GIFT OF MICHAEL REESE

\

ELECTROMAGNETIC THEORY.
BY
OLIVER HEAVISIDE,
N\
VOLUME I.
LONDON :
THE ELECTRICIAN" PRINTING AND PUBLISHING COMPANY
LIMITED.
SALISBURY COURT, FLEET STREET, B.C. [All Rights Reserved.]

%
Printed and Published by 'THK KLKCTRIOIAN" PRINTING AND PUBLISHING CO.,
1, 2 and 3, Salisbury Court, Fleet Street, London, E.C.

,

.

.

PREFACE.
THIS work was originally meant to be a continuation of the series "Electromagnetic Induction and its Propagation," published in The Electrician in 1885-6-7, but left unfinished. Owing, however, to the necessity of much introductory repetition, this plan was at once found to be impracticable, and was, by request, greatly modified. The result is something approaching a connected treatise on electrical theory, though without the strict formality usually associated with a treatise. As critics cannot always find time to read more
than the preface, the following remarks may serve to direct
their attention to some of the leading points in this volume. The first chapter will, I believe, be found easy to read,
and may perhaps be useful to many men who are accustomed
to show that they are practical by exhibiting their ignorance of the real meaning of scientific and mathematical methods
of enquiry.
The second chapter, pp. 20 to 131, consists of an outline scheme of the fundamentals of electromagnetic theory from the Faraday-Maxwell point of view, with some small modifications and extensions upon Maxwell's equations. It is done
in terms of my rational units, which furnish the only way ot
carrying out the idea of lines and tubes of force in a consistent and intelligible manner. It is also done mainly in terms of vectors, for the sufficient reason that vectors are the main subject of investigation. It is also done in the duplex form I introduced in 1885, whereby the electric and
A

IV.

PREFACE.

magnetic sides of electromagnetism are symmetrically exhibited and connected, whilst the "forces" and "fluxes" are the objects of immediate attention, instead of the potential functions which are such powerful aids to obscuring and complicating the subject, and hiding from view useful and sometimes important relations.
The third chapter, pp. 132 to 305, is devoted to vector
algebra and analysis, in the form used by me in my former
papers. As I have at the beginning and end of this chapter
stated my views concerning the unsuitability of quaternions for physical requirements, and my preference for a vector
algebra which is based upon the vector and is dominated by
vectorial ideas instead of quaternionic, it is needless to say
more on the point here. But I must add that it has been gratifying to discover among mathematical physicists a considerable and rapidly growing appreciation of vector algebra on these lines; and moreover, that students who had found
quaternions quite hopeless could understand my vectors very
well. Regarded as a treatise on vectorial algebra, this chapter has manifest shortcomings. It is only the first rudiments
of the subject. Nevertheless, as the reader may see from the
applications made, it is fully sufficient for ordinary use in the mathematical sciences where the Cartesian mathematics
is usually employed, and we need not trouble about more
advanced developments before the elements are taken up. Now, there are no treatises on vector algebra in existence yet, suitable for mathematical physics, and in harmony with the Cartesian mathematics (a matter to which I attach the greatest importance). I believe, therefore, that this chapter
may be useful as a stopgap. The fourth chapter, pp. 306 to 466, is devoted to the
theory of plane electromagnetic waves, and, being mainly
descriptive, may perhaps be read with profit by many who
are unable to tackle the mathematical theory comprehen-
sively. It may be also useful to have results of mathematical

PREFACE.

V.

reasoning expanded into ordinary language for the benefit of
mathematicians themselves, who are sometimes too apt to
work out results without a sufficient statement of their
meaning and effect. But it is only introductory to plane waves. Some examples in illustration thereof have been crowded out, and will probably be given in the next volume. I have, however, included in the present volume the application of the theory (in duplex form) to straight wires, and also an account of the effects of self-induction and leakage, which are of some significance in present practice as well as in possible future developments. There have been some very queer views promulgated officially in this country concerning the speed of the current, the impotence of selfinduction, and other material points concerned. No matter
how eminent they may be in their departments, officials need
not be scientific men. It is not expected of them. But should they profess to be, and lay down the law outside their knowledge, and obstruct the spreading of views they cannot understand, their official weight imparts a fictitious importance to their views, and acts most deleteriously in propagating error, especially when their official position is held up as a screen to protect them from criticism. But in other countries
there is, I find, considerable agreement with my views.
Having thus gone briefly through the book, it is desirable to say a few words regarding the outline sketch of electro-
magnetics in the second chapter. Two diverse opinions have been expressed about it. On the one hand, it has been said
to be too complicated. This probably came from a simple-
minded man. On the other hand, it has been said to be too
simple. This objection, coming from a wise man, is of weight, and demands some notice.
Whether a theory can be rightly described as too simple depends materially upon what it professes to be. The pheno-
mena involving electromagnetism may be roughly divided
into two classes, primary and secondary. Besides the main
A2

VI.

PREFACE.

primary phenomena, there is a large number of secondary ones, partly or even mainly electromagnetic, but also trenching
upon other physical sciences. Now the question arises whether
it is either practicable or useful to attempt to construct a theory of such comprehensiveness as to include the secondary phenomena, and to call it the theory of electromagnetism. I think not, at least at present. It might perhaps be done ii
the secondary phenomena were thoroughly known ; but their theory is so much more debatable than that of the primary phenomena that it would be an injustice to the latter to too closely amalgamate them. Then again, the expression of the
theory would be so unwieldy as to be practically useless ; the major phenomena would be apparently swamped by the minor. It would, therefore, seem best not to attempt too much, but to have a sort of abstract electromagnetic scheme for the primary phenomena only, and have subsidiary extensions thereof for the secondary. The theory of electromagnetism is then a primary theory, a skeleton framework corresponding to a possible state of things simpler than the real in innumerable details, but suitable for the primary effects, and
furnishing a guide to special extensions. From this point of
view, the theory cannot be expressed too simply, provided it be a consistent scheme, and be sufficiently comprehensive to serve for a framework. I believe the form of theory in the second chapter will answer the purpose. It is especially
useful in the duplex way of exhibiting the relations, which is
clarifying in complicated cases as well as in simple ones. It
is essentially Maxwell's theory, but there are some differences. Some are changes of form only ; for instance, the rationalisation effected by changing the units, and the substitution ol the second circuital law for Maxwell's equation of electromotive force involving the potentials, etc. But there is one
change in particular which raises a fresh question. What is Maxwell's theory? or, What should we agree to understand
by Maxwell's theory ?

PREFACE.

vii.

The first approximation to the answer is to say, There is

Maxwell's

book

as

he

wrote

it ;

there

is

his

text, and there

are his equations : together they make his theory. But when we come to examine it closely, we find that this answer is

unsatisfactory. To begin with, it is sufficient to refer to

papers by physicists, written say during the twelve years

following the first publication of Maxwell's treatise, to see
that there may be much difference of opinion as to what his

theory is. It may be, and has been, differently interpreted by

different men, which is a sign that it is not set forth in a per-

fectly clear and unmistakeable form. There are many obscuri-

ties and some inconsistencies. Speaking for myself, it was

only by changing its form of presentation that I was able to
see it clearly, and so as to avoid the inconsistencies. Now

there is no finality in a growing science. It is, therefore,

impossible to adhere strictly to Maxwell's theory as he gave it

to the world, if only on account of its inconvenient form. But it is clearly not admissible to make arbitrary changes in
it and still call it his. He might have repudiated them

utterly. But if we have good reason to believe that the

theory as stated in his treatise does require modification to

make it self-consistent, and to believe that he would have

admitted the necessity of the change when pointed out to him,

then I think the resulting modified theory may well be called

Maxwell's.

Now this state of things is exemplified by his celebrated

circuital law defining the electric current in terms of magnetic

force. For although he did not employ the other, or second

circuital law, yet it may be readily derived from his equation

of

electromotive

force ;

and when

this

is

done, and

the

law

made a fundamental one, we readily see that the change it

suffers in passing from the case of a stationary to that of a

moving medium should be necessarily accompanied by a similar change in the first, or Maxwell's circuital law. An

independent formal proof is unnecessary ; the similarity of

Vlll.

PREFACE.

form and of the conditions of motion show that Maxwell's

auxiliary term in the electromotive force, viz., VqB (the motional electric force), where q is the velocity of the medium and B the induction, requires the use of a similar auxiliary
term in the first circuital law, viz., VDq, the motional
magnetic force, D being the displacement. And there is yet another change sometimes needed. For whilst B is circuital,

so that a convective magnetic current does not appear in
D the second circuital equation, is not always circuital, and
convective electric current must therefore appear in the first circuital equation. For the reason just mentioned, it is the
theory as thus modified that I consider to represent the true Maxwellian theory, with the other small changes required to
make a fit. But further than this I should not like to go, because, having made a fit, it is not necessary, and because it would be taking too great a liberty to make additions without the strongest reason to consider them essential.
The following example, which has been suggested to me
by remarks in Prof. Lodge's recent paper on " Aberration Problems," referring to a previous investigation of Prof. J. J. Thomson, will illustrate the matter in question. It is known
V that if be the speed of light through ether, the speed

through a stationary transparent body, say water, is V//A, if p
is the refractive index. Now what is the speed when the

water is itself moving in the same direction as the light

waves ? This is a very old problem. Fresnel considered that

the

external

ether

was

stationary,

and

that

the

ether

was

2 /a

times as dense in the water as outside, and that, when

moving, the water only carried forward with it the extra ether

it contained (or equivalently). This makes the speed of

light

referred

to

the

external

ether

be

+2
V//* v(l -ft~ ),

if v

is the speed of the water. The experiments of Fizeau and

Michelson have shown that this result is at least approxi-

mately true, and there is other evidence to support FresnePs

hypothesis, at least in a generalised form. But, in the case

PREFACE.

is.

of water, the additional speed of light due to the motion of

the

water

might

be

^v

instead

of

-2
(1 fir ) v,

without

much

disagreement. Now suppose we examine the matter electro-

magnetically, and enquire what the increased speed through

a moving dielectric should be. If we follow Maxwell's

equations literally, we shall find that the extra speed is |r,

provided i?/V is small. This actually seems to corroborate

the experimental results. But the argument is entirely a

deceptive one. Maxwell's theory is a theory of propagation through a simple medium. Fundamentally it is the ether,
but when we pass to a solid or liquid dielectric it is still to be regarded as a simple medium in the same sense, because the

only change occurring in the equations is in the value of one or both ethereal constants, the permittivity and inductivity
practically only the first. Consequently, if we find, as above, that when the medium is itself moved, its velocity is not

superimposed upon that of the velocity of waves through the medium at rest, the true inference is that there is something wrong with the theory. For all motion is relative, and it is

an axiomatic truth that there should be superimposition of
velocities, so that V//* + v should be the velocity in the above

case according to any rational theory of propagation through

a simple

medium, the

extra velocity being

the

full

v t

instead

of Jv. And, as a matter of fact, if we employ the modified

or corrected circuital law above referred to, we do obtain full

superimposition of velocities. This example shows the importance of having a simply
expressed and sound primary theory. For if the auxiliary

hypotheses required to explain outstanding or secondary phe-
nomena be conjoined to an imperfect primary theory we shall surely be led to wrong results. Whereas if the primary theory be good, there is at least a chance of its extension by auxiliary hypotheses being also good. The true conclusion from Fizeau and Michelson's results is that a transparent medium like

water cannot be regarded as (in the electromagnetic theory)

X.

PREFACE.

a simple medium like the ether, at least for waves of light, and that a secondary theory is necessary. Fresnel's sagacious
speculation is justified, except indeed as regards its form of
expression. The ether, for example, may be identical inside
and outside the body, and the matter slip through it without sensibly affecting it. At any rate the evidence that this is the case preponderates, the latest being Prof. Lodge's experiments with whirling discs, though on the other hand must not be forgotten the contrary conclusion arrived at by Michelson as to the absence of relative motion between the earth and sur-

rounding ether. But if the ether be stationary, Fresnel's speculation is roughly equivalent to supposing that the mole-' cules of transparent matter act like little condensers in increasing the permittivity, and that the matter, when in motion, only carries forward the increased permittivity. But however
this matter may be finally interpreted, we must have a clear
primary theory that can be trusted within its limits. Whether Maxwell's theory will last, as a sufficient and satisfactory primary theory upon which the numerous secondary deve-
lopments required may be grafted, is a matter for the future
to determine. Let it not be forgotten that Maxwell's theory

is only the first

step

towards

a

full

theory

of

the

.ether ;

and,

moreover, that no theory of the ether can be complete that

does not fully account for the omnipresent force of gravi-

tation.

There is one other matter that demands notice in conclu-

sion. It is not long since it was taken for granted that the
common electrical units were correct. That curious and

obtrusive constant 4?r was considered by some to be a sort of blessed dispensation, without which all electrical theory would fall to pieces. I believe that this view is now nearly extinct, and that it is well recognised that the 4?r was an unfortunate
and mischievous mistake, the source of many evils. In plain English, the common system of electrical units involves an irrationality of the same kind as would be brought into the

PREFACE.

X i.

metric system of weights and measures, were we to define
the unit area to be the area, not of a square with unit side,
but of a circle of unit diameter. The constant TT would then

obtrude itself into the area of a rectangle, and everywhere it should not be, and be a source of great confusion and
inconvenience. So it is in the common electrical units, which are truly irrational. Now, to make a mistake is easy and natural to man. But that is not enough. The next thing is to correct it. When a mistake has once been started,
it is not necessary to go on repeating it for ever and ever with cumulative inconvenience.

The B. A. Committee on Electrical Standards had to do

two kinds of work. There was the practical work of making standards from the experimentally found properties of matter (and ether). This has been done at great length, and with much labour and success. But there was also the theoretical

work of fixing the relations of the units in a convenient,

rational, and harmonious manner. This work has not yet

been done. To say that they ought to do it is almost a

Who platitude.

else should do it ? To say that there is

not at present sufficient popular demand for the change does

not seem very satisfactory. Is it not for leaders to lead ?
And who should lead but the men of light and leading who

have practical influence in the matter ?

Whilst, on the one hand, the immense benefit to be gained

by rationalising the units requires some consideration to fully

appreciate, it is, on the other hand, very easy to overestimate

the difficulty of making the change. Some temporary incon-

venience is necessary, of course. For a time there would be two sorts of ohms, &c., the old style and the new (or rational).

But it is not a novelty to have two sorts of ohms. There have been several already. Eemember that the number of standards in present existence is as nothing to the number

going to be made, and with ever increasing rapidity, by reason

of the enormously rapid extension of electrical industries.

XII.

PREFACE.

Old style instruments would very soon be in a minority, and then disappear, like the pins. I do not know that there is a more important practical question than this one of rationalising the units, on account of its far-reaching effect, and
think that whilst the change could be made now with ease (with a will, of course), it will be far more troublesome if
put off until the general British units are reformed; even though that period be not so distant as it is customary to believe. Electricians should set a good example.
The reform which I advocate is somewhat similar to the important improvement made by chemists in their units about a quarter of a century ago. One day our respected master informed us that it had been found out that water
was not HO, as he had taught us before, but something
H else. It was henceforward to be 20. This was strange
at first, and inconvenient, for so many other formulae had to be altered, and new books written. But no one questions the wisdom of the change. Now observe, here, that the chemists, when they found that their atomic weights were wrong, and their formulae irrational, did not cry " Too late,"
ignore the matter, and ask Parliament to legalise the old erroneous weights ! They went and set the matter right.
Verb. sap.
DECEMBER 16, 1893.

CONTENTS.

[The dates within brackets are the dates of first publication.}

CHAPTER I.

INTRODUCTION.

SECTION.

PAGE.

1 7 [Jan. 2, 1891.] Preliminary Remarks

1

813 [Jan. 16, 1891.] On the nature of-Anti-Mathematicians and

of Mathematical Methods of Enquiry ...

...

7

14 16 [Jan. 30, 1891.] Description of some Electromagnetic Results

deduced by Mathematical Reasoning

... 14

CHAPTER II.

OUTLINE OF THE ELECTROMAGNETIC CONNECTIONS.

SECTION.

(Pages 20 to 131.)

PAGE.

20 [Feb. 13, 1891.] Electric and Magnetic Force ; Displacement and

Induction ;

Elastivity and Permittivity, Inductivity and Reluc-

tivity

20

21 Electric and Magnetic Energy

...

...

... 21

22 Eolotropic Relations

22

23 Distinction between Absolute and Relative Permittivity or

Inductivity

23

24

Dissipation of Energy.

The

Conduction-current ;

Conductivity

and Resistivity. The Electric Current ...

...

...

... 24

25 Fictitious Magnetic Conduction - current and Real Magnetic

Current

25

26 [Feb. 27, 1891.] Forces and Fluxes

25

27 Line-integral of a Force. Voltage and Gaussage

26

28 Surface-integral of a Flux. Density and Intensity

27

29 Conductance and Resistance ...

...

...

...

...

... 27

30 Permittance and Elastance ...

...

... 28

31 Permeance, Inductance and Reluctance ...

29

32 Inductance of a Circuit

30

XIV.

CONTENTS.

SECTION.

PAGE.

33 [March 13, 1891.] Cross-connections of Electric and Magnetic

Force. Circuital Flux. Circulation ...

...

...

... 32

34 First Law of Circuitation ...

...

33

35 Second Law of Circuitation ...

...

...

...

35

36 Definition of Curl

35

37 Impressed Force and Activity

...

...

...

...

... 36

38 Distinction between Force of the Field and Force of the Flux ... 37

39 [March 27, 1891.] Classification of Impressed Forces

38

40 Voltaic Force

39

41 Thermo-electric Force

40

42 Intrinsic Electrisation

41

43 Intrinsic Magnetisation

...

...

...

...

...

... 41

44 The Motional Electric and Magnetic Forces. Definition of a

Vector-Product

42

45 Example. A Stationary Electromagnetic Sheet ...

...

... 43

46 [April 17, 1891.] Connection between Motional Electric Force

and

" Electromagnetic

Force

"

...

...

...

...

... 44

47 Variation of the Induction through a Moving Circuit ...

... 45

48 Modification. Circuit fixed. Induction moving equivalently ... 46

49 The Motional Magnetic Force

48

50 The " Magneto-electric Force "

...

49

51 Electrification and its Magnetic Analogue. Definition of " Diver-

gence"

...

...

...

...

...

...

...

... 49

52 A Moving Source equivalent to a Convection Current, and makes

the True Current Circuital

51

53 [May 1, 1891.] Examples to illustrate Motional Forces in a Moving

Medium with a Moving Source. (1.) Source and Medium with

a Common Motion. Flux travels with them undisturbed ... 53

A 54 (2.) Source and Medium in Relative Motion.

Charge suddenly

jerked into Motion at the Speed of Propagation. Generation

of a Spherical Electromagnetic Sheet ; ultimately Plane. Equa-

tions of a Pure Electromagnetic Wave ...

...

...

... 54

55 (3.) Sudden Stoppage of Charge. Plane Sheet moves on.

Spherical Sheet generated. Final Result, the Stationary

Field

57

56 (4.) Medium moved instead of Charge. Or both moved with

same Relative Velocity

...

...

...

...

...

... 58

57 (5.) Meeting of a Pair of Plane Sheets with Point-Sources. Can-

celment

of

Charges ; or

else

Passage

through

one

another ;

different results. Spherical Sheet with two Plane Sheet

Appendages ...

...

...

...

...

...

...

... 59

58 (6.) Spherical Sheet without Plane Appendages produced by

sudden jerking apart of opposite Charges

...

...

