . .' The words in italics constitute the earliest expression of the second law of thermodynamics. The rest of the statement is founded on the erroneous principle of the conservation of heat or caloric, which found acceptance in Carnot's time, and we are not concerned with it. All heat engines, as Carnot noticed, in doing work, not only abstract heat from a source of heat ; but give up a portion of it to a region (condenser or surrounding atmosphere) where the temperature is lower than that of the source of heat. In practice it is found to be im- possible to consume heat from a source in doing work, without giving up some of it to a condenser, or something, at a lower temperature. It is true that a limited amount of work can be done simply at the expense of heat taken from a source without giving heat to any other body, as for example during the expansion of a gas. But an expansion cannot be extended indefinitely, and actual engines are machines which necessarily work in a cyclic fashion, and during some part of the cycle heat is always rejected. Were it not for this sort of limitation of the converti- bility of heat into work, the practicability of propelling ships at the expense of the heat in the surrounding sea might be contemplated. Following Planck, we shall provisionally regard the second law as equivalent to the statement : It is impossible to construct an engine which i. repeats periodically a cycle of operations, ii. raises a weight, iii. takes heat from a source of heat and does nothing else. The kind of machine which this axiom declares to be an im- possibility is called by Ostwald a perpetuum mobile of the 288 § 16-2] SECOND LAW OF THERMODYNAMICS 289 second kind to distinguish it from another type of impossible machine, namely one which simply does work gratis, or without the consumption of energy at all, and which may be called a perpetuum mobile of the first kind. § 16-2. Carnot's Cycle In order to make use of this axiom, we shall study an ideal type of heat engine first described in Carnot's treatise. It consists of a cylinder. A, (Fig. 16-2) and a piston, B, both made of material which is thermaUy perfectly insulating. The base, C, of the cylinder, is made of conducting material. Further, the piston can slide in the cylinder without any frictional resist- ance whatever. It is connected with ideal frictionless machinery, so as to enable it to raise a weight. The source of heat, X, / / B [4ZZ22ZZ / / V////////A Z Fig. 16-2 at the temperature ^2 (expressed in terms of some arbitrary scale) is supposed to be a perfectly conducting block of material, with a practically infinite heat capacity. There is a similar block of material, Y, at a lower temperature, ^1, which we shall call the refrigerator. A block of thermaUy perfectly insulating material, Z, can be used at certain stages in the periodic work- ing of the engine to cover the lower end of the cylinder. iVo assumjptions are made concerning the nature of the working sub- stance, except that it must be capable of exerting a pressure on the piston. It may be a gas, a mixture of water and its vapour, or anything else which might be used to operate an actual engine. Let us suppose the engine to begin work with its working substance in the state represented by the point 1 on the indicator diagram (Fig. 16-21). The base of the cylinder is covered by the source of heat, X, (Fig. 16-2), and the load is so adjusted that the upthrust on the piston exceeds by an infinitesimal amount 290 THEORETICAL PHYSICS [Ch. XIII the force necessary to balance the downward thrust due to the load. Under these circumstances the substance expands iso- thermally at the temperature t^. After a suitable expansion, corresponding to the point 2 on the diagram, the source is re- moved and the cylinder covered by the slab Z. The working substance now expands adiabatically, its temperature being steadily reduced till it reaches the state 3 on the diagram corre- sponding to the temperature ^i. The slab Z is now removed and the block Y brought into contact with the base of the cylinder. An infinitesimal readjustment of the load is now made, so that the piston descends with extreme slowness. The working substance is now compressed reversibly and isothermally. This is allowed to continue tiU it reaches the state 4, and then the block Y is replaced by Z and the compression is continued adiabatically till the substance reaches its original state. We may define the effici- ency of an engine as the work done during a cycle divided by the corresponding quantity of ^ heat taken from the source. O In the case of the reversible Fig. 16-21 engine just described, the work done during a cycle is equal (§ 15-4) to the area, W, of the closed curve (1, 2, 3, 4) on the indicator diagram (Fig. 16-21). We have therefore W Efficiency =-^ (16-2) where Q^ is the heat supplied by the source at the temperature t^. Since the working substance returns to its original state at the end of the cycle, the first law (15*6) requires that Q,-Q, = W {16-201) and hence Efficiency = ^'-^' .... (16-21) where Qi is the heat rejected to the refrigerator at the tem- perature ti. § 16-3. Cabnot's Peinciple We shall now prove that the axiom of § 16-1 leads to the con- sequence that all reversible engines working between the same temperatures, ^a and ti, have the same efficiency ; or, in other words, that the efficiency of a reversible engine depends on the §16-3] SECOND LAW OF THERMODYNAMICS 291 temperatures of the source and the refrigerator and on nothing else. Let us suppose that, of two reversible engines A and B, working between the temperatures t^ and ti, A has the greater efficiency and let us provisionally suppose further that both engines take the same quantity of heat O2 from the same source during a cycle, and that they use the same refrigerator. We have then TT > 7r » by liypothesis, and therefore W^>W^, ..... (16-3) where TF^ and W^ represent the work done during a cycle by the engines A and B respectively. It foUows from (16*201) that A rejects to the refrigerator a smaller quantity of heat during a cycle than does B. Let us now imagine the two engines to be coupled together by ideal machinery (i.e. frictionless machinery), so that A drives B backwards and makes it exactly reverse its normal operations in such a way that the two engines complete their cycles in equal times. This is possible because of the reversible character of B, and because of the inequality {16-3). The circumstance that during certain stages of this compound cycle, work is actually being done on the engine A, or indeed on both engines at the same time, need cause us no difficulty. We have only to think of the ideal machinery as suitably con- trolled by a fly-wheel with an enormous moment of inertia. It is clear that the ' source ' at the temperature t^ will now change in a way which is exactly periodic, the period being equal to that of either engine (say t) ; since during such a period A removes Q2 units of heat from it, while B restores the same amount to it. The ' refrigerator ' on the other hand has more heat abstracted from it by B during the period, t, than is restored to it by A. Of the work, W^, done by A, the portion, W^, is used in driving B backwards, and the balance, Wj^ — W^, may be applied to raise a weight. The combination of A and B and the ' source ' at the temperature ^2 constitutes an engine which i. repeats periodically a cycle of operations, ii. raises a weight, iii. takes heat from a source of heat (in this case from what, in the normal working of A and B, has been caUed the ' refrigerator ' ) and does nothing else. This is in conflict with the axiom of § 16-1, and therefore the hypothesis that the engine A has a greater efficiency than B is an untenable one. They must have the same efficiency. 