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With the ytt4&k&s Compliments.

MATTER AND MOTION

MATTER AND MOTION/

BY THE LATE

J. CLERK JV^^XWELL

M.A., LL.D.EDIN., F.R.SS.L. & E.

HONORARY FELLOW OF TRINITY COLLEGE, AND PROFESSOR OF

EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE

REPRINTED : WITH NOTES AND APPENDICES BY

SIR JOSEPH LARMOR, F.R.S., M.P.

FELLOW OF ST JOHN'S COLLEGE, AND

LUCASIAN PROFESSOR OF MATHEMATICS

LONDON :

SOCIETY FOR PROMOTING

CHRISTIAN KNOWLEDGE

NEW YORK : THE MACMILLAN CO.

1920

PREFACE (1877)

PHYSICAL SCIENCE, which up to the end of the eighteenth

century had been fully occupied in forming a conception

of natural phenomena as the result of forces acting

between one body and another, has now fairly entered

on the next stage of progress that in which the energy

of a material system is conceived as determined by the

configuration and motion of that system, and in which

the ideas of configuration, motion, and force are

generalised to the utmost extent warranted by their

physical definitions.

To become acquainted with these fundamental ideas,

to examine them under all their aspects, and habitually

to guide the current of thought along the channels of

strict dynamical reasoning, must be the foundation of

the training of the student of Physical Science.

The following statement of the fundamental doctrines

of Matter and Motion is therefore to be regarded as

an introduction to the study of Physical Science in

general.

NOTE

IN this reprint of Prof. Clerk Maxwell's classical

tractate on the principles of dynamics, the changes have

been confined strictly to typographical and a few verbal

improvements. After trial, the conclusion has been

reached that any additions to the text would alter the

flavour of the work, which would then no longer be

characteristic of its author. Accordingly only brief

footnotes have been introduced: and the few original

footnotes have been distinguished from them by

Arabic numeral references instead of asterisks and other

marks. A new index has been prepared.

A general exposition of this kind cannot be expected,

and doubtless was not intended, to come into use as a

working textbook : for that purpose methods of syste-

matic calculation must be prominent. But as a reasoned

conspectus of the Newtonian dynamics, generalizing

gradually from simple particles of matter to physical

systems which are beyond complete analysis, drawn

up by one of the masters of the science, with many

interesting side-lights, it must retain its power of sug-

gestion even though parts of the vector exposition may

now seem somewhat abstract. The few critical footnotes

and references to Appendices that have been added may

help to promote this feature of suggestion and stimulus.

The treatment of the fundamental principles of

dynamics has however been enlarged on the author's

own lines by the inclusion of the Chapter "On the

Equations of Motion of a Connected System" from

vol. ii of Electricity and Magnetism. For permission to

make use of this chapter the thanks of the publishers

are due to the Clarendon Press of the University of

Oxford.

viii NOTE

With the same end in view two Appendices have

been added by the editor. One of them treats the

Principle of Relativity of motion, which has recently

become very prominent in wider physical connexions,

on rather different lines from those in the text. The other

aims at development of the wider aspects of the Prin-

ciple of Least Action, which has been asserting its

position more and more as the essential principle of con-

nexion between the various domains of Theoretical

Physics.

These additions are of course much more advanced

than the rest of the book : but they will serve to complete

it by presenting the analytical side of dynamical science,

on which it justly aspires to be the definite foundation

for all Natural Philosophy.

The editor desires to express his acknowledgment

to the Cambridge University Press, and especially to

Mr J. B. Peace, for assistance and attention.

J.L.

BIOGRAPHICAL NOTE

JAMES CLERK MAXWELL was born in Edinburgh in 1831,

the only son of John Clerk Maxwell, of Glenlair, near

Dalbeattie, a family property in south-west Scotland to

which the son succeeded. After an early education at

home, and at the University of Edinburgh, he pro-

ceeded to Cambridge in 1850, first to Peterhouse,

migrating afterwards to Trinity College. In the

Mathematical Tripos of 1854, the Senior Wrangler was

E. J. Routh, afterwards a mathematical teacher and

investigator of the highest distinction, and Clerk Max-

well was second: they were placed as equal soon after

in the Smith's Prize Examination.

He was professor of Natural Philosophy at Aberdeen

from 1856 to 1860, in King's College, London from

1860 to 1865, and then retired to Glenlair for six years,

during which the teeming ideas of his mind doubtless

matured and fell into more systematic forms. He was

persuaded to return into residence at Cambridge in

1871, to undertake the task of organizing the new

Cavendish Laboratory. But after a time his health

broke, and he died in 1879 at the age of 48 years.

His scientific reputation during his lifetime was

upheld mainly by British mathematical physicists,

especially by the Cambridge school. But from the time

that Helmholtz took up the study of his theory of

electric action and light in 1870, and discussed it in

numerous powerful memoirs, the attention given abroad

to his work gradually increased, until as in England it

became the dominating force in physical science.

Nowadays by universal consent his ideas, as the

mathematical interpreter and continuator of Faraday,

rank as the greatest advance in our understanding of

the laws of the physical universe that has appeared

x . BIOGRAPHICAL NOTE

since the time of Newton. As with Faraday, his pro-

found investigations into nature were concomitant with

deep religious reverence for nature's cause. See the

Life by L. Campbell and W. Garnett (Macmillan, 1882).

The treatise on Electricity and Magnetism and the

Theory of Heat contain an important part of his work.

His Scientific Papers were republished by the Cam-

bridge University Press in two large memorial volumes.

There are many important letters from him in the

Memoir and Scientific Correspondence of Sir George

Stokes, Cambridge, 1904.