... 60

59 (7.) Collision of Equal Charges of same Name

61

60 (8.) Hemispherical Sheet. Plane, Conical, and Cylindrical Boun-

daries

61

61 General Nature of Electrified Spherical Electromagnetic Sheet ... 63

CONTENTS

XV.

SECTION.

PAOE.

62 [May 29, 1891.] General Remarks on the Circuital Laws.

Ampere's Rule for deriving the Magnetic Force from the

Current. Rational Current-element ...

... 64

> 63 The Cardinal Feature of Maxwell's System. Advice to anti-

Maxwellians . . .

...

...

...

...

... 66

64 Changes in the Form of the First Circuital Law

67

65 Introduction of the Second Circuital Law

68

66 [July 3, 1891.'] Meaning of True Current. Criterion

70

67 The Persistence of Energy. Continuity in Time and Space and

Flux of Energy

72

68 Examples. Convection of Energy and Flux of Energy due to an

active Stress. Gravitational difficulty

74

69 [July 17, 1891.] Specialised form of expression of the Con-

tinuity of Energy ...

...

...

...

...

...

... 77

70 Electromagnetic Application. Medium at Rest. The Poynting

Flux

78

71 Extension to a Moving Medium. Full interpretation of the

Equation of Activity and derivation of the Flux of Energy ... 80

72 Derivation of the Electromagnetic Stress from the Flux of

Energy. Division into an Electric and a Magnetic Stress ... 83

73 [July 31, 1891.] Uncertainty regarding the General Application

of the Electromagnetic Stress

...

...

...

...

... 85

74 The Electrostatic Stress in Air

87

75 The Moving Force on Electrification, bodily and superficial.

Harmonisation

...

...

...

...

... 90

76 Depth of Electrified Layer on a Conductor

...

...

... 91

77 [Aug. 21, 1891.] Electric Field disturbed by Foreign Body.

Effect of a Spherical Non-conductor

...

...

93

78 Dynamical Principle. Any Stress Self-equilibrating

...

... 95

79 Electric Application of the Principle. Resultant Action on Solid

Body independent of the Internal Stress, which is statically

indeterminate. Real Surface Traction is the Stress Difference 96

80 Translational Force due to Variation of Permittivity. Harmoni-

sation with Surface Traction

...

...

...

98

81 Movement of Insulators in Electric Field. Effect on the Stored

Energy

...

...

...

...

...

...

...

... 99

82 Magnetic Stress. Force due to Abrupt or Gradual Change of

Inductivity. Movement of Elastically Magnetised Bodies ... 100

82A [Sept. 4, 1891.] Force on Electric Current Conductors. The

Lateral Pressure becomes prominent, but no Stress Discon-

tinuity in general ...

...

...

...

... 102

83 Force on Intrinsically Magnetised Matter. Difficulty. Maxwell's

Solution probably wrong. Special Estimation of Energy of a

Magnet and the Moving Force it leads to

... 103

84 Force on Intrinsically Electrised Matter

106

85 Summary of the Forces. Extension to include varying States in

a Moving Medium ...

...

107

XVI.

CONTENTS.

SECTION.

PAGE.

86 [Sept. 25, 1891.] Union of Electric and Magnetic to produce

Electromagnetic Stress. Principal Axes

...

...

... 109

87 Dependence of the Fluxes due to an Impressed Forcive upon

its Curl only. General Demonstration of this Property ... 110

88 Identity of the Disturbances due to Impressed Forcives having

A the same Curl. Example :

Single Circuital Source of

Disturbance

112

89 Production of Steady State due to Impressed Forcive by crossing of Electromagnetic Waves. Example of a Circular Source. Distinction between Source of Energy and of Dis-

turbance ...

...

...

...

...

...

...

... 113

90 [Oct. 16, 1891.] The Eruption of "47r"s

116

91 The Origin and Spread of the Eruption

117

92 The Cure of the Disease by Proper Measure of the Strength of

Sources

119

93 Obnoxious Effects of the Eruption

120

94 A Plea for the Removal of the Eruption by the Radical Cure 122

95 [Oct. 30, 1891.] Rational v. Irrational Electric Poles

123

96 Rational v. Irrational Magnetic Poles

125

APPENDIX A. [Jan. 23, 1891.]
THE ROTATIONAL ETHER IN ITS APPLICATION TO ELECTROMAQNETISM ... 127

CHAPTER III.

THE ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS.

SECTION.

(Pages 132 to 305.)

PAGE.

97 [Nov. 13, 1891.] Scalars and Vectors ...

132

98 Characteristics of Cartesian and Vectdrial Analysis

133

99 Abstrusity of Quaternions and Comparative Simplicity gained

by ignoring them...

...

...

...

...

...

... 134

100 Elementary Vector Analysis independent of the Quaternion . . . 136

101 Tait v. Gibbs and Gibbs v. Tait

137

102 Abolition of the Minus Sign of Quaternions

138

103 [Dec. 4, 1891.] Type for Vectors. Greek, German, and Roman

Letters unsuitable. Clarendon Type suitable. Typographical

Backsliding in the Present Generation...

...

...

... 139

L04 Notation. Tensor and Components of a Vector. Unit Vectors

"

of Reference

142

L05 The Addition of Vectors. Circuital Property

143

L06 Application to Physical Vectors. Futility of Popular Demon-

107

strations. Barbarity of Euclid ..

...

...

[Dec. 18, 1891.] The Scalar Product of Two Vectors.

and Illustrations ...

...

...

...

...

Notation

...

...

147 148

108 Fundamental Property of Scalar Products, and Examples ... 151

109 Reciprocal of a Vector

;.

... 155

CONTENTS.

Xvii.

SECTION.

PAGE.

110 Expression of any Vector as the Sum of Three Indppendent

Vectors

155

111 {Jan. 1, 1892.] The Vector Product of Two Vectors. Illus-

trations

156

112 Combinations of Three Vectors. The Parallelepipedal Property 158 113 Semi- Cartesian Expansion of a Vector Product, and Proof of

the Fundamental Distributive Principle

...

...

... 159

114 Examples relating to Vector Products ...

...

...

... 162

115 [Jan. 29, 1892.} The Differentiation of Scalars and Vectors ... 163

116 Semi -Cartesian Differentiation. Examples of Differentiating

Functions of Vectors ...

...

...

...

...

... 165

117

Motion along a Curve in Space.

Tangency

and

Curvature ;

Velocity and Acceleration

...

...

..

167

118 Tortuosity of a Curve, and Various Forms of Expansion

169 . . .

119 [Feb. 12, 1892.} Hamilton's Finite Differentials Inconvenient

and Unnecessary

... 172

120 Determination of Possibility of Existence of Differential Co-

efficients

174

121 Variation of the Size and Ort of a Vector

176

122 [March 4, 1892.} Preliminary on V. Axial Differentiation.

Differentiation referred to Moving Matter ...

178

123 Motion of a Rigid Body. Resolution of a Spin into other Spins 180

124 Motion of Systems of Displacement, &c. ...

...

...

... 183

125 Motion of a Strain-Figure

185

126 [March 25, 1892.} Space-Variation or Slope VP of a Scalar

Function ...

...

...

...

...

...

...

... 186

127 Scalar Product VD. The Theorem of Divergence

188

123 Extension of the Theorem of Divergence

...

...

... 190

WE, 129 Vector Product

or the Curl of a Vector. The Theorem

of Version, and its Extension ...

...

...

...

... 191

130 [April 8, 1892.} Five Examples of the Operation of V in

Transforming from Surface to Volume Summations

194

131 Five Examples of the Operation of V in Transforming from

Circuital to Surface Summations

...

...

...

... 197

132 Nine Examples of the Differentiating Effects of V

199

133 [May 13, 1892.} The Potential of a Scalar or Vector. The

Characteristic Equation of a Potential, and its Solution ... 202

1 34 Connections of Potential, Curl, Divergence, and Slope. Separa-

A tion of a Vector into Circuital and Divergent Parts.

Series

135 136 137 138 139 140

of Circuital Vectors

A [May 27, 1892.}

Series of Divergent Vectors

The Operation inverse to Divergence ...

...

The Operation inverse to Slope

The Operation inverse to Curl

Remarks on the Inverse Operations

...

...

...

...

"
Integration by parts."

Energy Equivalences in the Circuital

206 209 212 213 214 215

Series

, 216

XV111.

CONTENTS.

SECTION.

PAGE.

141 [June 10, 1892.} Energy and other Equivalences in the Diver-

142 143 144 145
146
147 148 149 150 151

gent Series

The Isotropic Elastic Solid. Relation of Displacement to Force

through the Potential ...

._

The Stored Energy and the Stress in the Elastic Solid. The

Forceless and Torqueless Stress

...

...

...

...

Other Forms for the Displacement in terms of the Applied

Forcive ...

...

...

...

...

...

...

...

[June 24, 1892.} The Elastic Solid generalised to include

Elastic, Dissipative and Inertial Resistance to Translation,

Rotation, Expansion, and Distortion ...

...

Electromagnetic and Elastic Solid Comparisons. First Ex-

ample : Magnetic Force compared with Velocity in an In-

compressible Solid with Distortional Elasticity

...

...

Second Example : Same as last, but Electric Force compared

with Velocity
[July 15, 1892.} Third Example: A Conducting Dielectric

compared with a Viscous Solid. Failure
Fourth Example : A Pure Conductor compared with a Viscous

Liquid. Useful Analogy

...

...

...

...

...

Fifth Example : Modification of the Second and Fourth

...

A Sixth Example : Conducting Dielectric compared with an

Elastic Solid with Translational Friction

217 219 221 224
226
232 234 234 236 239 240

152 Seventh Example : Improvement of the Sixth ...

...

...

A 153 Eighth Example : Dielectric with Duplex Conductivity com-

pared with an Elastic Solid with Translational Elasticity and

Friction. The singular Distortionless Case ...

...

..

153A [July 29, 1892.} The Rotational Ether, Compressible or In-

compressible

...

...

...

...

...

...

...

154 First Rotational Analogy : Magnetic Force compared with

Velocity 155 Circuital Indeterminateness of the Flux of Energy in general 156 Second Rotational Analogy : Induction compared with

Velocity 157 [Aug. 5, 1892.} Probability of the Kinetic Nature of Magnetic

Energy

158 Unintelligibility of the Rotational Analogue for a Conducting

Dielectric when Magnetic Energy is Kinetic ...

...

...

159 The Rotational Analogy, with Electric Energy Kinetic, extended to a Conducting Dielectric by means of Translational

Friction

21Q
241 243 245 247 249 250 252
253

160 161 162

[Sept. 2, 1892.} Symmetrical Linear Operators, direct and in-

verse, referred to the Principal Axes ...

...

...

...

Geometrical Illustrations. The Sphere and Ellipsoid. Inverse

Perpendiculars and Maccullagh's Theorem ...

...

...

Internal Structure of Linear Operators. Manipulation of

several when Principal Axes are Parallel

...

...

...

256 259 262

CONTENTS.

Xix.

SECTION.

PAGE.

163 [Sept. 16, 1802.] Theory of Displacement in an Eolotropic

Dielectric. The Solution for a Point-Source...

...

... 264

164 Theory of the Relative Motion of Electrification and the

Medium. The Solution for a Point-Source in steady Recti-

linear Motion. The Equilibrium Surfaces in general

269 . . .

165 [Sept. 30, 1892.'] Theory of the Relative Motion of Mag-

netification and the Medium ...

...

...

...

... 274

166 Theory of the Relative Motion of Magnetisation and the

Medium. Increased Induction as well as Eolotropic Dis-

turbance

277

167 [Oct. 21, 1892.] Theory of the Relative Motion of Electric

Currents and the Medium

...

...

... 281

168 The General Linear Operator 169 The Dyadical Structure of Linear Operators ... 170 Hamilton's Theorem

283 285 ... 287

171 [Nov. 18, 1892.] Hamilton's Cubic and the Invariants con-

cerned

289

172 173
174 175

The Inversion of Linear Operators

...

...

...

...

Vector Product of a Vector and a Dyadic. The Differentiation

of Linear Operators

...

...

...

...

...

...

[Dec. 9, 1892.] Summary of Method of Vector Analysis

...

Uosuitability of Quaternions for Physical Needs. Axiom :

Once a Vector, always a Vector

...

...

...

...

293
295 297
301

CHAPTER IV.

THEORY OF PLANE ELECTROMAGNETIC WAVES.

SECTION.

(Pages 306 to 466.)

PAGE.

176 [Dec. 30, 1892.] Action at a Distance versus Intermediate
Agency. Contrast of New with Old Views about Electricity 306

177 General Notions about Electromagnetic Waves. Generation of

Spherical Waves and Steady States ...

...

...

... 310

178 [Jan. 6, 1893.] Intermittent Source producing Steady States

and Electromagnetic Sheets. A Train of S.H. Waves

... 314

179 Self-contained Forced Electromagnetic Vibrations. Contrast

with Static Problem

316

H 180 Relations between E and in a Pure Wave. Effect of Self-

KR Induction. Fatuity of Mr. Preece's "

law "

320

181

[Jan. 27, 1893.]

Wave-Fronts ;

their

Initiation

and

Progress

321

182 Effect of a Non-Conducting Obstacle on Waves. Also of a

Heterogeneous Medium ...

...

..

...

...

... 323

183 Effect of Eolotropy. Optical Wave- Surfaces. Electromagnetic

versus Elastic Solid Theories ...

..

...

...

... 325

A 184

Perfect Conductor is a Perfect Obstructor, but does not

absorb the Energy of Electromagnetic Waves

...

... 328

XX.

CONTENTS.

SECTION.
185 [Feb. 24, 1893.] Conductors at Low Temperatures 186 Equilibrium of Radiation. The Mean Flux of Energy 187 The Mean Pressure of Radiation

PAGE. 330
... 331 334

188 Emissivity and Temperature

.,.

... 335

189 [March 10, 1893.] Internal Obstruction and Superficial Con-

duction

337

190 The Effect of a Perfect Conductor on External Disturbances.

Reflection and Conduction of Waves ...

...

340

L91 [March 24, 1893.] The Effect of Conducting Matter in

Diverting External Induction ...

..

...

...

... 344

192 Parenthetical Remarks on Induction, Magnetisation, Induc-

tivity and Susceptibility

349

193 [April 7, 1893.] Effect of a Thin Plane Conducting Sheet on

a Wave. Persistence of Induction and Loss of Displacement 353

194 The Persistence of Induction in Plane Strata, and in general.

Also in Cores and in Linear Circuits ..

...

...

... 357

195 [April 21, 1893.] The Laws of Attenuation of Total Displace-

ment and Total Induction by Electric and Magnetic Conduc-

tance

360

196 The Laws of Attenuation at the Front of a Wave, due to

Electric and Magnetic Conductance ...

...

...

... 564

197 The Simple Propagation of Waves in a Distortionless Con-

ducting Medium ...

...

..

...

...

...

... 366

198 [May 5, 1893.] The Transformation by Conductance of an

Elastic Wave to a Wave of Diffusion. Generation of Tails.

199 200
201
202

Distinct Effects of Electric and Magnetic Conductance

...

Application to Waves along Straight Wires

[May 26, 1893.] Transformation of Variables from Electric

and Magnetic Force to Voltage and Gaussage

...

...

Transformation of the Circuital Equations to the Forms in-

volving Voltage and Gaussage ...

...

...

...

...

[June 9, 1893.] The Second Circuital Equation for Wires in

Terms of V and C when Penetration is Instantaneous

...

369 374 378 381 386

203 204

The Second Circuital Equation when Penetration is Not Instantaneous. Resistance Operators, and their Definite

Meaning ...

...

...

...

...

...

...

...

Simply Periodic Waves Easily Treated in Case of Imperfect

Penetration

...

...

...

...

...

...

...

390 393

205 [July 7, 1893.} Long Waves and Short Waves. Iden'tity of

Speed of Free and Guided Waves

... 395

206 The Guidance of Waves. Usually Two Guides. One sufficient,

though with Loss. Possibility of Guidance within a Single

Tube

399

207 Interpretation of Intermediate or Terminal Conditions in the

Exact Theory

401

208 [Aug. 25, 1893.] The Spreading of Charge and Current in a

long Circuit, and their Attenuation

403

CONTENTS.

xxi.

SECTION.

PAGE.

209 The Distortionless Circuit. No limiting Distance get by it

when the Attenuation is ignored

...

...

...

... 409

210 [Sept. 15, 1893.} The two Extreme Kinds of Diffusion in one

Theory

411

211 The Effect of varying the Four Line- Constanta as regards Dis-

tortion and Attenuation

...

...

... 413

212 The Beneficial Effect of Leakage in Submarine Cables

417

213 [Oct. 6, 1893.} Short History of Leakage Effects on a Cable

Circuit

420

214 Explanation of Anomalous Effects. Artificial Leaks

424

215 [Oct. 20. 1893.] Self-induction imparts Momentum to Waves.

and that carries them on. Analogy with a Flexible Cord ... 429

216 Self-induction combined with Leaks. The Bridge System of

Mr. A. W. Heaviside, and suggested Distortionless Circuit ... 433

217 [Nov. 3, 1893} Evidence in favour of Self-induction. Con-

dition of First- Class Telephony. Importance of the Magnetic

Eeactance

437

218 Various ways, good and bad, of increasing the Inductance of

Circuits

441

219 [Nov. 17, 1893} Effective Resistance and Inductance of a Combination when regarded as a Coil, and Effective Conductance and Permittance when regarded as a Condenser ... 446

220 221
222

Inductive Leaks applied to Submarine Cables

General Theory of Transmission of Waves along a Circuit with

or without Auxiliary Devices ...

...

...

...

...

Application of above Theory to Inductive Leakance ...

...

447
449 453

APPENDIX B. A GRAVITATIONAL AND ELECTROMAGNETIC ANALOGY.

Parti. [July 14, 1893.} Part II. [Aug. 4,1893.}

455 463

UNIV3

CHAPTER I.

INTRODUCTION.

1. Preliminary Remarks. The main object of the series of

articles of which this is the first, is to continue the work entitled *' Electromagnetic Induction and its Propagation," commenced in The Electrician on January 3, 1885, and continued to the 46th Section in September, 1887, when the great pressure on space and the want of readers appeared to necessitate its

abrupt discontinuance. (A straggler, the 47th Section, appeared

December 31, 1887.) Perhaps there were other reasons than

We those mentioned for the discontinuance.

do not dwell in

the Palace of Truth. But, as was mentioned to me not long since, " There is a time coming when all things shall be found
out." I am not so sanguine myself, believing that the well in

which Truth is said to reside is really a bottomless pit. The particular branch of the subject which I was publishing
in the summer of 1887 was the propagation of electromagnetic

waves along wires through the dielectric surrounding them. This is itself a large and many-sided subject. Besides a general treatment, its many-sidedness demands that special

cases of interest should receive separate full development. In

general, the mathematics required is more or less of the charac-

ter sometimes termed transcendental. This is a grandiloquent
word, suggestive of something beyond human capacity to find

out ;

a word to

frighten timid

people into

believing

that it

is

all speculation, and therefore unsound. I do not know where

transcendentality begins. You can find it in arithmetic. But

never mind the word. What is of more importance is the fact

that the interpretation of transcendental formulae is sometimes

2

ELECTROMAGNETIC THEORY.

CU. I.

very laborious. Now the real object of true naturalists, in Sir W. Thomson's meaning of the word, when they employ

mathematics to assist them, is not to make mathematical exer-

cises (though that may be necessary), but to find out the con-

nections of known phenomena, and by deductive reasoning, to obtain a knowledge of hitherto unknown phenomena. Any-

thing, therefore, that aids this, possesses a value of its own wholly

apart from immediate or, indeed, any application of the kind

commonly termed practical. There is, however, practicality in

theory as well as in practice.

