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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 subjectmatter 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 .
Coordinate 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 Coordinates . . . . . .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 Coordinates
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 Coordinates ...... 97
Principle of Energy . . . . . . .100
Equations of Hamilton and Lagrange . . . .102
Illustrations. Cyclic Coordinates .... 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 subdivisions 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 welldefined 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 oldfashioned
' 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 coplanar, 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 coordin
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 coordinates to
some point P, its components are the coordinates 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.
§ 21. 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. 21). 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 coordinates 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,^ .... (210)
and B^=BJ + B/ + B,^
we find on equating the two expressions for t^
Fig. 21
ABcos0=:^A+^A+^A . . (211)
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 = . . ._ . (212)
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 coordinates, 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 coordinates. We have here an example of an
invariant, i.e. of a quantity which has the same numerical
value whatever system of coordinates 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. 21) we have
vector product = AB sin 9
We shaU usually abbreviate this expression by writing it in the
form [AB].
Squaring both sides of (211) we have
A2B2  A2B2 sin^ d = {A,B, F A,B^ \ A,B,Y
or, by (210)
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,)^ . . . .(213)
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 coordinates, 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 righthanded 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 rightangles 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 coordinate axes about the
origin so that the vectors A and B lie in the XY plane in the way
indicated in Fig. 211. 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
coordinate 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
righthanded 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
(214)
Clearly this determinant is an invariant, since a scalar product
is an invariant. If again we imagine the axes of coordinates to
be turned about the origin till the vectors A and B lie in the
XY plane, as in Fig. (211), 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 . . . (215)
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])
§22]
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 righthanded 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
coplanar is :
=
(216
§ 22. COORDIKATE TRANSFORMATIONS
Let X, Y, Z and X', Y', Z' be two sets of rectangular axes
of coordinates with a common origin ; and let P be any
point, the coordinates of which
are x, y, z and x' , y\ z' in the
two systems respectively (Fig.
22). 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 coordinates
of P in the system X, Y, Z,
to find x\y\z\ its coordinates 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' coordinate 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 jriyy '\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' .... (221)
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 : —
. (222)
Mathematically a vector may be defined as a set of three
quantities v^hich transform according to the rules em
bodied in (222).
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
. (223)
§23] EUCLIDEAN TENSOR ANALYSIS 11
The correctness of these equations is obvious, since in each
case the lefthand 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 = ... (224)
and so on. These equations follow since the lefthand 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 (225;
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,),
§ 23. 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 antisymmetrical. 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 antisymmetrical 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.
§24] EUCLIDEAN TENSOR ANALYSIS 13
Let ^ be any quantity which varies continuously from point to
point. Then by a wellknown theorem of the differential calculus,
dx' dx dx' dy dx' dz dx'
But by (222)
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 . . . (241)
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, . . . . . (242)
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 ^ = . . . . . (2431)
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^^*
§24] EUCLIDEAN TENSOR ANALYSIS 15
A useful formula, frequently used in electromagnetic theory,
is the following : —
div [AB] = (B curl A)  (A curl B) . (244)
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. lefthand side,
or the X component of the lefthand side, in full.
(curl curl A>  ^ 1^^'  ^^4  ^ i^^"  ^^'
icurlcurl A}, g^ g gg —
{curl curl A}. = ^{^ + ^)  (g^^ + ^^),
and if we add and subtract ;r— ^ on the righthand 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
(3001
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 3dimensional character of the region
over which the integration extends.
From the definition of div A (§ 24) 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 crosssection dy dz . For this restricted
volume we have
nr
I ^^^ ^y ^^ = dy dz ^dx
=^dydz{{A,),(A,),} . . . .(3003)
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.
§ 31. 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) . (31)
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) (312)
This result, known as Green's theorem, was published in
1828 by George Green in an epochmaking 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
§ 31] 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 . . , (315)
dn )
In this formula the surface integral is extended over the
outer surface ahc of Fig. 31, 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.
§ 32. 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
§33] 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
+ ^^ +
?).... (32)
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,} . . (321
Green's theorem, and the formulae deduced from it, naturally
admit of a similar extension. We can, for example, deduce the
equation
F^==^^^^^'dxdydz . . . (322
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).
§ 33. 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) = . . . . (33)
the integration being now extended over the bounding surface
(abed J Fig. 33). 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) = . (3301)
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. 33
Fig. 331
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) = (3302)
ah ef
From (3301) and (3302) 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. 331) 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. 331) and
all end at a common point (2 in Fig. 331). 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
§ 33] 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. 331 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. 332). 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 coordinate of the point 1 (Fig. 332) be x ; the
X coordinate of the point 3 will be a; + dx, and as we pass from
3 to 4 we realize that the X coordinate 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 coordinate 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
coordinate of the same point, and it follows that
ddx = ddx (3314)
This means that the operations d and d are interchangeable,
at any rate when applied to the coordinates.
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. 333.
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 § 24 and equations (3'31).
The part of the integral involving A^ only is
2
^((32 dx — dx dz) — ~\dx dy — dy dx) ,
Fig. 333
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.
§33] 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 righthand 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)
(332)
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,
(402)
+ 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(Tc^Mr. . (404)
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 . . (405)
— 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.
§ 41. 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 (401)
+ 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 . • .} . (41)
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. 41
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. 41 ).
§41] 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. 411).
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,::^~{cosn7tl),
34
Therefore
and
THEORETICAL PHYSICS
[Ch. II
= 0,
A,= 
A,= 
and so on.
2 "
M)
4
4
¥^'
4
Therefore
jcos <J^ + 32 cos 3^ + — cos 5^ f . . .} (412)
The sums of the series
(41) and (412) are
equal when ^ ^ <[ itt.
If
f{cl>) = l, ^<0,
Fig. 412
(Fig. 412) we find the expansion
sin
If
and
(Fig. 413)
M)
<j> + sin B<j> + sin B<j> + . . .1
im =  1,
(413)
^ 2'
^>I'
i'l<^<i'
we obtain
4(1 1
= j cos —  COS 3^ +  cos 50 h
(4.14)
7p
7T
Fig. 413
§41] THEOREMS OF GAUSS, GREEN AND STOKES 35
As a further example take the case, illustrated in Fig. 414,
where
Fio. 414
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 ^  32 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. 415), for which we find the expansion
2 _ 4cos 2(f) cos 4^ cos 6(^
^ S rT3~" 35 57 '*
/(^)
(4.16)
Fig. 415
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 onevalued 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
 = 11 + 11 + ...
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 ''^'"' • • • • (417)
!/ = — 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 ei4> — iei20 _J_ l.QiS'i^ h . . .
2 3
§ 42. 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
§42] 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 nonvanishing
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^+ ... (42)
the coefficients being given by
"^^iHvEy^'
— TT
71 = 1, 2, 3 ... 00.
^„= j/(.)j!iLi)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) .... (421)
and since the integral relations (4*02) now have the form
j E^(r)EJr)dr = I, n = m,
= 0, 71=^ m . . . (422)
the coefficients c^ are given by
+ 77
On = ^^nr)E,(r)dr, n = 0, I, 2 . . . oo . (423)
— 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
(socalled) 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 evenlydivided
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 9806 cm. sec."^ in latitude 45°, and varies from
978 cm. sec.~2 at the equator to 9834 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 '.
§ 51. 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. 51),
^ ^^.^ — 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 coordinates of C. And if we take some
neighbouring point C with coordinates {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 (§ 24) 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) (512)
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,. . . .(5121)
If we replace W hj — V and use the letter T for , we have
T + F = To + Fo . . . . (5122)
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.
§ 52. Centre of Mass
Imagine a number of particles, the masses of which are
mi, m2, mg, . . . m^, and their coordinates {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, (52)
Mz = Em^Zg,
where M = Em^,
the summation being extended over all the particles of the
§53] 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 (521)
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.
§ 53. Path of Projectile
Let a particle of mass m be projected from the origin of rect
angular coordinates 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 { ^^ (531)
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 .... (5311)
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 coordinates so that this vertical plane is the
XY plane, and the Z coordinate 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 . . (532)
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 = :: '
(5321)
§54]
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.
§ 54. 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
(54)
dh
dt^
F.^.
Fig. 54
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] (5411)
(§ 21) 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. 541),
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 . (5412)
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 § 21). Therefore the scalar product
(Sir) is equal to zero, that is
Q.^x +Q,yy +Sl,z =0 , . . . (542)
This is the equation of a plane passing through the origin, i.e.
through the attracting centre.
