Electromagnetic Theory, Vol. 2 (1899) — part 1 of 31

ELECTROMAGNETIC

THEORY

Oliver Heaviside

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ELECTROMAGNETIC THEORY.

BY

OLIVER HEAVISIDE.

VOLUME I.

LONDON:

"THE ELECTRICIAN" PRINTING AND PUBLISHING COMPANY,

LIMITED,

9ALISBURT COVBT, FLRT STBin, B.C.

[AU Right$ Mesened.]

BOimn MAK I) ii)i3

Printeil and Pu)»llH]ied by "TIIK KLKOTRICIAN ' PRINTING AND PUBI.ISIIINO CO., LIMITED, 1, 2, and 3, Salisbury Court, Fleet Street, London, K.C.

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PREFACE.

Tms work was originaUy meant to be a . conjini^tion of ;the series "Electromagnetic Induction and its Propagatksn," pnblished in Ths Electrician in 1885-6-7, but left nnfinished.

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 theozyi 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 scientLQc and mathematical methodni of enquiry.

The second chapter, pp. 20 to 181, 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 of carrying out the idea of lines and tubes of force in a con- sistent and inteUigible manner. It is also done mainly in terms of erectors, for the sufficient reason that vectors are the main subject of investigation. It is also done in the duplex ton I introduced - m 1865, whereby the electric and

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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 importuit relations.

The third chapter, pp. 1H2 to 805, 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 klgebra 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, tliis 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. 800 to 46G, is devoted to the theory of plane electromagnetic waves, and, being nudnly 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 liow 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 electromagnotism may be rougbly divided into two classes, primary and secondary. Besides the main

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PREFACE.

primary phenomena, there is a large number of secondary ones, partly or even mainly eleotromagnetio, but also trenching

upon other physical sciences. Now the question arises whether it is either practicable or useful to attempt to construct a theoiy of such coraprehensiveness as to include the secondary phenomena, and to call it the theory of eleotromagnetism. I think not, at least at present. It might perhaps be done if the secondary phenomena were thoroughly known ; but their theory is so much more debatable than that of the primary phenomena tliat 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 eleotromagnetism is then a primaiy 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 of 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 ?

PKBFACE.

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 ioUowing the first publication of Maxwell's treatise, to see that there may be mach 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 arc many obscuri- ties and some inconsistencies. Speaking for myself, it was only by changing its form of presentation that I was able to Bee 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

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form and of the conditions of motion show that Maxwell's auxiliary term in the electromotive force, \nz., VqB (the motional electric force), where q is the velocity of the medium and B the induction, requires the use of a similar auxihary term in the first circuital law, viz., VDq, the motional magnetic force, D heing the displacement. And there is yet another change sometimes needed. For whilst B is circuital, so thiit 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. ]>ut 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 preWous investigation of Prof. J. J. Thomson, will illustrate the matter in question. It is known that if V be the speed of hght through ether, the speed through a stationary transparent body, say water, is Y//a, if fi 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 /x^ 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 hght referred to the external ether be V//x + ?-(l -/x-2), if r 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 Fresnel's hypothesis, at least in a generalised form. But, in the case

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of water, the additional speed of light due to the motion of the water might be instead of (l-fjr^)Vf 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 Jr, provided r/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. FHrndamentally 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/fi+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, mstead of Jr. And, as a matter of fact, if we employ the modified or corrected circuital law above referred to, we do obtain full snperimposition 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 Mzeau and Michelson's results is that a transparent medium Uke water cannot be regarded as (in the electromagnetic theory)

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PREFACE.

a simple medium like the ether, at least for waves of light, and that a secondary theory is necessary. Fresnel's sagadous 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 shp through it without sensibly affecting it. At any rate the evidence that this is the ease 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 httle 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 Umits. 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 etlier can be complete that does not fully account for the omnipresent force of gravi- tation.

