Electromagnetic Theory, Vol. 1 (1893) — part 1 of 31

GIFT OF MICHAEL REESE

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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.]

% 0 Printed and Published by

'THK KLKCTRIOIAN" PRINTING AND PUBLISHING CO.,

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, . .

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 some- thing 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 modifi- cations 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 con- sistent 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

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IV. PREFACE.

magnetic sides of electromagnetism are symmetrically ex- hibited 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 con- siderable 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 chap- ter 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 applica- tion 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 con- cerning the speed of the current, the impotence of self- induction, 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 impor- tance 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

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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 innu- merable 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 rationalisa- tion effected by changing the units, and the substitution ol the second circuital law for Maxwell's equation of electro- motive 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 the second circuital equation, D 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 that if V 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 /a2 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 V//* + v(l -ft~2), 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 (1 - fir2) 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 vt 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 increas- ing 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. Xi.

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 platitude. Who 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 rational- ising 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 else. It was henceforward to be H20. 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, 189 1.] Preliminary Remarks 1

8—13 [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. (Pages 20 to 131.)

SECTION. 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

54 (2.) Source and Medium in Relative Motion. A 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 82 A [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

the same Curl. Example : — A 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.

(Pages 132 to 305.)

SECTION. 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- strations. Barbarity of Euclid .. ... ... ... ... 147

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

and Illustrations ... ... ... ... ... 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

129 Vector Product WE, 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-

tion of a Vector into Circuital and Divergent Parts. A Series

of Circuital Vectors 206

135 [May 27, 1892.} A Series of Divergent Vectors 209

136 The Operation inverse to Divergence ... ... 212

137 The Operation inverse to Slope 213

138 The Operation inverse to Curl 214

139 Remarks on the Inverse Operations ... ... ... ... 215

140 Integration " by parts." Energy Equivalences in the Circuital

Series , 216

XV111. CONTENTS.

SECTION. PAGE.

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

gent Series 217

142 The Isotropic Elastic Solid. Relation of Displacement to Force

through the Potential ... ._ 219

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

Forceless and Torqueless Stress ... ... ... ... 221

144 Other Forms for the Displacement in terms of the Applied

Forcive ... ... ... ... ... ... ... ... 224

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

Elastic, Dissipative and Inertial Resistance to Translation, Rotation, Expansion, and Distortion ... ... 226

146 Electromagnetic and Elastic Solid Comparisons. First Ex-

ample : Magnetic Force compared with Velocity in an In- compressible Solid with Distortional Elasticity ... ... 232

147 Second Example : Same as last, but Electric Force compared

with Velocity 234

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

compared with a Viscous Solid. Failure 234

149 Fourth Example : A Pure Conductor compared with a Viscous

Liquid. Useful Analogy ... ... ... ... ... 236

150 Fifth Example : Modification of the Second and Fourth ... 239

151 Sixth Example : A Conducting Dielectric compared with an

Elastic Solid with Translational Friction 240

152 Seventh Example : Improvement of the Sixth ... ... ... 21Q

153 Eighth Example : A Dielectric with Duplex Conductivity com-

pared with an Elastic Solid with Translational Elasticity and Friction. The singular Distortionless Case ... ... .. 241