§ 543. 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
'""SJ^ = Fsy,
'dt^
§543] INTRODUCTION TO DYNAMICS 49
By a procedure similar to that used to deduce equations (5*41 )
we get :
^sik^  x,^\ = ^sF..  ^.P., . (543)
dt [ ^ dt ^dt
VsiT \ — ^s^ sy Us^ &
'dt[ 'dt ""'dt
which may be expressed more briefly in the following form :
n, = [r,FJ .... (5431)
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 (5432)
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
righthand 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
§ 55. 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 coordinates. Since,
as we have already proved (§ 54), the particle moves in a plane,
we shall place our coordinates so that this plane is the XY
plane. The Z coordinate of the particle will remain constant
and equal to zero. We now introduce polar coordinates, 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
§55] INTRODUCTION TO DYNAMICS 51
particle (see § 51). 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 (§§ 51 and
54). 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 = jrdd.
Equation (5*51) now becomes
J^(±y+J^+B^E . . .(5511)
2mr^\dd/ 2mr^ r
We now introduce a new variable u defined by
u= (5512)
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. + « = ^ • • • • (552)
The general solution of (552) is
■» =  ^ + B cos (e  ^) . . . (553)
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. 55).
Fig. 55
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. 551 ) 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.
§55] 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
^ = .... (5532)
mB a
The force exerted on one another by two gravitating particles is
^ (5533)
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. 551
and if we use axes of coordinates 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 .... (5534)
by (5533). It follows from (5532) that
QM^^^Il .... (5535)
We have seen (§ 54) 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— (5536)
T
, (554)
54 THEORETICAL PHYSICS [Ch. Ill
if T is the period of a complete revolution. From (5*535) and
(5*536) we get
Johannes Kepler (15711630) inferred from the observational
data accumulated by Tycho Brahe (15461601) 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^rii=0 .... (555)
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 coordinates. 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
§55] 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 semimajor 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 (552).
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 (559)
where R and rj are constants of integration. We may write
this equation in the form
Q^ + mC
mB
l+.cosj^/l+^
6
(5591)
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
(5592)
V'
1 + £COS {(9  7f}
where
The essential difference between the orbit represented by (5531)
and that represented by (5592) 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 Coordinates
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 coordinates,
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 coordin
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 § 54 (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 coordinates 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,)
. . . (602)
In these equations we may, just as in § 543, regard the forces
as of external origin, since the forces of internal origin con
tribute nothing to Z{yF^ — zFy), etc. The righthand members
of (6 '02) are the components of the applied torque or couple.
§ 61] 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.
§ 61. 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] (61)
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.
§ 62. 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, . (62)
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 coordinate axes
so that the axes and C are
in the XZ plane and perpen
dicular to the X axis (Fig. 62).
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 coordinate of the particle and Xq is the X
coordinate 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^ (621)
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
§62] INTRODUCTION TO DYNAMICS 61
the centre of mass, which we shall suppose is at the origin of
rectangular coordinates. 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^ . (622)
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 . . (623)
where M is the mass of the body and i, tj and C are X, Y and Z
coordinates. 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 . . . (6235)
where A, B and G have not necessarily the same values as in
(623). A, B and G (6235) 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 (6235) 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 (624)
/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 (624) 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' ' (6241)
^ G^
The length of the perpendicular, P, from the origin to the
tangent plane at (' t]' ^') is, by a wellknown rule,
J_
M
V
A^ B^ (72
or P == , ^  , by . . (6241)
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)
§64] 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.
§ 63. 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^^^ . . . : (63)
The righthand 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 (6301)
are the components of a tensor of the second rank. It can
easily be proved that this is the case (see § 23). It is called
the momental tensor and is more appropriately represented
in the following way :
V V i. (6302)
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} • • (6303)
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).
§ 64. 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 coordinates 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} . . . (641)
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 (642)
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]
§65] INTRODUCTION TO DYNAMICS 65
and the displacement of P. Let the coordinates 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)} • (643)
Therefore
% = ^ox + '^ix • • . • (6435)
Where v is the velocity of the particle, Vq the velocity of P and
^,, = J"(z2„)^%2/„) . . (6436)
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 . (644)
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 .... (645)
where Vq is the velocity of the centre of mass and co is the angular
velocity relative to the centre of mass.
§ 65. The Pendulum
The pendulum is usually a rigid body mounted so that it
can turn freely about a fixed horizontal axis, O, (Fig. 65),
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. 65
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
(65)
(6501)
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,
^65] 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
(651
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 wellknown
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 (6522)
The general solution of this equation can be put in the form
l9 = ^ cos (co^  ^) . . . . (6523)
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 = — (6524)
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,
§65]
or since <c = sm
INTRODUCTION TO DYNAMICS
«y
= T„(l+isin2),
T,=To(l+j) . .
since the difference between the squares of sin
(6526)
 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. 651, are
^7^ = Qx.
m
dp
Y
mg + a, (653)
Fig. 651
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 =
(6531
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 £ (654)
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 (6541)
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 £ =  ? (6542)
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
. . (6543)
Let 0', from which s is measured, coincide with the origin.
0, so that X = when y = ; and suppose that the particle,
§65] 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) (655)
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 coordinates. Let
PJ, Py and P/ be the components of the same vector
referred to a second set of rectangular coordinates, the origin,
0', of which coincides with 0, the origin of the first system.
Therefore, by (222)
Px ^^ ^iiPx ~i" (^2iPy ~r CisiPz
and ^ = {a,,P; + a,iP/ + a„P/} . . (7)
We shall suppose the first set of coordinates 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 coordinates 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
coordinates relatively to the moving coordinates,
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) . . (7002)
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^ , . . (701 )
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 (222) 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 coordinates of the end point of the line. Equations (7*01)
apply at the instant when the two coordinate 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 coordinate systems. Clearly the
components of the angular velocity of the body have the same
values in both sj^stems of coordinates 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 coordinates ; 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 (701).
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 coordinates) 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, . . . (702)
When the applied couple vanishes these equations become
,dco^
dt
^^ = iB (^H^.
b"^^ = {G  A)co,co,,
C^ = {A  B)m,cOy . . . (7021)
The equations (7*02) and (7*021) are the well known dynamical
equations of Euler.
On multiplying (702) 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^ . . . (704)
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 "^ ^^ ~ ^^^'''^' = . . . . (705)
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„ = . (7052)
The constant
iCB){CA)
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)) . . . (7053)
76 THEOHETICAL PHYSICS [Ch. IV
where R and >S' are small real constants, (/> and xp are constants, and
coo . . (7054)
J
{G~A){GB)
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
(7055)
cos p COS q
in consequence of (7054).
It follows that p and q differ by an odd multiple of — and
the solutions (7053) may consequently be put in the form
0)^ = R cos (at — cf)),
C0y==8 sin [at cf>) ... (7056)
the first of the equations (7055) 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 coordinates 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
semiaxes 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 Vly^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^ . . . . (706)
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^ {CA)A'
{AB)A m,'^
(C B)C coo'
(7072)
2 _ (^ B)(GA)^_^ ,
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 (707) reduces, as of course it should, to that
already found for this special case (equations 7*056 and 7*057).
§ 71. 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. 71), 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
§71]
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. 71
CO,
CO
■X, COy
CO
y, (^z
CO
—z
(71
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^=^^ . . . (712)
Therefore i^ = ^ (7121)
Similarly, by combining (6*41) and (7*1) we obtain
Ax'' + By^ + Cz^ = 2T^ . . . (713)
or M=2T^ . . .(7131)
It follows that — is a constant, namely
and consequently
CO
P
(7132)
^^V2TM\ . . _ .(7.133)
a
It is therefore constant and its direction cosines (711) 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 (711) gives us
AH^ { B^y^ ^ CV = ^ . . . (714)
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 . . . . (715)
determines the polhode.
If we multiply (714) by p^ and (715) by M and subtract,
we get
(p2^2 _ MA)x^ + tp2^2 _ MB)y^ + (p^C^  MC)z^ = (716)
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 (7132) that the
angular velocity about the instantaneous axis is proportional to
§ 71] 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 semiaxes of the ellipsoid of inertia are
Suppose,
A>B>G,
then
f'^i
In one extreme case
I
(717)
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 . (718)
Reference to (7*17) will show that the righthand 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 semiaxes of any of the ellipses is
lB(A
\ C{A
B{A~B)\
[Ch. IV
(7181
Similarly, we can show that the projections of the polhodes
on the XY plane are the ellipses,
(^2 _ AC)x^ + (52  BG)y^ = ^. MG . (7182)
and ratio of the semiaxes being in this case
V
A(G A)
(7183)
B{G B)
This result should be compared with (7057).