The characteristic portrait here reproduced, perhaps

for the first time, is from a carte de visile photograph

taken probably during his London period.

J. L.

CONTENTS

CHAPTER I

INTRODUCTION

ART. PAGE

1 Nature of Physical Science i

2 Definition of a Material System 2

3 Definition of Internal and External .... 2

4 Definition of Configuration 2

5 Diagrams 3

6 A Material Particle 3

7 Relative Position of two Material Particles ... 4

8 Vectors 4

9 System of Three Particles 5

10 Addition of Vectors 5

1 1 Subtraction of one Vector from another ... 6

12 Origin of Vectors 6

13 Relative Position of Two Systems 7

14 Three Data for the Comparison of Two Systems . . 7

15 On the Idea of Space 9

16 Error of Descartes 9

17 On the Idea of Time n

18 Absolute Space 12

19 Statement of the General Maxim of Physical Science . 13

CHAPTER II

ON MOTION

20 Definition of Displacement . . . , . . . 15

21 Diagram of Displacement 15

22 Relative Displacement 16

23 Uniform Displacement 17

24 On Motion 18

25 On the Continuity of Motion ... . 18

26 On Constant Velocity .... . 19

27 On the Measurement of Velocity when Variable . 19

28 Diagram of Velocities . . . ... 20

29 Properties of the Diagram of Velocities . 21

30 Meaning of the Phrase "At Rest" . . . 22

31 On Change of Velocity 22

32 On Acceleration . 23

33 On the Rate of Acceleration ... . . 24

34 Diagram of Accelerations . . . ... . 25

35 Acceleration a Relative Term . . . * . . 25

xii CONTENTS

CHAPTER III

ON FORCE

ART. PAGE

36 Kinematics and Kinetics 26

37 Mutual Action between Two Bodies Stress . . 26

38 External Force 26

39 Different Aspects of the same Phenomenon . . . 27

40 Newton's Laws of Motion ...... 27

41 The First Law of Motion 28

42 On the Equilibrium of Forces 30

43 Definition of Equal Times 31.

44 The Second Law of Motion 32

45 Definition of Equal Masses and of Equal Forces . 32

46 Measurement of Mass 33

47 Numerical Measurement of Force 35

48 Simultaneous Action of Forces on a Body ... 36

49 On Impulse 37

50 Relation between Force and Mass 38

51 On Momentum 38

52 Statement of the Second Law of Motion in Terms of

Impulse and Momentum . . . . . . 39

53 Addition of Forces . . . . . . . 39

54 The Third Law of Motion 40

55 Action and Reaction are the Partial Aspects of a Stress 40

56 Attraction and Repulsion . . . ' . . . 41

57 The Third Law True of Action at a Distance . . 42

58 Newton's Proof not Experimental ..... 42

CHAPTER IV

ON THE PROPERTIES OF THE CENTRE OF MASS

OF A MATERIAL SYSTEM

59 Definition of a Mass- Vector ...... 44

60 Centre of Mass of Two Particles . . . . 44

6 1 Centre of 'Mass of a System 45

62 Momentum represented as the Rate of Change of a

Mass- Vector 45

63 Effect of External Forces on the Motion of the Centre

of Mass 46

64 The Motion of the Centre of Mass of a System is not

affected by the Mutual Action of the Parts of the

System 47

65 First and Second Laws of Motion .... 48

66 Method of treating Systems of Molecules . .- . 48

CONTENTS xiii

ART. PAGE

67 By the Introduction of the Idea of Mass we pass from

Point- Vectors, Point Displacements, Velocities,

. Total Accelerations, and Rates of Acceleration, to

Mass- Vectors, Mass Displacements, Momenta, Im-

pulses, and Moving Forces 49

68 Definition of a Mass- Area 50

69 Angular Momentum 51

70 Moment of a Force about a Point 51

71 Conservation of Angular Momentum .... 52

CHAPTER V

ON WORK AND ENERGY

72 Definitions 54

73 Principle of Conservation of Energy .... 54

74 General Statement of the Principle of the Conservation

of Energy 55

75 Measurement of Work 56

76 Potential Energy 58

77 Kinetic Energy 58

78 Oblique Forces 60

79 Kinetic Energy of Two Particles referred to their

Centre of Ma'ss 61

80 Kinetic Energy of a Material System referred to its

Centre of Mass 62

81 Available Kinetic Energy 63

82 Potential Energy 65

83 Elasticity 65

84 Action at a Distance 66

85 Theory of Potential Energy more complicated than

that of Kinetic Energy 67

86 Application of the Method of Energy to the Calculation

of Forces 68

87 Specification of the [Mode of Action] of Forces . . 69

88 Application to a System in Motion .... 70

89 Application of the Method of Energy to the Investigation

of Real Bodies j(B

90 Variables on which the Energy depends . . . 71

91 Energy in Terms of the Variables 72

92 Theory of Heat 72

93 Heat a Form of Energy . 73

94 Energy Measured as Heat . . . ... 73

95 Scientific Work to be done 74

96 History of the Doctrine of Energy . . . 75

97 On the Different Forms of Energy . . , . . 76

CONTENTS

CHAPTER VI

RECAPITULATION

ART. PAGE

98 Retrospect of Abstract Dynamics . . . . . 79

99 Kinematics 79

100 Force .79

101 Stress 80

102 Relativity of Dynamical Knowledge .... 80

103 Relativity of Force 81

104 Rotation . . 83

105 Newton's Determination of the Absolute Velocity of

Rotation . . . . .'.:. . . . 84

106 Foucault's Pendulum 86

107 Matter and Energy 89

108 Test of a Material Substance 89

109 Energy not capable of Identification .... 90

no Absolute Value of the Energy of a Body unknown . 90

in Latent Energy 91

112 A Complete Discussion of Energy would include the

whole of Physical Science 91

CHAPTER VII

THE PENDULUM AND GRAVITY

113 On Uniform Motion in a Circle 92

114 Centrifugal Force 93

115 Periodic Time 93

116 On Simple Harmonic Vibrations . . ... . 94

117 On the Force acting on the Vibrating Body . . 94

118 Isochronous Vibrations 95

119 Potential Energy of the Vibrating Body . . , . 96

1 20 The Simple Pendulum . . . ..... 96

121 A Rigid Pendulum 98

122 Inversion of the Pendulum 100