292 THEORETICAL PHYSICS [Ch. XIH We have restricted ourselves to the case of engines taking the same quantity Q2 from the source during a cycle. We can however easily prove that the efficiency of a reversible engine is independent of the quantity of heat taken from the source during a cycle. Suppose we have a reversible engine working round the cycle abed (Fig. 16-3) between the isothermals t^ and ti. Let q be the quantity of heat taken from the source at ti, and w the work done during a cycle. Its efficiency is therefore w/q. If the engine be adjusted so as to work round the cycle hefc between the same isothermals t^ and t^, as before, its efficiency will not be altered provided it still takes the same quantity of heat q from the source. It follows that the work done during a cycle is also the same as before, i.e. the two areas abed and befc are each equal to w. Now let the engine be adjusted to work round the cycle aefd. Its effici- ency is equal to the area of the closed loop aefd divided by the heat it abstracts from the source. That is to say, it is equal to 2w/2q = w/q. So that doubling the quan- tity of heat it takes from Fi^- 16-3 the source does not affect its efficiency. A very obvi- ous extension of this proof leads to the conclusion that if the engine is adjusted so as to modify in any way whatever the quantity of heat it removes from the source during a cycle of operations its efficiency will not be affected and Garnot's prin- ciple is established. § 16-4. Kelvin's Work Scale of Temperature Carnot's principle enables us to define a scale of temperature which is quite independent of the nature of any of the physical quantities, or of the apparatus used in measuring temperatures. If we consider a number of reversible engines, all of which work between the same temperatures ^2 Q^nd ti, which we may suppose, for the present, to be measured in terms of some arbitrary scale, we have or ^2 = ^ = ^' = etc (16-4) ' "2 --2 .... (16-42) § 16-4] SECOND LAW OF THERMODYNAMICS 293 This means that if a substance in expanding isothermally, at the temperature t^, absorbs a quantity of heat, Q2 ; and in expanding isothermally at another temperature, t^, between the same two adiabatics, absorbs the quantity of heat Qi, the ratio ^ is independent of the nature of the substance, and also of the pair of adiabatics chosen, and depends solely on the tem- peratures ^2 and ti. In what foUows we shall usually employ this result as an axiom, in place of the axiom (16*1). It may be regarded as equivalent to the second law of thermodynamics. We shall now define a scale of temperature by the equation wrk ^''-''^ We can show that the scale so defined is independent of the particular substance which may absorb the quantities of heat Q2 and Qi, when expanding isothermally between the same pair of adiabatics. The ratio of the same two temperatures on the scales defined by different substances, using (16*41), is the same for all substances, i.e. because of (16'4). Now we have agreed that the temperature difference between saturated steam at normal pressure and melt- ing ice at the same pressure shall be numerically 100, therefore we get, when we apply (16*42) to these two temperatures, To + 100 _ T,' + 100 _ To" + 100 _ T, To' To'' ^''" where To, To, Tq", etc., represent the temperature of the melting ice on the scales defined by different substances. We see that To == To' = To" = etc. . . . It is clear, therefore, that the temperature of melting ice, measured on a scale defined in this way, is independent of the properties of the thermometric substance involved. We can now show very simply that this is true of any other temperature, for since T ^r_ ^T^ ^ To To' To" ^^" or, using the result just obtained, T ^T^ ^^ = To~To~ 'To ~ ^ """ therefore T = T' =T" == etc. 294 THEORETICAL PHYSICS [Ch. XIII It is obvious that if we use this scale of temperature, the efficiency of a reversible engine is expressed by where T^ and T^ are the temperatures of source and refrigerator respectively. The scale we have just described, and which we owe to Lord Kelvin, may be described in another way. Let us imagine any pair of adiabatics, abed and efgh (Fig. 16*4) of some substance constructed, and also the isothermals corresponding to the temperatures of steam and melting ice, which we may conveniently number 100° C. and 0° C. Now construct iso- ^ thermals to divide the area bfgc into 100 equal parts, the area of each of which we may call <^. If we num- ber them in order 1°, 2°, 3° . . . 99° C. and continue them below 0° C. and above 100° C. in the same way, that is, so as to have the same area, <^, between con- secutive isothermals and ^ the pair of adiabatics, we shall have the Kelvin scale of temperature, except for the trivial difference that we have numbered the temperatures from that of melting ice as a zero. This is obviously the case since (16'41) gives us Qi = <t>T, (16-43) where ^ is the same constant for the same pair of adiabatics, and therefore or if we apply this to the steam and ice isothermals \ V kV"* ^^ V \ N^^P^/OOT. ■\>^r Fig. 16-4 therefore or Gsteam " ^ice = 100 cj>, I Vsteam Vice </> 100 area bfgc loo (16-431 Equations (16-41) indicate that the zero isothermal on the Kelvin scale is characterized by the property that no heat is § 16-4] SECOND LAW OF THERMODYNAMICS 295 absorbed by the substance in passing from one adiabatic to another at this temperature. A reversible engine working round a cycle bounded by two adiabatics and the isothermals T and zero would consume all the heat absorbed at the temperature T in doing work, since none is rejected to the refrigerator at the temperature zero. Since the first law requires that more work than is equivalent to the heat supplied cannot be done in a Carnot cycle we must conclude that the zero on the Kelvin scale is the lowest of all temperatures. It is called the absolute zero. The Kelvin, or work scale of temperature as it is some- times called, is not the only absolute scale of temperature. There is an infinite number of such scales. We may for example define a scale of temperature by laying down that the efficiency of a Carnot engine, working in a cycle bounded by any two adiabatics, and by a pair of isothermals which are very close together, is proportional to the temperature difference between the isothermals.^ This means, if we use Q to represent tempera- tures on this scale, f = ..., where dQ is the excess of heat absorbed over that rejected, and a is a constant. We have therefore — = add, or T = Ce''\ where C is a constant of integration. We may choose such a value for the constant a as will make the temperature difference between melting ice and steam 100, and for the constant, (7, a value which wiU make one temperature, say that of melting ice, the same on both scales. If we do this, a is given by or and C is fixed by gaTice We see that the temperature corresponding to the Kelvin abso- ^ This scale was in fact proposed by Kelvin before the work scale. 20 rp -*■ steam = glOOa^ a = 100 '^^ -*■ steam , ^ice = CeaJ-ioe c y,ce 296 THEOHETICAL PHYSICS [Ch. XIII lute zero is represented by minus infinity on the new scale. There is a certain appropriateness about this ; since the so-called absolute zero is very difficult to approach, and indeed there is reason to suspect that it is a temperature which is unattainable. § 16-5. The Work Scale and the Gas Scale The real merit of the work scale, and the reason for preferring it to any other of the possible alternatives, lie in the fact that it is identical with the perfect gas scale, and therefore approximates very closely to the temperatures as given by a gas thermometer containing hydrogen or some gas differing little from a perfect gas. The temperatures as given by such a thermometer there- fore require only very small corrections to convert them to the work scale. We can prove this in the following way : li Qz and Qi represent the quantities of heat absorbed by a substance in expanding isothermally and reversibly from one adiabatic to another at the temperatures T^ and Ti respectively, that is say if Qz represents the heat absorbed by a substance expands from the point 1 to the point 2 (Fig. 16-21), and Qi that absorbed during an expansion from the point 4 to the point 3, then, as we have seen, This is true for any substance and therefore true for a perfect gas. In the case of a perfect gas, however, RTz' log ^^ ^^ !!i .... (16-5) ^^ RT^ log -^ by (15*41), since the internal energy does not change. Here Tz and T^' represent on the gas scale the same temperatures as Tz and T^ respectively. If we apply (15'96) to the adiabatic passing through 1 and 4 (Fig. 16-21), we have log T,' + (r - 1) log V, = log T,' + (7 - 1) log V,, and by applying it to the adiabatic through 2 and 3, we have log T^ + (r - 1) log vz = log T,' + (r - 1) log ^3. Subtracting the first of these equations from the second, we get log -' = log -', §16-6] SECOND LAW OP THERMODYNAMICS 297 and therefore equation (16*5) becomes hence which means that the two scales of temperature are identical. § 16-6. Entropy We shall now introduce a quantity to distinguish the adia- batics — the term is used in the restricted sense explained in § 15-3 — on the p, v diagram, just as temperature distinguishes the isothermals. This quantity is called entropy, a term intro- duced by Clausius (see § 13-4), to whom the conception of en- tropy is due. We may assign the value zero to the entropy of an arbitrarily chosen adiabatic, e.g., the adiabatic through the point PqVq, where ^o is the normal pressure, and Vq the volume of the substance at normal pressure and tem- perature ; just as on the Centi- grade scale we assign the value zero to the temperature of the isothermal through the same point. Having adopted an entropy scale, ^, it becomes obvious that the state of a substance (or system) in equilibrium will be determined by the corresponding values of T and <^, since each pair of values T, <j) is associated uniquely with a corresponding point _p, v on the p, V diagram ; and it will be helpful sometimes to employ a, T, (f) diagram instead oi a p, v or other diagram. The most convenient scale for ^ is that already defined by (16*43) or (16*431). If in Fig. 16-4, abed is the adiabatic of zero entropy, the area defined by (1 6*431), with the + or — sign, according