The projections on the XZ plane are the hyperbolas
[A^  AB)x^ + (C2  GB)z^ = ^ MB (7184)
§ 72. Efler's Angular Coordinates
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
§72]
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. 72). 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. 721). 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. 72). 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. 72). 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' coordinate 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)
§ 73. 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 (§ 72) gives us
L = mgh sin d cos 0,
M = —mgh sin d sin 0,
iV^ 0.
§73] 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 ^ = ByJ + (A — 0)G>/a„
= Cf^^ + (B  A)co^m,. . (73)
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 (§ 72) 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 . . . . (732)
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 (72),
C\ cos 6 ^ \ ^\ = a constant,
I dt dt)
or finally
coseg + ^^=co„ .... (733)
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 . . (7345)
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 , ^/^ (7351)
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^ .... (737)
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
§ 73] 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. 72), 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,) = {aM{lf,^)(abf,)\
Since ^ is a positive constant,
/(_00) =  CO,
/(+ OO) = + 00,
and further
/(l) = (a + 6)^
f(+l) = (ab)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. 73. 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. 73) 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. 73
J"(/^)
The three cases considered are illustrated by {a), (b) and (c)
respectively in Fig. 731, which exhibits the curve traced out by
the centre of mass on the sphere with centre, 0, and radius h.
Fig. 731
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
§73] 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 subcases. 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. 73. 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. 73 {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. 732, 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. 732
. 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. 733) 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.
§ 74. 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. 72. 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 . . (801)
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 = ... (802)
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 (803)
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^ = . . . . (8032)
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 . . . . (8034)
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
(8041)
^''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 coordinates be at the centre of the
§81] 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 (8032) and (8034) 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 coordinates. The conditions for equili
brium are expressed by equations (801), 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
(801) 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 (§ 61).
§ 81. 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
§82] 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).
§ 82. Generalized Coordinates
We shall now introduce the generalized coordinates of § 6.
The rectangular coordinates of any particle, s, of a system may
be expressed in terms of the generalized coordinates, 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 coordinate, 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^. . (82)
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, dqi_, dq^, . . .
dq^^ of a virtual displacement of the system are obviously quite
98 THEORETICAL PHYSICS [Ch. V
arbitrary, since the generalized coordinates 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 (§ 53).
(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 . (8221)
' (^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^ .... (8222)
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 (822) can be written in the form
2^ =^igi +^92^2 + . . Mn • • • (823)
or briefly . 2T = pjq^
where p, = Q,4^ + ^^2^2 + . • • + Qan% • • (8*24)
or i>, = Qa^q^'
§82] PRINCIPLES OF DYNAMICS 99
The quantity p^ is the generalized momentuin correspond
ing to the coordinate q^. Differentiating 2T partially with
respect to g„, we get (see § 64),
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 subdeter
minants or minors having its sign so adjusted that, for example
101 =QM\.i + QM\..+ . . . +QM\^n (8251)
then JJ„^=1^ (8252)
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^ (828)
in which it is easily seen that
We may term ^1, (1)2, etc. the generalized forces corresponding
to the coordinates q^, q^, etc.
6 =I^{f ^ 4 F ^' \ F ~'\ (8281)
§ 83. 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 (16291695). In § 51 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.
§83] 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 (§ 51), a function of the coordinates 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 ■ • • • (84)
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 . 8402
dV
or  ^qa>
This must be equal to the corresponding increase of the kinetic
energy, T, Therefore
dT = ^^dq^ (841)
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) ... (842)
=tv
+ '>
d{T + V) 
'>■
d(T + V) _
dt
dv
dt ■
§84] 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 (822) or (826), we get from (8421)
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
(8423)
cp^ cp^
and therefore by (827)
dH _dT
dp, dp,
dH
'^  dp.
dq, dH
dt dp.
or .^ _ ^ (843)
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 823)
dT dT
^^=f>+f> • • • («^^^
DT
Subtracting the last of these from the first, and replacing—
by q, (827), we get
dT==pMa^dq^, . . . .(8441)
and the second equation (8*44) may be expressed in the form
dT=pMa + ^^dq^ . . . (8442)
dT
since Pa=;^—' Hence by comparison of (8*441) and (8*442)
dq^
we find
f: = ^ (845)
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 (8451)
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 (841) we have
dt dq^ dt
therefore
dT dp^ D{T + V) dq, _
dp, dt Dq, dt
§85]
PRINCIPLES OF DYNAMICS
105
and replacing ^— by ■— (827), we obtain
\dt ~^ dqj dt '
and consequently also
/dp^ _^_L\dq^ ^
\ dt dqj dt
(See 8451).
This suggests, though it does not prove, the equations
dt dq^
and
dt dq^
(846)
(8461)
Their validity will be established in § 86. The equations 846
together with (843) are known as Hamilton's canonical
equations. The equations (8461) are the equations of
Lagrange and are usually written in the form
dt\dqj dq^
(8462)
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
§ 85. Illustrations. Cyclic Coordikates
As a first illustration we may take the case of the compound
pendulum § 65. Here we have one q, which is, conveniently,
the angle 0, Fig. 65.
The energy equation is (6501)
i/C^y  Mgh cos 6> = ^ . . . (85)
and system is conservative. The corresponding p is
^=W='dt • • • • (8501)
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, § 55, of a particle moving under
the influence of a central force — . We get (see 5*51) for the
Hamiltonian function
H=f + /^+? .... (851)
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,
§85] 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 (852)
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 coordinates 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 coordinates are termed cyclic coordinates.
108 THEORETICAL PHYSICS
The Lagrangian function derived from (551) 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 § 73) gives us
+ iC {^ + cos dip}^ + Mgh cos d = E . . (853)
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 (8532)
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 (731), (7*32) and (7'33) which we have found already.
The preceding examples illustrate conservative systems, in
§86] 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.
§ 86. 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 coordinates 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. 86).
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 (811), if applied to the type
Fig. 86
110 THEORETICAL PHYSICS [Ch. V
of djmamical system dealt with in §§84 and 85, 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. 86) 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. 331
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 . . . . (861)
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 § 33. 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^ '
§86] 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, (§ 82),
dV
dql
dq^ = (8612)
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^
(862)
<i) "" <i)^
jr+^j:+d/a:+Jjt:j
pc^doo
Fig. 861
With the aid of (8*62) we may now express (8*612) in the form
(XiX „
\^M.)
dt
m„
and therefore (§ 84)
\dt J 'dt \dtj 'dt \dtj
/dx,\^ , /dy\^ , /dz\^\ddt , 37.
0,
<^w!'
ST 2T'^ + dV ^^c
dt dt
and by (8421
l^ipM.}ST2Tf + SV§St
= 0,
0.
112
Now
THEORETICAL PHYSICS
dr ^ dt dt '
[Ch. V
therefore we find
'^^ipM.mST2T'^ + 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 = . . (863)
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 (863) 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
(8631)
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. . . (8632)
Padq. + (t t,)dE = d^
or, if we use the symbol A for the integral on the right,
Pjqa + (i ii)dE = dA ... (8633)
The function A is one of those to which the term action is
applied and (8633) indicates that it may be expressed as a
function of the g's and E, and therefore
dA
Pa =.
t. =
dq:
M
dE
(8634)
§86] 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 = (8636)
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 coterminous 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 = .... (864)
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 (865)
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 coterminous (in space and
time) at the lower limit only we get from (8*63), dropping the
upper index, 2,
Pa^qa ESt = dS (866)
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 . . . . (8662)
We therefore find
2
J (i>A + g«^i'.  g^^l.  ^%)rf« = . . (867)
I
But we have proved, (8*43), that
dH
qa = ^,
therefore (8*67) becomes
2
j (pJqa  ^f^^y^ = 0.
§86]
PRINCIPLES OF DYNAMICS
or
i(4^''af*'>' = »'
by (8662).
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 = . . . . (8675)
and if we substitute for E and the p's the expressions in (8661)
we get Hamilton's partial differential equation
. (868)
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 .... (8681)
116 THEORETICAL PHYSICS [Ch. V
§ 87. Jacobi's Theobem.
Hamilton's principal function, S, defined by (8*65), is a
function of the coordinates, 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 coordinates, 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
§ 87] 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, = (871)
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^(8711)
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^
. . . . (8712)
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^
. . . . (8713)
By (87 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 (8712),
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 (868) 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
§87] PRINCIPLES OF DYNAMICS 119
variables but takes account of the g's contained in the ;:'s,
./8^\ dq.dq^ dq^
Hence by (87 B),
d^S dH d^S dH d^S
dq^dt djti dqidqi dn^ dq^dq^
+ 1^^+^^ = 0.(8.715)
On comparing (8714) and (8*715) we find
djii _ dH
W ~ ~ dqi'
and we can establish the validity of the remaining equations
(8713) 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 (8633).