123 Illustration of Kater's Pendulum . . ... . 100

124 Determination of the Intensity of Gravity . . . 101

125 Method of Observation 102

126 Estimation of Error 103

CHAPTER VIII

UNIVERSAL GRAVITATION

127 Newton's Method . . 105

128 Kepler's Laws , . 105

129 Angular Velocity .106

130 Motion about the Centre of Mass . . . ; .. , . 106

CONTENTS xv

ART. PAGE

131 The Orbit . 107

. 107

. 108

. 109

no

. Ill

112

132 The Hodograph

133 Kepler's Second Law

134 Force on a Planet

135 Interpretation of Kepler's Third Law .

136 Law of Gravitation

137 Amended Form of Kepler's Third Law

138 Potential Energy due to Gravitation

139 Kinetic Energy of the System

140 Potential Energy of the System . . . .114

141 The Moon is a Heavy Body 115

142 Cavendish's Experiment 116

143 The Torsion Balance 117

144 Method of the Experiment 118

145 Universal Gravitation 119

146 Cause of Gravitation 120

147 Application of Newton's Method of Investigation . 121

148 Methods of Molecular Investigations .... 122

149 Importance of General and Elementary Properties . 122

[CHAPTER IX]

ON THE EQUATIONS OF MOTION OF A CON-

NECTED SYSTEM 123

APPENDIX I

THE RELATIVITY OF THE FORCES OF NATURE 137

APPENDIX II

THE PRINCIPLE OF LEAST ACTION . . .145

INDEX I62

Portrait of Prof. CLERK MAXWELL . . Frontispiece

MATTER AND MOTION

CHAPTER I

INTRODUCTION

i . NATURE OF PHYSICAL SCIENCE

PHYSICAL SCIENCE is that department of knowledge

which relates to the order of nature, or, in other words,

to the regular succession of events.

The name of physical science, however, is often

applied in a more or less restricted manner to those

branches of science in which the phenomena considered

are of the simplest and most abstract kind, excluding

the consideration of the more complex phenomena, such

as those observed in living beings.

The simplest case of all is that in which an event

or phenomenon can be described as a change in the

arrangement of certain bodies. Thus the motion of the

moon may be described by stating the changes in her

position relative to the earth in the order in which they

follow one another.

In other cases we may know that some change of

arrangement has taken place, but we may not be able

to ascertain what that change is.

Thus when water freezes we know that the molecules

or smallest parts of the substance must be arranged

differently in ice and in water. We also know that this

arrangement in ice must have a certain kind of sym-

metry, because the ice is in the form of symmetrical

crystals, but we have as yet no precise knowledge of

the actual arrangement of the molecules in ice. But

whenever we can completely describe the change of

2 INTRODUCTION [CH.

arrangement we have a knowledge, perfect so far as it

extends, of what has taken place, though we may still

have to learn the necessary conditions under which

a similar event will always take place.

Hence the first part of physical science relates to the

relative position and motion of bodies.

2. DEFINITION OF A MATERIAL SYSTEM

In all scientific procedure we begin by marking out a

certain region or subject as the field of our investiga-

tions. To this we must confine our attention, leaving

the rest of the universe out of account till we have

completed the investigation in which we are engaged.

In physical science, therefore, the first step is to define

clearly the material system which we make the subject

of our statements. This system may be of any degree

of complexity. It may be a single material particle, a

body of finite size, or any number of such bodies, and

it may even be extended so as to include the whole

material universe.

3. DEFINITION OF INTERNAL AND EXTERNAL

All relations or actions between one part of this sys-

tem and another are called Internal relations or actions.

Those between the whole or any part of the system

and bodies not included in the system are called Exter-

nal relations or actions. These we study only so far as

they affect the system itself, leaving their effect on

external bodies out of consideration. Relations and

actions between bodies not included in the system are

to be left out of consideration. We cannot investigate

them except by making our system include these other

bodies.

4. DEFINITION OF CONFIGURATION

When a material system is considered with respect

to the relative position of its parts, the assemblage of

relative positions is called the Configuration of the

system.

i] CONFIGURATION 3

A knowledge of the configuration of the system at a

given instant implies a knowledge of the position of

every point of the system with respect to every other

point at that instant.

5. DIAGRAMS

The configuration of material systems may be repre-

sented in models, plans, or diagrams. The model or

diagram is supposed to resemble the material system

only in form, not necessarily in any other respect.

A plan or a map represents on paper in two dimen-

sions what may really be in three dimensions, and can

only be completely represented by a model. We shall

use the term Diagram to signify any geometrical figure,

whether plane or not, by means of which we study the

properties of a material system. Thus, when we speak

of the configuration of a system, the image which we

form in our minds is that of a diagram, which completely

represents the configuration, but which has none of the

other properties of the material system. Besides dia-

grams of configuration we may have diagrams of velocity,

of stress, etc., which do not represent the form of the

system, but by means of which its relative velocities or

its internal forces may be studied.