There is one other matter that demands notice in condu- • sion. It is not long since it was taken for granted that the common electrical units were correct. That curious and

obtrusive constant 1- 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 4ir was an unfortunate and mischievous mistake, the source of many evils. In plain English, the common system of electrical units involves an irrationahty of the same Idnd as would be brought into the

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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 drole of unit diameter. The constant ir would then obtrade 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 kind>5 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 leader^ 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. Kemember 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.

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PBBFACE.

Old style instruments would yary soon ])e 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 com'se), 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 tJieir units about a f^uarter of a century ago. One J;iy 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 HgO. 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 formula^ irrational, did not cry " Too late," ignore the matter, and ask Parliament to legaUse the old erroneous weights I They went and set the matter right. Verb, nap,

DECEMiiKi: IG, 1898.

CONTEXTS.

.i— ^

[The data VfUhin brackets nre the dates of firit publication,]

CHAPTER I.' ' INTRODUCTION.

8KCTia». TAGB.

1 — 7 [/cm. 3, 1891.] Preliminary Remarks ... ... .. 1

8 — 13 [Jan. IS, 1891.] On the nature of Anti-Mathematiciana and

of Mathematical Methods of Enquiry 7

14—16 [Jan. 30, 1S91.] Dcscriptionof some Electromagnetic Results

deduced by Mathematical Reasoning 14

CHAPTP^Pv IT.

OUTLINE OF THE ELECTROMAGNETIC CONNECTIONS.

(Pages 20 to 13L)

SECTION. PAGI.

20 [FA. IS, 1891.] Electric and Magnetic Force ; Displacement and Induction ; Elaativity and Permittivity, Inductivity and Reluc-

t.l\l^^aa* ••• ••»

20

21

Electric iuul Magnetic Energy

21

22

Eolotropic Helations ...

22

23

Distinction between Absolute and lielative Permittivity or

IlldllCt'lVlt^

23

24

Dissipation of Energy. The Conduction-current; Conductivity

and Resistivity. Tlie Electric Current

24

25

Fictitious Magnetic Conduction 'current and Real Magnetic

Current ... ... ... ... ... ... ... ...

25

26

[Frh. sr. ISni.] Forces ami Fluxes

25

27

Line- integral of a Force. Voltage and Gaussoge ...

26

28

Snrface-integral of a Flux. Denrity 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

CONTBMTS.

BHXION. PAQE.

33 [Mareh IS, ISOIJ] Crcm-ooiinectioiu of Electric and Magnetic

Force. Circuital Flux. €Srcuitati<m 32

34 First Law of Circuitation ».. 35

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 S7t 1891.] Claanfication of Impreaaed 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 48

45 Example. A Stationaiy Electromagnetic Sheet 43

46 lApril 17, L901.1 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

^ The Motional Magnetic Force 48

60 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 Ciicuital ... ... ... ... ... 51

63 yUay 1, 1S91.] Examples to illustrate Motional Forces iu a Moving

Medium with a Moving Source. (1.) Source and Medium with a Common Motion. Flux travels with them undisturbed ... 63

64 (2.) Source and Medium in Relative Motion. A Charge suddenly

jerked into Motion at the Speed of Propac^atinn. Generation of a Spherical Electromagnetic Sheet ; ultimately Tlane. 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

f^^^d «•• ••■ e** 67

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

same Relative Velocity ... ... ... ... . . ... 58

67 (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

68 (6.) Sj^ierical Sheet without Plane Appendages produced by

sudden jerking apart of opposite Charges 60

59 (7.) Collision of E<]u;\l Charges of .^ame Name ... ... ... 61

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

c[&I*168»«« •■• ••• 6X

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

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SECTION. PAOI.

62 [May J9, 1S9I.] General Remarks on the Circuital Laws.