as the corresponding Q is positive or negative, will be the entropy of the substance when it is in any of the states represented by points on the adiabatic efgh. Or more generally the difference, <f)2 — cj)i, oi the entropies associated with two adiabatics is equal to the area on the p, v diagram enclosed between the adiabatics and any pair of isothermals, the corresponding temperatures of 4> 1 2 a. /S s r T, <P Fig. 16-6 which, on the work scale, differ by unity. Consequently (16-6) 298 THEORETICAL PHYSICS [Ch. XIII is equal to the area of the closed curve on the indicator diagram (e.g. 1, 2, 3, 4 in Fig. 16-21) of a Carnot cycle, between the temperatures T^ and T^ and the adiabatics 0i and <^2- So that the rectangular area, a^yd, on the T, <f) diagram (Fig. 16-6) is equal to the corresponding area on the p, v, or indicator diagram ; and it follows that the area of any closed curve on the^, v diagram is equal to the area of the corresponding curve on the T, (j> diagram ; since the former can be regarded as built up of infinitesimal elements formed by an infinite number of isothermals and adia- batics, while the latter can be regarded as built up of corre- sponding infinitesimal rectangles. § 16-7. Entropy and the Second Law of Thermodynamics According to the definition of entropy which we have adopted (16-43) ^2-^i=|- ..... (16-7) where ^i and ^2 are the entropies of a substance in two different equilibrium states 1 and 2. Q is the quantity of heat, positive or negative, that must be supplied to the substance in a reversible way along any isothermal whatever from a point on the adiabatic through 1 to the corresponding point on the adiabatic through 2, and T is the temperature of this isothermal on Kelvin's work scale. The possibility of expressing the entropy difference between two adiabatics in this way (16*7) is clearly a consequence of the second law and the adoption of Kelvin's work scale. Conversely we may deduce the second law (as expressed in §16-4) from the statement (16-7). For consider any pair of adiabatics with the entropies ^1 and ^2 (^2!><^i). Then ^.-^. =|^=|-;, by (16-7), where Q2 is the heat communicated to the system during a reversible isothermal change from the adiabatic 1 to the adia- batic 2 at the temperature T^, and Qi has a corresponding mean- ing for such an isothermal change at the temperature T^. Now consider any other pair of adiabatics, of the same or any other system, with entropies (f>i and ^2'- We have If If V2 Vi in which the significance of Q2 and Qi is obvious. It follows that Q2 V2 __ -j_2 § 16-7] SECOND LAW OF THERMODYNAMICS 299 But this is the statement of the second law of thermodynamics as given in § 16-4. Consequently (16*7) is equivalent to the second law. Let A and B be two neighbouring points on the^, v diagram, and let AC and BC be an isothermal through A, and an adiabatic through B respectively ; their point of intersection being C (Fig. 16-7). The net amount of heat communicated to the substance during the reversible cycle ABCA is equal to the area ABC, i.e. Area ABC = dQ^j, -f dQ^^ + dQ^^, or Area ABC = dQ^^ + ^Qga^ since BG is an adiabatic. In the limit when B and G approach TJ- Fig. 16-7 very near to A, the area ABG becomes vanishingly small by comparison with dQ^^ or dQ(j^, since it ultimately diminishes in the same way (AB)^ or (AG)^ ; whereas dQ^^ or dQcj^ diminish as AB or AG. Therefore dQAB + dQcA = 0, or dQ^B = (^Qag^ in the limit. Dividing both sides by the temperature, T, corre- sponding to the isothermal through A, we get dQAB _ dQAG T T ' The right-hand side of this equation represents, according to (16*7), the increase in entropy when the substance changes (reversibly) from the state A to the state B, We may therefore write dQAB d<t>AB = T or, simply #=¥ (16-71 300 THEORETICAL PHYSICS [Ch. XIII Consequently the increase in entropy of a substance, or system, in changing reversibly from a state 1 to another state 2 is expressed by 2 9^.-9^.= 1^. . . . {16-711) 1 and the value of the integral is clearly independent of the path joining the points 1 and 2 on the p, v diagram. An alternative expression for ^2 — ^1 is 02 - 01 = J ^ .... (16-72) 1 which, for the special case of constant volume, reduces to 2 <t>,-h= \^ ... (16-721) 1 We shall adopt (16'71) as a final statement of the second law.^ § 16-75. Entropy of a Gas For the unit mass of a perfect gas we have dQ _ dT . j.dV -^ ~ ^^~T T or d<j> = c^d log T + Rd log V T V and therefore (^ = c, log — + (c^ - cj log — . . (16-75) if we agree that shall be zero when the temperature and volume are Tq and Vq respectively. § 16-8. Properties of the Entropy Function. Thermodynamics and Statistical Mechanics. It is well to bear in mind that the systems with which we are dealing are characterized by an equation of state which expresses a distinctive variable, the temperature, as a function of the pressure and volume, when the system is in equilibrium. There are also systems in which there are other variables ^/i^u 2/2^25 . . . ys^s^ ' ' ' besides (or instead of) p and F. The ^ As we have seen, the dQin (16*71) and (16*71 1) is not any dQ, hut the special increment associated with a reversible process. No such cautionary- remark is necessary about (16*72) or (16*721) because dU + pdV repre- sents just this particular increment dQ that is in question. §16-8] SECOND LAW OF THERMODYNAMICS 301 external work done during a reversible change in such a system is expressed by Lpc^F or S y^dx^. It will be convenient to caU such systems thermodynamic systems, and we shall use the term closed system for one which does not interact in any way with thermodynamic systems outside it. A reversible process in a thermodynamic system is merely a limit that actual pro- cesses may approach — sometimes quite closely — but these latter are essentially irreversible. Mere transfer of heat — ajpart from volume changes — simply increases the internal energy of one part, a, of a system at the expense of that of another, ^ ; the con- sequent (algebraic) increment of entropy being, according to (16-721), dU^ ^ dU^ T. T, or since dU^ = -d\ the increment of entropy is dU^ dU. T. T, This is necessarily a positive quantity since, if dU^ is positive, T^ must be greater than T^ and, if dU^ is negative, T^ must be greater than T^. If an irreversible process in a closed system is associated with a change in volume, the internal energy of the system is bound to be greater when the final volume is reached than it would have been had the change occurred reversibly. If it were an expansion, for instance, the resisting pressure would be less at each stage of the process than would be the case during reversible expansion. Less external work is done therefore in a given irreversible increase in volume than when the same expansion occurs reversibly, with the consequence that in the former case the final value of the internal energy is greater. Similarly during an irreversible diminution in volume the external pressure is greater at each stage than that operating when the same diminution in volume is brought about reversibly, and again the final value of the internal energy is greater in the case of the irreversible process. Let Uq be the final value of the internal energy when the given increase in volume occurs reversibly and U its value when it occurs irreversibly, then u CdU and J -^ 302 THEORETICAL PHYSICS [Ch. XIII is necessarily positive. But this integral, according to (16*721) represents the amount by which the entropy at the end of the irreversible process exceeds that at the end of the reversible process. In the latter process there is no change in entropy, consequently the irreversible process is necessarily accom- panied by an increase in the entropy of the system. This result is quite general. In the words of Clausius : Die Energie der Welt ist constant. Die Entropy der Welt strebt einem Maximum zu. It follows from the foregoing discussion that the necessary and sufficient condition for the equilibrium of a closed thermo- dynamic system is : when some small change, d — for example a small change SV due to a slight readjustment of the external pressure — is made in the state of the system, dcl> = (16-8) where (f> is the total entropy of the system. The condition is necessary because reversible changes, which as we have seen consist of successive equilibrium states, are characterized by (f) = constant, and it is sufficient, because no departure from equilibrium is possible unless We have now brought to light the essential identity of the entropy, <^, of a thermodynamical system and the function represented by ip in Chapter XI ; and a brief comparison of statistical mechanics and thermodynamics will not be out of place here. Thermodynamics rests on two main principles, which we may conveniently call the principles of energy and of entropy. It is characteristic of its methods that no hypotheses concerning the nature of heat or the microscopic or sub-microscopic consti- tution of materials or systems are employed. Thermodynamics therefore enables us to arrive at reliable