The

very useful word

" practi-

cian " has lately come into use. It supplies a want, for it is

evident the moment it is mentioned that a practician need not

be

a

practical

man ;

and that,

on the other hand, it may happen

occasionally that a man who is not a practician may still be

quite practical.

2. Now, I was so fortunate as to discover, during the

examination of a practical telephonic problem, that in a certain

case of propagation along a conducting circuit through a con-

ducting dielectric, the transcendentality of the mathematics

automatically vanished, by the distorting effects on an electro-

magnetic wave, of the resistance of the conductor, and of the

conductance of the dielectric, being of opposite natures, so that

they neutralised one another, and rendered the circuit non-

distortional or distortionless. The mathematics was reduced,

in the main, to simple algebra, and the manner of transmission

of disturbances could be examined in complete detail in an

elementary manner. Nor was this all. The distortionless

circuit could be itself employed to enable us to understand the

inner meaning of the transcendental cases of propagation, when

the distortion caused by the resistance of the circuit makes the

mathematics more difficult of interpretation. For instance, by

a study of the distortionless circuit we are enabled to see not

only that, but also why, self-induction is of such great import-

ance in the transmission of rapidly-varying disturbances in

preserving their individuality and preventing them from being

attenuated to nearly nothing before getting from one end of a

long

circuit to

the

other ;

and

why copper wires are so success-

ful in, and iron wires so prejudicial to effective, long-distance

telephony. These matters were considered in Sections 40 to

INTRODUCTION.

3

45 (June, July, August, 1887) of the work I have referred to, and Sections 46, 47 contain further developments.

3. But that this matter of the distortionless circuit has,

directly, important practical applications, is, from the purely scientific point of view, a mere accidental circumstance. Perhaps a more valuable property of the distortionless circuit is, that it is the Royal Road to electromagnetic waves in general,
especially when the transmitting medium is a conductor as well as a dielectric. I have somewhat developed this matter in the

Phil. Mag., 1888-9. Fault has been found with these articles

that they are hard to read. They were harder, perhaps, to write. The necessity of condensation in a journal where space is so limited and so valuable, dealing with all branches of physical science, is imperative. What is an investigator to do, when

he can neither find acceptance of matter in a comparatively elementary form by journals of a partly scientific, partly tech-
nical type, with many readers, nor, in a more learned form, by a
purely scientific journal with comparatively few readers, and little space to spare ? To get published at all, he must condense greatly, and leave out all explanatory matter that he
possibly can. Otherwise, he may be told his papers are more
fit for publication in book form, and are therefore declined.

There is a third course, of course, viz., to keep his investigations to himself. But that does not answer, in a general way,

though it may do so sometimes. It is like putting away seed

mummy in a

case, instead of planting it, and letting it take its

chance of growing to a useful plant. There is nothing like publication and free criticism for utility. I can see only one good excuse for abstaining from publication when no obstacle
presents itself. You may grow your plant yourself, nurse it
carefully in a hot-house, and send it into the world full-grown. But it cannot often occur that it is worth the trouble taken.

As for the secretiveness of a Cavendish, that is utterly inex-
cusable ; it is a sin. It is possible to imagine the case of a man
being silent, either from a want of confidence in himself, or from disappointment at the reception given to, and want of
appreciation of, the work he gives to the world ; few men have an unbounded power of persistence ; but to make valuable discoveries, and to hoard them up as Cavendish did, without any
B2

4

ELECTROMAGNETIC THEORY.

CH. I.

valid reason, seems one of the most criminal acts such a man
could be guilty of. This seems strong language, but as Prof. Tait
tells us that it is almost criminal not to know several foreign
languages, which is a very venial offence in the opinion of others, it seems necessary to employ strong language when the criminality is more evident. (See, on this point, the article in The Electrician, November 14, 1890. It is both severe and logical.)

4. I had occasion, just lately, to use the word " naturalist."

The matter involved here is worthy of parenthetical considera-

tion.

Sir

W.

Thomson

does

not

like

" physicist,"

nor,

I

think,

"scientist" either. It must, however, be noted that the

naturalist, as at present generally understood, is a student of
living nature only. He has certainly no exclusive right to so excellent a name. On the other hand, the physicist is a

student of inanimate nature, in the main, so that he has no

exclusive right to the name, either. Both are naturalists. But

their work is so different, and their type of mind also so

different, that it seems very desirable that their names should

be

differentiated, and

that

" naturalist," comprehending

both,

should be subdivided. Could not one set of men be induced to call

We themselves organists?

have organic chemistry, and organisms,

and organic

science ;

then why

not

organists 1

Perhaps, how-

ever, organists might not care to be temporarily confounded

with those members of society who earn their living by setting a

cylinder in rotatory motion. If so, there is another good name,

viz., vitalist, for the organist, which would not have any ludic-

rous association. Then about the other set of men. Are they

not essentially students of the properties of matter, and therefore materialists? That "materialist" is the right name is

obvious at a glance. Here, however, a certain suppositions

evil association of the word might militate against its adoption.

But this would be, I think, an unsound objection, for I do not

think there is, or ever was, such a thing as a materialist, in

the supposed evil sense. Let that notion go, and the valuable

word "materialist" be put to its proper use, and be dignified

by association with an honourable body of men.

Buffon, Cuvier, Darwin, were typical vitalises. Newton, Faraday, Maxwell, were typical materialists.

INTRODUCTION.

5

All were naturalists. For my part I always admired the old-

fashioned term "natural philosopher." It was so dignified, and

raised up visions of the portraits of Count Rumford, Young,

Herschel, Sir H. Davy, &c., usually highly respectable-looking

elderly gentlemen, with very large bald heads, and much

wrapped up about the throats, sitting in their studies ponder-

ing calmly over the secrets of nature revealed to them by their

experiments. There are no natural philosophers now-a-days.
How is it possible to be a natural philosopher when a Salvation

Army band is performing outside ; joyously, it may be, but not

most melodiously ?

But

I

would not

disparage

their

work ;

it

may be far more important than his.

5. Returning to electromagnetic waves. Maxwell's inimitable theory of dielectric displacement was for long gene-
rally regarded as a speculation. There was, for many years, an
almost complete dearth of interest in the unverified parts of Maxwell's theory. Prof. Fitzgerald, of Dublin, was the most
prominent of the very few materialists (if I may use the word) who appeared to have a solid faith in the electromagnetic theory
of the ether ; thinking about it and endeavouring to arrive at an idea of the nature of diverging electromagnetic waves, and how to produce them, and to calculate the loss of energy by radiation.
An important step was then made by Poynting, establishing the
formula for the flow of energy. Still, however, the theory wanted experimental proof. Three years ago electromagnetic waves were nowhere. Shortly after, they were everywhere. This was due to a very remarkable and unexpected event, no less than the experimental discovery by Hertz, of Karlsruhe (now of Bonn), of the veritable actuality of electromagnetic
waves in the ether. And it never rains but it pours ; for whilst Hertz with his resonating circuit was working in Germany
(where one would least expect such a discovery to be made, if one judged only by the old German electro-dynamic theories), Lodge was doing in some respects similar work in England, in connection with the theory of lightning conductors. These researches, followed by the numerous others of Fitzgerald and Trouton, J. J. Thomson, &c., have dealt a death-blow to the electrodynamic speculations of the Weber-Clausius type (to mention only the first and one of the last), and have given to Maxwell's

G

ELECTROMAGNETIC THEORY.

C1I. I.

theory just what was wanted in its higher parts, more experimental basis. The interest excited has been immense, and the theorist can now write about electromagnetic waves without incurring the reproach that he is working out a mere paper theory. The speedy recognition of Dr. Hertz by the Royal Society is a very unusual testimony to the value of his researches.
At the same time I may remark that to one who had care-
fully examined the nature of Maxwell's theory, and looked into its consequences, and seen how rationally most of the phenomena of electromagnetism were explained by it, and how it
furnished the only approximately satisfactory (paper) theory of
light known ; to such a one Hertz's demonstration came as a matter of course only it came rather unexpectedly.

6. It is not by any means to be concluded that Maxwell spells Finality. There is no finality. It cannot even be accurately said that the Hertzian waves prove Maxwell's dielectric theory completely. The observations were very rough indeed, when compared with the refined tests in other parts of electrical science. The important thing proved is that electromagnetic waves in the ether at least approximately in accordance with Maxwell's theory are a reality, and that the Faraday-Maxwellian method is the correct one. The other kind of electrodynamic speculation is played out completely. There will be plenty of room for more theoretical speculation, but it must now be of
the Maxwellian type, to be really useful.

7. In what is to follow, the consideration of electromagnetic
waves will (perhaps) occupy a considerable space. How much

depends entirely upon the reception given to the articles.

Mathematics is at a discount, it seems. Nevertheless, as the

subject is intrinsically a mathematical one, I shall not scruple

to employ the appropriate methods when required. The reader

whose scientific horizon is bounded entirely by commercial con-

siderations may as well avoid these articles. Speaking without

prejudice, matter more to his taste may perhaps be found under

the heading TRADE NOTICES.* Sunt quos curricula.

I shall, however, endeavour to avoid investigations of a com-

plex

character ;

also,

when the methods and terms used are not

* Referring to The Electrician, in which this work first appeared.

INTRODUCTION.

7

generally known I shall explain them. Considering the lapse of time since the discontinuance of E.M.I, and its P. it would
be absurd to jump into the middle of the subject all at once. It . must, therefore, be gradually led up to. I shall, therefore, in the next place make a few remarks upon mathematical investigations in general, a subject upon which there are many popular delusions current, even amongst people who, one would think, should know better.

8. There are men of a certain type of mind who are never

wearied with gibing at mathematics, at mathematicians, and at

mathematical methods of inquiry. It goes almost without say-
ing that these men have themselves little mathematical bent.

I believe this to be

a

general

fact ;

but,

as a

fact,

it does

not

explain very well their attitude towards mathematicians. The
reason seems to lie deeper. How does it come about, for in-

stance, that whilst they are themselves so transparently ignorant

of the real nature, meaning, and effects of mathematical investi-

gation, they yet lay down the law in the most confident and

self-satisfied manner, telling the mathematician what the nature

of his work is (or rather is not), and of its erroneousness and inutility, and so forth 1 It is quite as if they knew all about it.

It reminds one of the professional paradoxers, the men who

want to make you believe that the ratio of the circumference

to the diameter of a circle is 3, or 3*125, or some other nice

easy number (any but the right one) ; or that the earth is flat,

or that the sun is a lump of ice ; or that the distance of the

moon is exactly 6 miles 500 yards, or that the speed of the

current varies as the square of the length of the line. They,

too, write as if they knew all about it ! Plainly, then, the

anti-mathematician must belong to the same class as the

paradoxer, whose characteristic is to be wise in his ignorance,
whereas the really wise man is ignorant in his wisdom. But this matter may be left for students of mind to settle. What

is of greater importance is that the anti-mathematicians some-

times do a deal of mischief. For there are many of a neutral

frame of mind, little acquainted themselves with mathematical

methods, who are sufficiently impressible to be easily taken in

by the gibers and to be prejudiced thereby ; and, should

they possess some mathematical bent, they may be hindered

8

ELECTROMAGNETIC THEORY.

CH. I.

We by their prejudice from giving it fair development.

cannot all be Newtons or Laplaces, but that there is an

immense amount of moderate mathematical talent lying latent

in the average man I regard

as a fact ;

and even

the moderate

development implied in a working knowledge of simple alge-

braical equations can, with common-sense to assist, be not

only the means of valuable mental discipline, but even be of

commercial importance (which goes a long way with some

people), should one's occupation be a branch of engineering for

example.

9. " Mathematics is gibberish." Little need be said about

this statement. It is only worthy of the utterly illiterate.
" What is the use of it ? It is all waste of time. Better be

doing something useful. Why, you might be inventing a new

dynamo in the time you waste over all that stuff." Now,

similar remarks to these I have often heard from fairly intelli-

gent and educated people. They don't see the use of it, that is

plain. That is nothing ; what is to the point is that they con-

clude that it is of no use. For it may be easily observed that

the parrot-cry "What's the use of it?" does not emanate in a

humble spirit of inquiry, but on the contrary, quite the reverse.

You can see the nose turn up.

But what is the use of it, then ? Well, it is quite certain that

if a person has no mathematical talent whatever he had really

better

be

doing

something

" useful,"

that

is

to say, something

else than mathematics, (inventing a dynamo, for instance,) and

not be wasting his time in (so to speak) trying to force a crop

of wheat on the sands of the sea-shore. This is quite a personal

question. Every mind should receive fair development (in

good directions) for what it is capable of doing fairly well.

People who do not cultivate their minds have no conception of

what they lose. They become mere eating and drinking and

money-grabbing machines. And yet they seem happy ! There

is some merciful dispensation at work, no doubt. " Mathematics is a mere machine. You can't get anything

out of it that you don't put in first. You put it in, and then

just grind it out again. You can't discover anything by mathematics, or invent anything. You can't get more than a

pint out of a pint pot." And so forth.

INTRODUCTION.

9

It is scarcely credible to the initiated that such statements
could be made by any person who could be said to have an intellect. But I have heard similar remarks from really talented men, who might have fair mathematical aptitude themselves, though quite undeveloped. The fact is, the statements contain at once a profound truth, and a mischievous fallacy. That the
fallacy is not self-evident affords an excuse for its not being
perceived even by those who may (perhaps imperfectly) recog-
nise the element of truth.. But as regards the truth mentioned, I doubt whether the caviller has generally any distinct idea of it either, or he would not express it so contemptuously
along with the fallacy.

10. By any process of reasoning whatever (not fancy) you

cannot get any results that are not implicitly contained in the

material with which you work, the fundamental data and their

connections, which form the basis of your inquiry. You may

make mistakes, and so arrive at erroneous results from the

most correct data. Or the data may be faulty, and lead to erroneous conclusions by the most correct reasoning. And in

general, if the data be imperfect, or be only true within cer-

tain limits hardly definable, the results can have but a limited
application. Now all this obtains exactly in mathematical

reasoning. It is in no way exempt from the perils of reasoning

in general. But why the mathematical reasoning should be

singled out for condemnation as mere machine work, dependent

upon what the machine is made to do, with a given supply of

material, is not very evident. The cause lies deep in the

nature of

the anti-mathematician ;

he

has

not recognised that

all reasoning must be, in a sense, mechanical, else it is not

sound reasoning at all, but vitiated by fancy.

Mathematical reasoning is, fundamentally, not different

from reasoning in general. And as by the exercise of the

reason discoveries can be made, why not by mathematical

reasoning? Whatever were Newton and the long array of

mathematical materialists who followed him doing all the time?

Making discoveries, of course, largely assisted by their mathe-

matics. I say nothing of the pure mathematicians. Their

discoveries are extensions of the field of mathematics itself a

perfectly limitless field. I refer only to students of Nature on

10

ELECTROMAGNETIC THEORY.

CH. I.

its material side, who have employed mathematics expressly for the purpose of making discoveries. Some of the unmathe-

matical believe that the mathematician is merely engaged in

counting

or

in

doing

long

sums ;

this

probably

arises

from

reminiscences of their schooldays, when they were flogged

over fractions. Now this is only a part of his work, a some-

times necessary and very disagreeable part, which he would

willingly hand over to a properly trained computator. This

part of the work only concerns the size of the effects, but

it is the effects themselves to which I refer when I speak of

discoveries.

11. Mathematics is reasoning about quantities. Even if
qualities are in question, it is their quantities that are subjected to the mathematics. If there be something which cannot be
reduced to a quantity, or more generally to a definite function, no matter how complex and involved, of any number of other quantities which can be measured (either actually, or in imagination), then that something cannot be accurately reasoned about, because it is in part unknown. Not unknown in the sense in which a quantity is said to be unknown in algebra, when it is virtually known because virtually expressible in terms of known quantities, but literally unknown by the absence of sufficient quantitative connection with the known. Thus only the known can be accurately reasoned about. But this includes, it will be observed, everything that can be deduced from the known, without appeal to the unknown. The un-
known is not necessarily unknowable ; fresh knowns may make the former unknowns become also known. The distinction is
a very important one. The limits of human knowledge are ever shifting. But there must be an ultimate limit, because
we are a part of Nature, and cannot go beyond it. Beyond this limit, the Unknown becomes the Unknowable, which it is
of little service to discuss, though it will always be a favourite subject of speculation. But whatever is in this Universe can be (or might be) found out, and therefore does not belong to the unknowable. Thus the constitution of the middle of the
sun, or of the ether, or the ultimate nature of magnetisation, or of universal gravitation, or of life, are not unknowable ; and this statement is true, even though they should never be dis-

INTRODUCTION.

11

covered. There are no inscrutables in Nature. By Faith only
can we go beyond as far and where we please.
Human nature, or say a man, is a highly complex quantity. We are compelled to take him in parts, and consider this or

that quality, and imagine it measured and brought into proper

connection with other qualities and external influences. Yet

a man, if we only knew him intimately enough, could be

formularised, and have his whole life-history developed. Even

the universe itself, if every law in action were thoroughly

known, could have its history, past, present, and tp come,

formularised down to the minutest particulars, provided no

discontinuity or special act of creation occur. But even the

special act of creation could be formularised, and its effects

deduced, if we knew in what it consisted. And special acts of

creation might be going on continuously, involving continuous

changes in the laws of nature, and could be formularised, if

the acts of creation were known, or the so to speak law of

the discontinuities. The case is somewhat analogous to that

of impressed forces acting upon a dynamical system. The

behaviour of the system is perfectly definite and formularisable

so long as no impressed forces act, and ceases to be definite if

unknown impressed forces act. But if the forces be also

known, then the course of events is again definitely formularis-

able. The assumption of a special act of creation, either now

We or at any time, is merely a confession of ignorance.

have

We no evidence of any such discontinuities.

cannot prove that

there have never been any; nor can we prove that the sun will

not rise to-morrow, or that the clock will not wind itself up

again when the weight has run down.

12. Nearly all the millions, or rather billions, of human beings who have peopled this earth have been content to go
through life taking things as they found them, and without any desire to understand what is going on around them. It is exceedingly remarkable that the scientific spirit (asking how it is done), which is so active and widespread at the present day, should be of such recent origin. With a few exceptions, it hardly existed amongst the Ancients (who would be more appropriately termed the Youngsters). It is a very encouraging fact for evolutionists, leading them to believe that the evolution of

12

ELECTROMAGNETIC THBOUT.

CH. I.

man is not played out ; but that man is capable, intellectually,
of great development, and that the general standard will be far higher in the future than at present.
Now, in the development of our knowledge of the workings of Nature out of the tremendously complex assemblage of phenomena presented to the scientific inquirer, mathematics plays in some respects a very limited, in others a very important part. As regards the limitations, it is merely necessary to refer to the sciences connected with living matter, and to the ologie^ generally, to see that the facts and their connections are too indistinctly known to render mathematical analysis practicable, to say nothing of the complexity. Facts are of not much use, considered as facts. They bewilder by their number and their apparent incoherency. Let them be digested into theory, however, and brought into mutual harmony, and it is another matter Theory is the essence of facts. Without theory scientific knowledge would be only worthy of the mad-
house.
In some branches of knowledge, the facts have been so far refined into theory that mathematical reasoning becomes applicable on a most extensive scale. One of these branches is Electromagnetism, that most extensive science which presents such a remarkable two-sidedness, showing the electric and the
magnetic aspects either separately or together, in stationary
conditions, and a third condition when the electric and mag-
netic forces act suitably in dynamical combination, with equal development of the electric and magnetic energies, the state of electromagnetic waves.
It goes without saying that there are numerous phenomena connected with electricity and magnetism which are very imperfectly understood, and which have not been formularised, except perhaps in an empirical manner. Such is particularly the case where the sciences of Electricity and Chemistry meet. Chemistry is, so far, eminently unmathematical (and therefore a suitable study for men of large capacity, who may be nearly destitute of mathematical talent but this by the way), and it appears to communicate a part of its complexity and vagueness to electrical science whenever electrical phenomena which we can study are accompanied by chemical changes. But generally speaking, excepting electrolytic phenomena and other compli-

INTRODUCTION.