Amp^'s Rule for derinng th« Magnetic Font from the Current. Bational Current-dement ... 64

63 The Cardinal Feature of ICazwell'a System. AdTioe to anti-

Maxwellians ... ... ... ... ... ... ».< ... 66

64 Changes in the Fomi of the First Circuital Law ... 67

65 Introfluction of the Second Circuital Law 68

66 [July 3, JS9L] Meaning of True Current. Criterion 70

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

Flux of Energy ... ... ... ... ... ... ... 72

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

active Stres.s. Gravitational difficulty ... 74

69 [July 17, JS91.] Specialised form of expreeaion of the Con-

tinuity of Energy ... ... ... ... ... 77

70 Electiomagnetic Application. Medium at Rest. The Poyuting

««9 ««« 73

71 Extennon to a Moving Medium. Full interpretation of the

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

72 Derivation of the Electromagnetic Stress from the Flux of

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

73 [July ISOl."] Uncertainty regarding the General Application

of the Electromagnetic Stress 8&

74 The Electrostatic Stren in Air 87

75 Tlie Moving Faroe on Electrification, bodily and superficiaL

^bnmcnusation ... ... ... ••. ... •■. ... 90*

76 Depth f)f Electrified Layer on a Conductor 91

77 [Aufj. ?/, 1S9J.] Electric Field dinturbed by Forngn Body,

Efiect of a Spherical Non-conductor ... ... ... ... OS-

78 Dynamical Principle. Any Stress Self-equilibrating 95-

79 Electric Application of the Principle. Resultant Aotitm on Solid

Body independent of the Internal Stress, which is statically indeterminate. Beal Surface Traction is the Stress Diffisrenoe 96

80 Translational Force due to Variation of Permittivity. Harmoni-

sation with Surface Traction ... ... ... ... ... 9fr

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

JEoGr^iy ••• ■•• »••

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

Inductivity. Movranent of Elastically Magnetised Bodies ... 100 82a ISept. 4, 1891.} Force on Electric Current Conductors. The Lateral Pressure becomes prominenti but no Stress DiBcon- tinuity in general ... ... ... ... ... ... ... 102

83 Force on Intrinsically Magnetised Matter. Difficulty. Ma.wvell's

Solution probably wrong. Special Estimation of Energy of a Magnet and the Moving Force it leads to 103

84 Force on Intrinsically Electonsed Matter 106

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

a Moving Medium ... ... ... ... ... ... ... lOT

CONTEXTS.

SECTION, PAGE.

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

Electromagaetac Street. Principal Axes 109

87 ' Dependence of the* Fluxes due to an Impressed Forcive upon

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

88 Identity of the Disturbances due to Impressed Forcivcs having '" the' Aame Curl. Example: — A Single Circuital Source of

• r)isturV>!ince  ...       ...       ...       ...       ...       ...  112 
  

89 I'ruduction of Steady State due to Impressed Forcive by

crossing of Electromagnetic Waves. Example of a Circular Soilrce. Distinction between Source 'of Energy and of Dis- turbanoe ... ... ... ... ... ... ... 113

90 [Oct. K;, ISOl.] 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 ECTects of the Eruptioh' 120

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

95 [Oct. 30, 1S91.] Rational v. Irrational Electric Poles 123

96 Rational v. Irrational Magnetic Poles 125

APPENDIX A. [Jan. J3, 1891.] The Botational Ethbb ur its Appucatiox to Elbctrom aonbtisu ... 127

CHAPTEK III.,

THE ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS.

(Pages 132 to 305.)

SECTIOS. PAGK.

97 [Nov. 13, 1S9L] Scalars and Vectors ... 132

98 Characteristics of Carteman and Vectorial Analysis 133

99 Abstrusity of Quaternions and Comparative Simplidty 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 [Uec. 4' IS'Jl.] Type for Vectors. Greek, German, and Roman

Letters unsuitable. Clarendon ll^pe suitable. Typographical

Backsliding in the Presmt Generation 139

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

of Reference ... ... ... ... 142

105 The Addition of Vector.''. Circuital Proi)erty 143

106 Application to Physical Vectors. Futility of Popular Demon-

strationcf. Barbarity oi Euclid .. ... ... ... ... 147

107 [Dec.lS,lS91.] The Scalar Product of Two Vectors. Notation

and Illustrations « 148

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

109 Reciprocal of a Vector 155

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' CONTENTS.