conclusions — reliable because of the proved reliability of the two main principles — which are quite independent of the (sub-microscopic) constitu- tion of materials and of the nature of the processes occurring in them. Statistical mechanics accomplishes something more than this. It starts out from the hypothesis that the special form of energy called heat is identical with mechanical energy ^ and bases the first law of thermodjoiamics on the mechanical principle of conservation of energy ; while the second law of thermodjoiamics and the entropy function emerge as statistical ^ This does not necessarily mean ' mechanical ' in the restricted Newtonian or Hamiltonian sense. § 16-8] SECOND LAW OF THERMODYNAMICS 303 properties of assemblages of vast numbers of mechanical systems which interact on one another in a random fashion. BIBLIOGRAPHY S. Cabnot : Reflexions sur la Puissance Mo trice du Feu et sur les Machines propres a developper cette Puissance, 1824. R. Clausitjs : The Mechanical Theory of Heat. (English translation by W. R. Browne. Macmillan, 1879.) W. Thomson (Lord Kelvin) : On an absolute thermometric scale founded on Carnot's theory of the motive power of heat, etc. (Phil. Mag., Vol. 33, p. 313, 1848.) Max Planck : Thermodynamik. CHAPTER XIV THE APPLICATION OF THERM ODYNAMICAL PRINCIPLES § 16-9. General Formulae for Homogeneous Systems WHEN a substance has an equation of state which is a relation between T, p and V, we have seen that its entropy, cj), is a quantity which is uniquely deter- mined by any two of these variables, i.e. (f) = function {T, F), or (/> = function {p, V), and it follows, if we write d<f> = AdT + BdV .... (16-9) that «-(lf.X ■ • • • <■'■"»' (see the formula (15*92) ). Now, as we have seen, we may also write dQ, the quantity of heat communicated reversibly to the substance in a similar way : dQ = A'dT + B'dV ; but we may not in this case infer -■ = m. These equations would imply that Q is a function of T and F. We have seen however that this is not the case. In fact, Q, the algebraic sum of the quantities of heat that may have been communicated to a substance, may have any value whatever while the independent variables that determine its state remain unchanged. We have only to recall the fact that, after complet- ing any Carnot cycle, the variables T and F, for example, re- 304 §16-9] THERMODYNAMICAL PRINCIPLES 305 assume their original values, while Q may have increased by a perfectly arbitrary quantity determined by the dimensions of the cycle (§16-2). Such a differential as dcf) is called a perfect, or complete differential. From (16'901) we derive the equation iwl = iSl — <»■"> which will serve us as a useful rule, when we meet with ex- pressions like (16*9), which are complete differentials. Writing (15-94) in the form we have # 4(S)/^ 4{©. +^'^^- Therefore, by (16-91), T^iKdV/T '^^] ~^ T dTdV ^ TKdTjr U-^)^+p}=t(%\ . . . (16-93) TdVdT (dip Substituting this result in (16-92) we have ■'«=(S),"+^(IV>'' or, if we are dealing with the unit mass of the substance/ dQ = c,dT + T(^^dV . . . • (16-94) In this equation it is of course understood that dQ is com- municated reversibly. If we further subject it to some condition, X, which might, for example, be constant volume, or constant pressure, and divide both sides by dT, we get '■-.+^(|.).©.- ■ ■<"••"> where c^, means the specific heat of the substance measured under the condition x. Let us now apply the same method when the independent variables are T and p. We find, since dU and dV are perfect differentials, 306 THEORETICAL PHYSICS [Ch. XIV or Assuming the unit mass of the material, this may obviously be written For d<f) we have On applying the rule of (16*91 ), we easily get and hence, by (16*961), dQ = c,dT - tI^^'J dp . . . (16-971) If again we suppose the reversible communication of heat dQ to be subject to some condition, x, and divide both sides by dT we get -0(i). • ■ <"■'"• When the condition, x, is that of constant pressure, the formula (16*95) leads to ^P\ /S^\ (16-98) and we arrive at precisely the same result from (16*972), when X means constant volume. We can readily verify that this result is in agreement with (15*95), which applies to a perfect gas ; for in this case the equation of state is pV = RT, and consequently ^m.'"-' whence it follows that (16*98) becomes §16-9] THERMODYNAMICAL PRINCIPLES 307 We may express Cp — Cv in terms of such quantities as the coefficient of expansion of the substance and its isothermal elasticity, which are more immediate results of experimental measurement than are the {dp \ (dV\ quantities \q^j^ or \Qm] • By definition the coefficient of expansion, a, IS and therefore aV = (^)^ ...... (16-981) fdv\ We may get rid of ( ^ ) in the formula by the following device : o=(gX.©.(i).- Consequently (|,)^ = _ g)^(?| or (^\ = ae, (16-982) since, by definition, the isothermal elasticity, e^, is Now substituting the expressions (16-981) and (16-982) in (16-98) we get Cp - c* = T6j.a2F (16-983) where V is the volume of a gram of the material. It will be observed that the product on the right (if different from zero) is essentially positive. Hence Cp — c« is always positive or zero. We derive the formulae which express adiabatic relationships by making c^. zero in (16'95), or in (16*972). The condition x is now simply the condition <f) = constant, consequently '■ = - <i).(»). ■ • • '■'•"> These equations reduce to c„ = - rae^d^) . . . (16-992) and c. = TaV\ (S), • • • • ^''-''^^ 308 THEORETICAL PHYSICS [Ch. XIV Dividing the latter by the former we get y or y = ^_i =^A (16-994) c., e a result we have already established for the special case of a perfect gas. § 17. Application to a v. d. Waals Body By differentiating the equation of state, /dV\ B and we obtain i -^ \ — /dV\ ^ li Consequently, on substituting in (16-98), J?2 P 2a( F-6)^ ' 73 T which becomes, if we neglect small quantities of the second order, 1- 2« -K'+m) ■ ■ ^''^ RTV For an adiabatic expansion of a v. d. Waals body we find from (16-99) TR /dV\ KdTJ ' c., = (F-6) or dropping the subscript, ^, = c,dT + RT- ^^ V § 17-1] THERMODYNAMICAL PRINCIPLES 309 Therefore = c^-— + R ^ T V -h' consequently = c^d log T -{- Rd log (V — h), and, if we may take c^ to be a constant we find, on integration, T'^(y — b)^ = constant, R or T(F -6^ = constant . . . (17-01) /dp\ If we divide (16*94) by T, and substitute for v^)^ the expression appropriate for a v. d. Waals body, we find dQ dT „ dV T ^'T ^""7 -b' dT ^ dV Therefore d<f> = Cv-jff + R T ^ ^ V -6' T {V —b) and = c. log y + 2? log _^ . . . (17-02) where Tq and Vq are the temperature and volume at which we have agreed the entropy shall be zero. § 17-1. Thermodynamic Potentials There is a number of functions which are prominent in the application of thermodynamical principles to special problems, and which on account of their properties are caUed thermo- dynamic potentials. Consider, for instance, any reversible process taking place at constant temperature and pressure. We have dQ ==dU -{-pdV, and therefore Td^ = dU + pdV. If now T and p are constant during the process, > d(Tcl>) =dU -\- d{pV), and consequently =d{U -Tcl>+pV}. . . (17-1) In such reversible processes therefore the function U — T(f> -\- pV remains constant. This function, which we shaU represent by the letter /, is commonly called the thermodynamic potential. Its increment df can be written df = (dU - Tdcl> + pdV) - <i>dT + Vdp, or since the terms in brackets are collectively zero, df = - cj^dT + Vdp . . . . (17-11) This is obviously a perfect differential, because the differentials of the terms which make up / are themselves perfect differentials. and 310 THEORETICAL PHYSICS [Ch. XIV It follows therefore that ''-(IX <"■"' It is to this property that the function / owes the name * potential.' A similar function is the free energy of a system, which we shall represent by the letter F. We arrive at it naturally in inquiring about the external work done by a system during a reversible process taking place at constant temperature. Start- ing out from Td<t>=dU +pdV, we have pdV = - dU -\- Td<l>, or, when the temperature is constant, pdV == -d{U -Tcl>} == -dF ..... . (17-13) Or the external work done during such a process is done at the expense of the quantity F=U -Tcf> (17-14) The increment dF of F is dF = (dU - Td<i> + :pdV) - <t>dT - pdV, and as the expression in brackets is zero, dF = - 4>dT - pdV. This is also a perfect differential, and consequently _ ,_/3^ -^=CSl ■ ■ ■ ■ ^''"''^ If we substitute for ^ in (17-14) the equivalent expression in (17-141) we have ^=^+^(SX • • • -^''-''^ This result is known as the Gibbs-Helmholtz formula. If instead of the variables p and F, the equation of state contains other corresponding variables, y and x, (17-15) becomes ^ = ^ + <i). • • • • ^''-'''^ The Gibbs-Helmholtz formula finds its chief applications in cases where y (ot p) is constant at constant temperature, i.e. is independent of a; (or V). dF\ A' ■dF § 17-2] THERMODYNAMICAL PRINCIPLES 311 If we write the equation Td^ =dU + pdV in the form dU = Tdcf> - pdV .... (17-16) we see that ?7 is a function which has similarities with / and F ; and also that T = f ;:r7- I , The increment, dU, of the internal energy of a system is equal to the quantity of heat, dQ, which would have to be communicated to it, at constant volume, to produce the increment dU. This consideration suggests still another function, namely one which has the property that its increment is equal to the quantity of heat supplied (reversibly) to the system under the condition of constant pressure. Now since dQ = dU-\-pdV, this condition leads to dQ = d(U +pV). If therefore we represent the function we are inquiring about by G, we have G = U -{- pV (17-17) It is called enthalpy {ddXno)^ = warmth, heat). We find for dG, dG == {dU - Tdcl> + pdV) + Tdcj> + Vdp, or dG = Tdct> + Vdp (17-171) Consequently T = (—-] , 'dG \dp 'SI ■ ■ ■ <"•■«) § 17-2. Maxwell's Thermodynamic Relations By applying the rule (16'91) to each of the differentials df, dG, dF and dU we obtain the four equations (^)* = wl- '^> (a=-(a.