If E is constant (conservative system)
S =A E X time .... (872)
dS „ dS dA
and = _ ^ =
dt dq, dq^
Let us take the constant a^ (87 A) to be E ; then Hamilton's
differential equation (868) becomes
'^^ ^ a, (873)
K« '■)
From (872) we get
as _9^_^
dai dai
or ^, = f£ _ «, by (87 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{utx) ... . . (901)
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^ . . . (9012)
so that at a fixed point on the cord
^ = ^ cos [cot — const.) . . . (9013)
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.  ^) . . (9014)
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 (9012) 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 (901) 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= <■
(903)
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
11=° "•«"'
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. 901) 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. 901
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
, (9051)
u
±J\ (9052)
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 . (907)
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(xut)
= 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. 902
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. 902). 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. 902) and
m
F{x) = 0.
Therefore f{x) differs from zero for values of x between a and c
§9] WAVE PROPAGATION 127
(Fig. 902) 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, (908)
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) (9081)
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 . . (9082)
u
in which ^^iVg and B^N^ (of 9081) 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 . . . (909)
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 BernoulliFourier 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{Lx) = f{L+x), .... (9091)
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/Lx \dt / L+x
(9092)
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. 903), and assume
the initial velocities to be zero, i.e. F(x) = 0.
^
The appropriate Fourier expansion (see § 41) 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 . . (9092)
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.
§ 91. Waves with Varying Amplitude
The type of wave represented by equation (9*01), which
we may term a onedimensional 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 onedimensional 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(utx) (91)
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) . , . . (911)
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.
§ 92. 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
coordinates 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 coordinates
X', Y\ Z' with the same origin as Z, 7, Z (§ 2:2), so that
^ =f{ut (Ix' + my' + nz')} . . . (92)
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,
§92] 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^ 2V72
dt
(921)
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(utr) .... (9221)
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>) . . . (923)
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.
§92] 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 .... (924)
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 § 92) 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 ^ = ^ "^'(^^^ .... (925)
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 .
§ 93. 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(^  ,) . . (93)
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 . (9301
(t X
or w = A' cos 2ti\  — 
where A' = 2A cos 27eia()^  iA(^)^ . (9302)
93]
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. 93.
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. 93
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)
(931
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.
§ 94. 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
coordinate. On the other hand the phase, in the case of a plane
sinusoidal wave (see 9*02), may be put in the form
^Ki5'
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
§94] WAVE PROPAGATION 139
where ac is a constant of suitable dimensions. It is usual to
represent — by Ji, so that
h
E =  . (94)
S
='63
The phase velocity of the wave will evidently be
u=^~ (941)
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? (942)
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^ (943)
where c is a universal constant with the dimensions of a velocity.
We shall have from (942)
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(^lJ)~* .... (944)
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
§ 95. 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 coordinates 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) (95)
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) (9501)
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 + ^, (951)
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;) . (952)
Now it follows from (951) that
ii — i = Xi — X { ai — a,
and we have therefore
ii i = x,x+ pjx, X) + p(y, y) + g^(^i z) (9521)
and corresponding expressions for 77 1 — ?^ and f 1 — C
In these equations, xi_ — x, yi — y and zs_ — 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, . . . .
. (9522)
From (9521) and (9522) we get
/^ , da\ . da . da
«.='l+'.('+l)+'l'
^. = '4 +'. +<'+!) ■
. (9523)
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
§ 95] 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), . . . (953)
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^ (I2I, ^2^, C2 f), . . .(9531)
p3 = (1^3 — i, rjs — ?7, Cs — C).
The vectors r will determine a parallelopiped the volume of
which is (§ 21)
^1x5 ^lyj 'Is
(9532)
(9533)
' 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 (9523) for the ^'s in (9533)
we get
(' + S) + ^'S + '4? '4x + '^^{' + 1) + ^'
dy
^8^^
da
X ^Q^ *"'% "^ '''''^■' ""^^
da
'■dz
dx ■ ''"
dy dy
''dy
V^dy)^''^dz'
^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
(9534)
(9535)
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)) . (9536)
 . , ^ > Increment in volume
dlv (a, (i, y) = pgj, ^^^ ^^j^^g
(954)
(955)
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 (§23). 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
§ 95] 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 (9561) for x, y and s, we shall get equations of the form
y = /^2il + /^22^ + i^asC, . . . (9562)
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^ = () . . . (957)
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^ = , . .(9571)
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.
§ 96. AisALYSis 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
a2 + 6^2 __ cj2 _!_ 2/9/C + 2^C + ^Un = ^' . (9601)
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^ . . . (9602)
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 (9601)
or (9602) 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 coordinate axes are
parallel to the principal axes of strain, equations (956) or
(9561) take the form :
<
■+s>
^ = 2/(1 + 1), . . . . (9603)
<'+!
)■
§ 96] ELASTICITY 147
the ^, — , ^, etc., vanishing. Similarly equations (9*562)
ox oy cy
become
ox
 *^ .... (9604)
(' 4)
0?/,
SO that the equation (9*602) of the strain ellipsoid is
J^2 ..2 ^2
+ , ^ c... . + . . .. = ^' • (9605)
('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,xq^z, (961)
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, . . . . (962)
 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 (9622),
«■='•(■+ 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 (961). The rest is
§96] 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 (9641)
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,, .... (965)
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 ... (966)
In Fig. 96 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 coordinate
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 coordinate
axes, (966) becomes
8,,x^ + S,,y^ + S,,z^ = M . . (9662)
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 (9662) 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 9661). 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. 96 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
§ 96] 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, (966) or (9'662), obtained by assigning to M every
positive and negative value, are separated by the asymptotic cone
^..x^ + S^^jV^ + S,,z^ = . . . (967)
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^ = . (9671)
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. 961, 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. 961
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
§97]
ELASTICITY
153
coincidence with rs, the faces p's and q'r will be inclined to
ps and qr by an amount
= 2£ = 2e, (969)
(see Fig. 962). The angle ^ = 2e is usually taken as a measure
of the simple shear. If the sides, jps and sr, of the unsheared
Fig. 962
Fig. 963
block are parallel to the coordinate axes, it is evident that
(j> = Apsp' + Arsr' (Fig. 963),
(9691
or
<^ = ?? + ?^
dy dx
The physical meanings of aU the components of the strain tensor
are now evident.
§ 97. 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. 97) 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
crosssection, 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. 971) 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. 971 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 (97)
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 (9701)
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. 971) is directed.
By definition, therefore,
P.n= t.n (9702)
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. 972). 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.
§97]
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
(9702). 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,) . . . (971)
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 socaUed 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. 972
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
(972)
156 THEORETICAL PHYSICS [Ch. VII
The quantities
^xx^ '^xy^ ^xzi
^j/aj' ^w ^i/a'
t.x, %y, i.. (9721)
constitute a tensor of the second rank, as the form of the
equations (972) 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. 973 973) 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 § 23.
§ 98. 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
§ 98] 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 lefthand 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 righthand 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\ . . . (981)
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 . . . (982)
which is the equation of a quadric surface. It is called the stress
quadric. Obviously a suitable rotation of the coordinate axes
reduces (982) to the simpler form
T^x^{Tyyy^ + T,,z^ = M , . . (9821)
The new coordinate 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 coordinates, 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,,) (983)
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 crosssection to be an equilateral right
angled triangle (aoh, Fig. 98), 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
§99]
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. 98
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
§ 99. Force and Stress
We shall next consider the resultant force exerted on the
^yx=t^ds
Fig. 99
medium within a closed surface in consequence of a state of
stress. Its X component is (see Fig. 99)
or
160
(§ 97).
THEORETICAL PHYSICS
This is equivalent (§ 32) to
dx dy dz
F =
■^ X
dx dy dz
[Ch. VII
. (99)
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 . . . (9901)
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
(991
dx
dy
dz
dx dy dz
(991
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), (9911)
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 coordinates 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. 991. The X com
djc
Fig. 991
§ 99] 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 § 95 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
rlp
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 (16351703)
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
(§ 98). 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 (§ 96) 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 .... (10011)
. ^ 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 (1002) become
» ('?)!+<'+ IF
3jdy'
dy'
Eliminating ^ we find
dy
and for ratio, ^ = ~ J^/
dy/ dx
_ %k — "In
^ ~ 2(3F+T)
The constant
(1004)
_ _^nk_ (10041)
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 (10041) enables us to find the bulk modulus, k, from
the experimentally determined values of Y and n.