13

cations (e.g., the transport of matter in rarefied media when electrical discharges occur), the phenomena of electromagnetisni are, in the main, remarkably well known, and amenable to mathematical treatment.

13. Ohm (a distinguished mathematician, be it noted)

brought into order a host of puzzling facts connecting electro-

motive force and electric current in conductors, which all pre-

vious electricians had only succeeded in loosely binding together

qualitatively under some rather vague statements. Even as

late as 20 years ago, "quantity" and "tension" were much used

by men who did not fully appreciate Ohm's law. (Is it not

rather remarkable that some of Germany's best men of genius

should have been, perhaps, unfairly treated ?

Ohm ;

Mayer ;

Reis ;

even von Helmholtz has mentioned the difficulty he

had

in getting recognised. But perhaps it is the same all the
world over.) Ohm found that the results could be summed

up in such a simple law that he who runs may read it, and a

schoolboy now can predict what a Faraday then could only

guess at roughly. By Ohm's discovery a large part of the

domain of electricity became annexed to theory. Another

large part became virtually annexed by Coulomb's discovery of the law of inverse squares, and completely annexed by Green's

investigations. Poisson attacked the difficult problem of in-

duced magnetisation, and his results, though differently

expressed, are still the theory, as a most important first

approximation. Ampere brought a multitude of phenomena

into theory by his investigations of the mechanical forces between conductors supporting currents and magnets. Then

there were the remarkable researches of Faraday, the prince of

experimentalists, on electrostatics and electrodynamics and the

induction of currents. These were rather long in being brought

from the crude experimental state to a compact system, ex-

my pressing the real essence. Unfortunately, in

opinion,

Faraday was not a mathematician. It can scarcely be doubted

that had he been one, he would have been greatly assisted in
his researches, have saved himself much useless speculation, and would have anticipated much later work. He would, for instance, knowing Ampere's theory, by his own results have

readily been led to Neumann's theory, and the connected

14

ELECTROMAGNETIC THEORY.

CH. 1.

work of Helmholtz and Thomson. But it is perhaps too much to expect a man to be both the prince of experimentalists and
a competent mathematician. Passing over the other developments which were made in
the theory of electricity and magnetism, without striking new departures, we come to about 1860. There was then a collection of detached theories, but loosely connected, and embedded in a heap of unnecessary hypotheses, scientifically valueless, and entirely opposed to the spirit of Faraday's ways of thinking,
and, in fact, to the spirit of the time. All the useless hypotheses had to be discarded, for one thing ; a complete and
harmonious theory had to be made up out of the useful re-
mainder, for another; and, in particular, the physics of the subject required to be rationalised, the supposed mutual attractions or repulsions of electricity, or of magnetism, or of elements of electric currents upon one another, abolished, and electromagnetic effects accounted for by continuous actions through
a medium, propagated in time. All this, and much more, was done. The crowning achievement was reserved for the heavensent Maxwell, a man \v ^e fame, great as it is now, has, com-
paratively speaking, yet to come.

*'"'

14. It will have been observed that I have said next to

nothing upon the study of pure, mathematics ; this is a matter with which we are not concerned. But that I have somewhat

dilated (and I do not think needlessly) upon the advantages attending the use of mathematical methods by the materialist to assist him in his study of the laws governing the material universe, by the proper co-ordination of known and the discovery of unknown (but not unknowable) phenomena.
It was discovered by mathematical reasoning that when an electric current is started in a wire, it begins entirely upon its skin, in fact upon the outside of its skin ; and that, in conse-
quence, sufficiently rapidly impressed fluctuations of the current keep to the skin of the wire, and do not sensibly penetrate to

its interior.

Now very few (if any) unmathematical electricians can

understand this fact ;

many of

them neither understand it nor

believe it. Even many who do believe it do so, I believe, simply

because they are told so, and not because they can in the least

INTRODUCTION.

15

feel positive about its truth of their own knowledge. As an

eminent practician remarked, after prolonged scepticism, " When

Sir

W.

Thomson

says

so,

who

can

doubt

" it ?

What a world of

worldly wisdom lay in that remark !

Now I do admire this characteristically stubborn English way

of being determined not to be imposed upon by any absurd

theory that goes against all one's most cherished convictions,

and which cannot be properly understood without mathe-

matics. For without the mathematics, and with only the sure knowledge of Ohm's law and the old-fashioned notions

concerning the function of a conducting wire to guide one, no

one would think of such a theory. It is quite preposterous

from this point of view.

Nevertheless,

it

is true ;

and

the

view was not put forward as a hypothesis, but as a plain matter

of fact.

The case in question is one in which we can be very sure of

all the fundamental data of any importance, and the laws con-

We cerned.

can, for instance, by straightforward experiment,

especially with properly constructed induction balances ad-

mitting of exact interpretation of resu '; readily satisfy our-
selves that a high degree of accuracy must obtain not merely

for Ohm's law, but also for Jbo,xiday's law of E.M.F. in circuits,

and even in iron for Poisson's law of induced magnetisation,

We within certain limits.

have, therefore, all the conditions

wanting for the successful application of mathematical reasoning of a precise character, and justification for the confidence that mathematicians can feel in the results theoretically deduced in
a legitimate manner, however difficult it may be to give an
easily intelligible account of their meaning to the unmathe-

matical.

This, however, I will say for the sceptic who has the courage of his convictions, and writes openly against what is, to him, pure nonsense. He is doing, in his way, good service in the cause of truth and the advancement of scientific knowledge, by

stimulating interest in the subject and causing people to inquire and read and think about these things, and form their own
judgment if possible, and modify their old views if they should be found wanting. Nothing is more useful than open and free

criticism, and the truly earnest and disinterested student of

science always welcomes it.

1C

ELECTROMAGNETIC THEORY.

CH. I.

15. The following may assist the unmathematical reader to
an understanding of the subject. It is not demonstrative, of course, but is merely descriptive. If, however, it be translated into mathematical language and properly worked out, it will be found to be demonstrative, and to lead to a complete theory of the functions of wires in general.
Start with a very long solenoid of fine wire in circuit with a source of electrical energy. Let the material inside the solenoid
be merely air, that is to say, ether and air. If we examine the
nature of the fluctuations of current in the coil in relation to
the fluctuations of impressed force on it, we find that the cur-
rent in the coil behaves as if it were a material fluid possessing
inertia and moving against resistance. The fanatics of Ohm's
law do not usually take into account the inertia. It is as if the
current in the coil could not move without simultaneously
setting into rotation a rigid material core filling the solenoid, and free to rotate on its axis.
If we take the air out of the solenoid and substitute any other non-conducting material for a core the same thing happens; only the inertia varies with the material, according to its mag-
netic inductivity.
But if we use a conducting core we get. new phenomena, for we find that there is no longer a definite resistance and a definite inertia. There is now frictional resistance in the core, and this increases the effective resistance of the coil. At the
same time the inertia is reduced.
On examining the theory of the matter (on the basis of
Ohm's law and Faraday's law applied to the conducting core) we find that we can now account for things (in our analogy) by supposing that the rigid solid core first used is replaced by a
viscous fluid core, like treacle. On starting a current in the coil it cannot now turn the core round bodily at once, but only
its external portion. In time, however, if the source of current be steadily operating, the motion will penetrate throughout the
viscous core, which will finally move as the former rigid core did.
If, however, the current in the coil fluctuate in strength very rapidly, the corresponding fluctuations of motion in the core
will be practically confined to its skin. The effective inertia is reduced because the core does not move as a rigid body ; the effective resistance is increased by the viscosity generating heat.-

INTRODUCTION.

17

Now, returning to the solenoid, we have perfect symmetry

with respect to its axis, since the core is supposed to be exceed-

ingly long, and uniformly lapped with wire. The situation of

the source of energy, as regards the core itself, is plainly on its

boundary, where the coil is placed ; and it is therefore a matter

of common-sense that in the communication of energy to the

core either when the current is steady or when it varies, the

transfer of energy takes place transversely, that is, from the

boundary to the axis, in planes perpendicular to the axis, and

therefore perpendicular to the current in the core itself. This

is confirmed by the electromagnetic equations.

But the electromagnetic equations go further than this, and

assert that the transference of energy in any isotropic electrical

conductor always takes place across the lines of conduction cur-

rent, and not merely in the case of a core uniformly lapped

with wire, where it is nearly self-evident that it must be so. This

is a very important result, being the post-finger pointing to a clear

understanding of electromagnetism. Passing to the case of a

very long straight round wire supporting an electric current, we

are bound to conclude that the transference of energy takes place

transversely, not longitudinally ; that is, across the wire instead

of along it. The source of energy must, therefore, first supply

the dielectric surrounding the wire before the substance of the

wire

itself

can

be

influenced ;

that

is,

the

dielectric

must

be

the real primary agent in the electromagnetic phenomena con-

nected with the electric current in the wire.

Beyond this transverse transference of energy, there does not, however, at first sight, appear to be much analogy between the case of the solenoid with a core and the straight wire in a dielectric. The source of energy in one case is virtually brought right up to the surface of the core in a uniform manner. But in
the other case the source of energy the battery, for instance
may be miles away at one end of the wire, and there is no
immediately obvious uniform application of the source to the skin of the wire. But observe that in the former case the

magnetic force is axial, and the electric current circular, whilst
in the latter case the electric current (in the straight wire) is
axial and the magnetic force circular. Now an examination
of the electromagnetic equations shows that the conditions of propagation of axial magnetic force and circular current are the

18

ELECTROMAGNETIC THEORY.

CH. I.

same as those for axial current and circular magnetic force.
We therefore further conclude that (with the exchanges made)
the phenomena concerned in the core of the solenoid and in the long straight wire are of the same character.
Furthermore, if we go into detail, and consider the influence of the surroundings of the wire (which go to determine the value of the inductance of the circuit) we shall find that not only is the character of the phenomena the same, but that they may be made similar in detail (so as to be represented by
similar curves, for example).
The source of energy, therefore, is virtually transferred instantly from its real place to the whole skin of the wire, over which it is uniformly spread, just as in the case of the con-
ducting core within a solenoid.

16. So far we can go by Ohm's law and Faraday's law of E.M.F., and, if need be, Poisson's law of induced magnetisation (or its modern equivalent practically). But it is quite impossible to stop here. Even if we had no knowledge of electrostatics and of the properties of condensers, we should, by the above course of inquiry, be irresistibly led to a theory of transmission of electrical disturbances through a medium surrounding the wire, instead of through the wire. Maxwell's theory of dielectric displacement furnishes what is wanted to explain results which are in some respects rather unintelligible when deduced in the above manner without reference to elec-

trostatic phenomena.
We learn from it that the battery or other source of energy

acts upon the dielectric primarily, producing electric displace-

ment and

magnetic

induction ;

that

disturbances

are

propa-

gated through the dielectric at the speed of light ; that the manner of propagation is similar to that of displacements and

motions in an incompressible elastic solid; that electrical

conductors act, as regards the internal propagation, not as

conductors but rather as obstructors, though they act as con-

ductors in another sense, by guiding the electromagnetic waves along definite paths in space, instead of allowing them

to be immediately spread away to nothing by spherical enlargement at the speed of light ; that when we deal with

steady states, or only slowly varying states, involving immensely

INTRODUCTION.

19

great wave length in the dielectric, the resulting magnetic

phenomena are just such as would arise were the speed

of propagation infinitely great instead of

being

finite ;

that

if we make our oscillations faster, we shall begin to get

signs of propagation in the manner of waves along wires, with,

however, great distortion and attenuation by the resistance of

the wires that if we make them much faster we shall obtain ;

a comparatively undistorted transmission of waves (as in long-

distance telephony over copper wires of low resistance) ; and that if we make our oscillations very fast indeed, we shall have

practically mere skin conduction of the waves along the wires

at the speed of light (as in some of Lodge's lightning-conductor

experiments, and more perfectly with Hertzian waves).
Now all these things have been worked out theoretically, and,

as is now well known, most of them have been proved experi-

mentally ; and yet I hear someone say that Hertz's experiments

don't prove anything in particular !

Lastly, from millions of vibrations per second, proceed to

billions, and we come to light (and heat) radiation, which are,

in Maxwell's theory, identified with electromagnetic disturb-

ances. The great gap between Hertzian waves and waves of

light has not yet been bridged, but I do not doubt that it will

be done by the discovery of improved methods of generating

and observing very short waves.

c2

CHAPTER 11.

OUTLINE OF THE ELECTROMAGNETIC
CONNECTIONS.

Electric

and

Magnetic

Force ;

Displacement and

Induction ;

Elastivity and Permittivity, Inductivity and Reluctivity.

20. Our primary knowledge of electricity, in its quantita-,

tive aspect, is founded upon the observation of the mechanical

forces experienced by an electrically charged body, by a mag-

netised body, and by a body supporting electric current. In

the study of these mechanical forces we are led to the more

abstract ideas of electric force and magnetic force, apart from

electrification, or magnetisation, or electric current, to work

upon

and

visible effects.

^produce

The conception of fields

of force nr J irally follows, with the mapping out of space

A by means o. -ines or tubes of force definitely distributed.

further and very important step is the recognition that the two

vectors, electric force and magnetic force, represent, or are

capable of measuring, the actual physical state of tl^e medium

concerned, from the electromagnetic point of view, when taken

in conjunction with other quantities experimentally recognis-

able as properties of matter, showing that different substances

are affected to different extents by the same intensity of electric

or magnetic force. Electric force is then to" be conceived as

producing or being invariably associated with a flux, the electric

displacement ; and similarly magnetic force as producing a

second flux, the magnetic induction.

D E If be the electric force at any point and

the displace-

ment, we have

D = cE:

(1)

OUTLINE OP ELECTROMAGNETIC CONNECTIONS.

21

H and similarly, if be the magnetic force and B the induction,

then

B = /*H

(2)

Here the ratios c and p represent physical properties of the medium. The one (ft), which indicates capacity for supporting
magnetic induction, is its inductivity ; whilst the other, indicating the capacity for psrmitting electric displacement, is its
permittivity (or permittancy). Otherwise, we may write

E^-iD,

(3)

H = ^B;

(4)

and

now the

ratio

c"1 is

the

elastivity and

1
fir

is the

reluctivity

(or reluctancy). Sometimes one way is preferable, sometimes

the other.

Electric and Magnetic Energy.

21. All space must be conceived to be filled with a medium

which can support displacement and induction. In the former

aspect only it is a dielectric. It is, however, equally necessary
to consider the magnetic side of the matter, and we may, without coining a new word, generally understand by a dielectric a medium which supports both the fluxes mentioned.
Away from matter (in the ordinary sense) the medium concerned is the ether, and p and c are absolute constants. The

presence of matter, to a first approximation, merely alters the value of these constants. The permittivity is al ys increased,
far as is known. On the other hand, the inductivity may be

either increased or reduced, there being a very small increase

or decrease in most substances, but a very large increase in a

few, the so-called magnetic metals. The range within which

the proportionality of flux to force obtains is then a limited one,

which, however, contains some important practical applications.

D When the fluxes vary, their rates of increase B and are the

E velocities corresponding to the forces

and H, provided no

other effects are produced. The activity of E is ED, and
H that of is HB. The work spent in producing the fluxes

(not counting what may be done simultaneously in other ways)

is, therefore
U = TE dD,

... B

T =| H rfB,

(5)

22

ELECTROMAGNETIC THEORY.

CH. IT.

U where

is the electric and T the magnetic energy per unit

volume.

When /* and c are constants, these give

..... U - JED - JcE*,

(6)

..... T = JHB = J/JP,

(7)

to express the energy stored in the medium, electric and mag-
netic respectively.
When p and c are not constants, the previous expressions (5)
will give definite values to the energy provided there be a definite relation between a force and a flux. If, however, there be no definite relation (which means that other circumstances than the value of the force control the value of the flux), the energy stored will not be strictly expressible in terms simply of the force and the flux, and there will be usually a waste of energy in a cyclical process, as in the case of iron, so closely studied by Ewing. This does nob come within the scope of a precise mathematical theory, which must of necessity be a sort of skeleton framework, with which complex details have to be separately adjusted in the most feasible manner that presents
itself.
Eolotropic Relations.

22. But a precise theory nevertheless admits of considerable extension from the above with //, and c regarded as scalar constants. All bodies are strained -more or less, and are thereby usually made eolotropic, even if they be not naturally eolotropic. The force and the flux are not then usually concurrent, or identical in direction. But at any point in an eolotropic substance there are always (if force and flux be proportional) three mutually perpendicular axes of concurrence the principal axes when we have (referring to displacement)

if the c's are the principal permittivities, the E's the corre-

sponding effective components of the electric force, and the

D's those of the displacement.

E If

be parallel to a principal

axis, so is D. In general, by compounding the force compo-

nents, we obtain E the actual force, and by compounding the

D flux components obtain

the displacement to correspond,

which can only concur with E in the above-mentioned special

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

23

cases if the principal permittivities be all different. But should

D a pair be equal, then E and

concur in the plane containing

the equal permittivities, for the permittivity is the same for

any axis in this plane.

The energy stored is still half the scalar product of the \

force and the flux, or JED, understanding that the scalar

product of two vectors, which is the product of their tensors (or magnitudes) when they concur, is the same multiplied by

the cosine of their included angle in the general case.

Vector-analysis is, I think, most profitably studied in the

concrete application to physical questions, for which, indeed, it

is specially adapted. Nevertheless, it will be convenient, a

little later, to give a short account of the very elements of the

subject, in order not to have to too frequently interrupt our

electromagnetic arguments by mathematical explanations. In the meantime, consider the dielectric medium further.

Distinction between Absolute and Relative Permittivity or
Inductivity.
23. The two quantities c and /* are to be regarded as known data, given over all space, usually absolute constants ; but when the simpler properties of the ether are complicated by the presence of matter, then varying in value from place to place in isotropic but heterogenous substances ; or, in case of eolotropy, the three principal values must be given, as well
as the direction of their axes, for every point Considered. Keeping to the case of isotropy, the ra^'os c/c and /X//AO of the permittivity and inductivity of a body to that of the standard ether are the specific inductive capacities, electric and magnetic
respectively ; and are mere numerics, of course. They do not
express physical properties themselves except in the limited
sense of telling us how many times as great something is in
one case than in another. This is an important point. It is like the difference between density and specific gravity. It is possible to so choose the electric and magnetic units that /*= 1,
c=l in ether; then /A and c in all bodies are mere numerics.
But although this system (used by Hertz) has some evident recommendations, I do not think its adoption is desirable, at least at present. I do not see how it is possible for any medium to have less than two physical properties effective in

24

ELECTROMAGNETIC THEORY.

CH. II.

the propagation of waves. If this be admitted, I think it may

also be admitted to be desirable to explicitly admit their exist-

ence and symbolise them (not as mere numerics, but as physical

magnitudes in a wider sense), although their precise interpre-

tation may long remain unknown.
H If, for example, be imagined to be the velocity of a sub-

stance,

then

2
JfiH

is

its kinetic

energy,

and

/x

its

density.