6. A MATERIAL PARTICLE

A body so small that, for the purposes of our investi-

gation, the distances between its different parts may be

neglected, is called a material particle.

Thus in certain astronomical investigations the planets,

and even the sun, may be regarded each as a material

particle, because the difference of the actions of different

parts of these bodies does not come under our notice.

But we cannot treat them as material particles when we

investigate their rotation. Even an atom, when we

consider it as capable of rotation, must be regarded as

consisting of many material particles.

The diagram of a material particle is of course a

mathematical point, which has no configuration.

4 INTRODUCTION [CH.

7. RELATIVE POSITION OF TWO MATERIAL PARTICLES

The diagram of two material particles consists of two

points, as, for instance, A and B.

The position of B relative to A is indicated by the

direction and length of the straight line AB drawn

from A to B. If you start from A and travel in the

direction indicated by the line AB and for a distance

equal to the length of that line, you will get to B.

This direction and distance may be indicated equally

well by any other line, such as ab, which is parallel

and equal to AB. The position of A with respect to

B is indicated by the direction and length of the line

BA, drawn from B to A, or the line ba, equal and

parallel to BA.

It is evident that BA = AB.

In naming a line by the letters at its extremities,

the order of the letters is always that in which the line

is to be drawn.

8. VECTORS

The expression AB, in geometry, is merely the

name of a line. Here it indicates the operation by

which the line is drawn, that of carrying a tracing

point in a certain directionj:or a certain distance. As

indicating an operation, AB is called a Vector, and

the operation is completely defined by the direction

and distance of the transference. The starting point,

which is called the Origin of the vector, may be any-

where.

To define a finite straight line we must state its

origin as well as its direction and length. All vectors,

however, are regarded as equal which are parallel (and

drawn towards the same parts) and of the same magni-

tude.

Any quantity, such, for instance, as a velocity or a

i] VECTORS 5

t. - . 4

force*, which has a definite direction and a definite

magnitude may be treated as a vector, and may

be indicated in a diagram by a straight line whose

direction is parallel to the vector, and whose length

represents, according to a determinate scale, the mag-

nitude of the vector.

9. SYSTEM OF THREE PARTICLES

Let us next consider a system of three particles.

Its configuration is represented by a diagram of

three points, A, B, C.

The position of B with respect to D. C

A is indicated by the vector AB, I //

and that of C with respect to B by A/ I

the vector EC. A *- 'B

It is manifest that from these data, Fig. i.

when A is known, we can find B and

then C, so that the configuration of the three points is

completely determined.

The position of C with respect to A is indicated by

the vector AC, and by the last remark the value of AC

must be deducible from those of AB and BC.

The result of the operation AC is to carry the

tracing point from A to C. But the result is the same

if the tracing point is carried first from A to B and

then from B to C, and this is the sum of the operations

AB + BC.

10. ADDITION OF VECTORS

Hence the rule for the addition of vectors may be

stated thus: From any point as origin draw the suc-

cessive vectors in series, so that each vector begins at

the end of the preceding one. The straight line from

the origin to the extremity of the series represents the

vector which is the sum of the vectors.

* A force is more completely specified as a vector localised in

its line of action, called by Clifford a rotor; moreover it is only

when the body on which it acts is treated as rigid that the point

of application is inessential.

6 INTRODUCTION [CH.

The order of addition is indifferent, for if we write

BC + AB the_pperation indicated may be performed

by drawing AD parallel and equal to BC, and then

joining DC, which, by Euclid, I. 33, is parallel and

equal to AB, so that by these two operations we arrive

at the point C in whichever order we perform them.

The same is true for any number of vectors, take

them in what order we please.

1 1 . SUBTRACTION OF ONE VECTOR FROM ANOTHER

To express the position of C with respect to B in

terms of the positions of B and C with respect to A,

we observe that we can get from B to C either by

passing along the straight line BC or by passing from

B to A and then from A to C. Hence

= AC + BA since the order of addition is indifferent

= AC AB since AB is equal and opposite to BA .

Or the vector BC, which expresses the position of C

with respect to B, is found by subtracting the vector of

B from the vector of C, these vectors being drawn to

B and C respectively from any common origin A.

12. ORIGIN OF VECTORS

The positions of any number of particles belonging

to a material system may be defined by means of the

vectors drawn to each of these particles from some one

point. This point is called the origin of the vectors,

or, more briefly, the Origin.

This system of vectors determines the configura-

tion of the whole system; for if we wish to know

the position of any point B with respect to any other

point A, it may be found from the vectors OA and OB

by the equation

AB=OB-OA.

i] RELATIVE POSITION 7

We may choose any point whatever for the origin,

and there is for the present no reason why we should

choose one point rather than another. The configura-

tion of the system that is to say, the position of its

parts with respect to each other remains the same,

whatever point be chosen as origin. Many inquiries,

however, are simplified by a proper selection of the

origin.

13. RELATIVE POSITION OF Two SYSTEMS

If the configurations of two different systems are

known, each system having its own

origin, and if we then wish to include \

both systems in a larger system,

having, say, the same origin as the _,

first of the two systems, we must

ascertain the position of the origin of

the second system with respect to that of the first, and

we must be able to draw lines in the second system

parallel to those in the first.

Then by Article 9 the position of a point P of the

second system, with respect to the first origin, O, is

represented by the sum of the vector O'P of that point

with respect to the second origin, O', and the vector OO'

of the second origin, O', with respect to the first, O.

14. THREE DATA FOR THE COMPARISON OF

Two SYSTEMS

We have an instance of this formation of a large

system out of two or more smaller systems, when two

neighbouring nations, having each surveyed and

mapped its own territory, agree to connect their sur-

MATTER AND MOTION

MATTER AND MOTION/

BY THE LATE

J. CLERK JV^^XWELL

M.A., LL.D.EDIN., F.R.SS.L. & E.