166 THEORETICAL PHYSICS [Ch. VII
§ 101. 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^
§ 101] 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 . . . . (10101)
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 (1011)
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 (1012)
§ 102. 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 socalled 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 (10201)
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 § 92
enables us to infer, therefore, that when a small strain is pro
duced in an elastic solid two waves will travel outwards from
§ 102]
ELASTICITY
169
the centre of disturbance, a longitudinal (or dilatational) wave
with a velocity
V
f
» +
('+?)
. (1021
and a transverse (or distortional) wave with a velocity
10211)
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
^ (10212)
4
The expressions (1021) and (10211) 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 (10201) 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. 102) represent an element of the rod dx in length
parallel to the X axis and suppose x to be the coordinate 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 crosssection 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
(1022)
§ 103] 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.
§ 103. 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 coordinates we have
X
a = w,
r
y = ^w, .... (10301)
r
172 THEORETICAL PHYSICS [Ch. VII
where r = {x, y, z) and x, y and z are the coordinates (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
(rr> + PT • • (^"^^^^
diy {a, l^,y)=w + ^^ . . . (1031)
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 coordinates 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; = . . (1034)
dr"" dr ^ ^
§ 103] 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 = . . (10341)
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^^ (1035)
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 (1002)
and on substituting for w the expression (1035) we get
p^ = UA ^ . . . . (1036)
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\^ (10361)
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 (1035) for w in equation (1031).
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.
§ 104. 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 crosssectional 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
§ 105] 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 coordinate 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} . (104)
This expression represents the strain energy per unit volume in
terms of the principal elongations e, / and g and the moduli
k and n,
§ 105. Equation of Contintjity. Prevision of Relativity
It will be remembered that a distinction was made between
the coordinates (x, y, z), which refer to the positions of portions
or elements of the medium in its undisplaced or undeformed
condition, and the coordinates (, r], C) which refer to actual
or instantaneous positions at some instant t. In the present
paragraph we are concerned with the latter coordinates 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
acrosssectional 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. 105
If
{QC, dS).
By the theorem of Gauss (§ 3) this is equal to
[ OX dy dw '
(105)
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
10501)
(1051)
This result is true for any volume and therefore true when
the volume is simply the element dx dy dz. We may therefore
§105] ELASTICITY 177
drop the symbols of integration and so obtain the important
result
d(Qu) _^ dJQv) _^ dJQw) _j_ gg _ Q ^ ^ (1052)
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 (§ 99), the in
stantaneous coordinates of the part of the medium considered
and the t^.^, t^y, t^.^ are functions of these coordinates. On the
d^a
other hand the a, in ^^^, on the right is regarded as a function
ot
of t and the coordinates 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
coordinates 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 053)
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. 105) lead us to the
conclusion that
XmgUg'Vg'dSy
is the X component of the momentum which crosses the boundary
dSy (see Fig. 973) 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) _ ^ no54)
dx dy dz dt ^ ^
§ 105] 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 fourdimensional
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) = . (1055)
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
§ 106. 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
§ 106] 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 coordinates 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 coordinates of a particle of the medium at the instant t,
or as the instantaneous coordinates of the particle. It is how
ever sometimes convenient to use as independent variables the
coordinates (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 (§ 105) that this limiting value is
dt dx dy dw
182
THEORETICAL PHYSICS
[Ch.
VIII
Therefore
Dy)
Dt
dy)
~dt '
ox
dy
dw
(10702)
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.
§ 106] 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 lefthand 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('^^^—] — ^rii'^)  (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 . . (10721)
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.
108] HYDRODYNAMICS
[ence in such a case
dx
dy
185
«,= _i(,+^,) =2 . . (1073)
This means that if at any instant ^o a velocity potential ^o exists
there will always be a velocity potential ^ = ;^ + 0o.
§ 108. 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
. , . . (108)
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. 108. The case
{a) represents irrotational motion, the true rotational motion
Fig. 108
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).
§ 109. Theorem of Bernoulli
Turning to the first of Euler's equations (10'7), let us subtract
from it the identical equation
or 54^ == % + '^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,, (109;
§ 109] 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>^ . (10901)
If the motion is irrotational as well, co^ = My = co^ = and
consequently
77 + F 4 Jc2 = constant . . . (1091)
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 (10902)
Multiply equations (10901) respectively by dx, dy and dz and
make use of (10902). We find
— d(n + F + Jc2) = 2A {uwcOy — uvco^
+ VUCO^ — VWCO^ ~\ WVCOg, — WUCOy }.
The righthand 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 . . . (1091)
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. 109 (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 crosssection 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. 109
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. 109.
The stream lines converge in the neighbourhood of the orifice
(see Fig. 109 (6)) until at a place C, a short distance outside
the vessel, the crosssectional 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 crosssectional 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 § 51 and Fig. 51). 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 coordinates) 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 § 31. Applying the method of § 31 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/_\
(1103)
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 (1104)
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
§ 111. 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 (31) 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 (§31) ^^ means differentiation in the direction of the
dn
outward normal to the surface.
§ 112. 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 coordinates ;
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 (11201)
§ 112] 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
^ = .... . (11202)
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 (1101). 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^'« .... (1122)
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
^='# (1123)
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 coordinates.
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
§ 112] HYDRODYNAMICS 195
screw which is travelling along the Z axis in a positive sense.
The coordinates r, and ^ are known as polar coordinates
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> . . . (11242)
Introducing the new coordinates 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^ .... (1125)
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 onehalf the mass of
the fluid flUing a volume equal to that of the sphere.
196 THEORETICAL PHYSICS [Ch. VIII
§ 113. 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^' . . . (11311)
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)
§ 113] 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 . . . (11331)
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
^•)
(1134)
z .
In deep water therefore, where  is very great, the expression
for the velocity approximates to
V 27t
(1135)
§ 114. 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 crosssectional 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.
§ 114] 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 =
^^^4f
. . .
. (1141)
which becomes, if co is constant,
q = to+4 . . . . (11411)
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 (11412)
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=^^" .... (11413)
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. 108, (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.
§ 114] 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) = .... (1143)
when the surface integral is extended over the closed surface
and dS has the direction of the outward normal. If now the
Fig. 114
closed surface be part of a vortex tube bounded by two cross
sectional surfaces A and B (Fig. 114), 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 crosssectional faces A and B, and so
f [(todS) + f f(todS) =0 (11431)
A B
The vector dS, having in (11431) 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 (11431) becomes
[[(todS) = rr(todS) . . (11432)
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
§ 115. 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 § 105, 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' . . (n52..
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.
§ 116] MOTION IN VISCOUS FLUIDS 205
§ 116. 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 coordinates, 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, (1161)
 Il + fc\7'w = 0.
It follows at once that p is constant over any crosssection 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^) . . . (lleil)
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 lefthand member of the equation is
a function of z only, and the righthand member a function of
r only. It follows that
dz
t<L(r^^\=G . . . (11621)
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^) .... (1163)
4f^
§ 116] MOTION IN VISCOUS FLUIDS 207
We shaU now deduce an expression for the volume of liquid
passing any crosssection of the tube per second. The area of
the part of the crosssection 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 crosssection per
second,
R
I*
Q = 27t\ wrdr.
R
G
Consequently Q = 27i [^(r^  Rh)dr,
or Q = ^ ..... (1164)
From the first of the equations (11'621) we have
Q^ P^ i>i .... (11641)
i
where pi and ^2 ^^^ the pressures at two crosssections separated
by the distance I, the liquid flowing from 1 to 2. On substi
tuting in (11*64) we obtain the weUknown formula of Poiseuille
Q = ili:zJP^ .... {1165)
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) (1166)
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. 116, 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. 116 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^
§ 117] 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 crosssection to another. The volume — call it
W — flowing per second through the crosssection 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 crosssections. 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)
§ 117. 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 coordinates — 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
(11522) 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^^ (1172)
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)
§ 117] 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. . . . . (1173)
Substitution in (11*72) leads to
dp ^v72JL
5   'a,'*
while substitution in the divergence equation,
div c = 0,
leads to
^' = V2^ .... (11732)
We proceed further by adopting the simplest method of satisfy
ing (11731) and (11732), namely
p =  pS7^4>,
SJhjo^^O (1174)
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$ .... (11741)
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^=— (11742)
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 12 suggests putting
^ = ^^ + 5^ + ^! • • • • (1175)
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 § 112). Consequently
p = /.V^i • • (11751)
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^ '
§ 117] 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 + #^ • • • (1176)
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 (11762)
214 THEORETICAL PHYSICS [Ch. IX
On substituting these values for A, a and b we get
^ = <'«*^' + ^ (1177)
P=^^ (1178)
OJ)^^ (11781)
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 . . (11782)
(Fig. 117). 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) . . . (1179)
Now
, /,, , ^n'\dw , /,, 2n'\/du , dv\
§ 117] MOTION IN VISCOUS FLUIDS
by § 115; or
since div c = ; and
. jZw . du\
Replacing n' by /x we get
4. =  i> + 2/^^,
/dw , du\
215
(11791)
Fig. 117
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 (1178)
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 .... (11792)
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
§ 118. 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).