And if be a torque, then c"1 (the elastivity) is the corre-

sponding coefficient of elasticity, the rigidity, or gwm'-rigidity,

as the case may be ; whilst c is the coefficient of compliance, or

the compliancy ;

and

2
JcE

is

the stored

energy of the

strain.

Dissipation of Energy.

The Conduction-current Conduc;

tivity and Resistivity. The Electric Current.

w 24. Besides influencing the values of .the ether constants

above described, we have also to admit that in certain kinds

of matter, when under the influence of electric force, energy is

dissipated continuously, besides being stored. These are called
electrical conductors. When the conduction is of the simplest

(metallic) type, the waste of energy takes place at a rate pro-

portional to the square of the electric force. Thus, if Q x be

the Joulean waste,

->r

Q^/jE^EC,

(8)

if

C= E

(9)

This new flux C is the conduction current, and k is the conduc-

tivity (electric). Its reciprocal is the resistivity.
The termination -ivity is used in connection with specific

properties. It does not always sound well at first, but that

wears off. Sometimes the termination -ancy does as well.

The conductivity Jc is constant (at one temperature), or is a

linear operator, as in

the previous cases with respect

to

and
//,

c.

The dissipation of

energy does

not

-
imply its destruction,

but simply its rejection or waste, so far as the special electromagnetic affairs we are concerned with. The conductor is

heated, and the heat is radiated or conducted away. This is also (most probably) an electromagnetic process, but of a

different order. Only in so far as the effect of the heat alters

the conductivity, &c., or, by differences of temperature, causes

OUTLINE OP ELECTROMAGNETIC CONNECTIONS.

25

thermo-electric force, are we concerned with energy wasted

according to Joule's law.

The activity of the electric force, when there is waste, as well

as storage, is

E(C + 6) = Q1 + U

(10)

The sum D + is the electric current, when the medium is at rest. When it is in motion, a further term has sometimes to
be added, viz., the convection current.

Fictitious Magnetic Conduction-current and Real Magnetic
Current.

25. If a substance were found which could not support magnetic force without a continuous dissipation of energy, such & substance would (by analogy) be a magnetic conductor. Let,

for instance,

K = <?H,

(11)

K then is the density of the magnetic conduction current, and
the rate of waste of energy is

Q2

=

HK

=

2
<7H

(12)

The activity of the magnetic force is now

H(K + B) = Q2 + T

(13)

Compare this equation with (10).
K + B.

The magnetic current is

As there is (I believe) no evidence that the property symbolised by g has any existence, it is needless to invent a special name for it or its reciprocal, but to simply call g the magnetic ' conductivity. The idea of a magnetic current is a very useful

one, nevertheless.

The

magnetic current

B is

of

course

real ;

K it is the part that is speculative. It plays an important

part in the theory of the transmission of waves in conductors.

Forces and Fluxes.

26. So far we have considered the two forces, electric and

magnetic, producing four fluxes, two involving storage and two

waste of energy, and we have defined the terminology when the

We state of things at a point is concerned.

reckon forces per

26

ELECTROMAGNETIC THEORY.

CII. IK

unit length, fluxes per unit area, and energies or wastes per unit volume. It is thus a unit cube that is referred to, whose edge, side, and volume are utilised. But a unit cube does not mean a cube whose edge is 1 centim. or any other concrete
length ; it may indeed be of any size if the quantities concerned
are uniformly distributed throughout it, but as they usually vary from place to place, the unit cube of reference should be imagined to be infinitely small. The next step is to display the equivalent relations, and develop the equivalent terminology, when any finite volume is concerned, in those cases that admit of the same simple representation in the form of linear equa-
tions.

Line-integral of a Force. Voltage and Gaussage.

27. The line-integral of the electric force from one point to

another along a stated path is the electromotive force along

that path ; this was abbreviated by Fleemiug Jenkin to E.M.F.
He was a practical man, as well as a practician. When ex-

pressed in terms of a certain unit called the volt, electromotive

force may be, and often is, called the voltage. This is much better than " the volts." I think, however, that it may often

be conveniently termed the voltage irrespective of any par-

We ticular unit.

might put it in this way. Volta was a

distinguished man who made important researches connected

with electromotive force, which is, therefore, called voltage,

whilst a certain unit of voltage is called a volt. At any rate,

we may try it and see how it works.

The line-integral of the magnetic force from one point to

another along a stated path is sometimes called the magneto-

motive force. The only recommendation of this cumbrous

term is that it is correctly correlated with the equally cum-

brous electromotive force. Magnetomotive force may be called

the gaussage [pr. gowsage], after Gauss, who distinguished him-

self in magnetic researches; and a certain unit of gaussage

may be called a gauss [pr. gowce]. I believe this last has

already been done, though it has not been formally sanctioned.

Gaussage may also be experimented with. The voltage or the gaussage along a line is the sum of the

effective electric

or

magnetic forces

along

the

line ;

the effec-

tive force being merely the tangential component of the real

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

27

force. Thus, electric force is the voltage per unit length, and magnetic force the gaussage per unit length along lines of force.

Surface-integral of a Flux. Density and Intensity.

28. Next as regards the fluxes, when considered with reference to any area. The flux through an area is the sum

of

the

effective

fluxes

through

its

elementary units of

area ;

the effective flux being the normal component of the flux or

We the component perpendicular to the area.

do not, I think,

need a number of new words to distinguish fluxes through a

surface from fluxes per unit surface. Thus, we may speak of

the induction through a surface (or through a circuit bounding

it) ; or of the current through a surface (as across the section of a wire); or of the displacement through a surface (as in aeon-

denser), without any indefiniteness, meaning in all cases the

surface integral of the flux in question.

In contradistinction to this, it may be sometimes convenient

to speak of the density of the current, or of the induction, or

of the displacement, that is, the amount per unit area. Similarly, we may sometimes speak of the intensity of the

electric or 'magnetic force, using "density" for a flux and " intensity " for a force.

It may be observed by a thoughtful reader that there is a

good deal of the conventional in thus associating one set of vectors with a line, and another set with a surface, and other

quantities with a volume. It is, however, of considerable practical utility to carry out these distinctions, at least in a mathe-

matical treatment. But it should never be forgotten that

electric force, equally with displacement, is distributed through-

out volumes, and not merely along lines or over areas.

Conductance and Resistance.
29. Conductivity gives rise to conductance, and resistivity to resistance. For explicitness, let a conducting mass of any
A shape be perfectly insulated, except at two places, and B, to
be conductively connected with a source of voltage. Let the
voltage established between A and B through the conductor be
V, and let it be the same by any path. This will be the case when the current is steady. Also let G be this steady current,
in at A and out at B. We shall have

28

ELECTROMAGNETIC THEORY.

CH. II.

..... V = RC, C = KV,

(1)

R K where and are constants, the resistance and the conduct-
ance, taking the place of resistivity and conductivity, when voltage takes the place of electric force, and the current that of current-density. The activity of the impressed voltage is

..... VC = RC2 = KV 2

. . (2)

and represents the Joulean waste per second in the whole con-
ductor, or the volume-integral of EC or of &E 2 before con-
sidered.

Permittance and Elastance.

30. Permittivity gives rise to permittance, and elastivity to elastance. To illustrate, for the conductor, substitute a

nonconducting dielectric, leaving the terminals and external
arrangements as before. We have now a charged condenser.

Displacement, i.e., the time-integral of the current, takes the
place of current in the last case, and we now have

D = SV,

.... V = S-!D,

(3)

D if is the displacement, S the permittance, and its reciprocal
the elastance of the condenser. This elastance has been called
the stiffness of the condenser by Lord Rayleigh. It is the elastic resistance to displacement. The displacement is the measure of the charge of the condenser.
The total energy in the condenser is

(4)

i.e., half the product of the force (total) and the flux (total),

between and at the terminals ; it is also the volume-integral of

the

energy-density,

or

2
J2cE .

As the dielectric is supposed to be a non-conductor, the cur-

rent is I) or SV, and only exists when the charge is varying.
But it may also be conducting. If so, let the conductance be K, making the conduction current be C = KV. The true current (that is, the current) is now the sum of the conduction
and displacement currents. Say,

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

29"

This is the characteristic equation, of a condenser. It comes to the same thing if the condenser be non-conducting, but be shunted by a conductance, K. In a conducting dielectric the permittivity and the conductivity are therefore in parallel arcr as it were. It was probably by a consideration of conduction in a leaky condenser that Maxwell was led to his inimitable theory of the dielectric, by which he boldly cut the Gordian knot of electromagnetic theory.
The activity of the terminal voltage we find by multiplying,
(5) by V, giving

vr=vc-fvi>,

2+

\ SV2 .... (6)

representing the waste in Joulean heating and the rate of increase of the electric energy. Each of these quantities is the sum of the same quantities per unit volume throughout the substance concerned.

Permeance, Inductance and Reluctance.
31. Permeability gives rise to permeance, inductivity to inductance, and reluctivity to reluctance.
The formal relation of reluctance to reluctivity with magnetic force and induction, is the same as that of resistance to resistivity with electric force and conduction current, or of elastance to elastivity with electric force and displacement.
Permeance is the reciprocal of reluctance. In this sense I have used it, though only once or twice. Prof. S. P. Thompson has also used tfae word in this sense in his Cantor Lectures
with good effect. If we replace our illustrative conductor by an inductor,
supporting magnetic induction, and suppose it surrounded by imaginary matter of zero inductivity, and have an impressed gaussage instead of voltage at the terminals, we shall have a flux of induction which will, if the force be weak enough, vary
H as the force. If be the gaussage and B the induction enter-
ing at the one and leaving at the other terminal, the ratio
H/B is the reluctance, and the reciprocal B/H is the permeance. The energy stored is

30

ELECTROMAGNETIC THEORY.

CH. II.

When the relation of flux to force is not linear, we can still

usefully employ the analogy with conduction current or with
H displacement by treating the ratio B/H as a function of or

of B ;

as witness the improved and

simplified way of

consider-

ing the dynamo in recent years. I must, however, wonder at the persistence with which the practicians have stuck to " the

lines," as they usually term the flux in question.
I am aware that the use of the name induction for this flux,

which I have taken from Maxwell, is in partial conflict with an

older use. But it is seldom, if ever, that these uses occur to-

.gether, for one thing ; another thing is that the older (and often vague) use of the word induction has very largely ceased

of late years. It was not without consideration that induction was adopted and, to harmonise with it, inductance and induc-

tivity were coined.

Inductance of a Circuit.

32. The meaning of inductance has sometimes been mis-

conceived. It is not a synonym for induction, nor for self-induction, but means " the coefficient of self-induction," sometimes
abbreviated to " the self-induction." It is essentially the same

as permeance, the reciprocal of reluctance, but there is a prac-

tical distinction. Consider a closed conducting circuit of one

turn of wire, supporting a current Cr As will later appear,

this G! is also the gaussage. That is, the line -integral of the

magnetic force in any closed circuit (or the circuitation of the

force) embracing the current once is Cr

Let also B be the A

induction through the circuit of Cr Then

BX-LA, ...... (8)

where, by what has already been explained, LH is the permeance of the magnetic circuit, a function of the distribution of inductivity and of the form and position of the conducting core. The magnetic energy is
JBA-iW ..... (9)
H by using the first expression in (7), remembering that
there is now represented by C^ and then using (8). This

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

31

energy " of the current " resides in all parts of the field, only (usually) a small portion occupying the conductor itself.
Now substitute for the one turn of wire a bundle of wires,
N in number, of the same size and form. Disregarding small
differences due to the want of exact correspondence between
the bundle and the one wire, everything will be the same as before if the above Cj means the total current in the bundle. But if the same current be supported by each wire, practical
convenience in respect to the external connections of the coil
requires us to make the current in each wire the current. Let
this be C, so that Cj = NO. Then we shall have, by (8),

B i

=

(L1N)C

.

.

. . (8a)

to express the flux

of induction ;

and by

(9)

and

(Sa),

...... (9o)

N if L =

2L 1}

to

express

the

energy.

This L is the inductance

N of the coil. It is 2 times the permeance of the magnetic

circuit.

B N Again, regarding the coil as a single circuit, : is the

induction through it

that

is,

B x

through

each

winding.

Calling

this total B, we have, by (8a),

.... B

=

B 1

N=(L

N
1

2

)C

=

LC,

(86)

which harmonises properly with (9a). The difference between inductance and permeance, therefore,
merely depends upon the different way of reckoning the current in the coil. With one winding only, they are identical. I
should here observe that I am employing at present rational
units. Their connection with the Gaussian units will appear later. It would only serve to obscure the subject to bring in 47T, that arbitrary and unnecessary constant which has puzzled
so many people.
It will be seen that the distinction between permeance and inductance is a practical necessity, in spite of their fundamental
identity. But which should be which ? On the whole, I
prefer it as above stated, especially to connect with self-induction. Regarding permeability itself, it would seem that this
name is more particularly suitable to express the ratio /*//z of

32

ELECTROMAGNETIC THEORY.

CH. II.

the inductivity of a medium to that of ether, which is, in factr

consistent with the original meaning, I believe, as used by
Sir W. Thomson in connection with his " electromagnetic definition " of magnetic force. But to inductivity, as before-

mentioned, a wider significance should be attached. As has

been more particularly accentuated by Prof. Riicker, we really

do not know anything about the real dimensions of ft and c ; or, more strictly, we do not know the real nature of the

electromagnetic mechanism, so that ft and c are very much

what we choose to make them, by assumptions. The two prin-

= cipal systems are the so-called electrostatic, in which c 1 in

ether,

and

the

electromagnetic,

in

which

=
ft

1

in ether.

But

with these specialities we have no further concern at present.

Cross-connections of Electric and Magnetic Force. Circuital
Flux. Circuitation.
33. The two sets of quantities, the electric and magnetic forces, with their corresponding fluxes and currents, and the connected products and ratios, may be considered quite independently of one another, without any explicit connection being stated between the electric set and the magnetic set, whether they coexist or not. But to have a dynamical electromagnetic theory, we require to know something more, viz., the cross-
connections or interactions between E and H. Or, in another
form, we require to know how an electric field and a magnetic-
field mutually influence one another. One of these interactions has been already partially men-
tioned, though only incidentally, in stating the meanings of permeance and inductance. It was observed that the electric current in a simple conductive circuit was measured by the gaussage in the corresponding magnetic circuit.
A word has been much wanted to express in a convenient
and concise manner the property possessed by some fluxes and other vectors of being distributed in closed circuits. This want has been recently supplied by Sir W. Thomson's introduction of the word " circuital " for the purpose.* Thus electric current is a circuital flux, and so is magnetic induction. The fundamental basis of the property is that as much of a circuital flux enters
* "Mathematical and Physical Papers," Vol. III., p. 451.

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

33

any volume at some parts of its surface as leaves it at others, so that the flux has no divergence anywhere. This qualification,

"anywhere," should be remembered, for a flux which may

diverge locally, as, for instance, electric displacement, is not

circuital in general, though even electric displacement may be

circuital sometimes. Further, as a flux need not be distributed

throughout a volume, but may be confined to a surface, or to

a line, we have then specialised meanings of circuital and of

divergence. Or a volume-distribution and a surface or line-

distribution of a flux may be necessarily conjoined, without,

however, any departure from the essential principle concerned.

The word " circuital," which will be often used, suggested to

me

the word

" circulation,"

to

indicate

the

often-occurring

operation of a line-integral in a closed circuit ; as, for instance,

in the estimation of circuital voltage or gaussage. Now, in the
case of a moving fluid, Sir W. Thomson called the line-integral
of the velocity in a closed circuit the " circulation." This is curiously like " circulation." But " circulation " seems to have

too specialised a meaning to be suitable for application to any vector, and I shall employ " circulation." The operation of

circuitation is applicable to any vector, whether it be circuital

or not.

First Law of Circuitation.

34. Now in the case of a simple conductive circuit, we have

two circuital fluxes. There is a circuital conducting core sup-

porting an electric current, and there is a circuital flux of in-

duction through the conductive circuit. In the electric circuit

we have Ohm's law,

E = RC,

(1)

E R where

is the circuital voltage, C the current, and the re-

sistance. And in the magnetic circuit we have a formally

similar relation,

H=L-IB,

(2)

H where is the circuital gaussage, B the induction, and Lr1 the

reluctance. Or,

B = LH,

(3)

where L is the inductance (or the permeance, when there is
only one turn of wire). D

34

ELECTROMAGNETIC THEORY.

CH. II.

Now, the cross-connection in this special case is implied in
H the assertion that and G are the same quantity, when mea-
sured in rational units The expression of the law of which
this is an illustration is contained in any of the following alter-

native statements.

The line-integral of the magnetic force in any closed circuit

measures the electric current through any surface bounded by

the circuit. Or,

The circuitation of the magnetic force measures the electric

current through the circuit. Or,

The electric current is measured by the magnetic circuita-

tion, or by the circuital gaussage.

The terminology of electromagnetism is in a transitional

state at present, owing to the change that is taking place in

popular ideas concerning electricity, and the unsuitability of the

old terminology, founded upon the fluidity of electricity, for a

comprehensive view of electromagnetism. This is the excuse

for so many new words and forms of expression. Some of

them may find permanent acceptation.

The above law applies to any circuit of any size or shape,

and irrespective of the kind of matter it passes through, meaning by " circuit " merely a closed line, along which the gauss-
age is reckoned. By " the current " is to be understood the

current ;

not merely the

conduction

current alone,

or the dis-

placement current alone, but their sum (the convection current

term will be considered separately).

It is also necessary to understand that a certain convention

is implied in the statement of the law, regarding positive

senses of translation and rotation when taking line and surface

integrals. Look at the face of a watch, and imagine its circum-

ference to be the electric circuit. The ends of the pointers

travel in this circuit in the positive sense, if you are looking

through the circuit along its axis in the positive sense. Also, you are looking at the negative side of the circuit. Thus, when the current is positive in its circuit, the magnetic induction goes through it in the positive direction, from the negative side to the
positive side. Otherwise, the positive sense of the current in a circuit and the induction through it are connected in the same
way as the motions of rotation and translation of a nut on an ordinary right-handed screw. This is the " vine " system used

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

35

by all British writers; but some continental writers use the "hop" system, in which the rotation is the other way, for the same translation. It is useless trying to work both systems, and when one comes across the. left-handed system in papers, it is, perhaps, best to marginally put the matter straight, and
then ignore the text.

Second Law of Circuitation.

35. The other cross-connection required is a precisely

similar relation between voltage and magnetic current, with,

however, a change of sign. Thus :

The negative line-integral of the electric force in any circuit

(or the electric circuitation) measures the magnetic current

through the circuit. Or,

The voltage in any circuit measures the magnetic current

through the circuit taken negatively. Or,

Magnetic current is measured by the circuital voltage re-

versed ;

and

other alternative

equivalent statements.

Definition of Curl.

36. In the above laws of circuitation the currents are the
concrete currents (surface-integrals), and the forces also the
concrete voltage or gaussage. When we pass to the unit
volume it is the current-density that is the flux. The circuitation of the force is then called its " curl." Thus, if J be the
G electric current and the magnetic current, the two laws are

curlH^J,

(4)

-cur!E 1

=

G,

(5)

H where E and x

are the
x

electric

and

magnetic

force

of

the

We field.

may now say concisely that

The electric current is the curl of the magnetic force.
The magnetic current is the negative curl of the electric force. There is nothing transcendental about " curl." Any man who understands the laws of circuitation also understands
what " curl " means, though he may not himself be aware of his knowledge, being like the Frenchman who talked prose for many years without knowing it. The concrete circuitation is

sufficient for many problems, especially those concerning linear

D2

36

ELECTROMAGNETIC THEORY.