HONORARY FELLOW OF TRINITY COLLEGE, AND PROFESSOR OF

EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE

REPRINTED : WITH NOTES AND APPENDICES BY

SIR JOSEPH LARMOR, F.R.S., M.P.

FELLOW OF ST JOHN'S COLLEGE, AND

LUCASIAN PROFESSOR OF MATHEMATICS

LONDON :

SOCIETY FOR PROMOTING

CHRISTIAN KNOWLEDGE

NEW YORK : THE MACMILLAN CO.

1920

PREFACE (1877)

PHYSICAL SCIENCE, which up to the end of the eighteenth

century had been fully occupied in forming a conception

of natural phenomena as the result of forces acting

between one body and another, has now fairly entered

on the next stage of progress that in which the energy

of a material system is conceived as determined by the

configuration and motion of that system, and in which

the ideas of configuration, motion, and force are

generalised to the utmost extent warranted by their

physical definitions.

To become acquainted with these fundamental ideas,

to examine them under all their aspects, and habitually

to guide the current of thought along the channels of

strict dynamical reasoning, must be the foundation of

the training of the student of Physical Science.

The following statement of the fundamental doctrines

of Matter and Motion is therefore to be regarded as

an introduction to the study of Physical Science in

general.

NOTE

IN this reprint of Prof. Clerk Maxwell's classical

tractate on the principles of dynamics, the changes have

been confined strictly to typographical and a few verbal

improvements. After trial, the conclusion has been

reached that any additions to the text would alter the

flavour of the work, which would then no longer be

characteristic of its author. Accordingly only brief

footnotes have been introduced: and the few original

footnotes have been distinguished from them by

Arabic numeral references instead of asterisks and other

marks. A new index has been prepared.

A general exposition of this kind cannot be expected,

and doubtless was not intended, to come into use as a

working textbook : for that purpose methods of syste-

matic calculation must be prominent. But as a reasoned

conspectus of the Newtonian dynamics, generalizing

gradually from simple particles of matter to physical

systems which are beyond complete analysis, drawn

up by one of the masters of the science, with many

interesting side-lights, it must retain its power of sug-

gestion even though parts of the vector exposition may

now seem somewhat abstract. The few critical footnotes

and references to Appendices that have been added may

help to promote this feature of suggestion and stimulus.

The treatment of the fundamental principles of

dynamics has however been enlarged on the author's

own lines by the inclusion of the Chapter "On the

Equations of Motion of a Connected System" from

vol. ii of Electricity and Magnetism. For permission to

make use of this chapter the thanks of the publishers

are due to the Clarendon Press of the University of

Oxford.

viii NOTE

With the same end in view two Appendices have

been added by the editor. One of them treats the

Principle of Relativity of motion, which has recently

become very prominent in wider physical connexions,

on rather different lines from those in the text. The other

aims at development of the wider aspects of the Prin-

ciple of Least Action, which has been asserting its

position more and more as the essential principle of con-

nexion between the various domains of Theoretical

Physics.

These additions are of course much more advanced

than the rest of the book : but they will serve to complete

it by presenting the analytical side of dynamical science,

on which it justly aspires to be the definite foundation

for all Natural Philosophy.

The editor desires to express his acknowledgment

to the Cambridge University Press, and especially to

Mr J. B. Peace, for assistance and attention.

J.L.

BIOGRAPHICAL NOTE

JAMES CLERK MAXWELL was born in Edinburgh in 1831,

the only son of John Clerk Maxwell, of Glenlair, near

Dalbeattie, a family property in south-west Scotland to

which the son succeeded. After an early education at

home, and at the University of Edinburgh, he pro-

ceeded to Cambridge in 1850, first to Peterhouse,

migrating afterwards to Trinity College. In the

Mathematical Tripos of 1854, the Senior Wrangler was

E. J. Routh, afterwards a mathematical teacher and

investigator of the highest distinction, and Clerk Max-

well was second: they were placed as equal soon after

in the Smith's Prize Examination.

He was professor of Natural Philosophy at Aberdeen

from 1856 to 1860, in King's College, London from

1860 to 1865, and then retired to Glenlair for six years,

during which the teeming ideas of his mind doubtless

matured and fell into more systematic forms. He was

persuaded to return into residence at Cambridge in

1871, to undertake the task of organizing the new

Cavendish Laboratory. But after a time his health

broke, and he died in 1879 at the age of 48 years.

His scientific reputation during his lifetime was

upheld mainly by British mathematical physicists,

especially by the Cambridge school. But from the time

that Helmholtz took up the study of his theory of

electric action and light in 1870, and discussed it in

numerous powerful memoirs, the attention given abroad

to his work gradually increased, until as in England it

became the dominating force in physical science.

Nowadays by universal consent his ideas, as the

mathematical interpreter and continuator of Faraday,

rank as the greatest advance in our understanding of

the laws of the physical universe that has appeared

x . BIOGRAPHICAL NOTE

since the time of Newton. As with Faraday, his pro-

found investigations into nature were concomitant with

deep religious reverence for nature's cause. See the

Life by L. Campbell and W. Garnett (Macmillan, 1882).

The treatise on Electricity and Magnetism and the

Theory of Heat contain an important part of his work.

His Scientific Papers were republished by the Cam-

bridge University Press in two large memorial volumes.

There are many important letters from him in the

Memoir and Scientific Correspondence of Sir George

Stokes, Cambridge, 1904.