(^) (--) known as Maxwell's thermodynamic relations. They are given in the order in which Maxwell himself gave them.^ 1 J. Clerk Maxwell : Theory of Heat. 21 312 THEORETICAL PHYSICS [Ch. XIV § 17-3. The Experiments of Joule and Kelvin and the Realization of the Work Scale of Temperature In Joule's historic experiments on the expansion of gases into a previously exhausted region (§ 15-7) the temperature changes could not be determined even approximately on account of the relatively enormous heat capacity of the surrounding vessel and medium, as compared with that of the gas itself. The experiments only sufficed to show that such temperature changes were relatively small. Kelvin devised an experimental method which evaded the difficulties of the earlier experiments, and which he, in collaboration with Joule,^ successfully applied to a number of gases. The gas under experiment was forced by pressure through a porous plug of cotton wool which occupied a short length (between 2 and 3 inches) of a long tube. This latter was immersed in water maintained at a constant tem- perature. The part of the tube containing the porous plug was made of box- wood, 1 J inch in internal diameter. The box- wood being a bad conductor, and the temperature gradients small, no appreciable transfer of heat occiirred between the expanding r AJ Fig. 17-3 gas and the surrounding medium. A sensitive thermometer, placed immediately behind the porous plug, gave the temperature of the gas on emerging from the plug. Only the box-wood part of the tube was thermally insulating, so that before expansion the gas had the temperature of the surrounding water. The pressures on both sides of the plug were maintained constant, on one side atmospheric pressure and various pressures up to several atmospheres on the other. We may visualize the porous plug as a diaphragm, A (Fig. 17-3), with a minute aperture in it. The gas expands through the aperture from a region of constant high pressure, pi, into a region of constant low pressure, ^2- If Fi be the volume of the unit mass of the gas at the pressure, pi, and Fa its volume at the lower pressure, p^, the external work done on the unit mass of the gas wiU be ^iFi, and that done by it, PzY^- Consequently the net amount of work done by the unit mass of the gas in expanding wiU be ViVi -i?iFi, or Ai>F. If therefore the gas does not deviate appreciably * Joule : Scientific Papers , Vol. II, p. 217. § 17-3] THERMODYNAMICAL PRINCIPLES 313 from Boyle's law over the range of pressures to which it is subjected in the experiments, the net external work done by it (or on it) is zero ; in which case the experiment is essentially Joule's original experiment in a slightly different form. In general however there will be small, but appreciable, deviations from Boyle's law, so that the temperature change accompanying the expansion of the gas in passing through the porous plug will be partly due to this. As there is no appreciable transfer of heat between the gas and the surrounding medium we have A{U+pV)=0 or A^ = (17-3) The corresponding condition in the original Joule experiment, assuming no transfer of heat, is AU = (17-31) The successive steps in the application of thermodynamical principles to the two experiments are given in parallel columns below. Most of them are fairly obvious and are therefore given without detailed explanation : ^{(W^^ 0, Joule-Kelvin. TL^ + V/^p = m AT + (f) \dp/T + VAP = 0, .dp/T Ap = 0, Ai> = 0, Applying Maxwell's relation (a) (17-2) tC-I) KdTj, ,_ AT ^~ AP' c,AT+\V If Ap = 0. c,S + \V t(E\ dT/, 0. (17-32: TA4>- Joule. AU==0, -pl^V = 0. dcf> '■©.A" +(a-^ -PAV = 0, - \P AF = 0, AF = 0, Applying Maxwell's relation [y] (17-2) dp AT-\p If CvV ■Tm\ AT AF = 0. = 0. (17-321 314 THEORETICAL PHYSICS [Ch. XIV It should be observed that, in these two formulae, the tem- perature is expressed in terms of the work scale ; while in the experiments themselves the temperatures or temperature differ- ences were determined by mercury in glass thermometers. Imagine the temperatures, in the Joule-Kelvin experiments, to be expressed in terms of the constant pressure gas scale of the gas under experiment. This is practicable, since it is only neces- sary to compare the thermometer actually used with the gas thermometer. Let T' be the temperature on the constant pres- sure gas scale as defined by pV = BT', p being constant, and B chosen as explained in § 15'1. We have consequently The specific heat, c^, and the ratio, i, when expressed in terms of the scale T', may be represented by c'^ and i' respectively. It is clear that because specific heat is a quantity with the temperature as denominator, whereas it constitutes the numerator in the quantity I, and in the product of the two the peculiarities of the scale of temperature actually used cancel out. We may now re- write (17'32), and mutatis mutandis (17'321), in the manner shown below, and obtain results which enable us to use the observations in the Joule-Kelvin and the Joule experiment to correct the readings of the constant pressure and constant volume gas thermometer (containing the gas experimented on) respectively, so as to get temperatures on the work scale. or c'r+ Joule-Kelvin. 1 KdT'J^dT Applying (17-33) c' r + \v -T- y dT' T dT dT T dT T'\l-{ c\r 0, (17-34) Joule. =•■'■- {.-(|r,).fh». Applying the formula anal- ogous to (17-33) \ p J V*' '-'■«■-•■/ §17-3] THERMODYNAMICAL PRINCIPLES 315 Taking the left-hand formula (17*34) and integrating over the range of temperature from that of melting ice to that of saturated water vapour at normal pressure, we have To'+lOO To To' where Tq and Tq' are the temperatures of melting ice on Kelvin's scale and on the constant pressure gas scale respectively. The integral on the right is made up of observable quantities only, and can be evaluated. Calling it Tq, we have T. = ^^ ..... (17.35) Similarly for any other temperature, T\ on the constant pressure gas scale and the corresponding temperature, T, on Kelvin's work scale, we have , T / dT' log ?r = 1/ _ / ai To J^a.jn^^^j* To' Calling the integral on the right t, we have lOOe^ , . . . . (17-36) This formula will also serve for the constant volume ther- mometer if for T and Tq we substitute the values of the corre- sponding integrals obtained from (17*341). Such an application would, however, have no practical value if we had to rely on estimates of rj' derived from experiments of the original Joule type. The theory of the Joule-Kelvin experiment (and of the Joule experiment) applies not only to gases, but to any sort of fluid up to equations (17*32) and (17*321). Joule and Kelvin found a cooling effect (i positive) for aU the gases they experimented on, except hydrogen, for which they observed a very small rise in temperature (| negative). They found the change in tem- perature to be proportional to the drop in pressure, pi — Pz, and inversely proportional to the square of the absolute tem- perature. The cooling effect is of course the basis of the methods of liquefying air which are most extensively used at the present time. Gases like hydrogen and helium, which in the ordinary way exhibit a heating effect, are found at sufficiently low tem- peratures to be cooled. There is therefore a temperature of 316 THEORETICAL PHYSICS [Ch. XIV inversion at which | changes sign, i.e. becomes zero. From equation (17'32) we learn that, when 1 = 0, 1=©. ■ ■ • • • "'-' If we represent the equation of state of the gas by f(T,V,p)=0 (17-38) differentiate it with respect to T, keeping p constant, and then /dV\ V equate the expression thus found for (^^j? to — , ( 17*37), we get an equation ^(T, V,p)=0 .... (17-39) connecting T, V and p, which is true for all states of the gas for which i = 0. We may eliminate one of the variables, p for example, from (17*39), by using the equation of state (17*38), and we thus obtain an equation which gives us the temperature of inversion in terms of the volume. Its graphical representation on Si TV diagram is called the curve of inversion. For a V. d. Waals body /dV\ R \dT/p ~ ( a\ 2a, ^^ , ' and therefore by (17*37) we have F R [p + tO ~~Y^^ "^^ where Tt is the temperature of inversion. On eliminating p we easily find The following null method of realizing the Kelvin work scale is of interest, though it may not be of practical importance. Imagine we have found empirically the equation of state of a gas, f{T'Vp) =0 (17-392) where T' is the temperature in terms of the constant pressure scale of the gas in question, and likewise the equation of the curve of inversion, g{T'V) =0 (17-393) Differentiating (17-392) with respect to T\ keeping p constant, we obtain an equation dT' Multiplying both sides of this by -y^ we find for states of the gas repre- sented by points on the curve of inversion, by (17-37), V dT' §17-5] THERMODYNAMICAL PRINCIPLES 317 We can eliminate p and F, by means of (17'392) and (17*393) and thus obtain ~=UT')dT' (17-394) which, since the function /g is known, enables us to find T in terms of T\ § 17-4. Heterogeneous Systems We now turn to systems in which two or more states of aggregation, or phases, are in equilibrium with one another. The simplest example is that of a liquid in equilibrium with its vapour. For the range of temperatures below the critical temperature of the substance there exists for each temperature a definite pressure, usually called the saturation pressure of the vapour, but more appropriately called the equilibrium pressure, under which the liquid and its vapour are in equi- librium. This is represented by the horizontal lines such as BF in Fig. 12-51. According to the theory of v. d. Waals the isothermals have the shape illustrated by ACDEG, assuming the whole of the substance to be in one state of aggregation at any given pressure or volume. This is supported by the fact that the portions BC and EF can be experimentally realized. The question arises : What is the situation of the horizontal line BF relatively to the curved line ACDEG ? During the reversible passage of the substance from the state A to the state G, there is a definite increase in its entropy, determined, as we have seen, solely by the positions of the points A and G on the diagram. If therefore the passage occurs isothermally the quantity of heat communicated to the substance wiU be just the same for either of the alternative paths ABFG or ACEG. On the other hand the increase in the internal energy is also the same for both paths, since this too is determined solely by the positions of the points A and G on the diagram. Thus it follows, by the first law of thermodynamics, that the work G done, pdv, is the same in both cases, and this means that the A area BCD is equal to the area DEF. § 17-5. The Triple Point When we plot the pressures associated with each of the horizontal lines BF (Fig. 12-51) against the corresponding tem- peratures we get such a curve as OA in Fig. 15-31. For a given point on such a curve, the function / (§ 17-1) has the same value 318 THEORETICAL PHYSICS [Ch. XIV for a gram of the liquid as for a gram of the vapour, since the conversion from one state to the other takes place reversibly at constant pressure and temperature (17*1). Therefore /.=/, (17-5) and since /^ and /^ are definite functions of the independent variables jp and T, equation (17*5) is the equation of the curve OA. Similar remarks apply to the equilibria between liquid and solid and solid and vapour, represented by the curves OB and OC. The equations of the three curves are therefore Jv ^^ Jh J I "^ J Si fs=f. (17-51) The point of intersection of OA and OB, being common to both curves, satisfies both of the first two of these equations, and hence for this point Js /v which shows that it is a point on 00. In other words the three curves intersect in one point, as the figure has anticipated. This is called the triple point. § 17-6. Latent Heat Equations Consider two neighbouring points on OA (Fig. 15-31). By (17.5) and /, +(4 =fi + <^fu and therefore df^ = dfi. Consequently by (17*11) - cl>,dT + V,dp = - cfyjdT + V,dp, or (cl,,-<f>,)dT = (V,-r^dp. Now cl>,-cl^, = L/T, by (16-7), where L is the latent heat of evaporation. Therefore i = (F, -F,)r^ .... (17-6) This is known as Clapeyron's equation. It is important to remember that this formula implies the use of absolute units. For example p means force per unit area, force being measured by rate of change of momentum ; work is measured by the product of force and distance and L is measured in work units of energy reckoned per gram of the substance. It is of course immaterial what are the precise fundamental units which have been adopted, whether pound, § 17-6] THERMODYNAMICAL PRINCIPLES 319 foot, Fahrenheit degree, etc., or gram, centimetre, centigrade degree, etc. As an illustration consider the equilibrium between ice (solid) and water (liquid). The latent heat of fusion is approximately 80 X 4-2 X 10' ergs per gram ; Vi —Vs = — 0-09 c.c. per gram. At normal atmospheric pressure, i.e. 1,014,000 dynes per cm.^, the equi- librium temperature (so-called melting point) is 273 on Kelvin's scale. Therefore 80 X 4-2 X 10' = - 0:09 x 273 x —,, dT where dT is the elevation of the melting point of ice due to the elevation. dp, of the pressure. Hence dT — = — 7-3 X 10"' approximately, or the melting point is lowered by 0-0073° per each atmosphere increase in pressure. The Gibbs-Helmholtz formula (17 '15) provides an alternative way of deriving Clapeyron's equation. For a gram of the vapour and Uquid respectively, Hence F„ - Fi = U, - Ui + ^ f ^^'gy ^'\ ' Now in this case Fv -Fi = -p{Vv -Vi), therefore -p{Vv - Vi) = U, -Ui -{Vv - Vi)T^, or U. -Ui+ piVr, - F0= {Vv - Vi)T^. This is Clapeyron's equation, since the left-hand member is obviously identical with the latent heat. Finally it will be noted that this equation is a special case of the more general formula (16*94). Let US now turn to the variation of the latent heat with temperature. It is convenient to make use of the constant pressure lines of the substance (e.g. water and water vapour) on a T, <f> diagram (Fig. 17-6). Starting at a point A, imagine heat to be communicated to the liquid reversibly at constant pressure. The entropy and temperature wiU both increase until a point B is reached for which the temperature is the equilibrium temperature of the liquid and its vapour for the particular pressure chosen. The reversible communication of heat is now associated with reversible vaporization, the temperature remaining A ' 320 THEORETICAL PHYSICS [Ch. XIV constant till the whole of the liquid is vaporized. This stage is represented by the horizontal line BC. Beyond the point, C, the curve will again ascend, as shown by CD. For a slightly higher pressure we have a corresponding curve A'B'C'D'. We may represent the equilibrium temperatures and latent heats corresponding to BC and B'C hj T, T -\- dT and L, L + dL ^ respectively. The broken line BB' represents, for dif- ferent temperatures, states ) of the liquid in which it is in equilibrium with its vapour. Similarly CC represents saturated vapour at different temperatures. Consider now the heat com- / municated to the substance O ^ when it is taken round the Fig. 17-6 cycle BB'C'CB. Let Cj be the specific heat of the liquid when in equilibrium with its vapour, and Cg that of the saturated vapour. The net amount of heat communicated during the cycle is obviously c^dT + L +dL - c4T - L or {c,-c,)dT+dL .... (17-61) But we have already seen (§ 16-3) that this is equal to the area of the closed loop, i.e. to dT X {BC), or dT(cf>, - <!>,), or, finally, ^ ..... . (17-62) On equating (17*61) and (17*62) we get c.-c.=§-§ .... (17.63) § 17-7. The Phase Rule Turning again to the equilibrium between two phases of a single constituent, e.g. water, we have seen that we can represent it by a curve on a ^ J' diagram ; for liquid and vapour the curve OA (Fig. 15-31). Within the limits between which these phases can exist we may have equilibrium at any temperature we like to choose ; but having once fixed the temperature there is only one pressure under which equilibrium is possible. Or on the §17-7] THERMODYNAMICAL PRINCIPLES 321 other hand we may choose any pressure we like, but there will then be only one temperature at which equilibrium is possible. We say the system of liquid and vapour has one decree of freedom. The equilibrium of three simultaneous phases is represented by a single point, 0, on the diagram. In this case there are no degrees of freedom at all. In the case of only one phase, e.g. liquid, there are obviously two degrees of freedom. Over the range of pressures and temperatures for which this phase can exist we may choose both arbitrarily and independently of one another. These facts are instances of a simple general rule due to Willard-Gibbs, and known as the Phase Rule. It may be stated in the following form : F -\-P = C -\-2 (17-7) where F is the number of degrees of freedom when P phases are in equilibrium, the number of constituents being C. As an illustration of the case of two constituents, let us take water and a soluble salt. Consider the phases, ice, solid