SECTION. PAOl.

110 Expression of any Vector as the Sum of Three Independent

Vectors ... ... ... ... ... ... ... ... 155

111 [Jan. i, 180^.] The Vector Product of Two Vectors. Illua-

tffttuMu ••• ••• ••• •.• ••• 156

112 Combinations of Three Vecton. The Parallelepipedal Property 158

113 Semi- Cartesian Expansion of a Vector Product, and Proof of

the FundamentAl Distributive Principle 159

114 Examples relating t<i Vector Products ... ... ... ... 162

115 [Jan. JO, IS'JJJ] The Ditlereutiatiou of Scalars and X'ectors ... 163

116 Semi- Cartesian Uilil-reutiation. Examples of Difierentiatiug

Functions' of Vectors 165

117 Motion along a Curve in Space. Tangen<7 and Cunrature;

Velocity and Acceleration 167

118 Tortuosity of a Curve, and Various Forms of Expansion ... 160

119 [Feb. IJy IS'JJ.] Hamilton's B^inite Differentiala Inconvenient

and Unnecessary ... < 172

120 Determination of Possibili^ of Existence of Difl^rential Co-

efficients • . ... ... •*• .•• >•• ... 174

121 Variation of the Siae and Ort of a Vector 176

122 XMarch 4, ISO J.] 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, 183

125 Motion of a Strain-Figure ... ... 185

126 [March 25, 189S.] Space-Variation or Slope VP of a Scalar

Function ... ... ... ... ... ... ... ... 186

127 Scalar Product VD. The Theorem of Divei^noe 188

128 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, 18£f£] Five Examples of the Operation of V in

Transforming from Surface to Volume Summations 194

131 Five fixamples of the Operation of V in Transforming from

Circuital to Surface Summations ... 197

132 Nine P'xamples of the l^ifTorentiating Effects of V 199

133 [May 13, ISO.'.-] The Potential of a Scalar or Vector. The

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

134 Connections of Potential, Curl, Divergence, and Slope. Separa-

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

of Circuital Vectors ... ... 206

135 [Ma7j j7, ISO.'.] A Series of Divergent Vectors' 209

156 The Operation inverse to Divejgence ... ... ... ... 212

137 The Operation inverse to Slope 213

138 The Operation inverse to Curl 214

199 fiemark's on the Inverse Operations 215

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

Series ••• ... ... ... ... ... 216

• • •

XVIU.

CONTENTS.

141 [June 10, 1892,] Entrgy and other EqnivaleiioM In the Diver-

gen t 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

Foroeleas and Torqueleas Streia 221

144 Other Forms for the Displaoement in terms of the Applied

ForciTe ... .• •• ... ... 284

146 [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 com}^>ared with a Viscous

Liquid. Useful Analogy 236

160 Fifth EiEample : Modification of the Second and Fourth ... 230

161 Sixth Example : A Cronducting Dielectric compared with an

Elastic Solid with IVanslational Friction ... ... ... 2^0

152 Seventh Example : Improvement of the Sixth ... ... ... 240

163 Eighth Example : A Dielectric with Duplex Conductivity com-

pared with an Elastic Solid with Translational Elasticity and

Friction. The singular Distortionlees Case • 241

163a [Jidy 29, 1892,] The Botational Ether, Compressible or In-

compresmble ... ... ... ... ... ••• ... 243

164 First Rotational Analogy : Magnetic Force compared with

Velocity ... ... 245

155 Circuital Indeterminateness of the Flux of Energy in general 247

166 Second Rotational Analogy : Induction compared with

^^Glocit*^ ••• ^« •■• ■«« 249*

167 [Aug, 5, 1892,] Ftobability of the Kinetic Nature of Magnetic

^fiiQCfl^^Q^ ••• ••• •••

168 Unintelligibility of the Rotational Analogue for a Conductinfc

Dielectric when Magnetic Energy is Kinetic 262