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Full text of "Theoretical physics"

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BOOK 530. W69 v. 1 c. 1 

WILSON # THEORETICAL PHYSICS 



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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 



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