salt, solution of salt in water, and water vapour. For two phases, e.g. solution and vapour, the rule gives, F + 2 =2 +2, or two degrees of freedom. This means that we may, for example, choose both pressure and temperature (within the limits between which these phases can exist) at will. Equilibrium will be always possible at some definite concentration of the solution, or we may adjust at will the concentration and temperature ; there will then be a definite pressure under which the two phases are in equilibrium. When three phases are in equilibrium, for example ice, solution and vapour, there is only one degree of freedom. We can establish the phase rule in the following way : In any revers- ible transference of one or more constituents from one phase to another (i.e. transference under equilibrium conditions) the function / for the whole system remains unaltered, if we keep the pressure and temperature constant (§ 17-1). Therefore Sf = (17-71) If there are P phases, / is a sum of contributions from each phase, or / =/ +r +r + . . . +r^ .... (i7-72) and consequently df = 8f + 8f' + (5f + . . . +SfP^ = . . (17-721) In any redistribution of the constituents among the P phases, let Sm^\ dm^'\ dm/", . . . Sm^^P^ represent the increments of constituent number 1 in the P phases respectively and dm^'i dm^", dm^'", . . . dm^^^^ those of constituent number 2 in the P phases respectively, and so on. 322 THEORETICAL PHYSICS [Ch. XIV Since the total masses mj, m^, . • . me, are given, dm/ + dm^" + dm^"' + . . . + dmi^^^ = 0, &m^' + dm^" + dm^'" + . . . + dm^^P'> = 0, dmo' + 8mo" + dm^''' + . . . + dmc^^^ = . (17-73) Taking any one of these equations, the first one for instance, we may- choose only P — 1 of the dm-^'s, arbitrarily, the remaining one being determined by the equation. So that altogether there are C(P — 1) dm's only which we may choose arbitrarily. Let us represent them by dx^, dx^y dx^y . . . dxc(p^i). The condition for equihbrium (17'71) or (17*721) now becomes §-Sx, + gfe, + . . . + ^,^-«.-« = 0, . (17-74) and since the die's are arbitrary, we have -^ = dx^ "' ^ = 0, dx^ 8f •^ =0 (17-75) ^^C(P-l) These C{P — 1) conditions are necessary and sufficient for the equi- hbrium of the P phases at some given pressm-e and temperature. Let us now consider how many data are required to fix the state of the system. To begin with we have the two data pressure and tem- perature. In addition to these we require the data fixing the constitution of each phase. For each phase C — I data are evidently necessary for given total masses of the C constituents, since the character of a phase is determined by the C —I ratios ma' mg' nio^ nil * '^1 ' ' ' * nil of the masses of the C constituents present in it. The constitution of the P phases is therefore determined by P(C — 1) data. Adding to these the 2 data, pressure and temperature, mentioned above, we require altogether 2 + P(0 - 1) data to completely describe the state of the system. We have already seen that C{P — 1) relations must exist between them, and there remain over consequently at our arbitrary disposal 2 +P{G -1) -C{P -1) or 2 - P + C factors. This means that we may choose 2 — P -|- C of the independent variables, which determine the state of the system, quite arbitrarily and still have P phases in equihbrixmi, i.e. P = 2 - P + C. § 17-8] THERMODYNAMICAL PRINCIPLES 323 § 17-8. Dilute Solutions A solution of a crystalline or other body, in water for example, has a lower equilibrium vapour pressure than the pure solvent. We can ex- plain this in the following way : Imagine two vessels A and B in an enclosure (Fig. 17'8), the former containing the pure solvent, the latter a dilute solution, and the rest of the enclosure only the vapour of the solvent. If the two levels a and h were initially coincident, the surface of the solution would function as a semi-permeable membrane, and vapour would condense into B until finally a difference in level, h, equivalent to the osmotic pressure, P, of the solution in B, became established (§ 12-4). When this equilibrium condition exists, the vapour pressure, CL - — h -^ 1 A Fig. 17-8 p% must be the same at all points in the horizontal plane, b, and if p be the pressure at the lower level, a, i.e. the equihbrium pressure between the vapour and the pure solvent, obviously P —p' = SgK where <5 is the vapour density, which we may take to be approximately imiform. On the other hand the osmotic pressure, P, is expressed by P = ggh, where q is the density of the solution, or, in the case of a dilute solution, the density of the solvent itself. Hence p —p'_d (17-8) The equiHbrivmi between the vapour and the pure solvent, and that between the vapour and the dilute solution, are represented by the curves AB and A'B' respectively in Fig. 17-81. Let T be the boiling point of the pure solvent, i.e. the equihbrium temperature for the solvent and its vapour, when the pressure is the normal pressure of 76 cm. of mercury. The boiling point of the solution will be T', a Uttle higher as the diagram explains. Now by Clapeyron's equation (17'6) L = VT (ai5) where L is the latent heat of the solvent, V is the volume of the unit mass of the vapour (we have neglected the volume of the unit mass of the 324 THEORETICAL PHYSICS [Ch. XIV liquid, since it is small by comparison). This equation may be written in the form l=It?-^' since Vd = 1. Combining this with equation {17*8), we get TP T'-T = -r (17-81) qL for the excess of the boiling point of a dilute solution over that of the pure solvent, and there is obviously an analogous formula for the excess of the equilibrium temperature of solution and solid solvent over that of liquid and solid solvent. The kinetic theory suggests (§ 12-8) that in a dilute solution the relation between osmotic pressure, volume of solution and temperature is identical with the perfect gas equation, to a first approximation at any rate. Therefore PV =RT (17-82) where P is the osmotic pressure, V the volume of a gram molecule of the dissolved substance, and R is the gas constant (8-315 x 10' ergs per ° C.) for a gram molecule (we are assuming that the ultimate particles of the dissolved substance in the solution are molecules, i.e. that it does not dissociate, nor associate). If a is the concentration, i.e. the quantity of dissolved substance per unit volume, and M its molecular weight, a and therefore ~ =RT (17-82) Combining this with (17-81) we find RT^ a ^'-^-Mll (17-83) a formula which enables an approximate estimate to be made of the molecular weight of a body from the elevation of the boiling point due to dissolving it in a suitable solvent. We have assumed that the dissolved body does not dissociate (nor associate). If each molecule in solution were to break up into two parts (ions), the osmotic pressure would of course be twice that which would result if no such dissociation occurred, and conversely if association of the molecules to form larger particles were to occur, the osmotic pressure would be correspondingly lower. This is the reason for the abnormally low osmotic pressures of colloidal solutions. In aqueous solutions of crystalline bodies, the ultimate particles in solution are always, or in most cases, either molecules of the dissolved substance, or ions into which it dissociates. In the case of cane sugar (and other non-electrolytic crystalline bodies) the osmotic pressure is quite close to that which we should calculate on the assumption that it occupied, in gaseous form, a volume equal to that of the (dilute) solution, without dissociating. In the case of common salt (and similar electrolytic bodies) the osmotic pressure in dilute solutions is approximately twice that of the equivalent solution of cane sugar, indicating that each molecule dissociates into two particles (ions). The phenomena of electrolysis furnish independent evidence in support of this view. A rough classification of bodies is §17-8] THERMODYNAMICAL PRINCIPLES 325 usually made into crystalloids and colloids. Cane sugar and common salt are examples of the former class. Their solutions are characterized by high osmotic pressure, which we explain by the supposition that the idtimate particles of the dissolved substance in solution are molecules or still smaller particles into which their molecules have broken up. Colloids on the other hand are substances the aqueous solutions of which have low osmotic pressures, the ultimate particles of such substances, when in solution, ranging from the order of magnitude of molecules at one extreme to Perrin's visible spherules at the other. BIBLIOGRAPHY Wlllard-Gibbs : Scientific Papers. (Longmans.) Clerk Maxwell : Theory of Heat. (Edition with corrections and additions by Lord Rayleigh. Longmans, 1897.) Max Planck: Thermodynamik. INDEX OF SUBJECTS Absolute zero, 275, 295 Action, 112 Adiabatic, 277 — change, 285 — strain, 167 Amplitude, 122, 131 Analytic functions, 162 Atomic weight, 253 Average kinetic energy of molecule, 229, 231, 232 — square of velocity, 230 — velocity, 230 Avogadro's law, 218, 221, 223 — number, 253 Bernoulli-Fourier