CH. II.

conductors in magnetic theory. But it does not suffice for mathematical analysis, and to go into detail we require to pass
from the concrete to the specific and use curl. How to mani-
pulate " curl " is a different matter altogether from clearly understanding what it means and the part it plays. The latter is open to everybody ; for the former, vector-analysis is most
suitable.
Let a unit area be chosen perpendicular to the electric cur-
H rent J. Its edge is then the circuit to which a belongs in (4).
The gaussage in this circuit measures the current-density. Similarly, regarding (5), the voltage in a unit circuit perpendicular to the magnetic current measures its density (negatively). In short, what circuitation is in general, curl is the same per unit area.

Impressed Force and Activity.

37. In the statement of the laws of circuitation, I have
intentionally omitted all reference to impressed forces. That there must be impressed forces is obvious enough, because a
dynamical system comprehending only the electric and magnetic stored energies and the Joulean waste, is only a part of
the dynamical system of Nature. We require means of show-
ing the communication of energy to or from our electromagnetic
system without having to enlarge it by making it a portion of a more complex system. Thus, taking it as it stands at present, the activity per unit volume we have seen to be
..... (6)

where the left side expresses the activity of the electric and the

magnetic force on the corresponding currents, and the right side

Q what results, viz., waste of energy, per second, and increase per

U second of the electric energy

and

the

magnetic

T ;

and,

as

there are supposed to be no impressed forces, if we integrate

through all space, we shall obtain

(7)

where 2 means summation of what follows it. Q Or, if be the

U total waste, and similarly

and T the total energies, .

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

37

meaning, that whatever energy there be wasting itself is derived solely from the electric or magnetic energy, which decrease
accordingly. This is the persistence of energy when there are no impressed forces.
Now, if there be impressed forces communicating energy at the rate A, the last equation must become

.... T ,

(8)

A and must be the sum of the activities of the impressed forces

f in the elements of volume, in whatever way space may be

divided into elements, large or small, and however we may

choose to reckon the impressed forces. There may be many

ways

of

doing

it ;

f may

sometimes, for

example, be

an ordi-

nary force, and v, the velocity to match, is then a translational

velocity. But for our immediate purpose, it is naturally con-

venient to reckon the impressed forces electrically and mag-

netically ; so that the corresponding velocities are the electric

We and magnetic currents.

shall then have, if e be the im-

pressed electric, and h the impressed magnetic force,

to represent their activity per unit volume, and in all space,
... (9)
We Instead of (7). This is the integral equation of activity.
cannot remove the sign of summation and make the same form do for the unit volume, for this would make every unit volume independent of the rest, and do away with all mutual action between contiguous elements and transfer of energy between
them. This matter will be returned to in connection with the
transference of energy.

Distinction between Force of the Field and Force of the
Flux.

H H 38. The distinction between and x

and between E and x

E We is often a matter of considerable importance.

have

(10)

(11)

38

ELECTROMAGNETIC THEORY.

CH. II.

H Now it is E and that are effective in producing fluxes. Thus

D H E is the force of the flux

and also of

and

;

is the force

of the flux B. On the other hand, in the laws of circuitation,

as above expressed, the impressed forces do not count at all j
so that we have, in terms of the forces E and H,

curl(H-h) = J, -curl (E-e) = G,

(12) (13)

equivalent to (4) and (5). To distinguish from the forces of

E H the fluxes, I sometimes call

and
l

the forces " of the field."
3

Of course they only differ where there is impressed force. As

the distribution of the energy, as well as of the fluxes, depends
upon E and H, it is usually best to use them in the formulae.

Classification of Impressed Forces.
39. The vectors representing impressed electric and magnetic force demand consideration as to the different forms they may assume. Their line-integrals are impressed voltage and gaussage. Their activities or powers are eJ and hG- respectively per unit volume, and in this statement we have a sort of definition of what is to be understood by impressed force. For, J being the electric current anywhere, if there be an impressed force e acting, the amount eJ of energy per unit volume is communicated to, or taken in by, the electromagnetic system per second ; and this should be understood to take place at the spot in question. It must then be either stored on the spot, or wasted on the spot, or be somehow transmitted away to other places, to be there stored or wasted, according to a law which will appear later on. Similarly as regards h and G-.
But this concerns only the reckoning of impressed force, and
is independent of its physical origin, which may be of several kinds. Thus under e we include
(1.) Voltaic force.
(2.) Thermo-electric force.
(3.) The force of intrinsic electrisation.
(4.) Motional electric force.
(5.) Perhaps due to various secondary causes, especially ia connection with strains.

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

39

And under h we include
(1.) The force of intrinsic magnetisation. (2.) Motional magnetic force. (3.) Perhaps due to secondary causes.

Voltaic Force.

40. Voltaic force has its origin in chemical affinity. This

is still a very obscure matter. For a rational theory of Chemistry,

one of the oldest of the sciences, we may have to wait long, in

spite of the activity of chemical research and of the develop-

ment of the suggestive periodic law. Yet Chemistry and

Electricity are so intimately connected that we cannot under-

stand either without some explanation of the other. Elec-

tricity is, in its essentials, a far simpler matter than Chemistry,

and it is possible that great light may be cast upon chemical

problems (and molecular physics generally) by previous dis-

coveries and speculations in Electricity. The very abstract nature

of Electricity is, in some respects, in its favour. For there is

considerable truth in the remark (which, if it has not been made

before, is now originated) that the more abstract a theory is,

the more likely it is to be true. For example, it may be that

Maxwell's theory of displacement and induction in the ether

is far more than a working theory, and is something very near

the truth, though we know not what displacement and induc-

tion are. But if we try to materialise the theory by inventing a

special mechanism we are almost certain to go wrong, however

useful the materialisation may be for certain purposes. No

one knows what matter is, any more than ether. But we do

know that the properties of matter are remarkably complex.

It is, therefore, a real advantage to get away from matter

when possible, and think of something far more simple and

We uniform in its properties.

should rather explain matter

in terms of ether, than go the other way to work.

However this be, we have the fact that definite chemical

changes involve definite voltages, and herein lies one of the

most important sources of electric current. Furthermore,

there is the remarkable connection between the quantity of

matter and the time-integral of the current (or quantity of

electricity) produced, involved in the law of electro-chemical

40

ELECTROMAGNETIC THEORY.

CI1. II.

equivalents, which is one of the most suggestive facts in physics, and must be a necessary part of the theory of the atom which is to come. That the energy of chemical affinity
may itself be partly electromagnetic is likely enough. That even conduction may be an electrolytic process is possible, in
spite of the sweet simplicity of Ohm's law and that of Joule. For these laws are most probably merely laws of averages. The well-known failure of Ohm's law (apparent at any rate) when the periodicity of electromagnetic waves in a conductor
amounts to billions per second may perhaps arise from the
period being too short to allow of the averages concerned in
Ohm's law to be established. If so, this may give a clue to the
required modification.
Thermo-electric Force.
41. Thermo-electric force has its origin in the heat of bodies, manifesting itself at the contact of different substances or between parts of the same substance differing in temperature.
Now heat is generally supposed to consist in the energy of
agitation of the molecules of bodies, and this is constantly being transferred to the ether in the form of radiant energy, i.e., electromagnetic vibrations of very great frequency, but in a thoroughly irregular manner. It is this irregularity that is
a general characteristic of radiation. Now the result of sub-
jecting conductors to electric force is to dissipate energy and to heat them. This is, however, an irreversible process. But when contiguous parts of a body are at different temperatures, a differential action on the ether results, whereby a continued effect of a regular type is produced, reversible with the current, and therefore formularisable as due to an intrinsic electric
force, the thermo-electric force. At the junction of different materials at the same temperature it is still the heat that is
the source of energy.
The theory of thermo-electric force due to Sir W. Thomson, based upon the application of the Second Law of Thermo-
dynamics (the First is a matter of course) to the reversible heat effects has been verified for conductive metallic circuits by the experiments of its author, and those of Prof. Tait and others. With some success the same principle has also been applied by von Helmholtz to voltaic cells, which are thermo-electric as well

OUTLINE OP ELECTROMAGNETIC CONNECTIONS.

41

as voltaic cells. There are wheels within wheels, and Ohm's law is merely the crust of the pie.

Intrinsic Electrisation.

42. Intrinsic electrisation is a phenomenon shown by most solid dielectrics under the continued action of electric force. It

is the manifestation of a departure from perfect electric elasti-

city, and is probably due to a molecular rearrangement, result-

ing in a partial fixation of the electric displacement, whereby it is rendered independent of the " external " electrising force.

Thus the displacement initially produced by a given voltage

slowly increases, and upon the removal of the impressed voltage

only the initial displacement will subside, if permitted, imme-

diately. The remainder has become intrinsic, for the time, and

may be considered due to an intrinsic electric force e.

If I be x

the intensity of intrinsic electrisation, and c the permittivity,

then

ii- ......... a)

Ij is the full displacement the force e can produce elastically, all external reaction being removed by short circuiting. It is
not necessarily the actual displacement. The phenomenon of
41 residual charge," "soakage," "absorption," &c., are accounted for by this e and its slow variations.
Maxwell attempted to give a physical explanation of this
phenomenon by supposing the dielectric to be heterogeneously conductive. This is perhaps not the most lucidly successful of
Maxwell's speculations. How far electrolysis is concerned in
the matter is not thoroughly clear.

Intrinsic Magnetisation.

43. Intrinsic magnetisation is, in some respects, a similar

phenomenon, due to a passage from the elastic to the intrinsic

form of induction externally induced in solid materials. Calling

the

intensity

of

intrinsic

magnetisation

I 2,

we

have

where h is the equivalent intrinsic magnetic force, and /* the
inductivity (elastically reckoned). In one important respect intrinsic induction is a less general
phenomenon than intrinsic displacement. There is no magnetic

42

ELECTROMAGNETIC THEORY.

CH. IL.

conductivity to produce similar results as regards the magnetic

current as there is electric conductivity as regards the electric

current. But if there were, then we could have a magnetic

" condenser,"

with

a

magnetically conductive

external

circuit,

and get our residual results to show themselves in it, quite

similarly in kind to, though varying in magnitude and perma-

nence from, what we find with an electric condenser.

The

analogue

of Maxwell's

explanation

of

"

"

absorption

would be heterogeneous magnetic conductivity. This is infi-

nitely more speculative than the other, which is sufficiently

doubtful.

Swing's recent improvement of Weber's theory of magnetism seems important. But as in static explanations of dynamical
phenomena the very vigorous molecular agitations are ignored,
We it is clear that we have not got to the root of the matter.
want another Newton, the Newton of molecular physics. Facts there are in plenty to work upon, and perhaps another heavenborn genius may come to make their meaning plain. Properties of matter are all very well, but what is matter, and why
their properties ? This is not a metaphysical inquiry, but con-
cerns the construction of a physical theory.

The Motional Electric and Magnetic Forces. Definition of a
Vector-Product.

44. The motional electric and magnetic forces are the

forces induced by the motion of the medium supporting the

fluxes. To express them symbolically, it will save much and

repeated circumlocution if we first define the vector-product

of a pair of vectors.

Let a and b be any vectors, and c their vector-product. This

is denoted by

c = Vab,

(3)

V the prefix

meaning "vector," or, more particularly here,

"vector-product." The vector c is perpendicular to the plane

of the vectors a and b, and its tensor (or magnitude) equals

the product of the tensor of a into the tensor of b into the

sine of the angle between a and b. Thus

c = absmO,

(4)

if the italic letters denote the tensors, and 6 be the included

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

43

angle of the vectors. As regards the positive sense of the

vector c, this is reckoned in the same way as before explained with regard to circuitation. Thus, when the tensor c is posi-

tive, a positive rotation about c in the plane of a and b will

carry a to b. If the time by a watch is three o'clock, and the

big hand be a and the little hand b, then the vector c is

directed through the watch from its face to its back. These

vector-products are of such frequent occurrence, and their

Cartesian representation is so complex, that the above concise

way of representing them should be clearly understood. On this understanding, then, we can conveniently say that

the motional electric force is the vector-product of the velocity

and the induction, and that the motional magnetic force is the

vector -product of the displacement and the velocity. Or, in

symbols, according to (3),

....... (5)

..... h = VDq,

. . (6)

where q. is the vector velocity.

A Example.

Stationary Electromagnetic Sheet.

45. It should be remembered that we regard the displacement and the induction as actual states of the medium, and therefore if the medium be moving, it carries its states with it. Besides this, it usually happens that these states are themselves being transferred through the medium (independently of its translational motion), so that the resultant effect on pro-
pagation, considered with respect to fixed space, is a combination
of the natural propagation through a medium at rest, and what we may call the convective propagation. Of course we could not expect the two laws of circuitation for a medium at rest to remain true when there is convective propagation.
The matter is placed in a very clear light by considering the
H very simple case of an infinite plane lamina of E and travel-
ling at the speed of light v perpendicularly to itself through a homogeneous dielectric. This is possible, as will appear later,
H when E and are perpendicular, and their tensors are thus
related :
. (7)

44

ELECTROMAGNETIC THEORY.

CH. IL

Or, vectorising v to v,

...... (8)

according to the definition of a vector-product, gives the directional relations as well as the numerical.
Now, suppose we set the whole medium moving the other way at the speed of light. The travelling plane electro-
magnetic sheet will be brought to rest in space, whilst the
medium pours past it. Being at rest and steady, the electric displacement and magnetic induction can cnly be kept up by
coincident impressed forces, viz. :
e = E, h = H
Now compare (8) with (5) and (6) ; consider the directions
carefully, and remember that the velocity q. is the negative of the velocity v, and we shall obtain the formulae (5), (6), which are thus proved for the case of plane wave motion, by starting with a simple solution belonging to a medium at rest.
The method is, however, principally useful in showing the necessity of, and the inner meaning of the motional electric and magnetic forces. To show the general application of (5) and (6) requires a more general consideration of the motional question, to which we now proceed.

Connection between Motional Electric Force and

"
Electromagnetic

Force."

A 46.

second way of arriving at the motional electric

force is by a consideration of the work done in moving a con-

ducting circuit in a magnetic field. It results from Ampere's

researches, and may be independently proved in a variety

of ways, that the forcive (or system of forces) acting upon

a conducting circuit supporting a current, may be accounted

for by supposing that every element of the conductor is subject to what Maxwell termed " the electromagnetic force." This is

a force perpendicular to the vector current and to the vector

induction, and its magnitude equals the product of their ten-

sors multiplied by the sine of the angle between them. In

short, the electromagnetic force is the vector-product of the

current and the induction. Or, by the definition of a vector-

product,

F = VCB, ...... (9)

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

45

if P is the force per unit volume, the current, and B the induction. Here F is the force arising from the stress in the

magnetic field. Its negative, say f, is therefore the impressed

mechanical force, or

f=VBO, ...... (10)

to be used when we desire to consider work done upon the

electromagnetic system.

The activity of f is fq, if q be the velocity ; or, by (10),

fq = qVBC

(11)

This is identically the same as
fq = CVqB,

..... (12)

by a fundamental formula in vector-analysis.* Here, on the

left side, the activity is expressed mechanically ; on the right

side, on the other hand, it is expressed electrically, as the

scalar product of the current and another vector, which is the

corresponding force; it is necessarily an electric force, and

necessarily impressed. So, calling it e, we have

e = VqB

(13)

again, to express the motional electric force.
It should be observed that we are not concerned in this
mode of reasoning with the explicit connection between e and C; and in this respect the process is remarkably simple. As it, however, rests upon a knowledge of the electromagnetic force, we depart from the method of deriving relations previously pursued. But, conversely, we may by (11) and (12) derive the electromagnetic force from the motional electric force.

Variation of the Induction through a Moving Circuit.

A 47. third method of arriving at (13) is by considering

the rate of change of the amount of induction through a

We moving circuit.

need not think of a conducting circuit,

but, more generally, of any circuit. Let it be moving in any

way whatever, changing in shape and size arbitrarily. The

induction through it is altering in two entirely distinct ways.

First, there is the magnetic current before considered, due to

the time-variation of the induction, so that, if the circuit were

* Proved, with other working formulae, in the chapter on the Algebra of Vectors.

46

ELECTROMAGNETIC THEORY.

CH. II.

at rest in its momentary position, we should have the second law of circulation true in its primitive form

-curlE = G,

(14)

when expressed for a unit circuit. But now, in addition, the

motion of the elements of the circuit in the magnetic field

causes, independently of the time-variation of the field, addi-

tional induction to pass through the circuit. Let its rate of

increase due to this cause be g per unit area. If, then, we
E assume that the circulation of the electric force (of the flux)

equals the rate of decrease of the induction through the circuit

always, whether it be at rest or in motion, the equation (14)

becomes altered to

..... -curlE = G + g,

(15)

where the additional g may be regarded as a fictitious magnetic
current. That it is also expressible as the curl of a vector is

obvious, because it depends upon the velocity of each part of the circuit, and is therefore a line-integral. Examination in detail shows that

g=-curlVciB,

(16)

BO that we have, by inserting (16) in (15),

.... -curl(E-e) = G,

(17)

the standard form of the second law of circuitation, when we
use (13) to express the impressed force.
The method by which Maxwell deduced (13) is substantially the same in principle ; he, however, makes use of an auxiliary function, the vector-potential of the electric current, and this
rather complicates the matter, especially as regards the physical
meaning of the process. It is always desirable when possible to keep as near as one can to first principles. The above may, without any formal change, be applied to the case of assumed
G magnetic conductivity, when involves dissipation of energy;
the auxiliary g in (15), depending merely upon 'the motion of the circuit across the induction, does not itself involve dissipation.

Modification. Circuit Fixed. Induction moving
ectuivalently.
48. Perhaps the matter may be put in a somewhat clearer
light by converting the case of a moving circuit into that of a

OUTLINE OP ELECTROMAGNETIC CONNECTIONS.

47

oircuit at rest, and then employing the law of circuitation in its primitive form. The moving circuit has at any instant a definite position. Imagine it to be momentarily fixed in that position, by stopping the motion of its parts. In order that the relation of the circuit to the induction should be the same
&s when it was moving, we must now communicate momentarily
to the lines of induction the identically opposite motion to the (abolished) motion of the part of the circuit they touch.
We now get equation (15), on the understanding that g means
the additional magnetic current through a fixed circuit due to a given motion of the lines of induction across its boundary, such motion being the negative of the (abolished) motion of the circuit. The matter, is, therefore, simplified in treatment. For, in the former way, the process of demonstrating (15) which I have referred to as an " examination in detail," is really considerably complex, involving the translation, rotation, and distortion of an elementary circuit (or equivalently for any circuit). Fixing the circuit does away with this, and we have merely to examine what happens at a single element of the circuit, as induction sweeps across it, in increasing the induction through the circuit, and then apply the resulting formula to every element.
In the consideration of a single element, it is immaterial what
the shape of the circuit may be; it may, therefore, be chosen to be

a unit square in the plane of the paper, one of whose sides, AB,
AB is the element of unit length. Now, suppose the induction at
is perpendicular to the plane of the paper, directed downwards, and that it moves from right to left perpendicularly across
A B AB. Let also from to be the positive sense in the circuit.
It is evident, without any argumentation, that the directions
chosen for q and B are the most favourable ones possible for

48

ELECTROMAGNETIC THEORY.