The characteristic portrait here reproduced, perhaps

for the first time, is from a carte de visile photograph

taken probably during his London period.

J. L.

CONTENTS

CHAPTER I

INTRODUCTION

ART. PAGE

1 Nature of Physical Science i

2 Definition of a Material System 2

3 Definition of Internal and External .... 2

4 Definition of Configuration 2

5 Diagrams 3

6 A Material Particle 3

7 Relative Position of two Material Particles ... 4

8 Vectors 4

9 System of Three Particles 5

10 Addition of Vectors 5

1 1 Subtraction of one Vector from another ... 6

12 Origin of Vectors 6

13 Relative Position of Two Systems 7

14 Three Data for the Comparison of Two Systems . . 7

15 On the Idea of Space 9

16 Error of Descartes 9

17 On the Idea of Time n

18 Absolute Space 12

19 Statement of the General Maxim of Physical Science . 13

CHAPTER II

ON MOTION

20 Definition of Displacement . . . , . . . 15

21 Diagram of Displacement 15

22 Relative Displacement 16

23 Uniform Displacement 17

24 On Motion 18

25 On the Continuity of Motion ... . 18

26 On Constant Velocity .... . 19

27 On the Measurement of Velocity when Variable . 19

28 Diagram of Velocities . . . ... 20

29 Properties of the Diagram of Velocities . 21

30 Meaning of the Phrase "At Rest" . . . 22

31 On Change of Velocity 22

32 On Acceleration . 23

33 On the Rate of Acceleration ... . . 24

34 Diagram of Accelerations . . . ... . 25

35 Acceleration a Relative Term . . . * . . 25

xii CONTENTS

CHAPTER III

ON FORCE

ART. PAGE

36 Kinematics and Kinetics 26

37 Mutual Action between Two Bodies Stress . . 26

38 External Force 26

39 Different Aspects of the same Phenomenon . . . 27

40 Newton's Laws of Motion ...... 27

41 The First Law of Motion 28

42 On the Equilibrium of Forces 30

43 Definition of Equal Times 31.

44 The Second Law of Motion 32

45 Definition of Equal Masses and of Equal Forces . 32

46 Measurement of Mass 33

47 Numerical Measurement of Force 35

48 Simultaneous Action of Forces on a Body ... 36

49 On Impulse 37

50 Relation between Force and Mass 38

51 On Momentum 38

52 Statement of the Second Law of Motion in Terms of

Impulse and Momentum . . . . . . 39

53 Addition of Forces . . . . . . . 39

54 The Third Law of Motion 40

55 Action and Reaction are the Partial Aspects of a Stress 40

56 Attraction and Repulsion . . . ' . . . 41

57 The Third Law True of Action at a Distance . . 42

58 Newton's Proof not Experimental ..... 42

CHAPTER IV

ON THE PROPERTIES OF THE CENTRE OF MASS

OF A MATERIAL SYSTEM

59 Definition of a Mass- Vector ...... 44

60 Centre of Mass of Two Particles . . . . 44

6 1 Centre of 'Mass of a System 45

62 Momentum represented as the Rate of Change of a

Mass- Vector 45

63 Effect of External Forces on the Motion of the Centre

of Mass 46

64 The Motion of the Centre of Mass of a System is not

affected by the Mutual Action of the Parts of the

System 47

65 First and Second Laws of Motion .... 48

66 Method of treating Systems of Molecules . .- . 48

CONTENTS xiii

ART. PAGE

67 By the Introduction of the Idea of Mass we pass from

Point- Vectors, Point Displacements, Velocities,

. Total Accelerations, and Rates of Acceleration, to

Mass- Vectors, Mass Displacements, Momenta, Im-

pulses, and Moving Forces 49

68 Definition of a Mass- Area 50

69 Angular Momentum 51

70 Moment of a Force about a Point 51

71 Conservation of Angular Momentum .... 52

CHAPTER V

ON WORK AND ENERGY

72 Definitions 54

73 Principle of Conservation of Energy .... 54

74 General Statement of the Principle of the Conservation

of Energy 55

75 Measurement of Work 56

76 Potential Energy 58

77 Kinetic Energy 58

78 Oblique Forces 60

79 Kinetic Energy of Two Particles referred to their

Centre of Ma'ss 61

80 Kinetic Energy of a Material System referred to its

Centre of Mass 62

81 Available Kinetic Energy 63

82 Potential Energy 65

83 Elasticity 65

84 Action at a Distance 66

85 Theory of Potential Energy more complicated than

that of Kinetic Energy 67

86 Application of the Method of Energy to the Calculation

of Forces 68

87 Specification of the [Mode of Action] of Forces . . 69

88 Application to a System in Motion .... 70

89 Application of the Method of Energy to the Investigation

of Real Bodies j(B

90 Variables on which the Energy depends . . . 71

91 Energy in Terms of the Variables 72

92 Theory of Heat 72

93 Heat a Form of Energy . 73

94 Energy Measured as Heat . . . ... 73

95 Scientific Work to be done 74

96 History of the Doctrine of Energy . . . 75

97 On the Different Forms of Energy . . , . . 76

CONTENTS

CHAPTER VI

RECAPITULATION

ART. PAGE

98 Retrospect of Abstract Dynamics . . . . . 79

99 Kinematics 79

100 Force .79

101 Stress 80

102 Relativity of Dynamical Knowledge .... 80

103 Relativity of Force 81

104 Rotation . . 83

105 Newton's Determination of the Absolute Velocity of

Rotation . . . . .'.:. . . . 84

106 Foucault's Pendulum 86

107 Matter and Energy 89

108 Test of a Material Substance 89

109 Energy not capable of Identification .... 90

no Absolute Value of the Energy of a Body unknown . 90

in Latent Energy 91

112 A Complete Discussion of Energy would include the

whole of Physical Science 91

CHAPTER VII

THE PENDULUM AND GRAVITY

113 On Uniform Motion in a Circle 92

114 Centrifugal Force 93

115 Periodic Time 93

116 On Simple Harmonic Vibrations . . ... . 94

117 On the Force acting on the Vibrating Body . . 94

118 Isochronous Vibrations 95

119 Potential Energy of the Vibrating Body . . , . 96

1 20 The Simple Pendulum . . . ..... 96

121 A Rigid Pendulum 98

122 Inversion of the Pendulum 100