§ 119. 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. 119 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. 119), their extremities will be uniformly distributed
over a sphere of radius c. It is to be understood that an element
§ 119]
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
""^^ (119)
n =
4:7Z
n'
(11902)
Interior
If we take for d£t the solid angle contained between the polar
angles 6, + d%, ^ and </> \ d(l) (§ 112) we shaU have
dSl = sin e dO d(l> . . . (11901)
and we may express (119) 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. 1191) the volume of
which is
c dt dS cos 6,
and therefore equal to
n' cdtdS cos 6 . (11903)
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. 1191
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 . . . (11904)
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
3dS.
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 (1191). The identification of
heat and kinetic energy will make the righthand 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+ (11912)
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^, 461 x 10*
and 492 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 coordinates 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 nonexistent.
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 (11911) that
pv = tNaT,
or, if we write k for !«,
pv =NkT (1201)
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 (1201) 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 (1201) in the form
pv=BT (12011)
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 = lNkT + constant .... (1202)
where iV is the number of molecules in a gram. The specific
heat of the gas at constant volume is therefore
or c, = jJ? (12021)
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 .... (12022)
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
§ 121] 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 . . . . (12023)
and consequently the ratio
r = ^ = 1 +  . . . . (1203)
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.
§ 121. 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 § 119. 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 (§ 119). Let us represent the velocities of the individual
molecules by points on a diagram (Fig. 121), 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. 121
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 . . . (121)
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. 121
which represent molecules. At all points on the surface of a
§ 121] 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 (1212)
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<^^' . . . . . (1213)
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(^) (12131)
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,
§121] KINETIC THEORY OF GASES 229
and by the second equation (12'131)
A jr*a* = 1
or A == a^Tz^ .... (1214)
If we use rectangular coordinates, the number of molecules in
the element of volume du dv dw of the representative space of
Fig. 121 is
NA^e(^'+^'+^'Hu dv dw . . . (1215)
or NA^e'^^'c^dc sm 6 dd dct> . . (12151)
if we use polar coordinates 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 (12131)
K = 2nmA^^n^as
or K = 2nmam~i.l7i^a~^,
and therefore
K = imai .... (12152)
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 12152) we
get from (12152)
and therefore
« = 2& (^2154)
We thus find for the number of molecules per unit volume
of the representative space
g . . . . (1216)
V
m^^
This is Maxwell's law of distribution of velocities.
Starting from (12151), 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 (12131) this is
c = 271 A^a^
and since A = a^7t~^
c = 27ia^7c~^a~^
or c = 27i~^a~^
Therefore
{cY = _ (1217)
But we have seen that
a7c
a
or fmc^ = I—,
a
and therefore
c2 = i .... (12171)
2a
so that (1217 and 12171)
Q77.
c^=^(c)^ . . . (12172)
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
§ 121] 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. 1191) 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 .... (12181)
For the kinetic energy passing through the unit area per second
we get in a similar way
^^nA^ f ^2^a.dx . . . (12182)
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 (12182) by (12181). This yields
K' =ma^ .... (12191)
On comparing this with K (12152), we see that
K' =\k (12192)
§ 122. 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.
§ 122] KINETIC THEORY OF GASES 233
c, which the molecule has travelled, by the number of collisions,
we get
Ao=V ..... (122)
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. 122, 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. 122
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
(1221
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 (12211)
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 (122 ) 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 (12211).
In arriving at (122) and (1222) 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 (1221), 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 (1223)
Therefore the number of collisions made by it in the unit time
will be
nna^ (12231)
§122] 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} . . (12232)
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 . . . (12233)
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^as
or since A = a%~*,
V = 2V2'\7i^a^nNai ... . (12234)
The total length of the paths is
Nc=N2n^ai . . . (12235)
by (1217), and therefore on dividing (12235) by (12234)
we find for the mean free path
A=i (1224)
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
178
102
095
065
§ 123. 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 (§ 115).
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 § 115 now simplifies to the single component
^'V,, and by (USl)
or r„ = ^J' (1231)
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
§ 123] 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. 1191, 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. 123
(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. 123) 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 . . . (12311)
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 § 119 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 . (12312)
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^^ . . . (12313)
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:^ . . . (12314)
Q dz ^ '
Subtracting (12313) from (12314) 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 = ^ ...... (1232)
The I in this formula will not be very different from X^ of (1222),
so that we obtain as an approximate expression for the viscosity
11=^ (12321)
or, by (1222)
[ji = — , .... (12322)
A rigorous calculation of /jl for spherical molecules leads to
[ji = 350 . . . pcX . . . (12323)
1
where X =
V2
na^n
and consequently a = — ^= . . . (12324)
§123] 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^^^ (1233)
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, . . . . (12331)
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% (12332)
where a is a constant in the neighbourhood of 25 for monatomic
gases, such as helium and argon, 19 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 = 25, 19 and 175 for mole
cules with one, two and three atoms respectively.
§ 124. 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 coordinate, 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
§ 124] KINETIC THEORY OF GASES 241
Following the method of calculating the viscosity in § 123, 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 (§ 123).
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
.... (12401)
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 _ _ (12402)
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
§ 124] 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
PiP2 = 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 nondiffusing
gas, or the pressure it would exert if it occupied the palladium
vessel alone. Similar phenomena are associated with diffusion
in liquids through semipermeable 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 semipermeable vessel over that outside, when equi
librium has ultimately been reached, is naturally associated with
the dissolved sugar and is called its osmotic pressure.
§ 125. 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 (§ 119).
Let us now examine some of the consequences which ensue when
§ 125]
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. 125)
will reach a plane CD, separ
ated from AB by the distance
cr/2. Let us replace the cylin
der BCDE of Fig. 1191 by the
cyhnder DCEF of Fig. 125,
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 recalculate 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. 125
This formula might
(12.5;
but
N
V  nam
3
(12501)
246 THEORETICAL PHYSICS [Ch. X
If now the cylinder DCEF (Fig. 125) were in the interior of
the gas, the space within it available for the centres of molecules
would be
V  %tGm
cdtdScosO 1 . . (12502)
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. 125). 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 ^^^ . . (12503)
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  33^
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
§ 125] 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 (1252)
we must amend (12*011) to read
p(V b) =RT (1253)
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 (1253) 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+^^{Vb)=BT . . . (1254)
which may also be written in the form
V^ (b\—\v^hV —=0 . (12541)
\ p / p p
The isothermals (constant temperature curves) according to
(12*54) or (12541) are diagrammatically illustrated in Fig. 1251.
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. 1251
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
§ 125] 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 (§ 174) which
indicate that the situation of this horizontal line (BE in Eig.
1251) 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. 1251) 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(^*) • • • (1255)
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 (1256)
We may find the critical pressure by substituting 36 (12*56)
for F in equation (12*55). This gives
P, = Ao (12561)
±0 2762 ^ '
Finally we get an expression for the critical temperature by
substituting the values (1256) and (12561) for the volume
and pressure respectively in van der Waals' equation (1254).
This will be seen to give
T, = ^ .... (12562)
' 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 (1257) express the theorem of
§125] 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. 1251) 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. 1252).
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. 1252
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. 1252) ; 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).
§ 126]
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 376.^
§ 126. 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 = lOOS.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 . , , . ,., ^,.,
(12322)
and we might add
V = =jTr , CL^UiVeiJ
LCXit tU
mc
• •
= :=<•
M = Nm,
.
(12172)
^ Quite recent experimental investigations of the relative masses of
the atoms of isotopes have led to a further very minute readjustment.
§ 127] 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 == 8315 X 10^ ergs per °C. (the same approximately for
all gases).
M = 2016 gram.