solution, 129 Bernoulli's theorem, 186, 187 Boltzmann's constant, 270 Boundary conditions, 125 Boyle's law, 167, 217, 218, 221, 283 deviations from, 252 Brownian movement, 255 Bucherer's experiment, 140 Bulk modulus, 163 Calorie, 280 Calorimeter, 280 Canonical distributions, 262, 263 — equations, 105, 114, 115 Carnot's cycle, 289 — engine, 289 -- principle, 290, 292 Central forces, 47 Centre of mass, 44 motion of, 45 Circulation, 200 Clapeyron's equation, 318 Closed system, 301 Coefficient of viscosity, 204 Colloids, 325 Complete integral, 116 - 22 327 Component, 5 Compound pendulum, 105 Condition for Thermodynamic equi- Ubrium, 302 Conservation of energy, 101, 280 Conservative system, 102 Constants, inherent, 46 — of integration, 46 Constraints, 92 Continuity, equation of, 175, 177 Continuous medium, 141 Contractile aether, 170 Correction of gas thermometer, 314, 315 Corresponding states, theorem of, 250, 251 Couple, 58, 59 Covering spheres, 245 Criteria of maxima and minima, 267 Critical point, 251 — pressure, 250 — temperature, 249 — voliune, 250 Crystalloids, 325 Curl, 13 Curve of inversion, 316 Cyclic co-ordinates, 105, 107 Degrees of freedom, 67, 109, 222, 321 Differential, 43 — equations of strain, 168 Diffusion, of gases, 240 — through minute apertures, 243 Diffusivity, 240, 242 Dilatation, 144, 163 — uniform, 151 Dilute solutions, 323 Displacement tensor, 144 Divergence, 13 — four-dimensional, 179 — of a tensor, 160 328 THEORETICAL PHYSICS Ecliptic, 91 Efficiency of heat engine, 290 Einstein's formula, 258 Elastic moduli, 162, 166 Elasticity, 141 Ellipsoid of inertia, 62 — of gyration, 62 Elliptic functions, 77 Elongation, 149 — quadric, 150 Energy, 42, 59, 101 — and mass, 140 — in a strained medium, 174 — kinetic and potential, 44 Entropy, 218, 270, 297 — and the second law, 298, 299 — constant, 270 — properties of, 300 — of a gas, 300 — scale, 297 Equation of continuity, 175, 177 Equations of Hamilton and La- grange, 102 — of Lagrange, 105, 114, 115 — of motion in a viscous fluid, 203 — of state, 275 Equinoctial points, 91 Equipartition of energy, 217, 218, 221, 223, 270, 271 Erg, 43 Euler's angular co-ordinates, 82, 83 — dynamical equations, 72, 74 — hydrodynamical equations, 181 Extension in phase, 262 Format's principle, 139 Fields, vector and tensor, 12 First law of thermodynamics, 280 Force, 39 — scale, 40 Formula of Stokes, 209 Formulae for homogeneous systems, 304 Fourier's expansion, 28, 31 — theorem, 31 Fundamental frequency, 129, 131 Gas constant, 275 General integral, 116 Generalized co-ordinates, 57, 97 — forces, 100 — momentum, 57, 99 — velocity, 57 Geometrical optics and dynamics, 138 Gradient, 14 Graham's law, 244 Gravitation, constant of, 53 Gravity, intensity of, 42 Green's theorem, 18, 19 Group of waves, 136 — velocity, 136, 137, 139 Gyroscope, 83, 84, 86 Hamiltonian function, 103, 106, 109 Hamilton's canonical equations, 105, 114 — characteristic function, 114, 115 — partial differential equation, 115, 119 — principal function, 114, 116 — principle, 114 Harmonics, 129 Heat, 280 Herpolhode, 80 Heterogeneous systems, 317 Homogeneous strain, 141 Hooke's law, 162 Hydrodynamical equations, 180 Impressed force, 93 Indicial equation, 173 Inertia, principal axes of, 62 Integrals, 116 — used in kinetic theory, 228 Internal energy, 282 Invariant, 8 Irreversible process, 302 Irrotational motion, 186 Isochores, 277 Isopiestics, 277 Isothermal, 277 — strain, 167 Jacobi's theorem, 116, 120 Joule's law, 283 Kepler's laws, 54 Kinetic energy in a fluid, 192 — energy of a rigid body, 63 — theory of gases, 217 Lagrange's hydrodynamical equa- tions, 182 Lagrangian function, 105, 109 Lamellar flow, 189, 190 INDEX OF SUBJECTS 329 Lamellar vector, 189 Laplace's equation, 190 solutions of, 191 Laplacian, 14 Latent heat equations, 318 Law of action and reaction, 39 — Charles, 221, 223, 275 — Clausius, 283 — distribution of velocities, 218 — partial pressures, 221 Laws of motion, 2, 41 Least action, principle of, 113 Liouville's theorem, 261 Longitudinal wave, 134, 135, 169 Loschmidt's number, 253 Mass, 39 — and energy, 140 — definition of, 40 — dependence on velocity, 140 — unit of, 41 Maxwell's law of distribution, 225, 229 — thermodynamic relations, 311 Mean free path, 232, 236 Mechanical equivalent, 101, 284 — wave, 140 Moduli of elasticity, 162 Modulus of a canonical distribution, 263, 267 — of rigidity, 163 — Young's, 165 Molecular collisions, 232 — weight, 253 Moment of a force, 49 — of inertia, 58, 60, 62 Momental ellipsoid, 62, 79 — tensor, 63 Momentum, 41 — angular, 47, 48, 49 Motion in viscous fluids, 203 — of a sphere in a fluid, 192, 209 Mutually interacting systems, 263 Nabla, 14 Newton's laws of motion, 39 Nutation, 87 Orthogonal functions, 36 complete system of, 38 Osmotic pressure, 244, 256, 323,^ 324 Partial pressures, law of, 221 Particle, equations of motion of, 43, 45 — path of, 46 Pendulum, 65, 89 — cycloidal, 69 Perfect gas, 275, 283 Perpetuum mobile, 101, 288, 289 Phase, 122, 138 — rule, 320 — space, 260 — velocity, 136, 137, 139 Plane waves, 132 Planetary motion, 50 Poiseuille's formula, 205, 207 Poisson's equation, 191 — ratio, 165 Polhode, 79, 80 Porous plug experiments, 312 Precession, 87 — of the equinoxes, 91 Principal axes, of strain, 146 of stress, 157 — elongations, 150 — function, 114 — tensions, 157 Principle of Carnot, 290 — of conservation of energy, 100, 101 — of d'Alembert, 95, 96, 109 — of least action, 113, 139 — of virtual displacements, 92 Probabilities, a priori, 264 Product, scalar, 5, 6, 8 — vector, 5, 7 Products of inertia, 60, 61 Projectile, path of, 45 Proper volume, 217 Pure strain, 144, 146, 148 Radial strain in a sphere, 171 Radius of gyration, 60 Rank, of tensor, 5 Ratio of elasticities, 285 — of specific heats, 225, 285 Reduced pressure, 250 Relativity, 175 — special, 179 Representative space, 260 Resultant, 4 Reversible cycle, 289 — expansion, 225, 276, 286 330 THEORETICAL PHYSICS Rotational and irrotational motion, 185, 186 Rotation of a vector, 13 Scalar, 4 Scale of temperature, Kelvin's, 292 Schroedinger's principle, 140 Second law of thermodynamics, 288 Semi-permeable membrane, 244 Shear, 151 Shearing stress, 159 Simple harmonic motion, 68, 129, 131 Specific heat, 224, 225, 282, 283 Spherical waves, 132, 134 Statistical equilibrium, 222, 262 — mechanics, 218, 259 Strain, 142 — differential equation of, 168 — ellipsoid, 146 — homogeneous, 141 — tensor, 149 Stream Line, 187 — momentum, 236 Stress, 153, 154 — principal axes of, 157 — quadric, 156, 157 — tensor, 156 Sutherland's formula, 239 Temperature, 272 — of inversion, 315 — scales of, 223, 274 Tensor, 5, 11 Theorem of Gauss, 16, 17, 18 — - of Stokes, 23, 28 Thermal conductivity, 239 and viscosity, 240 Thermodynamic diagrams, 276 — potentials, 309, 310, 311 — systems, 301 Thermodynamics and statistical mechanics, 300 Top, 84 Torque, 58 Transformations, 9 Transverse wave, 134, 136 velocity of, 169 Triple point, 317, 318 van der Waals' equation, 247 theory, 244 Vector, 4 Velocity of sound in a gas, 286 — of waves along a rod, 171 — of waves in deep water, 198 — potential, 184, 185, 189 Vena contracta, 188 Viscosity, 166, 204 — in gases, 236 Vortex, 198, 202 Wave along a stretched cord, 121 — equation, 123, 133 — front, 132 — length, 122 Weber's hydrodynamical equations, 184 Weight, 41 Work, 42, 59 — of reversible expansion, 278 Young's modulus, 165 INDEX OF NAMES Avogadro, 218, 223, 253 Bernoulli, 127, 129, 186, 187, 217 Black, 280 Boltzmaim, 218, 258 Boyle, 167, 217, 218, 223 Brook Taylor, 129 Brown, Robert, 255, 256 Bucherer, 140 Camot, 272, 285, 289 Charles, 221, 223 Clapeyron, 318 Clausius, 218, 303 Clement and Desomies, 285 Colding, 101 Crabtree, 91 Dalton, 221 Davy, 101 d'Alembert, 95, 96, 109, 126, 127, 129, 130 Dirichlet, 28 Einstein, 257, 258 Euclid, 3 Euler, 72, 74, 75, 82, 83, 84, 85, 91, 126, 180 Ferguson, A., 253 Format, 139 Fourier, 28, 30, 31, 129, 130 Fresnel, 169 GaHleo, 1, 2, 71 Gauss, 16, 18, 22, 23, 160 Gay Lussac, 282 Graham, 244 Gray, 91 Green, 18, 19, 22, 23 HamUton, 102, 105, 106, 114, 115, 116, 119, 120 Hehnholtz, 287 Hirn, 101, 281 Hooke, 162, 163 Huygens, 69, 71, 100 Jacobi, 116, 119, 120 Jager, 258 Jeans, 258 Joule, 101, 281, 282, 312 Kelvin, 169, 272, 282, 303, 312 Kepler, 54 Lagrange, 102, 105, 106, 114, 115, 180 Laplace, 190 Liouville, 261 Lorentz, 216 Loschmidt, 253 Love, 179 MacCuUagh, 169 Mach, 71, 287 Maupertuis, 113, 139 Maxwell, 1, 2, 12, 218, 225, 239, 311, 325 Mayer, 272, 285, 287 Mecklenburg, 258 Nemst, 272 Neumann, F., 169 Newton, 2, 3, 39, 41, 43, 71, 100 Ostwald, 288 Perrin, 255, 256, 257, 258 Planck, 287 Poinsot, 78, 91 Poiseuille, 205 Poisson, 165, 191 331 332 THEORETICAL PHYSICS Routh, 120 Rumford, 101, 281 Schrodinger, 140 Schuster, 2 Stokes, 23, 209, 216 Sutherland, 239 Thomson and Tait, 120 Thomson, James, 249 TorriceUi, 187 van der Waals, 244 Waterston, 217 Weber, H., 120 Webster, A. G., 91, 120 Whittaker, E. T., 120 Wiener, C, 255, 256 Willard, Gibbs, 170, 218, 262, 263, 271, 321, 325 Young, 165 Printed in Great Britain by Butler & Tanner Ltd., Frome and London '