123 Illustration of Kater's Pendulum . . ... . 100

124 Determination of the Intensity of Gravity . . . 101

125 Method of Observation 102

126 Estimation of Error 103

CHAPTER VIII

UNIVERSAL GRAVITATION

127 Newton's Method . . 105

128 Kepler's Laws , . 105

129 Angular Velocity .106

130 Motion about the Centre of Mass . . . ; .. , . 106

CONTENTS xv

ART. PAGE

131 The Orbit . 107

. 107

. 108

. 109

no

. Ill

112

132 The Hodograph

133 Kepler's Second Law

134 Force on a Planet

135 Interpretation of Kepler's Third Law .

136 Law of Gravitation

137 Amended Form of Kepler's Third Law

138 Potential Energy due to Gravitation

139 Kinetic Energy of the System

140 Potential Energy of the System . . . .114

141 The Moon is a Heavy Body 115

142 Cavendish's Experiment 116

143 The Torsion Balance 117

144 Method of the Experiment 118

145 Universal Gravitation 119

146 Cause of Gravitation 120

147 Application of Newton's Method of Investigation . 121

148 Methods of Molecular Investigations .... 122

149 Importance of General and Elementary Properties . 122

[CHAPTER IX]

ON THE EQUATIONS OF MOTION OF A CON-

NECTED SYSTEM 123

APPENDIX I

THE RELATIVITY OF THE FORCES OF NATURE 137

APPENDIX II

THE PRINCIPLE OF LEAST ACTION . . .145

INDEX I62

Portrait of Prof. CLERK MAXWELL . . Frontispiece

MATTER AND MOTION

CHAPTER I

INTRODUCTION

i . NATURE OF PHYSICAL SCIENCE

PHYSICAL SCIENCE is that department of knowledge

which relates to the order of nature, or, in other words,

to the regular succession of events.

The name of physical science, however, is often

applied in a more or less restricted manner to those

branches of science in which the phenomena considered

are of the simplest and most abstract kind, excluding

the consideration of the more complex phenomena, such

as those observed in living beings.

The simplest case of all is that in which an event

or phenomenon can be described as a change in the

arrangement of certain bodies. Thus the motion of the

moon may be described by stating the changes in her

position relative to the earth in the order in which they

follow one another.

In other cases we may know that some change of

arrangement has taken place, but we may not be able

to ascertain what that change is.

Thus when water freezes we know that the molecules

or smallest parts of the substance must be arranged

differently in ice and in water. We also know that this

arrangement in ice must have a certain kind of sym-

metry, because the ice is in the form of symmetrical

crystals, but we have as yet no precise knowledge of

the actual arrangement of the molecules in ice. But

whenever we can completely describe the change of

2 INTRODUCTION [CH.

arrangement we have a knowledge, perfect so far as it

extends, of what has taken place, though we may still

have to learn the necessary conditions under which

a similar event will always take place.

Hence the first part of physical science relates to the

relative position and motion of bodies.

2. DEFINITION OF A MATERIAL SYSTEM

In all scientific procedure we begin by marking out a

certain region or subject as the field of our investiga-

tions. To this we must confine our attention, leaving

the rest of the universe out of account till we have

completed the investigation in which we are engaged.

In physical science, therefore, the first step is to define

clearly the material system which we make the subject

of our statements. This system may be of any degree

of complexity. It may be a single material particle, a

body of finite size, or any number of such bodies, and

it may even be extended so as to include the whole

material universe.

3. DEFINITION OF INTERNAL AND EXTERNAL

All relations or actions between one part of this sys-

tem and another are called Internal relations or actions.

Those between the whole or any part of the system

and bodies not included in the system are called Exter-

nal relations or actions. These we study only so far as

they affect the system itself, leaving their effect on

external bodies out of consideration. Relations and

actions between bodies not included in the system are

to be left out of consideration. We cannot investigate

them except by making our system include these other

bodies.

4. DEFINITION OF CONFIGURATION

When a material system is considered with respect

to the relative position of its parts, the assemblage of

relative positions is called the Configuration of the

system.

i] CONFIGURATION 3

A knowledge of the configuration of the system at a

given instant implies a knowledge of the position of

every point of the system with respect to every other

point at that instant.

5. DIAGRAMS

The configuration of material systems may be repre-

sented in models, plans, or diagrams. The model or

diagram is supposed to resemble the material system

only in form, not necessarily in any other respect.

A plan or a map represents on paper in two dimen-

sions what may really be in three dimensions, and can

only be completely represented by a model. We shall

use the term Diagram to signify any geometrical figure,

whether plane or not, by means of which we study the

properties of a material system. Thus, when we speak

of the configuration of a system, the image which we

form in our minds is that of a diagram, which completely

represents the configuration, but which has none of the

other properties of the material system. Besides dia-

grams of configuration we may have diagrams of velocity,

of stress, etc., which do not represent the form of the

system, but by means of which its relative velocities or

its internal forces may be studied.