T = 273, if we chose the temperature of melting ice.
fi = 86 X 10"^ gram per cm. per sec.
b = 197 c.c. for a gram molecule and hence v = 4925.
When we substitute these data we find
a = 274 X 108 cm.,
A^ = 46 X 1023.
Obviously these numbers cannot be regarded as expressing any
thing better than the order of magnitude of a and A^.
§ 127. 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.
§ 128. 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{= 8315 x 10^ ergs per ° C.) is the gas constant for a
gram molecule. In this way Perrin found for N numbers vary
ing from 65 x 10^3 to 72 x lO^s.
§ 128] 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—  = — Sr + X,
dt^ dt '
m^. = >S^ + Z; . . . . (1282)
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 , ^
mxj = — Sx— + xX,
dt^ dt
and therefore
d / dx\ /dx\^ S d(x^) , „
Consequently f^^i.^)  .()^ =  f ^ + ^ (12821)
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 twothirds 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
(12822) becomes
S
or ^e=?^ ... (12823)
dt m m
This may be written
and therefore
s^=Ae^i . . . . (1283)
where ^ is a constant of integration. If t is sufficiently long,
the righthand side of this equation is not sensibly different
from zero, and we have
d^^ 2kT
and consequently
dt S '
— „ 2kT
S
or x^ = ^r ..... (1284)
_ 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 55 x lO^^ and
8 X 1023, his mean value being 688 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
§ 129. 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 • • (129)
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 (129) may be written
pdi.di, . . . diji,^, . . . di^, , (12901)
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 coordinates, 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,^^ . . . cZ2^,
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,
§ 129] 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 § 107 (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^, . . (12921)
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) (1302)
where / is any function, and H is the energy of a system ; for
if we represent ^ by /',
and r = 1^/' = iJ''
dp, dp'
§ 131] 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, § 121. 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 . . (1304)
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.
§ 131. 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
sl. 2. 3, . . .
E= 2] E,N, . . . . . (13i:
s1. 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, . . . . (13101)
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'
§ 131] 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 = — . . . (1313)
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%% .... (1314)
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,. . . (1315)
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
31. 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
s1. 2. 3. . .".
X ^sSfs =0,
s = l, 2, 3. . . .
2J ,5/, =0. ..... (1316)
s = l,2, 3, ...
Hence it follows that the most probable distribution is given by
log/3 +^^, +a = . . . . (1317)
where a and /5 are constants, and consequently
f,=Be^^^ .... (13171)
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^ . (13172)
where ^ is a constant, and hence
P = l (1318)
§ 133] STATISTICAL MECHANICS 267
The constant B can of course be expressed in terms of /3 (or 0),
since
S/, = 1 = 5Se^^^ . . . (13185)
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 (1319)
in consequence of (13*185) and (13*101).
§ 132. 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/jlog/, + log (l + ^^) I /, log/,],
[ore
Sy, =  Nz[sf, log/, + if, + a/,) log (l + j)].
Now when
dip = ip  iprr,,
this reduces to
which is essentially negative whatever the df^ may be, provided they are
small enough. Hence ipm is a maximum.
§ 133. 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, . . . S1. 2. 3, . . .
^ dB E.^
and consequently, on substituting in (13*30),
dxp^ = pdE,
or
(1331)
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 = (1332)
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 (1333)
§ 133] STATISTICAL MECHANICS 269
The conditions for statistical equilibrium are therefore
dE^ +dEs=0; . . . . (1334)
and, by (13'31).
J dEj,
dV>B = ^•
^B
On substituting for dy)j^ and df^ in equations (1334) we get
dE^_dE^_
^ Ob '
whence
0^ = 0^ (1335)
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
^+^ (1336)
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 121, and more
especially to equations (12), (1201) and (1216) will indicate
that we must identify S with hT. For the thermal equilibrium
of two assemblages (two gases for example)
Oa = O^. by (1335),^
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.
§ 134. 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=^ (134)
In consequence of this relationship h is often called the entropy
constant. It is also known as Boltzmann's constant.
§ 135. 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,^ + • . . (135)
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 coordinates 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 coordinate, s, is the same for all the co
kT
ordinates and equal to  or to — . The number of systems in
§ 135] 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 coordinate, 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.
(1351)
(13511)
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 (12131) 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 socalled 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 037 kilogrammetres per gramcalorie.
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
§151] 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 thermocouple ; 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
§ 1515. 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^ ...... (1515)
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.
§ 152] 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 (1516)
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
(1516) the form
pv = ET' (15171)
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.
§ 152. 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 (1516) and (15171) 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^' • • • (^52)
where a, b and E are constants characteristic of the particular gas.
276 THEORETICAL PHYSICS [Ch. XII
§ 153. 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. 153). 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
§ 153] 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. 1531
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 pressuretempera
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. 1531.
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 coexist.
§ 154. Work Done During Reversible Expansion
Let us imagine the substance to be contained in a cylinder
(Fig. 154) 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. 154 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 (154)
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 .... (15401)
^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. 153.
In the special case of the isothermal expansion of a gas, we
, . ^T'
have, smce p = ,
V
w = Rr[
J V
§ 154]
THERMODYNAMICS. FIRST LAW
279
or
W = ET' log ^
(1541)
or, since in this case
PlVl =P2V2,
W
Rriog^ . . . (15411)
The formulae are, of course, only approximately true for actual
gases. If we deal with a grammolecule 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 = 8315 X 107 ergs per °C.,
so that the work of expansion in such a case is given by
W = 8315 X 10^^' W !^^
If we use the practical unit of work, the joule, we have obviously
to give R the value 8315 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 198 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 (15401) 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. 1541) 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
§ 155. 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 (17281799) 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
001 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.
§ 156. 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.
§156] THERMODYNAMICS. FIRST LAW 281
This is a clear statement of the energy principle and Carnot's
puissance motrice is simply what we nowadays 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 4188 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=:dQdW . . . . (156)
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.
§ 157. Internal Energy of a Gas
Experiments carried out by GayLussac 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.
§ 158. 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
§ 159] 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.
§ 159. 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=dUipdv . . . . (1591)
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 (1592)
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 . . . (15941)
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 . . . . (15951)
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 „ „
§159] THERMODYNAMICS. FIRST LAW 285
from which we get, by substituting in ( 15*9 51),
J = 414 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^ + (yl)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 . . . (1596)
^Vi = constant . . (15961)
pvy = constant . . . (1597)
TV~^ = constant . . . (1598)
the two latter equations being obtained by eliminating T' and
V respectively in (15961) by the substitutions T' = pv/R and
V = RT/p.
The constant y, as defined above, is the ratio Cj,/c^,. Reference
to equations (1011) and (1012) 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
(15942) 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 = ±iidv,
dv
or dp =f(v),dv,
§ 16] THERMODYNAMICS. FIRST LAW 287
and therefore
dQ = &dv + Tpdv.
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 (1601)
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
§ 161. 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
§ 162]
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.
§ 162. 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. 162) 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. 162
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. 1621). The base of the cylinder is covered by
the source of heat, X, (Fig. 162), 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. 1621 engine just described, the work
done during a cycle is equal
(§ 154) to the area, W, of the closed curve (1, 2, 3, 4) on the
indicator diagram (Fig. 1621). We have therefore
W
Efficiency =^ (162)
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 {16201)
and hence
Efficiency = ^'^' .... (1621)
where Qi is the heat rejected to the refrigerator at the tem
perature ti.
§ 163. Cabnot's Peinciple
We shall now prove that the axiom of § 161 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
§163] 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^, ..... (163)
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
{163).
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 flywheel 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 § 161, 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. 163) 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^ 163 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.
§ 164. 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 (164)
' "2 2 .... (1642)
§ 164] 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, (1643)
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. 164
therefore
or
Gsteam " ^ice = 100 cj>,
I Vsteam Vice
</>
100
area bfgc
loo
(16431
Equations (1641) indicate that the zero isothermal on the
Kelvin scale is characterized by the property that no heat is
§ 164] 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
=
CeaJioe
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 socalled
absolute zero is very difficult to approach, and indeed there is
reason to suspect that it is a temperature which is unattainable.
§ 165. 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. 1621), 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 .... (165)
^^ 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. 1621), 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 ',
§166] SECOND LAW OP THERMODYNAMICS 297
and therefore equation (16*5) becomes
hence
which means that the two scales of temperature are identical.
§ 166. Entropy
We shall now introduce a quantity to distinguish the adia
batics — the term is used in the restricted sense explained in
§ 153 — on the p, v diagram, just as temperature distinguishes
the isothermals. This quantity is called entropy, a term intro
duced by Clausius (see § 134), 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. 164, 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. 166
which, on the work scale, differ by unity.