CH. II.

increasing the induction through the circuit, and that the rate

AB of its increase, so far as

alone is concerned, is simply qB,

the product of the tensors of the velocity q of transverse motion

and of the induction B. Further, if the velocity q be not wholly
transverse to B as described, but still be wholly transverse to

AB, we must take, instead of q, the effective transverse com-

ponent

q

sin

6 y

if

be the angle between q and B, making our

result to be qB sin 0. Now, this is the tensor of VqB, whose
A direction is from to B. The motional electric force in the

element AB is therefore from B to A, and is VBq, because it

is the negative circuitation which measures the magnetic cur-

rent through a circuit. Lastly, if the motion of B be not

wholly transverse to AB, we must further multiply by the

cosine of the angle between VBq and the element AB.

This merely amounts to taking the effective part of VBq along

the circuit. So, finally, we see that VBq fully represents the

AB impressed electric force per unit length in

when it is fixed,

and the induction moves across it, or that its negative

e = VqB

represents the motional electric force when it is the element
AB that moves with velocity q through the induction B. Now,
apply the process of circuitation, and we see that e is such that its curl represents the rate of increase of induction through the unit circuit due to the motion alone.
This may seem rather laboured, but is perhaps quite as much
to the point as a complete analytical demonstration, where one
may get lost in the maze of differential coefficients, and have
some difficulty in interpreting the analytical steps electromag-
netically.
The fictitious motion of the induction above assumed has
nothing to do with the real motion of the induction through the medium. If there be any, its effect is fully included in the term G, the real magnetic current.

The Motional Magnetic Force.

49. The motional magnetic force h may be similarly deduced. First we have the primitive form of the first law of

circuitation,

curlH=J,

(18)

OUTLINE OP ELECTROMAGNETIC CONNECTIONS.

49

when the unit circuit is at rest, where J is the complete electric

current-density, and next

..... curlH = J+j,

(19)

when

the

circuit

moves ;

where

the

auxiliary j is

a

fictitious

electric current equivalent to the increase of displacement

through the circuit by its motion only. Next show that

...... j = curlh,

(20)

and

...... h = VDq,

(21)

by

similar

reasoning

to

that

concerning

e ;

so that by insertion

in (19) the first law of circuitation is reduced to the standard

form

..... curl (H-h) = J,

(22)

with the special form of the impressed force h stated. Comparing the form of h with that of e we observe that
there is a reversal of direction in the vector-products, the flux being before the velocity in one and after it in the other. This arises from the opposite senses of circuitation of the electric and the magnetic force to represent the magnetic and electric currents.
The "Magneto-electric Force."
50. The activity of the motional h is found by multiplying it by the magnetic current, and is, therefore,

by the same transformation as from (11) to (12).
We conclude that VGD is an impressed mechanical force,
per unit volume, and, therefore, that VDG- is a mechanical force, that is, of the Newtonian type, arising from the electric stress. By analogy with the electromagnetic force it may be termed the magnetoelectric force, acting on dielectrics supporting displacement when the induction varies with the time. Of this more hereafter.
Electrification and its Magnetic Analogue. Definition of
"Divergence."
51. So far nothing has been laid down about electrification. But the laws of circuitation cannot be completed without including electrification and its suggested magnetic analogue.

50

ELECTROMAGNETIC THEORY.

CH. II.

Describe a closed surface in a dielectric, and observe the net amount of displacement leaving it. This, of course, means the
excess of the quantity leaving over that entering it. If the
net amount be zero, there is no electrification within the region bounded by the surface. If the amount be finite, there is just that amount of electrification in the region. This is independent altogether of its distribution within the region, and of the size and shape of the region.
More formally, the surface-integral of the displacement leaving any closed surface measures the electrification within it.
This being general, if we wish to find the distribution of electrification we must break up the region into smaller regions, and in the same manner determine the electrifications in them.

Carrying this on down to the infinitely small unit volume, we,

by the same process of surface-integration, find the volume-

density of the electrification. It is then called the divergence

of the displacement.

That is, in general, the divergence of any flux is the amount

of the flux leaving the unit volume.

And in particular, the divergence of the displacement

measures the density'^pf electrification.

Similarly, the divergence of the induction measures the

" magnetification," if

thu;e

is

any to

measure,

which

is a very

doubtful matter indeed. There is no evidence that the flux

induction has any divergence ; it is purely a circuital flux, so far as is certainly known, and this is most intimately connected with the other missing link in a symmetrical electromagnetic scheme, the (unknown) magnetic conductivity.
Divergence is represented by div, thus :

divD = />,

(1)

divB = o-,

(2)

if p and cr are the electrification and magnetification densities

respectively.
In another form, electrification is the source of displacement,
and magnetification the source of induction. How these fluxes
are distributed after leaving their sources is a perfectly indifferent matter, so far as concerns the measure of the strength of
the sources. In an isotropic uniform medium at rest, the fluxes naturally spread out uniformly and radially from point-sources

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

51

of displacement or of induction. The density of the fluxes then varies as the inverse square of the distance, because the concentric spherical surfaces through which they pass vary in area directly as the square of the distance. Thus

D-

? ~

B- * .... (3)

are the tensors of the displacement and induction at distance r from point-sources p and a-.
If the source be spread uniformly over a plane in a uniform
isotropic medium to surface-density p or cr, then, by the mere symmetry, we see that half the flux goes one way and half the
other, perpendicularly to the plane, so that

(4)

at any distance from the plane. But if we by any means make the source send all the flux one way only, then

at any distance.

D= />,

..... B = o-,

(5)

A Moving Source equivalent to a Convection Current, and
makes the True Current Circuital.
52. The above being merely to concisely explain the essential meaning of electrification in relation to displacement, and how it is to be measured, consider a pointsource or charge to be in motion through a dielectric at
rest. Starting with the charge at rest at one place, the
displacement is radial and stationary. When permanently at
rest in another place, the displacement is the same with reference to it. In the transition, therefore, the displacement has changed its distribution. There must, therefore, be electric current. Now, the only place where the dis-
placement diverges, however the source may be moving, is
at the source itself, and therefore the only place where the displacement current diverges is at the source, because it is the time-variation of the displacement. The displacement current is therefore circuital, with the exception of a missing
E2

52

ELECTROMAGNETIC THEORY.

CH. 1L

link at the moving charge. If we suppose that the charge p moving with velocity u constitutes a current Tip, that is, in

the same sense as the motion, and such that the volume-

integral of the current density is u/>, then the complete system of this " convection " current, and the displacement

current together form a circuital flux.

Thus, suppose the charge to be first outside a closed surface
and then move across it to its inside. When outside, if the

displacement goes through the surface to the inner region,
it leaves it again. On the other hand, when the charge is

inside, the whole displacement passes outward. Therefore,
when the charge is in the very act of crossing the surface, the

displacement through it outward changes from to /o, and this

is the time-integral of the displacement current outward whilst

the charge crosses. This is perfectly and simultaneously com-

pensated by the convection current, making the whole current

always circuital.
The electric current is, therefore, made up of three parts,

the conduction current, the displacement current, and the con-

vection current ;

thus,

(6)

p being the volume-density of electrification moving through the stationary medium with the velocity u.
If the medium be also moving at velocity q. referred tofixed space, we must understand by u above the velocity also referred to fixed space. The velocities q. and u are only the same when the medium and the charge move together. Thus it will come to the same thing if we stop the motion of the charge altogether, and let the medium have the motion equiva-
lent to the former relative motion.

Similarly, if there should be such a thing as diverging in-

duction,

or

the

"

"

magnetification

denoted

by

<r

above, then

we-

shall be obliged to consider a moving magnetic charge as con-

tributing to the magnetic current, making the complete mag-

netic current be expressed by

w if be the velocity of the magnetification of density o-.

OUTLINE OP ELECTROMAGNETIC CONNECTIONS.

53

Examples to illustrate Motional Forces in a Moving Medium with a Moving Source. (1.) Source and Medium with a Common Motion. Flux travels with them undisturbed.
53. In order to clearly understand the sense in which motion of a charge through a medium, or motion of the
medium itself, or of both together with respect to fixed space, is to be understood, and of the part played therein by the motional electric and magnetic forces, it will be desirable to
give a few illustrative examples of such a nature that their meaning can be readily followed from a description, without the mathematical representation of the results. It does not, indeed, often happen that this can be done with profit and
without much circumlocution. In the present case, however,
it is rather easier to see the meaning of the solutions from a description, than from the formulae.
In the first place, let us start with a single charge p at rest
A at any point in an infinite isotropic non-conducting dielectric
ether, for example which is also at rest. Under these
circumstances the stationary condition is one of isotropic radial
A displacement from the charge at according to the inverse-
square law, and there is nothing to disturb this distribution. Now, if the whole medium and the charge itself are supposed
to have a common motion (referred to an assumed fixed space,
in the background, as it were), no change whatever will take place in the distribution of displacement referred to the moving
charge. That this should be so in a rational system we may
conclude from the relativity of motion (the absolute motion of the universe being quite unknown, if not inconceivable) combined with our initial assumption that the electric flux (and the magnetic flux not here present) represent states of
the medium, which may be carried with it just as states
of matter are carried with matter in its motion. But as
the charge, and with it the displacement, move through space as a rigid body without rotation, the changing displacement at any point constitutes an electric current, and therefore would necessitate the existence of magnetic force, if we treated the first law of circuitation in its primitive form, referred to a stationary medium. Here, however, the motional magnetic force, which is ( 44, 49) the vector-product of the

54

ELECTROMAGNETIC THEORY.

CH. II.

displacement and the velocity of the medium, comes into play,

and it is so constructed as to precisely annul all magnetic

force under the circumstances, and leave the displacement

(referred

to

the

moving

medium)

unaffected ;

or,

in

another

H form, it changes the law of circuitation (curl = J) referred to

fixed space, so as to refer it in the same form to the moving

medium.

H The result is = 0, and D moves with the medium.

Similar remarks apply to other stationary states. They are unaffected by a common motion of the whole medium and the sources (or quasi-sources), and this result is mathematically obtained by the motional electric and magnetic forces.

A (2.) Source and Medium in Relative Motion.

Charge

suddenly jerked into Motion at the Speed of Propaga-

tion. Generation of a Spherical Electromagnetic Sheet ;

ultimately Plane. Equations of a Pure Electromagnetic

Wave.

54. But the case is entirely altered if the charge and the medium have a relative translational motion.

Start again with charge and medium at rest, and the displacement stationary and isotropically radial. Next, introduce
the fact (the truth of which will be fully seen later) that the
medium transmits all disturbances of the fluxes through itself at the speed (MC)~~, which" call -y; and let us suddenly set the charge moving in any direction rectilinearly through the medium at this same speed, v. Ths question is, what will

happen ?
A part of the result can be foreseen without mathematical

investigation ; the remainder is an example of the theory of the

simplest spherical wave given by me in "Electromagnetic Waves."

A Let (in Fig. 1) be the initial position of the charge when it

AC first begins to move, and let

be the direction of its sub-

AB sequent motion. Describe a sphere of radius

= vt ;

then,

at

the time t the charge has reached B. Now, from the mere fact

that the speed of propagation is v, it follows that the dis-

placement outside the sphere is undisturbed. It is clear that

there cannot be any change to the right of B, because the

charge has only just reached that place, and disturbances only

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

i>5

travel at the same speed as it is moving itself. Similar con-
siderations applied to the expanding sphere through this
A charge at every moment of its passage from B to will show
that no disturbance can have got outside the sphere. The
radial lines, therefore, represent the actual displacement, as
well as the original displacement, though of course, in the
latter case, they extended to the point A.
We have now to complete the description of the solution.
There is no displacement whatever inside the sphere BTCDF.

The displacement emanating from the charge at B, therefore,
joins on to the external displacement over the spherical surface.
We can say beforehand that it should do so in the simplest
conceivable manner, by the shortest paths. On leaving the pole B it spreads uniformly in all directions on the surface of
the sphere, and each portion goes the shortest way to the opposite pole D. But it leaks out externally on the way, in such a manner that the leakages are equal from equal areas. The displacement thus follows the lines of longitude.

56

ELECTROMAGNETIC THEORY.

CH. II.

This completes the case so far as the displacement is concerned. But the spherical surface constitutes an electromagnetic sheet, and corresponding to the displacement there

is a distribution of coincident, induction. This induction is

perpendicular to the tangential displacement, and therefore

follows the lines of latitude. Its direction is up through the

A paper above

(at E, for example), and down through the

A paper below (at F, for example). The tangential displace-

ment and induction surface-densities (or fluxes per unit area of

D the sheet), say,

and B , are connected by the equation

or,

= cv .

H E Or, if and

be the equivalent forces got by dividing by

= c

and

by

ft

respectively,

then,

since

2 fj.cv

1,

E = /xvH .
Or, expressing the mutual directions as well,

E = VB v;

where v is the vector velocity of the electromagnetic sheet at the place considered. These last are, in fact, as we shall see

later, the general equations of a wave-front or of a free wave,

which though it may attenuate as it travels, does not suffer

distortion by mixing up with other disturbances.
Now, as time goes on, the charge at B moves off to the right,
the electromagnetic sheet simultaneously expanding. The ex-

ternal

displacement,

therefore,

becomes

infinitesimal ;

likewise

D that on the

side of the shere. Practically, therefore, we

are finally left with a plane electromagnetic sheet moving

perpendicularly to itself at speed v, at one point of which is the moving charge, from which the displacement diverges

uniformly in the sheet, following, therefore, the law of the

inverse first power (instead of the original inverse square), accompanied by a distribution of induction in circles round the

axis of motion, varying in density with the distance according

to the same law, and connected with the displacement by the

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

57

above equations. In the diagram, AB has to be very great,
and the plane sheet is the portion of the spherical sheet round B, which is then of insensible curvature.

(3.) Sudden Stoppage of Charge. Plane Sheet moves on.
Spherical Sheet generated. Final Result, the Stationary
Field.
55. Having thus turned the radial isotropic displacement of the stationary charge into a travelling plane distribution,
let us suddenly reduce the charge to rest. We know that

FIG. 2.

after some time has elapsed, the former isotropic distribution will be reassumed ; and now the question is, how will this take

place ?

Let B in Fig. 2 be the position of the charge at the moment

B of stoppage and after. Describe a sphere of radius vt, with

for centre ;

then

the

point

C

is where the

charge

would have

got at time t after the stoppage had it not been stopped, and
the plane DOE would have been the position of the plane

58

ELECTROMAGNETIC THEORY.

CH. IL

electromagnetic sneer. Now, the actual state of things isdescribed by saying that :
(1.) The plane sheet DOE moves on quite unaltered,
except at its core C, where the charge has been taken out. (2.) The stationary radial displacement of the charge in its
new position at B is fully established within the sphere, with-
out any induction.
(3.) The internal displacement joins itself on to the external in the plane sheet, over the spherical surface, by leaking into it and then following the shortest route to the pole C. That is,
the tangential displacement follows the lines of longitude.
(4.) The induction in the spherical sheet is oppositely
directed to before, still, however, following the lines of latitude,
and being connected with the tangential displacement by the
former relations.
In time, therefore, the plane sheet and the spherical sheet go out to infinity, and there is left behind simply the radial displacement of the stationary charge.

(4.) Medium moved instead of Charge. Or both moved with same Relative Velocity.

56. Now, return to the case of 54, and referring to Fig. 1,

suppose it to be the charge that is kept at rest, whilst the

medium is made to move bodily past it from right to left at

We speed v, so that the relative motion is the same as before.

must now suppose B to be at rest, the charge being there origi-

A nally, and remaining there, whilst it is

that is travelling

from right to left, and the spherical surface has a motion com-
A pounded of expansion from the centre and translation with

it. Attending to this, the former description applies exactly.

The external displacement is continuously altering, and there

is electric current to correspond, but there is no magnetic

force (except in the spherical sheet), and this, is, as before said,

accounted for by the motional magnetic force.

The final result is now a stationary plane electromagnetic

sheet, as, in fact, described before in 45, where we considered

the displacement and induction in the sheet to be kept up

steadily by electric and magnetic forces impressed by the

motion

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

59

Now stop the motion of the medium, without altering
the position of the charge, and Fig. 2 will show the growth of

the radial stationary displacement, as in 55, as it is in fact the
same case precisely after the first moment.
We can in a similar manner treat the cases of motion and

stoppage of both charge and medium, provided the relative

speed be

always

the

speed

v t

however

different

from this may

be the actual speeds.

(5.) Meeting of a Pair of Plane Sheets with Point-Sources. Cancelment of Charges ; or else passage through one another; different results. Spherical Sheet with two Plane Sheet Appendages.
57. From the two solutions of 54, 55 (either of which may be derived from the other) we may deduce a number of
other interesting cases.
Thus, let initially a pair of equal opposite charges +p and
- p be moving towards one another, each at speed v through the medium (which for simplicity we may consider stationary), each with its accompanying plane electromagnetic sheet.
When the charges meet the two sheets coincide, the two dis-
placements cancel, leaving none, and the two inductions add,
doubling the induction. We have thus, momentarily, a mere
sheet of induction.
Now, if we can carry the charges through one another, without change in their motion, the two sheets will immediately reappear and separate. That is, the plane waves will pass through one another, as well as the charges.
But if the charges cancel one another continuously after their
first union, a fresh case arises. It is, given a certain plane sheet of induction initially, what becomes of it, on the understanding that there is to be no electrification ?
The answer is, that the induction sheet immediately splits into two plane electromagnetic sheets, joined by a spherical sheet, as in Fig. 3. For it is the same as the problem of stoppage in 55 with another equal charge of opposite kind moving the other way and stopped simultaneously, so that there is no
electrification ever after. Touching the sphere at the point F
in Fig. 2 is to be placed the additional plane wave, and the

60

ELECTROMAGNETIC THEORY.

CH. II.

internal displacement is to be abolished. That is to say, in
Fig. 3 the displacement converges uniformly to F in the plane sheet there, then flows without leakage to the opposite pole C
along the lines of longitude, and there diverges uniformly in the other plane sheet. Each displacement sheet has its corresponding coincident induction, according to the former formulae. They all move out to infinity, leaving nothing behind, as there
is no source left.

Fio. 6.
(6.) Spherical Sheet without Plane Appendages produced by sudden jerking apart of opposite Charges.
58. Similarly, let there be a pair of coincident or infinitely close opposite charges, with no displacement, and let them be suddenly jerked apart, each moving at the speed of propagation of disturbances. The result is simply a single spherical wave, without plane appendages, and without leakage of the displacement. The charges are at opposite poles, at the ends of the axis of motion, and the displacement just flows over the

OUTLINE OF ELECTROMAGNETIC; CONNECTIONS.