6. A MATERIAL PARTICLE

A body so small that, for the purposes of our investi-

gation, the distances between its different parts may be

neglected, is called a material particle.

Thus in certain astronomical investigations the planets,

and even the sun, may be regarded each as a material

particle, because the difference of the actions of different

parts of these bodies does not come under our notice.

But we cannot treat them as material particles when we

investigate their rotation. Even an atom, when we

consider it as capable of rotation, must be regarded as

consisting of many material particles.

The diagram of a material particle is of course a

mathematical point, which has no configuration.

4 INTRODUCTION [CH.

7. RELATIVE POSITION OF TWO MATERIAL PARTICLES

The diagram of two material particles consists of two

points, as, for instance, A and B.

The position of B relative to A is indicated by the

direction and length of the straight line AB drawn

from A to B. If you start from A and travel in the

direction indicated by the line AB and for a distance

equal to the length of that line, you will get to B.

This direction and distance may be indicated equally

well by any other line, such as ab, which is parallel

and equal to AB. The position of A with respect to

B is indicated by the direction and length of the line

BA, drawn from B to A, or the line ba, equal and

parallel to BA.

It is evident that BA = AB.

In naming a line by the letters at its extremities,

the order of the letters is always that in which the line

is to be drawn.

8. VECTORS

The expression AB, in geometry, is merely the

name of a line. Here it indicates the operation by

which the line is drawn, that of carrying a tracing

point in a certain directionj:or a certain distance. As

indicating an operation, AB is called a Vector, and

the operation is completely defined by the direction

and distance of the transference. The starting point,

which is called the Origin of the vector, may be any-

where.

To define a finite straight line we must state its

origin as well as its direction and length. All vectors,

however, are regarded as equal which are parallel (and

drawn towards the same parts) and of the same magni-

tude.

Any quantity, such, for instance, as a velocity or a

i] VECTORS 5

t. - . 4

force*, which has a definite direction and a definite

magnitude may be treated as a vector, and may

be indicated in a diagram by a straight line whose

direction is parallel to the vector, and whose length

represents, according to a determinate scale, the mag-

nitude of the vector.

9. SYSTEM OF THREE PARTICLES

Let us next consider a system of three particles.

Its configuration is represented by a diagram of

three points, A, B, C.

The position of B with respect to D. C

A is indicated by the vector AB, I //

and that of C with respect to B by A/ I

the vector EC. A *- 'B

It is manifest that from these data, Fig. i.

when A is known, we can find B and

then C, so that the configuration of the three points is

completely determined.

The position of C with respect to A is indicated by

the vector AC, and by the last remark the value of AC

must be deducible from those of AB and BC.

The result of the operation AC is to carry the

tracing point from A to C. But the result is the same

if the tracing point is carried first from A to B and

then from B to C, and this is the sum of the operations

AB + BC.

10. ADDITION OF VECTORS

Hence the rule for the addition of vectors may be

stated thus: From any point as origin draw the suc-

cessive vectors in series, so that each vector begins at

the end of the preceding one. The straight line from

the origin to the extremity of the series represents the

vector which is the sum of the vectors.

* A force is more completely specified as a vector localised in

its line of action, called by Clifford a rotor; moreover it is only

when the body on which it acts is treated as rigid that the point

of application is inessential.

6 INTRODUCTION [CH.

The order of addition is indifferent, for if we write

BC + AB the_pperation indicated may be performed

by drawing AD parallel and equal to BC, and then

joining DC, which, by Euclid, I. 33, is parallel and

equal to AB, so that by these two operations we arrive

at the point C in whichever order we perform them.

The same is true for any number of vectors, take

them in what order we please.

1 1 . SUBTRACTION OF ONE VECTOR FROM ANOTHER

To express the position of C with respect to B in

terms of the positions of B and C with respect to A,

we observe that we can get from B to C either by

passing along the straight line BC or by passing from

B to A and then from A to C. Hence

= AC + BA since the order of addition is indifferent

= AC AB since AB is equal and opposite to BA .

Or the vector BC, which expresses the position of C

with respect to B, is found by subtracting the vector of

B from the vector of C, these vectors being drawn to

B and C respectively from any common origin A.

12. ORIGIN OF VECTORS

The positions of any number of particles belonging

to a material system may be defined by means of the

vectors drawn to each of these particles from some one

point. This point is called the origin of the vectors,

or, more briefly, the Origin.

This system of vectors determines the configura-

tion of the whole system; for if we wish to know

the position of any point B with respect to any other

point A, it may be found from the vectors OA and OB

by the equation

AB=OB-OA.

i] RELATIVE POSITION 7

We may choose any point whatever for the origin,

and there is for the present no reason why we should

choose one point rather than another. The configura-

tion of the system that is to say, the position of its

parts with respect to each other remains the same,

whatever point be chosen as origin. Many inquiries,

however, are simplified by a proper selection of the

origin.

13. RELATIVE POSITION OF Two SYSTEMS

If the configurations of two different systems are

known, each system having its own

origin, and if we then wish to include \

both systems in a larger system,

having, say, the same origin as the _,

first of the two systems, we must

ascertain the position of the origin of

the second system with respect to that of the first, and

we must be able to draw lines in the second system

parallel to those in the first.

Then by Article 9 the position of a point P of the

second system, with respect to the first origin, O, is

represented by the sum of the vector O'P of that point

with respect to the second origin, O', and the vector OO'

of the second origin, O', with respect to the first, O.

14. THREE DATA FOR THE COMPARISON OF

Two SYSTEMS

We have an instance of this formation of a large

system out of two or more smaller systems, when two

neighbouring nations, having each surveyed and

mapped its own territory, agree to connect their sur-