Consequently
(166)
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. 1621) 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. 166) 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.
§ 167. Entropy and the Second Law of Thermodynamics
According to the definition of entropy which we have adopted
(1643)
^2^i= ..... (167)
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
§164) from the statement (167). For consider any pair of
adiabatics with the entropies ^1 and ^2 (^2!><^i). Then
^.^. =^=;, by (167),
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
§ 167]
SECOND LAW OF THERMODYNAMICS
299
But this is the statement of the second law of thermodynamics
as given in § 164. 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. 167).
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. 167
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 righthand 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
#=¥
(1671
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^. . . . {16711)
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 ^ .... (1672)
1
which, for the special case of constant volume, reduces to
2
<t>,h= \^ ... (16721)
1
We shall adopt (16'71) as a final statement of the second law.^
§ 1675. 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 — . . (1675)
if we agree that shall be zero when the temperature and volume
are Tq and Vq respectively.
§ 168. 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.
§168] 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
(16721),
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> = (168)
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 submicroscopic 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 (submicroscopic) 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.
§ 168] 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
§ 169. 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 .... (169)
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
§169] THERMODYNAMICAL PRINCIPLES 305
assume their original values, while Q may have increased by a
perfectly arbitrary quantity determined by the dimensions of
the cycle (§162).
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
(1594) in the form
we have # 4(S)/^ 4{©. +^'^^
Therefore, by (1691),
T^iKdV/T '^^] ~^ T dTdV ^ TKdTjr
U^)^+p}=t(%\ . . . (1693)
TdVdT
(dip
Substituting this result in (1692) we have
■'«=(S),"+^(IV>''
or, if we are dealing with the unit mass of the substance/
dQ = c,dT + T(^^dV . . . • (1694)
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 . . . (16971)
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^\ (1698)
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
§169] 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 = (^)^ ...... (16981)
fdv\
We may get rid of ( ^ ) in the formula by the following device :
o=(gX.©.(i).
Consequently (,)^ = _ g)^(?
or
(^\ = ae, (16982)
since, by definition, the isothermal elasticity, e^, is
Now substituting the expressions (16981) and (16982) in (1698) we get
Cp  c* = T6j.a2F (16983)
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^) . . . (16992)
and c. = TaV\
(S), • • • • ^''''^^
308 THEORETICAL PHYSICS [Ch. XIV
Dividing the latter by the former we get
y
or y = ^_i =^A (16994)
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 (1698),
J?2
P 2a( F6)^ '
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 (1699)
TR /dV\
KdTJ '
c., =
(F6)
or dropping the subscript, ^,
= c,dT + RT ^^
V
§ 171] 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 . . . (1701)
/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> = Cvjff + R
T ^ ^ V 6'
T {V —b)
and = c. log y + 2? log _^ . . . (1702)
where Tq and Vq are the temperature and volume at which we have
agreed the entropy shall be zero.
§ 171. 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}. . . (171)
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 . . . . (1711)
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 ..... . (1713)
Or the external work done during such a process is done at the
expense of the quantity
F=U Tcf> (1714)
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 (1714) the equivalent expression in
(17141) we have
^=^+^(SX • • • ^''''^
This result is known as the GibbsHelmholtz formula. If instead
of the variables p and F, the equation of state contains other
corresponding variables, y and x, (1715) becomes
^ = ^ + <i). • • • • ^'''''^
The GibbsHelmholtz 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
§ 172] THERMODYNAMICAL PRINCIPLES 311
If we write the equation
Td^ =dU + pdV
in the form dU = Tdcf>  pdV .... (1716)
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 (1717)
It is called enthalpy {ddXno)^ = warmth, heat). We find for dG,
dG == {dU  Tdcl> + pdV) + Tdcj> + Vdp,
or dG = Tdct> + Vdp (17171)
Consequently T = (—] ,
'dG
\dp
'SI ■ ■ ■ <"•■«)
§ 172. 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
§ 173. 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 (§ 157) 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. 173
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 boxwood
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. 173), 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.
§ 173]
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^ = (173)
The corresponding condition in the original Joule experiment,
assuming no transfer of heat, is
AU = (1731)
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,
JouleKelvin.
TL^ + V/^p =
m AT + (f)
\dp/T
+ VAP = 0,
.dp/T
Ap = 0,
Ai> = 0,
Applying Maxwell's
relation (a) (172)
tCI)
KdTj,
,_ AT
^~ AP'
c,AT+\V
If
Ap = 0.
c,S + \V
t(E\
dT/,
0.
(1732:
TA4>
Joule.
AU==0,
pl^V = 0.
dcf>
'■©.A"
+(a^
PAV = 0,
 \P
AF = 0,
AF = 0,
Applying Maxwell's
relation [y] (172)
dp
AT\p
If
CvV
■Tm\
AT
AF = 0.
= 0.
(17321
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 JouleKelvin 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 JouleKelvin 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+
JouleKelvin.
1 KdT'J^dT
Applying (1733)
c' r + \v T
y dT'
T dT
dT
T
dT
T'\l{
c\r
0,
(1734)
Joule.
=•■'■ {.(r,).fh».
Applying the formula anal
ogous to (1733)
\ p J
V*' ''■«■•■/
§173] THERMODYNAMICAL PRINCIPLES 315
Taking the lefthand 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^
, . . . . (1736)
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 JouleKelvin 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 (1738)
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 .... (1739)
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 (17392)
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 (17393)
Differentiating (17392) 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 (1737),
V dT'
§175] THERMODYNAMICAL PRINCIPLES 317
We can eliminate p and F, by means of (17'392) and (17*393) and
thus obtain
~=UT')dT' (17394)
which, since the function /g is known, enables us to find T in terms of T\
§ 174. 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. 1251. 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.
§ 175. The Triple Point
When we plot the pressures associated with each of the
horizontal lines BF (Fig. 1251) against the corresponding tem
peratures we get such a curve as OA in Fig. 1531. For a given
point on such a curve, the function / (§ 171) 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
/.=/, (175)
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. (1751)
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.
§ 176. Latent Heat Equations
Consider two neighbouring points on OA (Fig. 1531). 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 (167),
where L is the latent heat of evaporation. Therefore
i = (F, F,)r^ .... (176)
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,
§ 176] 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 42 X 10' ergs per gram ;
Vi —Vs = — 009 c.c. per gram.
At normal atmospheric pressure, i.e. 1,014,000 dynes per cm.^, the equi
librium temperature (socalled melting point) is 273 on Kelvin's scale.
Therefore
80 X 42 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
— = — 73 X 10"' approximately,
or the melting point is lowered by 00073° per each atmosphere increase
in pressure.
The GibbsHelmholtz 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 lefthand 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. 176). 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. 176 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 .... (1761)
But we have already seen (§ 163) that this is equal to the area
of the closed loop, i.e. to
dT X {BC),
or dT(cf>,  <!>,),
or, finally,
^ ..... . (1762)
On equating (17*61) and (17*62) we get
c.c.=§§ .... (17.63)
§ 177. 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. 1531). 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
§177] 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 WillardGibbs, and known as the Phase Rule.
It may be stated in the following form :
F \P = C \2 (177)
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 (§ 171). Therefore
Sf = (1771)
If there are P phases, / is a sum of contributions from each phase, or
/ =/ +r +r + . . . +r^ .... (i772)
and consequently
df = 8f + 8f' + (5f + . . . +SfP^ = . . (17721)
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^^^ = . (1773)
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, . (1774)
and since the die's are arbitrary, we have
^ =
dx^ "'
^ = 0,
dx^
8f
•^ =0 (1775)
^^C(Pl)
These C{P — 1) conditions are necessary and sufficient for the equi
hbrium of the P phases at some given pressme 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.
§ 178]
THERMODYNAMICAL PRINCIPLES
323
§ 178. 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 semipermeable 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
(§ 124). When this equilibrium condition exists, the vapour pressure,
CL 
—
h
^
1
A
Fig. 178
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
(178)
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. 1781. 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 (1781)
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 (§ 128) 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 (1782)
where P is the osmotic pressure, V the volume of a gram molecule of
the dissolved substance, and R is the gas constant (8315 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 (1782)
Combining this with (1781) we find
RT^ a
^'^Mll (1783)
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 nonelectrolytic
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
§178] 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
WlllardGibbs : 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
BernoulliFourier 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 coordinates, 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
— fourdimensional, 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 coordinates, 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 coordinates, 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
Semipermeable 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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