61

surface from one to the other symmetrically. There is the
usual induction B = /wD to match.
Fig. 3 also shows this case, if we leave out the plane sheets and
suppose the positive charge to be at F and the negative at C.
After a sufficient time, we have practically two widely separated plane electromagnetic sheets, although they are really
portions of a large spherical sheet.
Now, imagine the motion of the two charges to be reversed ;. if we simultaneously reverse the induction in the spherical sheet, without altering the displacement, it will still be an electromagnetic sheet, but will contract instead of expanding. It will go on contracting to nothing when the charges meet. If they are then stopped nothing more happens. But if the charges can separate again, the result is an expanding spherical
electromagnetic sheet as before.
(7.) Collision of Equal Charges of same Name.
59. If, in the case of colliding plane sheets with charges, 57, they be of the same name, then, on meeting, it is theinduction that vanishes, whilst the displacement is doubled.
That is, we have momentarily a plane sheet of displacement. If the charges be kept together thereafter, this plane
sheet splits into two plane electromagnetic sheets joined by a spherical sheet. At the centre of the last is the (doubled) charge 2/5, which sends its displacement isotropically to the surface of the sphere, where it is picked up and turned round towards one pole or the other. The equator of the sphere is the line of division of the oppositely flowing displacements. The displacement gets greater and greater as the poles are neared, the total amount reaching each pole being p (half the central charge), which then diverges in the plane sheet touching the pole.
The final result, when the waves have gone out to infinity,.
is, of course, merely the stationary field of the charge 2p.
(8.) Hemispherical Sheet. Plane, Conical and Cylindrical
Boundaries.
60. If a charge be initially in contact with a perfectly conducting plane, and be then suddenly jerked away from it at the speed v, the result is merely a hemispherical electromag-
netic shell. The negative charge, corresponding to the moving

62

ELECTROMAGNETIC THEORY.

CH. II.

point-charge, expands in a circular ring upon the conducting
plane, this ring being the equator of the (complete) sphere. This case, in fact, merely amounts to taking one-half of the
solution in 58, and then terminating the displacement
normally upon a conductor.
A In Fig. 4, is the original position of p on the conducting plane CAE, and when the charge has reached B the displace-
ment terminates upon the plane in the circle DF.

FIG. 4.

Instead of a plane conducting boundary, we may similarly

have conical boundaries, internal and external (or one conical

boundary alone), with portions of perfect spherical waves run-

ning along them at the speed v.

If the two conical boundaries have nearly the same angle,

and this angle be small, we have a sort of concentric cable

(inner and outer conductor with dielectric between), of continuously increasing thickness. The case of uniform thickness

is included as an

extreme

case ;

the

(portion

of

the)

spherical

wave then becomes a plane wave.

OUTLINE OP ELECTROMAGNETIC CONNECTIONS.

63

General Nature of Electrified Spherical Electromagnetic
Sheet.
61. The nature of a spherical electromagnetic sheet expanding or contracting at speed v, when it is charged in an arbitrary manner, may also be readily seen by the foregoing. So far, when there has been electrification on the sheet, it has
been a solitary point-charge or else a pair, at opposite poles. Now, the general case of an arbitrary distribution of electrification can be followed up from the case of a pair of charges not at opposite ends of a diameter, and each
of these may be taken by itself by means of an opposite
charge at the centre or externally, so that integration docs
the rest of the work. When there is a pair of equal charges
of opposite sign we do not need any external or internal complementary electrification, but we may make use of them argumentatively ; or we may let one charge leak outward, the other inward, and have the external and internal electrification in reality. The leakage should be of the isotropic character always. But the internal electrification need not be at the central point. It may be uniformly distributed upon a concentric sphere. This, again, may be stationary, or it may itself be in motion, expanding or contracting at the speed v. The external electrification, too, may be on a concentric spherical surface, which may be in similar motion. The sheets, too, may be of finite depth, and arbitrarily electrified, so that we have any volume-distribution of electrification moving in space in radial lines to or from a centre, accom-
panied by electromagnetic disturbances arranged in spherical
sheets.
There is thus a great variety of ways of making up problems of this character, the nature of whose solutions can be readily
pictured mentally.
Two charges, ql and q^ for example, on a spherical sheet. One way is to put - qt and q2 at the centre, and superpose the two solutions. Or the charges - (ql and q2) may be put externally, with isotropic leakage. Or part may be inside, and part outside. Or we may have no complementary electrification at all, but lead the displacement away into plane waves touching
the sphere.

64

ELECTROMAGNETIC THEORY.

CH. IL

Only when the total charge on the spherical sheet is zero can we dispense with these external aids (which to use depending

upon the conditions of the problem) ; then the displacement has sources and sinks on the sheet which balance one another.

The corresponding induction is always perpendicular to the

resultant tangential displacement, and is given by the above

D formula, or B = /*VvD , where

is the tangential displace-

ment in the sheet (volume density x depth).

One case we may notice. If the density of electrification be

uniform over the surface, there is no induction at all. That is,

it is not an electromagnetic sheet, but only a sheet of electrifi-

cation, without tangential displacement, and therefore without

induction.

So, with a condenser consisting of a pair of concentric shells
uniformly electrified, either or both may expand or contract
without magnetic force. This is, however, not peculiar to the
case of motion at speed v. Any speed will do. But in general,
if the speed be not exactly v, there result diffused disturbances. The electromagnetic waves are no longer of the same

pure type.

General Remarks of the Circuital Laws. Ampere's Rule for deriving the Magnetic Force from the Current.
Rational Current-element.

62. The two laws of circulation did not start into full

activity all at once. On the contrary, although they express

the fundamental electromagnetic principles concerned in the

most concise and clear manner, it was comparatively late in

the history of electromagnetism that they became clearly re-

We cognised and explicitly formularised.

have not here, how-

ever, to do the work of the electrical Todhunter, but only to

notice a few points of interest.

The first law had its first beginnings in the discovery of

Oersted that the electric conflict acted in a revolving manner,

and in the almost simultaneous remarkable investigations of

Ampere. It did not, however, receive the above used form of

expression. In fact, in the long series of investigations in

electro-dynamics to which Oersted's discovery, and the work of

Ampere, Henry, and Faraday, gave rise, it was customary to

consider an element of a conduction current as generating a

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

65

certain field of magnetic force. Natural as this course may
have seemed, it was an unfortunate one, for it left the question of the closure of the current open ; and it is quite easy to see now that this alone constituted a great hindrance to progress.
But so far as closed currents are concerned, in a medium of uniform inductivity, this way of regarding the relation between current and magnetic force gives equivalent results to those
obtained from the first law of circuitation in the limited form
suitable to the circumstances stated.
If C is the density of conduction current at any place, the corresponding field of magnetic force is given by

at distance r from the

current-element 0,

if

x be a unit vector l

H along r from the element to the point where is reckoned.

H The intensity of

thus follows the law of the inverse square

of the distance along any radius vector proceeding from the

current-element; but, in passing from one radius vector to

another, we have to consider its inclination to the axis of the

current by means of the factor sin (where is the angle

between r and the axis), involved in the vector product. Also,
H is perpendicular to the plane containing r and the axis of H the current-element, or the lines of are circles about this axis.
H But from the Maxwellian point of view this field of is that

corresponding to a certain circuital distribution of electric

current, of which the current-element mentioned is only a

part ; this complete current being related to the current-

element in the same way as the induction of an elementary

magnet is to the intensity of magnetisation of the latter.

Calling the complete system of electric current a rational

current-element, it may be easily seen that in a circuital distri-

bution of rational current-elements the external portion of the

current disappears by mutual cancelling, and there is left only

the circuital current made up of the elements in the older

We sense.

may, therefore, employ the formula (1) to calculate

without ambiguity the magnetic force of any circuital distri-

bution of current. This applies not merely to conduction

current (which was all that the older electricians reckoned),

but to electric current in the wider sense introduced by

p

66

ELECTROMAGNETIC THEORY.

CH. II.

Maxwell. But the result will not be the real magnetic force

unless the 'distribution of inductivity is uniform.

When /*

varies, we may regard the magnetic force of the current thus

obtained as an impressed magnetic force, and then calculate

what induction it sets up in the field of varying inductivity.

This may be regarded as an independent problem.

Tn passing, we may remark that we can mount from magnetic

current to electric force by a formula precisely similar to (1),

but subject to similar reservations.

The Cardinal Feature of Maxwell's System. Advice to
anti-Maxwellians.

63. But this method of mounting from current to magnetic force (or equivalent methods employing potentials) is quite unsuitable to the treatment of electromagnetic waves, and is

then usually of a quite unpractical nature. Besides that, the function " electric current " is then often a quite subsidiary
and unimportant quantity. It is the two fluxes, induction and displacement (or equivalently the two forces to correspond), that are important and significant; and if we wish to know the
electric current (which may be quite a useless piece of information) we may derive it readily from the magnetic force by

differentiation ;

the simplicity of

the

process being in striking

contrast to that of the integrations by which we may mount

from current to magnetic force.

To exemplify, consider the illustrations of plane and

spherical electromagnetic waves of the simplest type given in

53 to 61, and observe that whilst the results are rationally

and simply describable in terms of the fluxes or forces, yet to

describe in terms of electric current (and derive the rest from

it) would introduce such complications and obscurities as would

tend to anything but intelligibility.

Now, Maxwell made the first law of circuitation (not, how-

ever, in its complete form) practically the definition of electric current. This involves very important and far-reaching consequences. That it makes the electric current always circuital at once does away with a host of indeterminate and highly

speculative problems relating to supposititious unclosed currents. It also necessitates the existence of electric current in

perfect non-conductors or insulators. This has always been a

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

67

stumbling-block to practicians who think themselves practical. But Maxwell's innovation was really the most practical improvement in electrical theory conceivable. The electric current in a nonconductor was the very thing wanted to coordinate electrostatics and electrokinetics, and consistently
harmonise the equations of electromagnetism. It is the
cardinal feature of Maxwell's system, and, when properly followed up, makes the insulating medium the true medium in the transmission of disturbances, and explains a multitude of
phenomena that are inconceivable in any other theory (unless it be one of the same type). But let the theoretical recommendations (apart from modern experiments), which can only be appreciated after a pretty close study of theory, Maxwellian and otherwise, stand aside, and only let the tree be judged by its fruit. Is it not singular that there should be found people, the authors of works on Electricity, who are so intensely prejudiced against the Maxwellian view, which it would be quite natural not to appreciate from the theoretical standpoint, as to be apparently quite unable to recognise that the fruit has any good flavour or savour, but think it no better than Dead Sea fruit ? The subject is quite sufficiently difficult to render understandable popularly, without the unnecessary obstruction evidenced by a carping and unreceptive spirit. The
labours of many may be required before a satisfactory elementary presentation of the theory can be given. So much the
more need, therefore, is there for the popular writer to recognise the profound significance of the remarkable experimental work of late years, a significance he appears to have so sadly missed.
When that is done, then will be the time for an understanding
of Maxwell's views. Let him have patience, and believe that
it is not all speculative metaphysics because it is not to his
present taste. Never mind the ether disturbances playing pranks with the planets. They can take care of themselves.

Changes in the Form of the First Circuital Law.

64. Two or three changes I have made* in Maxwell's form

of the first circuital law. One is of a formal character, the

introduction of the h term to express the intrinsic force of

* "Electromagnetic Induction and its Propagation," The Electrician^

1885, January 3, and later : or reprint.

F2

68

ELECTROMAGNETIC THEORY.

CH. II.

magnetisation. This somewhat simplifies the mathematics, and places the essential relations more clearly before the eye. Connected with this is a different reckoning of the energy of an intrinsic magnet, in order to get consistent results.
The second change, which is not merely one of a formal character, but is an extension of an obligatory character, is the introduction of the h term to represent the motional magnetic force. In general problems relating to electromagnetic wavea it is equally important with the motional electric force.
The third change is the introduction (first done, I think, by Prof. Fitzgerald) of the term to explicitly represent the convection-current or electrification in motion as a part of the true current. Although Maxwell did not himself explicitly represent this, which was a remarkable oversight, he was strongly insistent upon the circuital nature of electric current, and would doubtless have seen the oversight the moment it was suggested to him. Now, there are spots on the sun, and I see no good reason why the many faults in Maxwell's treatise should be ignored. Tt is most objectionable to stereotype the work of a great man, apparently merely because it was so great an advance, and because of the great respect thereby induced. The remark applies generally ; to the science of Quaternions,
for instance, which, if I understand rightly, Prof. Tait would: preserve in the form given to it by Hamilton. In application
to Maxwell's theory, I am sure that it is in a measure to the
recognition of the faults in his treatise that a clearer view of. the theory in its broader sense is due.
Introduction of the Second Circuital Law.
65. The second circuital law, like the first, had an experimental origin, of course, and, like the first, was long in approximating to its present form much longer, in fact, though in a different manner. The experimental foundation was Faraday's recognition that the voltage induced in a conducting circuit was conditioned by the variation of the number of lines of force
through it. But, rather remarkably, mathematicians did not put this
straight into symbols for an elementary circuit, but went to work in a more roundabout way, and expressed it through the medium of an integration extended along a concrete circuit >

OUTLINE OF ELECTROMAGNETIC CONNECTIONS.

69

or else, of an equation of electromotive force containing a

function called the vector potential of the current, and another

potential, the electrostatic, working together not altogether in

the most harmoniously intelligible manner in plain English,

muddling one another. It is, I believe, a fact which has been

recognised that not even Maxwell himself quite understood how

We they operated in his

" general equations

of

propagation."

need not wonder, then, that Maxwell's followers have not found

it a very easy task to understand what his theory really meant,

and how to work it out. I had occasion to remark, some years

since, that it was very much Maxwell's own fault that his

views obtained such slow acceptance; and, in now repeating the

remark, do not abate one jot of my appreciation of his work,

which increases daily. For he devoted the greater part of his

treatise to the working out and presentation of results which

could be equally well done in terms of other theories, and gave

only a very cursory and incomplete exposition of what were

peculiarly his own views and their consequences, which are

of the utmost importance. At the same time, it is easily to be

recognised that he was himself fully aware of their importance,

by the tone of quiet confidence in which he wrote concerning

them.

Finding these equations of propagation containing the two

potentials unmanageable, and also not sufficiently comprehen-

sive,

I was obliged to dispense

with them ;

and,

going

back to

first principles, introduced* what I term the second circuital law

as a fundamental equation, the natural companion to the first.

The change is, I believe, a practical one, and enables us to con-

siderably simplify and clarify the treatment of general ques-

tions, whilst bringing to light interesting relations which were

formerly hidden from view by the intervention of the vector

potential A, and its parasites J and "VP.

Another rather curious point relates to the old German

electro-dynamic investigations and their extensions to endeavour

to include, supersede, or generalise Maxwell by anti-Maxwellian

methods. It would be slaying the slain to attack them; but

one point about them deserves notice. The main causes of the

variety of formulae, and the great complexity of the investiga-

tions, were first, the indefiniteness produced by the want of

* See footnote, p. 67.

70

ELECTROMAGNETIC THEORY.

CH. II.

circuitality in the current, and next the potential methods

employed.

[J. J.

Thomson's

" Report

on Electrical Theories "

contains an account of many. As a very full example, that

most astoundingly complex investigation of Clausius, in the
second volume of his " Mechanische Warmetheorie," may be

referred to.] But if the critical reader will look through these investigations, and eliminate the potentials, he will find, as a

useful residuum, the second circuital law; and, bringing it

into .full view, will see that many of these investigations are

purely artificial elaborations, devoid of physical significance;

gropings after mares' nests, so to speak. Now, using this law

in the investigations, it will be seen to involve, merely as a

matter of the mathematical fitness of things, the use of another,

viz., the first circuital law, and so to justify Maxwell in his

doctrine of the circuitality of the current. The useful moral

to be deduced is, I think, that in the choice of variables to ex-

press physical phenomena, one should keep as close as possible to those with which we are experimentally acquainted, and

which are of dynamical significance, and be on one's guard

against being led away from the straight and narrow path in

the pursuit of the Will-o'-the-wisp.

As regards the terms in the expression of the second law

which

stand

for

unknown

7
properties, the}

may

be

regarded

merely as mathematical extensions which, by symmetrizing the

equations, render the correct electric and magnetic analogies

plainer, and sometimes assist working out. But some other

extensions of meaning, yet to be considered, have a more sub-

stantial foundation.

Meaning of True Current. Criterion.
66. One of these extensions refers to the meaning to be attached to the term current, electric or magnetic respectively. As we have seen, electric current, which was originally conduction current only, had a second part, the displacement current, added to it by Maxwell, to produce a circuital flux; and further, when there is electrification in motion, we must, working to the same end, add a third term, the convection current, to preserve
circuitality.
A fourth term may now be added to make up the " true "
current, when the medium supporting the fluxes is in motion.

OUTLINE OP ELECTROMAGNETIC CONNECTIONS.

71

Thus, let us separate the motional electric and magnetic forces from all other intrinsic forces that go with them in the two

circuital laws (voltaic, thermo-electric, <feq., 39) ; denoting the
former by e and h, and the latter by e and h . The two equations of circuitation [(12), (13) 38, and (6), (7) 52)] now

read

curl(H-h -h) = J = C + D + u/>, . . . (1)

K - curl (E - e

-
e)

=G=

+ B + wcr.

. . (2)

Now transfer the e and h terms to the right side, producing

=0 curl (H-h ) = J

+ D + u/o + j,.

. . (3)

K G - curl (E - ec) = = + B + wo- + g, . . (4)

where j and g are the auxiliaries before used, given by

j = curl h,

g = - curl e.

It is the new vectors J and G which should be regarded as
the true currents when the medium moves.

As the auxiliaries j and g are themselves circuital vectors,

there may not at first sight appear to be any reason for further complicating the meaning of true current when separated into

component

parts,

for

the

current

J

is

circuital, and

so is J ,

which is of course the same as J when the medium is station-

ary. The extension would appear to be an unnecessary one, of

a merely formal character.

On the other hand, it is to be observed that from the

Maxwellian method of regarding the current as a function of

the magnetic force, the extended meaning of true current is not

a further complication, but is a simplification. For whereas
in the equation (2) we deduct from the force E of the flux

not only the intrinsic force e but also the motional force e

to obtain

the

effective force whose curl measures

the

current ;

on the other hand, in (4) we deduct only the intrinsic force.

Away, therefore, from the sources of energy which are inde-

pendent of the motion of the medium, the force whose curl is

taken in (4) is the force of the flux, which specifies the elec-

tric state of the medium, whether it be stationary or moving.

To show the effect of the change, consider the example of

56, in which the medium moves past the charge, and there is

continuously changing displacement outside a certain sphere.

If we consider J the true current, we should say there is elec-

72

ELECTROMAGNETIC THEORY.

CH. II.

trie current, although there is no magnetic force. But accord-

ing to the other way there is no true current, and the absence

of magnetic force implies the absence of true electric current.

But still this question remains, so far, somewhat of a con-

ventional one. Is there any test to be applied which shall

effectually discriminate between J and J as the true current ?

There would appear to be one and only one ; viz., that e being

an intrinsic force, its activity, whatever it be, say e x, expresses

the rate of communication of energy to the electromagnetic

system from a source not included therein, and not connected
/ with the motional force e. When the medium is stationary x

is J, and the question is, is x to be J or J (or anything else)

when the medium moves ?

Now this question can only be answered by making it a part

of a much larger and more important one ; that is to say, by a

comprehensive examination of all the fluxes of energy concerned

in the equations (1), (2), or (3), (4), and their mutual harmo-

nisation. This being done, the result is that e J is the activity

of e so that it is J ,

that is the

measure

of the true current,

with the simpler relation to the force of the flux; and it is

naturally suggested that in case of further possible extensions,

we should follow in the same track, and consider the true current to be always the curl of (H - h ), independently of the

make up of its component parts.

This is not the place for a full investigation of this complex

question, but the main steps can be given ; not for the exhibi-

tion of the mathematical working, which may either be taken

for granted, or filled in by those who can do it, but especially

with a view to the understanding in a broad manner of the

course of the argument. Up to the present I have used the

notation of vectors for the concise and plain presentation of

principles and results, but not for working purposes. This
course will be continued now. How to work vectors may form

the subject of a future chapter. It is not so hard when you

know how to do it.

The Persistence of Energy. Continuity in Time and Space and Flux of Energy.
67. The principle of the conservation or persistence of energy is certainly as old as Newton, when viewed from the